Subgroups — C₂×#22569117 #31720142

Order 17152.

2674 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864193	C₂	1	1	yes	maximal, characteristic
2	C₂×#16864161	C₂	1	2	yes	maximal, complemented
3	#22569117	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16863519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863515	C₂	1	2	yes	maximal, complemented
6	C₂×#16863513	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863512	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16863511	C₂	1	2	yes	maximal, complemented
9	C₂×#16863443	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#104×D₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#16863348	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#444×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#41×D₆₇	C₂²	1	2	yes	—
14	#16864193	C₄	1	8	yes	complemented
15	#16864193	C₂²	1	8	yes	—
16	C₂×#14524075	C₄	1	2	yes	complemented
17	C₂×#14524072	C₄	1	2	yes	complemented
18	C₂×#14524070	C₄	1	2	yes	complemented
19	C₂×#14524067	C₄	1	2	yes	complemented
20	C₂×#14524065	C₂²	1	1	yes	characteristic
21	C₂×#14524063	C₂²	1	2	yes	—
22	#16864161	—	2	8	no	—
23	#16864161	C₂²	1	16	yes	complemented
24	C₂×#14524051	—	2	2	no	—
25	C₂×#14524051	C₂²	1	2	yes	complemented
26	C₂×#14524050	C₂²	1	2	yes	complemented
27	C₂×#14524049	—	2	2	no	—
28	C₂×#14524049	C₂²	1	2	yes	complemented
29	C₂²×#11761611	—	2	2	no	—
30	C₂²×#11761611	C₂²	1	1	yes	characteristic
31	C₂²×#11761611	C₂²	1	1	yes	characteristic
32	C₂×M₁₆×D₆₇	—	2	2	no	—
33	C₂×M₁₆×D₆₇	C₂²	1	2	yes	—
34	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
35	#16863519	C₂²	1	8	yes	complemented
36	#16863515	C₂²	1	16	yes	complemented
37	#16863513	C₂²	1	8	yes	complemented
38	#16863512	C₂²	1	8	yes	complemented
39	#16863511	C₂²	1	16	yes	complemented
40	#16863443	C₂²	1	8	yes	complemented
41	C₂×#14523979	C₂²	1	1	yes	characteristic, complemented
42	C₂×#14523978	C₂²	1	2	yes	complemented
43	C₂×#14523971	C₂²	1	2	yes	complemented
44	C₂×#14523970	C₂²	1	2	yes	complemented
45	C₂×#14523969	C₂²	1	2	yes	complemented
46	#104×D₆₇	C₂²	1	16	yes	complemented
47	#16863348	C₂²	1	8	yes	complemented
48	C₂×#14523954	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
51	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
53	#444×C₆₇	C₂²	1	8	yes	complemented
54	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
55	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
56	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
57	C₂×#11761629	C₂×C₄	1	2	yes	—
58	C₂×#11761628	—	2	2	no	—
59	C₂×#11761628	C₂³	1	2	yes	—
60	C₂×#11761627	—	2	2	no	—
61	C₂×#11761627	C₂×C₄	1	2	yes	—
62	#41×D₆₇	—	2	4	no	—
63	#41×D₆₇	—	2	4	no	—
64	#41×D₆₇	C₂×C₄	1	8	yes	—
65	#41×D₆₇	C₂³	1	8	yes	—
66	C₂×Q₈×D₆₇	C₂×C₄	1	2	yes	—
67	C₂×D₄×D₆₇	—	2	2	no	—
68	C₂×D₄×D₆₇	C₂×C₄	1	2	yes	—
69	C₂²×C₄×D₆₇	—	2	2	no	—
70	C₂²×C₄×D₆₇	—	2	1	no	—
71	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
72	#14524075	C₂×C₄	1	16	yes	complemented
73	#14524072	C₂×C₄	1	16	yes	complemented
74	#14524070	C₂×C₄	1	16	yes	complemented
75	#14524067	C₂×C₄	1	16	yes	complemented
