Subgroups — C₂×#22569125 #31720151

Order 17152.

2642 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524065	C₂	1	1	yes	maximal, characteristic
2	C₂×#16864162	C₂	1	2	yes	maximal, complemented
3	#22569125	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16863517	C₂	1	4	yes	maximal, complemented
5	C₂×#16863512	C₂	1	4	yes	maximal, complemented
6	C₂×#16863412	C₂	1	2	yes	maximal, complemented
7	C₂×#16863351	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11761628	C₂²	1	2	yes	—
10	C₂×#14524065	C₄	1	2	yes	complemented
11	C₂×#14524065	C₄	1	2	yes	complemented
12	C₂×#14524065	C₄	1	8	yes	complemented
13	C₂×#14524065	C₂²	1	2	yes	—
14	C₂×#14524065	C₂²	1	2	yes	—
15	C₂×#14524065	C₂²	1	8	yes	—
16	C₂²×Dic₂₆₈	C₄	1	1	yes	characteristic, complemented
17	C₂²×#11761617	C₄	1	2	yes	complemented
18	C₂²×D₅₃₆	C₄	1	1	yes	characteristic, complemented
19	#16864162	—	2	8	no	—
20	#16864162	C₂²	1	16	yes	complemented
21	C₂×#14524051	C₂²	1	8	yes	complemented
22	C₂×#14524049	—	2	4	no	—
23	C₂²×#11761612	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×D₆₇	—	2	4	no	—
25	C₂×#14524002	C₂²	1	1	yes	characteristic
26	#16863517	C₂²	1	32	yes	complemented
27	#16863512	C₂²	1	32	yes	complemented
28	#16863412	C₂²	1	16	yes	complemented
29	C₂×#14523969	C₂²	1	8	yes	complemented
30	#16863351	C₂²	1	8	yes	complemented
31	C₂×#14523953	C₂²	1	2	yes	complemented
32	C₂×C₈×Dic₆₇	C₂²	1	2	yes	complemented
33	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
34	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#108×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#11761628	—	2	2	no	—
39	C₂×#11761628	—	2	4	no	—
40	C₂×#11761628	—	2	2	no	—
41	C₂×#11761628	—	2	4	no	—
42	C₂×#11761628	C₂×C₄	1	4	yes	—
43	C₂×#11761628	C₂×C₄	1	8	yes	—
44	C₂×#11761628	C₂³	1	4	yes	—
45	C₂×#11761628	C₂³	1	8	yes	—
46	C₂²×Dic₁₃₄	C₂×C₄	1	2	yes	—
47	C₂²×D₂₆₈	C₂×C₄	1	2	yes	—
48	C₂²×C₂²⋊D₆₇	—	2	2	no	—
49	C₂²×C₄×D₆₇	—	2	2	no	—
50	#14524065	C₂×C₄	1	8	yes	—
51	#14524065	C₂×C₄	1	8	yes	—
52	#14524065	C₂×C₄	1	16	yes	complemented
53	#14524065	C₂×C₄	1	16	yes	complemented
54	#14524065	C₂³	1	8	yes	—
55	#14524065	C₂³	1	8	yes	—
56	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
57	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
58	C₂×Dic₂₆₈	C₂×C₄	1	8	yes	complemented
59	C₂×#11761617	C₂×C₄	1	4	yes	—
60	C₂×#11761617	C₂×C₄	1	4	yes	—
61	C₂×#11761617	C₂×C₄	1	16	yes	complemented
62	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
63	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
64	C₂×D₅₃₆	C₂×C₄	1	8	yes	complemented
65	#14524051	—	2	16	no	—
66	#14524051	C₂³	1	32	yes	complemented
67	#14524049	—	2	32	no	—
68	C₂×#11761612	D₄	1	2	yes	—
69	C₂×#11761612	D₄	1	2	yes	—
70	C₂×#11761612	D₄	1	4	yes	—
71	C₂×#11761612	C₂³	1	4	yes	—
72	C₂×#11761611	—	2	8	no	—
73	M₁₆×D₆₇	—	2	32	no	—
74	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
75	C₂×C₈×D₆₇	—	2	8	no	—
76	#14524002	—	2	2	no	—
77	#14524002	C₂³	1	4	yes	—
78	C₂×C₄⋊Dic₆₇	—	2	1	no	—
79	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
80	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
81	#14523969	C₂³	1	32	yes	complemented
82	#14523953	C₂³	1	8	yes	complemented
83	C₈×Dic₆₇	C₂³	1	8	yes	complemented
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₆₇	—	2	2	no	—
86	C₂×M₁₆×C₆₇	—	2	4	no	—
87	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
88	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
89	C₂²×C₈×C₆₇	—	2	2	no	—
90	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