76	#14524065	C₂×C₄	1	4	yes	complemented
77	#14524065	C₂³	1	4	yes	—
78	#14524063	C₂×C₄	1	8	yes	complemented
79	#14524063	C₂³	1	8	yes	—
80	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic, complemented
81	C₂×#11761618	C₂×C₄	1	2	yes	complemented
82	C₂×#11761617	C₂×C₄	1	2	yes	complemented
83	C₂×#11761616	C₂×C₄	1	2	yes	complemented
84	C₂×#11761615	C₂×C₄	1	2	yes	complemented
85	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic, complemented
86	C₂×#11761613	C₂×C₄	1	2	yes	complemented
87	#14524051	—	2	16	no	—
88	#14524051	—	2	4	no	—
89	#14524051	C₂³	1	8	yes	complemented
90	#14524050	—	2	4	no	—
91	#14524050	C₂³	1	8	yes	complemented
92	#14524049	—	2	16	no	—
93	#14524049	—	2	4	no	—
94	#14524049	C₂³	1	8	yes	complemented
95	C₂×#11761612	—	2	2	no	—
96	C₂×#11761612	C₂³	1	2	yes	—
97	C₂×#11761611	—	2	2	no	—
98	C₂×#11761611	—	2	16	no	—
99	C₂×#11761611	—	2	2	no	—
100	C₂×#11761611	—	2	2	no	—
101	C₂×#11761611	C₂×C₄	1	2	yes	—
102	C₂×#11761611	C₂×C₄	1	2	yes	—
103	C₂×#11761611	C₂×C₄	1	4	yes	—
104	C₂×#11761611	D₄	1	2	yes	—
105	C₂×#11761611	D₄	1	2	yes	—
106	C₂×#11761611	D₄	1	4	yes	—
107	C₂×#11761611	C₂³	1	4	yes	—
108	C₂×#11761611	C₂³	1	4	yes	—
109	M₁₆×D₆₇	—	2	4	no	—
110	M₁₆×D₆₇	—	2	16	no	—
111	M₁₆×D₆₇	C₂³	1	8	yes	—
112	C₂²×C₆₇⋊C₈	—	2	2	no	—
113	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
114	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
115	C₂×C₈×D₆₇	—	2	2	no	—
116	C₂×C₈×D₆₇	—	2	2	no	—
117	C₂×C₈×D₆₇	—	2	2	no	—
118	C₂×C₄×Dic₆₇	—	2	1	no	—
119	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
120	C₄²×D₆₇	—	2	2	no	—
121	C₄²×D₆₇	C₂³	1	4	yes	—
122	#14523979	C₂³	1	4	yes	complemented
123	#14523978	C₂³	1	8	yes	complemented
124	#14523971	C₂³	1	8	yes	complemented
125	#14523970	C₂³	1	8	yes	complemented
126	#14523969	C₂³	1	8	yes	complemented
127	#14523954	C₂³	1	4	yes	complemented
128	C₈×Dic₆₇	C₂³	1	4	yes	complemented
129	C₈×Dic₆₇	C₂³	1	4	yes	complemented
130	C₂×#41×C₆₇	C₂³	1	2	yes	—
131	#135×C₆₇	C₂×C₄	1	4	yes	complemented
132	#135×C₆₇	C₂³	1	4	yes	—
133	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
134	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
135	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
136	#131×C₆₇	—	2	4	no	—
137	#131×C₆₇	C₂³	1	8	yes	complemented
138	C₂×M₁₆×C₆₇	—	2	2	no	—
139	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
140	C₂²×C₈×C₆₇	—	2	2	no	—
141	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
142	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
143	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
144	#108×C₆₇	C₂³	1	4	yes	complemented
145	#104×C₆₇	C₂³	1	8	yes	complemented
146	#97×C₆₇	C₂³	1	4	yes	complemented
147	C₂³×D₆₇	—	2	2	no	—
148	C₂³×D₆₇	#41	1	1	yes	characteristic
149	#11761629	—	2	4	no	—
150	#11761629	C₂²×C₄	1	8	yes	—
151	#11761628	—	2	2	no	—
152	#11761628	—	2	2	no	—
153	#11761628	—	2	8	no	—
154	#11761628	—	2	8	no	—
155	#11761628	C₂²×C₄	1	4	yes	—
156	#11761628	C₂⁴	1	4	yes	—
157	#11761627	—	2	8	no	—
158	#11761627	—	2	8	no	—
159	#11761627	—	2	4	no	—
160	#11761627	C₂²×C₄	1	8	yes	—
161	C₂×Dic₁₃₄	—	2	2	no	—
162	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
163	Q₈×D₆₇	—	2	4	no	—
164	Q₈×D₆₇	C₂²×C₄	1	8	yes	—
165	C₂×D₂₆₈	—	2	2	no	—
166	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
167	C₂×C₂²⋊D₆₇	—	2	2	no	—
168	C₂×C₂²⋊D₆₇	—	2	2	no	—
169	C₂×C₂²⋊D₆₇	—	2	2	no	—
170	D₄×D₆₇	—	2	8	no	—
171	D₄×D₆₇	—	2	8	no	—
172	D₄×D₆₇	—	2	4	no	—
173	D₄×D₆₇	C₂²×C₄	1	8	yes	—
174	C₂²×Dic₆₇	—	2	1	no	—
175	C₂²×Dic₆₇	—	2	2	no	—
176	C₂²×Dic₆₇	#41	1	1	yes	characteristic
177	C₂×C₄×D₆₇	—	2	2	no	—
178	C₂×C₄×D₆₇	—	2	2	no	—
179	C₂×C₄×D₆₇	—	2	2	no	—
180	C₂×C₄×D₆₇	—	2	8	no	—
181	C₂×C₄×D₆₇	—	2	8	no	—
182	C₂×C₄×D₆₇	—	2	4	no	—
183	C₂×C₄×D₆₇	—	4	4	no	—
184	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
185	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
186	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
187	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
188	C₂×C₄×D₆₇	#41	1	2	yes	—
189	C₂×C₄×D₆₇	C₂⁴	1	2	yes	—
190	Dic₂₆₈	C₂²×C₄	1	4	yes	complemented
191	#11761618	C₂²×C₄	1	8	yes	complemented
192	#11761617	C₂²×C₄	1	8	yes	complemented
193	#11761616	C₂²×C₄	1	8	yes	complemented
194	#11761615	C₂²×C₄	1	8	yes	complemented
195	D₅₃₆	C₂²×C₄	1	4	yes	complemented
196	#11761613	C₂²×C₄	1	8	yes	complemented
197	#11761612	—	2	2	no	—
198	#11761612	—	2	8	no	—
199	#11761612	C₂⁴	1	4	yes	—
200	#11761611	—	2	8	no	—
201	#11761611	—	2	8	no	—
202	#11761611	—	2	8	no	—
203	#11761611	C₂²×C₄	1	8	yes	—
204	#11761611	C₂²×C₄	1	8	yes	—
205	#11761611	C₂×D₄	1	8	yes	—
206	#11761611	C₂×D₄	1	8	yes	—
207	C₂×C₆₇⋊C₈	—	2	2	no	—
208	C₂×C₆₇⋊C₈	—	2	8	no	—
209	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
210	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
211	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
212	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
213	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
214	C₂×C₆₇⋊C₈	C₂⁴	1	2	yes	—
215	C₂×C₆₇⋊C₈	C₂⁴	1	2	yes	—
216	C₈×D₆₇	—	2	8	no	—
217	C₈×D₆₇	—	2	8	no	—
218	C₈×D₆₇	—	2	8	no	—
219	C₄×Dic₆₇	—	2	1	no	—
220	C₄×Dic₆₇	—	2	4	no	—
221	C₄×Dic₆₇	C₂⁴	1	2	yes	—
222	#41×C₆₇	—	2	2	no	—
223	#41×C₆₇	—	2	2	no	—
224	#41×C₆₇	C₂²×C₄	1	4	yes	—
225	#41×C₆₇	C₂⁴	1	4	yes	—
226	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
227	C₂×D₄×C₆₇	—	2	2	no	—
228	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
229	C₂²×C₄×C₆₇	—	2	2	no	—
230	C₂²×C₄×C₆₇	—	2	1	no	—
231	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
232	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
233	SD₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
234	D₈×C₆₇	C₂²×C₄	1	4	yes	complemented
235	M₁₆×C₆₇	—	2	2	no	—
236	M₁₆×C₆₇	—	2	8	no	—
237	M₁₆×C₆₇	C₂⁴	1	4	yes	—
238	C₂×C₈×C₆₇	—	2	2	no	—
239	C₂×C₈×C₆₇	—	2	8	no	—
240	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
241	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
242	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
243	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
244	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
245	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