98	#108×C₆₇	C₂³	1	8	yes	complemented
99	#108×C₆₇	C₂³	1	8	yes	complemented
100	C₂³×D₆₇	—	2	2	no	—
101	#11761628	—	2	8	no	—
102	#11761628	—	2	8	no	—
103	#11761628	—	2	8	no	—
104	#11761628	—	2	8	no	—
105	#11761628	—	2	4	no	—
106	#11761628	—	2	4	no	—
107	#11761628	—	2	8	no	—
108	#11761628	C₂²×C₄	1	8	yes	—
109	#11761628	C₂²×C₄	1	16	yes	—
110	#11761628	C₂⁴	1	8	yes	—
111	C₂×Dic₁₃₄	—	2	2	no	—
112	C₂×Dic₁₃₄	—	2	4	no	—
113	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
114	C₂×Dic₁₃₄	C₂²×C₄	1	8	yes	—
115	C₂×D₂₆₈	—	2	2	no	—
116	C₂×D₂₆₈	—	2	4	no	—
117	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
118	C₂×D₂₆₈	C₂²×C₄	1	8	yes	—
119	C₂×C₂²⋊D₆₇	—	2	4	no	—
120	C₂×C₂²⋊D₆₇	—	2	8	no	—
121	C₂×C₂²⋊D₆₇	—	2	4	no	—
122	C₂×C₂²⋊D₆₇	—	2	8	no	—
123	C₂²×Dic₆₇	—	2	2	no	—
124	C₂²×Dic₆₇	—	2	2	no	—
125	C₂×C₄×D₆₇	—	2	4	no	—
126	C₂×C₄×D₆₇	—	2	8	no	—
127	C₂×C₄×D₆₇	—	2	4	no	—
128	C₂×C₄×D₆₇	—	2	8	no	—
129	Dic₂₆₈	C₂²×C₄	1	8	yes	—
130	Dic₂₆₈	C₂²×C₄	1	8	yes	—
131	#11761617	C₂²×C₄	1	16	yes	—
132	#11761617	C₂²×C₄	1	16	yes	—
133	D₅₃₆	C₂²×C₄	1	8	yes	—
134	D₅₃₆	C₂²×C₄	1	8	yes	—
135	#11761612	C₂×D₄	1	8	yes	—
136	#11761612	C₂×D₄	1	8	yes	—
137	#11761611	—	2	32	no	—
138	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
139	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
140	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
141	C₈×D₆₇	—	2	32	no	—
142	C₄⋊Dic₆₇	—	2	4	no	—
143	C₂²⋊Dic₆₇	—	2	4	no	—
144	C₄×Dic₆₇	—	2	2	no	—
145	C₄×Dic₆₇	C₂⁴	1	4	yes	—
146	C₂⁴×C₆₇	#41	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	#41	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
157	M₁₆×C₆₇	—	2	8	no	—
158	M₁₆×C₆₇	—	2	4	no	—
159	M₁₆×C₆₇	C₂⁴	1	8	yes	—
160	C₂×C₈×C₆₇	—	2	4	no	—
161	C₂×C₈×C₆₇	—	2	8	no	—
162	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
165	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
168	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
169	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
170	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
171	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
173	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
176	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂³×C₆₇	C₄×D₄	1	2	yes	—
178	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂³×C₆₇	C₂×#41	1	2	yes	—
180	C₂²×D₆₇	—	2	4	no	—
181	C₂²×D₆₇	—	2	8	no	—
182	C₂²×D₆₇	—	8	4	no	—
183	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
186	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
200	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
201	C₂²⋊D₆₇	—	2	16	no	—
202	C₂²⋊D₆₇	—	2	8	no	—
203	C₂²⋊D₆₇	—	2	8	no	—
204	C₂×Dic₆₇	—	2	8	no	—
205	C₂×Dic₆₇	—	2	4	no	—
206	C₂×Dic₆₇	—	2	4	no	—
207	C₂×Dic₆₇	—	4	4	no	—
208	D₂₆₈	—	2	4	no	—
209	D₂₆₈	—	2	8	no	—
210	D₂₆₈	C₂³×C₄	1	8	yes	—
211	C₄×D₆₇	—	2	8	no	—
212	C₄×D₆₇	—	2	16	no	—
213	C₄×D₆₇	—	2	8	no	—
214	Dic₁₃₄	—	2	4	no	—
215	Dic₁₃₄	—	2	8	no	—
216	Dic₁₃₄	C₂³×C₄	1	8	yes	—
217	C₈×C₆₇	—	2	8	no	—
218	C₈×C₆₇	C₂³×C₄	1	2	yes	—
219	C₈×C₆₇	C₂³×C₄	1	2	yes	—
220	C₈×C₆₇	C₂³×C₄	1	2	yes	—
221	C₈×C₆₇	C₂³×C₄	1	2	yes	—
222	C₆₇⋊C₈	C₂²×D₄	1	8	yes	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
229	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
230	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
232	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
233	C₂×D₆₇	—	2	8	no	—
234	C₂×D₆₇	—	4	8	no	—
235	C₂×D₆₇	—	8	8	no	—
236	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