246	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
247	C₄²×C₆₇	—	2	1	no	—
248	C₄²×C₆₇	C₂⁴	1	2	yes	—
249	C₂³×C₆₇	—	2	2	no	—
250	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
251	C₂²×D₆₇	—	2	2	no	—
252	C₂²×D₆₇	—	2	8	no	—
253	C₂²×D₆₇	—	4	2	no	—
254	C₂²×D₆₇	—	8	4	no	—
255	C₂²×D₆₇	C₄×D₄	1	2	yes	—
256	C₂²×D₆₇	C₄×D₄	1	2	yes	—
257	C₂²×D₆₇	C₂×#41	1	2	yes	—
258	Q₈×C₆₇	—	2	2	no	—
259	Q₈×C₆₇	C₂³×C₄	1	4	yes	—
260	D₄×C₆₇	—	2	4	no	—
261	D₄×C₆₇	—	2	4	no	—
262	D₄×C₆₇	—	2	2	no	—
263	D₄×C₆₇	C₂³×C₄	1	4	yes	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	4	2	no	—
269	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
270	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
275	C₂²⋊D₆₇	—	2	4	no	—
276	C₂²⋊D₆₇	—	2	4	no	—
277	C₂²⋊D₆₇	—	2	8	no	—
278	C₂²⋊D₆₇	—	2	8	no	—
279	C₂×Dic₆₇	—	2	4	no	—
280	C₂×Dic₆₇	—	2	2	no	—
281	C₂×Dic₆₇	—	2	2	no	—
282	C₂×Dic₆₇	—	2	8	no	—
283	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
284	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
285	C₂×Dic₆₇	C₂×#41	1	2	yes	—
286	D₂₆₈	—	2	8	no	—
287	D₂₆₈	—	2	2	no	—
288	D₂₆₈	C₂³×C₄	1	4	yes	—
289	C₄×D₆₇	—	2	8	no	—
290	C₄×D₆₇	—	2	4	no	—
291	C₄×D₆₇	—	2	4	no	—
292	C₄×D₆₇	—	2	8	no	—
293	C₄×D₆₇	—	4	8	no	—
294	C₄×D₆₇	C₄×D₄	1	4	yes	—
295	C₄×D₆₇	C₄×D₄	1	4	yes	—
296	C₄×D₆₇	C₂³×C₄	1	4	yes	—
297	C₄×D₆₇	C₂²×D₄	1	4	yes	—
298	Dic₁₃₄	—	2	2	no	—
299	Dic₁₃₄	—	2	8	no	—
300	Dic₁₃₄	C₂³×C₄	1	4	yes	—
301	C₈×C₆₇	—	2	4	no	—
302	C₈×C₆₇	C₂³×C₄	1	2	yes	—
303	C₈×C₆₇	C₂³×C₄	1	2	yes	—
304	C₈×C₆₇	C₂²×D₄	1	4	yes	—
305	C₆₇⋊C₈	—	2	4	no	—
306	C₆₇⋊C₈	C₂³×C₄	1	4	yes	—
307	C₆₇⋊C₈	C₂²×D₄	1	2	yes	—
308	C₆₇⋊C₈	C₂²×D₄	1	2	yes	—
309	C₂²×C₆₇	—	2	2	no	—
310	C₂²×C₆₇	—	2	4	no	—
311	C₂²×C₆₇	—	4	4	no	—
312	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
313	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
314	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
315	C₂×D₆₇	—	2	4	no	—
316	C₂×D₆₇	—	2	4	no	—
317	C₂×D₆₇	—	4	2	no	—
318	C₂×D₆₇	—	4	2	no	—
319	C₂×D₆₇	—	4	4	no	—
320	C₂×D₆₇	—	8	8	no	—
321	C₂×D₆₇	C₂×C₄×D₄	1	4	yes	—
322	C₄×C₆₇	—	2	4	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
325	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
326	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
327	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
328	Dic₆₇	—	2	4	no	—
329	Dic₆₇	—	2	4	no	—
330	Dic₆₇	C₂×C₄×D₄	1	4	yes	—
331	C₂×#2956	—	67	1	no	maximal, Sylow
332	C₂×C₆₇	—	2	1	no	—
333	C₂×C₆₇	—	2	1	no	—
334	C₂×C₆₇	—	4	4	no	—
335	C₂×C₆₇	#2956	1	2	yes	complemented
336	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
337	D₆₇	—	2	4	no	—
338	D₆₇	—	4	4	no	—
339	C₂×#575	—	67	1	no	—
340	C₂×#568	—	67	2	no	—
341	#2956	—	67	16	no	—
342	C₂×#444	—	67	2	no	—
343	C₂×#444	—	67	1	no	—
344	C₂×#444	D₆₇	1	1	yes	characteristic, complemented
345	C₂×#443	—	67	1	no	—
346	C₂×#443	—	67	1	no	—
347	C₂×#443	—	67	2	no	—
348	C₂×#430	—	67	1	no	—
349	C₂²×#104	—	67	2	no	—
350	C₂×C₄×M₁₆	—	67	1	no	—
351	C₆₇	C₂×#2956	1	1	yes	characteristic, complemented, Sylow
352	C₂²×#41	—	67	2	no	—
353	#575	—	67	8	no	—
354	#575	—	67	8	no	—
355	C₂×#137	—	67	2	no	—
356	C₂×#137	—	67	2	no	—
357	C₂×AGL(1,ℤ/8)	—	67	2	no	—
358	C₂×AGL(1,ℤ/8)	—	67	2	no	—
359	C₂×#135	—	67	2	no	—
360	C₂×#135	—	67	1	no	—
361	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
362	#568	—	67	16	no	—
363	#568	—	134	8	no	—
364	C₂×#131	—	67	2	no	—
365	C₂×#131	—	67	2	no	—
366	C₂×#131	—	67	2	no	—
367	C₂×#131	—	134	2	no	—
368	C₂×#131	—	134	2	no	—
369	C₂×#131	C₂×D₆₇	1	2	yes	complemented
370	C₂²×M₁₆	—	67	1	no	—
371	C₂²×M₁₆	—	67	1	no	—
372	C₂²×M₁₆	—	67	2	no	—
373	C₂²×M₁₆	—	134	2	no	—
374	C₂²×M₁₆	—	134	2	no	—
375	C₂²×C₄²	—	67	1	no	—
376	#444	—	67	16	no	—
377	#444	—	67	8	no	—
378	#444	C₂×D₆₇	1	8	yes	complemented
379	#443	—	67	8	no	—
380	#443	—	67	16	no	—
381	#443	—	67	8	no	—
382	#430	—	67	8	no	—
383	C₂×#108	—	67	2	no	—
384	C₂×#108	—	67	1	no	—
385	C₂×#108	C₂×D₆₇	1	1	yes	characteristic, complemented
386	C₂×#104	—	67	2	no	—
387	C₂×#104	—	67	2	no	—
388	C₂×#104	—	67	2	no	—
389	C₂×#104	—	67	16	no	—
390	C₂×#104	C₂×D₆₇	1	2	yes	complemented
391	C₄×M₁₆	—	67	8	no	—
392	C₂×#97	—	67	1	no	—
393	C₂×#97	C₂×D₆₇	1	1	yes	characteristic, complemented
394	C₂×C₄×C₈	—	67	1	no	—
395	C₂×C₄×C₈	—	67	1	no	—
396	C₂×#41	—	67	8	no	—
397	C₂×#41	—	67	8	no	—
398	C₂×#41	—	67	2	no	—
399	C₂×#41	—	67	2	no	—
400	C₂×#41	—	67	2	no	—
401	C₂×#41	—	134	2	no	—
402	C₂×#41	—	134	2	no	—
403	C₂×#41	—	134	4	no	—
404	C₂×#41	—	134	4	no	—
405	C₂×#41	C₂²×D₆₇	1	2	yes	—
406	C₂²×Q₈	—	67	2	no	—
407	C₂²×D₄	—	67	2	no	—
408	C₂²×D₄	—	134	2	no	—
409	C₂³×C₄	—	67	1	no	—
410	C₂³×C₄	—	134	1	no	—
411	C₂³×C₄	—	134	2	no	—
412	#137	—	67	16	no	—
413	#137	—	67	16	no	—
414	AGL(1,ℤ/8)	—	67	16	no	—
415	AGL(1,ℤ/8)	—	67	16	no	—
416	#135	—	67	8	no	—
417	#135	—	67	8	no	—
418	#135	—	67	4	no	—
419	#135	—	67	4	no	—
420	#135	C₄×D₆₇	1	4	yes	complemented
421	#135	C₂²×D₆₇	1	4	yes	—
422	C₂×Q₁₆	—	67	2	no	—
423	C₂×Q₁₆	—	67	1	no	—
424	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic, complemented
425	C₂×SD₁₆	—	67	2	no	—
426	C₂×SD₁₆	—	67	2	no	—
427	C₂×SD₁₆	—	67	2	no	—
428	C₂×SD₁₆	C₄×D₆₇	1	2	yes	complemented
429	C₂×D₈	—	67	2	no	—
430	C₂×D₈	—	67	1	no	—
431	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic, complemented
432	#131	—	2	4	no	—
433	#131	—	67	8	no	—
434	#131	—	67	8	no	—
435	#131	—	67	8	no	—
436	#131	—	134	16	no	—
437	#131	—	134	4	no	—
438	#131	—	134	4	no	—
439	#131	—	134	16	no	—
440	#131	—	134	4	no	—
441	#131	C₂²×D₆₇	1	8	yes	complemented
442	C₂×M₁₆	—	2	2	no	—
443	C₂×M₁₆	—	67	2	no	—
444	C₂×M₁₆	—	67	2	no	—
445	C₂×M₁₆	—	67	4	no	—
446	C₂×M₁₆	—	67	2	no	—
447	C₂×M₁₆	—	67	2	no	—
448	C₂×M₁₆	—	67	4	no	—
449	C₂×M₁₆	—	67	4	no	—
450	C₂×M₁₆	—	67	4	no	—
451	C₂×M₁₆	—	67	2	no	—
452	C₂×M₁₆	—	67	8	no	—
453	C₂×M₁₆	—	134	2	no	—
454	C₂×M₁₆	—	134	16	no	—
455	C₂×M₁₆	—	134	2	no	—
456	C₂×M₁₆	—	134	2	no	—
457	C₂×M₁₆	—	134	16	no	—
458	C₂×M₁₆	—	134	4	no	—
459	C₂×M₁₆	—	134	2	no	—
460	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
461	C₂²×C₈	—	2	2	no	—
462	C₂²×C₈	—	67	1	no	—
463	C₂²×C₈	—	67	1	no	—
464	C₂²×C₈	—	134	2	no	—
465	C₂²×C₈	—	134	2	no	—
466	C₂²×C₈	—	134	2	no	—
467	C₂²×C₈	—	134	2	no	—
468	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
469	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
470	C₂×C₄²	—	67	1	no	—
471	C₂×C₄²	—	67	4	no	—
472	C₂×C₄²	—	134	2	no	—
473	C₂×C₄²	—	134	1	no	—
474	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
475	#108	—	67	8	no	—
476	#108	—	67	4	no	—
477	#108	C₂²×D₆₇	1	4	yes	complemented
478	#104	—	67	8	no	—
479	#104	—	67	8	no	—
480	#104	—	67	8	no	—
481	#104	C₂²×D₆₇	1	8	yes	complemented
482	#97	—	67	4	no	—
483	#97	C₂²×D₆₇	1	4	yes	complemented
484	C₄×C₈	—	67	4	no	—
485	C₄×C₈	—	67	4	no	—
486	C₂⁴	—	67	1	no	—
487	C₂⁴	—	134	2	no	—
488	#41	—	2	2	no	—
489	#41	—	2	2	no	—
490	#41	—	67	8	no	—
491	#41	—	67	8	no	—
492	#41	—	67	4	no	—
493	#41	—	67	4	no	—
494	#41	—	134	8	no	—
495	#41	—	134	8	no	—
496	#41	—	134	8	no	—
497	#41	—	134	8	no	—
498	#41	—	134	4	no	—
499	#41	—	134	4	no	—
500	#41	—	134	2	no	—
501	#41	—	134	2	no	—
502	#41	C₂×C₄×D₆₇	1	4	yes	—
503	#41	C₂³×D₆₇	1	4	yes	—
504	C₂×Q₈	—	67	8	no	—
505	C₂×Q₈	—	67	2	no	—
506	C₂×Q₈	—	134	2	no	—
507	C₂×Q₈	—	134	4	no	—
508	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	67	8	no	—
511	C₂×D₄	—	67	2	no	—
512	C₂×D₄	—	134	2	no	—
513	C₂×D₄	—	134	4	no	—
514	C₂×D₄	—	134	2	no	—
515	C₂×D₄	—	134	8	no	—
516	C₂×D₄	—	134	8	no	—
517	C₂×D₄	—	134	2	no	—
518	C₂×D₄	—	134	2	no	—
519	C₂×D₄	C₂×C₄×D₆₇	1	2	yes	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	268	4	no	—
523	C₂²×C₄	—	67	1	no	—
524	C₂²×C₄	—	67	2	no	—
525	C₂²×C₄	—	67	2	no	—
526	C₂²×C₄	—	67	2	no	—
527	C₂²×C₄	—	67	2	no	—
528	C₂²×C₄	—	67	2	no	—
529	C₂²×C₄	—	67	2	no	—
530	C₂²×C₄	—	134	4	no	—
531	C₂²×C₄	—	134	8	no	—
532	C₂²×C₄	—	134	2	no	—
533	C₂²×C₄	—	134	8	no	—
534	C₂²×C₄	—	134	2	no	—
535	C₂²×C₄	—	134	2	no	—
536	C₂²×C₄	—	134	1	no	—
537	C₂²×C₄	—	134	2	no	—
538	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
539	Q₁₆	—	67	8	no	—
540	Q₁₆	—	67	4	no	—
541	Q₁₆	C₂×C₄×D₆₇	1	4	yes	complemented
542	SD₁₆	—	67	8	no	—
543	SD₁₆	—	67	8	no	—
544	SD₁₆	—	67	8	no	—
545	SD₁₆	C₂×C₄×D₆₇	1	8	yes	complemented
546	D₈	—	67	8	no	—
547	D₈	—	67	4	no	—
548	D₈	C₂×C₄×D₆₇	1	4	yes	complemented
549	M₁₆	—	2	2	no	—
550	M₁₆	—	2	8	no	—
551	M₁₆	—	67	8	no	—
552	M₁₆	—	67	8	no	—
553	M₁₆	—	67	8	no	—
554	M₁₆	—	67	8	no	—
555	M₁₆	—	67	4	no	—
556	M₁₆	—	134	8	no	—
557	M₁₆	—	134	8	no	—
558	M₁₆	—	134	8	no	—
559	M₁₆	—	134	2	no	—
560	M₁₆	—	134	8	no	—
561	M₁₆	C₂³×D₆₇	1	4	yes	—
562	C₂×C₈	—	2	2	no	—
563	C₂×C₈	—	2	8	no	—
564	C₂×C₈	—	67	2	no	—
565	C₂×C₈	—	67	2	no	—
566	C₂×C₈	—	67	2	no	—
567	C₂×C₈	—	67	1	no	—
568	C₂×C₈	—	67	1	no	—
569	C₂×C₈	—	67	2	no	—
570	C₂×C₈	—	67	2	no	—
571	C₂×C₈	—	134	8	no	—
572	C₂×C₈	—	134	8	no	—
573	C₂×C₈	—	134	8	no	—
574	C₂×C₈	—	134	2	no	—
575	C₂×C₈	—	134	8	no	—
576	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
577	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
578	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
579	C₂×C₈	D₄×D₆₇	1	2	yes	—
580	C₂×C₈	D₄×D₆₇	1	2	yes	—
581	C₂×C₈	C₂³×D₆₇	1	2	yes	—
582	C₂×C₈	C₂³×D₆₇	1	2	yes	—
583	C₄²	—	2	1	no	—
584	C₄²	—	67	2	no	—
585	C₄²	—	134	4	no	—
586	C₄²	—	134	1	no	—
587	C₄²	C₂³×D₆₇	1	2	yes	—
588	C₂³	—	2	2	no	—
589	C₂³	—	268	2	no	—
590	C₂³	—	536	4	no	—
591	C₂³	—	67	2	no	—
592	C₂³	—	67	2	no	—
593	C₂³	—	67	2	no	—
594	C₂³	—	134	8	no	—
595	C₂³	—	134	2	no	—
596	C₂³	#41×D₆₇	1	1	yes	characteristic
597	Q₈	—	2	2	no	—
598	Q₈	—	67	4	no	—
599	Q₈	—	134	8	no	—
600	Q₈	—	134	2	no	—
601	Q₈	C₂²×C₄×D₆₇	1	4	yes	—
602	D₄	—	2	4	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	2	no	—
605	D₄	—	67	4	no	—
606	D₄	—	134	8	no	—
607	D₄	—	134	8	no	—
608	D₄	—	134	8	no	—
609	D₄	—	134	4	no	—
610	D₄	—	134	4	no	—
611	D₄	—	134	2	no	—
612	D₄	C₂²×C₄×D₆₇	1	4	yes	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	268	8	no	—
619	C₂×C₄	—	67	2	no	—
620	C₂×C₄	—	67	2	no	—
621	C₂×C₄	—	67	2	no	—
622	C₂×C₄	—	67	4	no	—
623	C₂×C₄	—	67	4	no	—
624	C₂×C₄	—	67	4	no	—
625	C₂×C₄	—	67	4	no	—
626	C₂×C₄	—	134	8	no	—
627	C₂×C₄	—	134	8	no	—
628	C₂×C₄	—	134	8	no	—
629	C₂×C₄	—	134	4	no	—
630	C₂×C₄	—	134	2	no	—
631	C₂×C₄	—	134	2	no	—
632	C₂×C₄	—	134	4	no	—
633	C₂×C₄	—	134	4	no	—
634	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
635	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
636	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
637	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
638	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
639	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
640	C₈	—	2	4	no	—
641	C₈	—	67	4	no	—
642	C₈	—	67	2	no	—
643	C₈	—	67	2	no	—
644	C₈	—	134	4	no	—
645	C₈	C₂²×C₄×D₆₇	1	2	yes	—
646	C₈	C₂²×C₄×D₆₇	1	2	yes	—
647	C₈	C₂×D₄×D₆₇	1	4	yes	—
648	C₂²	—	2	2	no	—
649	C₂²	—	2	4	no	—
650	C₂²	—	4	4	no	—
651	C₂²	—	268	2	no	—
652	C₂²	—	268	2	no	—
653	C₂²	—	268	4	no	—
654	C₂²	—	536	8	no	—
655	C₂²	—	67	4	no	—
656	C₂²	—	134	4	no	—
657	C₂²	—	134	4	no	—
658	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
659	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
660	C₂²	C₂×#41×D₆₇	1	1	yes	characteristic
661	C₄	—	2	4	no	—
662	C₄	—	4	2	no	—
663	C₄	—	67	4	no	—
664	C₄	—	134	4	no	—
665	C₄	—	134	4	no	—
666	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
667	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
668	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
669	C₄	C₂²×D₄×D₆₇	1	1	yes	characteristic
670	C₂	—	2	1	no	—
671	C₂	—	2	1	no	—
672	C₂	—	4	4	no	—
673	C₂	—	268	4	no	—
674	C₂	—	134	4	no	—
675	C₂	#22569117	1	2	yes	direct product factor
676	C₂	C₂×C₄×D₄×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#444×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#16864193	C₂
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×#444	D₆₇
hypocenter	C₆₇	C₂×#2956
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#444

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#16864161	C₂	semidirect
3. #22569117	C₂	direct (N × H)
4. C₂×#16863519	C₂	semidirect
5. C₂×#16863515	C₂	semidirect
6. C₂×#16863513	C₂	semidirect
7. C₂×#16863512	C₂	semidirect
8. C₂×#16863511	C₂	semidirect
9. C₂×#16863443	C₂	semidirect
10. C₂×#104×D₆₇	C₂	semidirect
11. C₂×#16863348	C₂	semidirect
12. C₂×#444×C₆₇	C₂	semidirect
14. #16864193	C₄	semidirect
16. C₂×#14524075	C₄	semidirect
17. C₂×#14524072	C₄	semidirect
18. C₂×#14524070	C₄	semidirect
19. C₂×#14524067	C₄	semidirect
23. #16864161	C₂²	semidirect
25. C₂×#14524051	C₂²	semidirect
26. C₂×#14524050	C₂²	semidirect
28. C₂×#14524049	C₂²	semidirect
35. #16863519	C₂²	semidirect
36. #16863515	C₂²	semidirect
37. #16863513	C₂²	semidirect
38. #16863512	C₂²	semidirect
39. #16863511	C₂²	semidirect
40. #16863443	C₂²	semidirect
41. C₂×#14523979	C₂²	semidirect
42. C₂×#14523978	C₂²	semidirect
43. C₂×#14523971	C₂²	semidirect
44. C₂×#14523970	C₂²	semidirect
45. C₂×#14523969	C₂²	semidirect
46. #104×D₆₇	C₂²	semidirect
47. #16863348	C₂²	semidirect
48. C₂×#14523954	C₂²	semidirect
49. C₂×C₈×Dic₆₇	C₂²	semidirect
50. C₂×C₈×Dic₆₇	C₂²	semidirect
52. C₂×#131×C₆₇	C₂²	semidirect
53. #444×C₆₇	C₂²	semidirect
54. C₂×#108×C₆₇	C₂²	semidirect
55. C₂×#104×C₆₇	C₂²	semidirect
56. C₂×#97×C₆₇	C₂²	semidirect
72. #14524075	C₂×C₄	semidirect
73. #14524072	C₂×C₄	semidirect
74. #14524070	C₂×C₄	semidirect
75. #14524067	C₂×C₄	semidirect
76. #14524065	C₂×C₄	semidirect
78. #14524063	C₂×C₄	semidirect
80. C₂×Dic₂₆₈	C₂×C₄	semidirect
81. C₂×#11761618	C₂×C₄	semidirect
82. C₂×#11761617	C₂×C₄	semidirect
83. C₂×#11761616	C₂×C₄	semidirect
84. C₂×#11761615	C₂×C₄	semidirect
85. C₂×D₅₃₆	C₂×C₄	semidirect
86. C₂×#11761613	C₂×C₄	semidirect
89. #14524051	C₂³	semidirect
91. #14524050	C₂³	semidirect
94. #14524049	C₂³	semidirect
122. #14523979	C₂³	semidirect
123. #14523978	C₂³	semidirect
124. #14523971	C₂³	semidirect
125. #14523970	C₂³	semidirect
126. #14523969	C₂³	semidirect
127. #14523954	C₂³	semidirect
128. C₈×Dic₆₇	C₂³	semidirect
129. C₈×Dic₆₇	C₂³	semidirect
131. #135×C₆₇	C₂×C₄	semidirect
133. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
134. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
135. C₂×D₈×C₆₇	C₂×C₄	semidirect
137. #131×C₆₇	C₂³	semidirect
144. #108×C₆₇	C₂³	semidirect
145. #104×C₆₇	C₂³	semidirect
146. #97×C₆₇	C₂³	semidirect
190. Dic₂₆₈	C₂²×C₄	semidirect
191. #11761618	C₂²×C₄	semidirect
192. #11761617	C₂²×C₄	semidirect
193. #11761616	C₂²×C₄	semidirect
194. #11761615	C₂²×C₄	semidirect
195. D₅₃₆	C₂²×C₄	semidirect
196. #11761613	C₂²×C₄	semidirect
232. Q₁₆×C₆₇	C₂²×C₄	semidirect
233. SD₁₆×C₆₇	C₂²×C₄	semidirect
234. D₈×C₆₇	C₂²×C₄	semidirect
335. C₂×C₆₇	#2956	semidirect
344. C₂×#444	D₆₇	semidirect
351. C₆₇	C₂×#2956	semidirect
369. C₂×#131	C₂×D₆₇	semidirect
378. #444	C₂×D₆₇	semidirect
385. C₂×#108	C₂×D₆₇	semidirect
390. C₂×#104	C₂×D₆₇	semidirect
393. C₂×#97	C₂×D₆₇	semidirect
420. #135	C₄×D₆₇	semidirect
424. C₂×Q₁₆	C₄×D₆₇	semidirect
428. C₂×SD₁₆	C₄×D₆₇	semidirect
431. C₂×D₈	C₄×D₆₇	semidirect
441. #131	C₂²×D₆₇	semidirect
477. #108	C₂²×D₆₇	semidirect
481. #104	C₂²×D₆₇	semidirect
483. #97	C₂²×D₆₇	semidirect
541. Q₁₆	C₂×C₄×D₆₇	semidirect
545. SD₁₆	C₂×C₄×D₆₇	semidirect
548. D₈	C₂×C₄×D₆₇	semidirect
675. C₂	#22569117	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16864193	2
C₂×#16864161	2
#22569117	2
C₂×#16863519	2
C₂×#16863515	2
C₂×#16863513	2
C₂×#16863512	2
C₂×#16863511	2
C₂×#16863443	2
C₂×#104×D₆₇	2
C₂×#16863348	2
C₂×#444×C₆₇	2
C₂×#2956	67

Subgroups — C2×#22569117 #31720142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569117 #31720142