237	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
238	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
239	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
240	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
241	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
242	Dic₆₇	—	2	8	no	—
243	Dic₆₇	—	4	4	no	—
244	C₂×#2957	—	67	1	no	maximal, Sylow
245	C₂×C₆₇	—	2	1	no	—
246	C₂×C₆₇	—	2	2	no	—
247	C₂×C₆₇	—	2	1	no	—
248	C₂×C₆₇	—	2	2	no	—
249	C₂×C₆₇	#2957	1	2	yes	complemented
250	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
251	D₆₇	—	4	8	no	—
252	C₂²×#135	—	67	1	no	—
253	C₂×#568	—	67	2	no	—
254	#2957	—	67	16	no	—
255	C₂×#444	—	67	4	no	—
256	C₂×#443	—	67	4	no	—
257	C₂²×#108	D₆₇	1	1	yes	characteristic, complemented
258	C₂×#421	—	67	2	no	—
259	C₂×#405	—	67	1	no	—
260	C₆₇	C₂×#2957	1	1	yes	characteristic, complemented, Sylow
261	C₂²×#41	—	67	2	no	—
262	C₂×#135	—	67	2	no	—
263	C₂×#135	—	67	2	no	—
264	C₂×#135	—	67	8	no	—
265	C₂×#135	—	67	2	no	—
266	C₂×#135	—	67	2	no	—
267	C₂×#135	—	67	8	no	—
268	C₂²×Q₁₆	—	67	1	no	—
269	C₂²×SD₁₆	—	67	2	no	—
270	C₂²×D₈	—	67	1	no	—
271	#568	—	67	16	no	—
272	#568	—	134	8	no	—
273	C₂×#131	—	67	8	no	—
274	C₂×#131	—	134	4	no	—
275	C₂²×M₁₆	—	67	1	no	—
276	C₂²×M₁₆	—	134	4	no	—
277	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
278	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
279	C₂×#117	—	67	1	no	—
280	#444	—	67	32	no	—
281	#443	—	67	32	no	—
282	C₂×#108	C₂×D₆₇	1	2	yes	complemented
283	C₂×#108	C₂×D₆₇	1	2	yes	complemented
284	C₂×#108	C₂×D₆₇	1	8	yes	complemented
285	#421	—	67	16	no	—
286	C₂×#104	—	67	8	no	—
287	#405	—	67	8	no	—
288	C₂×#97	—	67	2	no	—
289	C₂×C₄×C₈	—	67	2	no	—
290	C₂×#41	—	67	4	no	—
291	C₂×#41	—	67	8	no	—
292	C₂×#41	—	67	4	no	—
293	C₂×#41	—	67	8	no	—
294	C₂×#41	—	134	2	no	—
295	C₂×#41	—	134	4	no	—
296	C₂×#41	—	134	2	no	—
297	C₂×#41	—	134	4	no	—
298	C₂²×Q₈	—	67	2	no	—
299	C₂²×D₄	—	67	2	no	—
300	C₂²×D₄	—	134	2	no	—
301	C₂³×C₄	—	134	2	no	—
302	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
303	#135	—	67	8	no	—
304	#135	—	67	8	no	—
305	#135	—	67	16	no	—
306	#135	—	67	16	no	—
307	#135	—	67	8	no	—
308	#135	—	67	8	no	—
309	C₂×Q₁₆	—	67	2	no	—
310	C₂×Q₁₆	—	67	2	no	—
311	C₂×Q₁₆	—	67	8	no	—
312	C₂×SD₁₆	—	67	4	no	—
313	C₂×SD₁₆	—	67	4	no	—
314	C₂×SD₁₆	—	67	16	no	—
315	C₂×D₈	—	67	2	no	—
316	C₂×D₈	—	67	2	no	—
317	C₂×D₈	—	67	8	no	—
318	#131	—	67	32	no	—
319	#131	—	134	32	no	—
320	#131	—	134	16	no	—
321	C₂×M₁₆	—	2	2	no	—
322	C₂×M₁₆	—	2	4	no	—
323	C₂×M₁₆	—	67	2	no	—
324	C₂×M₁₆	—	67	2	no	—
325	C₂×M₁₆	—	67	4	no	—
326	C₂×M₁₆	—	67	4	no	—
327	C₂×M₁₆	—	134	8	no	—
328	C₂×M₁₆	—	134	32	no	—
329	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
330	C₂×M₁₆	C₂²×D₆₇	1	8	yes	—
331	C₂²×C₈	—	2	2	no	—
332	C₂²×C₈	—	67	2	no	—
333	C₂²×C₈	—	134	8	no	—
334	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
335	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
336	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
337	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
338	C₂²×C₈	C₄×D₆₇	1	4	yes	—
339	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
340	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
341	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
342	#117	—	67	4	no	—
343	#117	—	134	2	no	—
344	C₂×C₄⋊C₄	—	134	1	no	—
345	C₂×C₂²⋊C₄	—	134	1	no	—
346	C₂×C₄²	—	67	2	no	—
347	#108	C₂²×D₆₇	1	8	yes	complemented
348	#108	C₂²×D₆₇	1	8	yes	complemented
349	#104	—	67	32	no	—
350	#97	—	67	8	no	—
351	C₄×C₈	—	67	8	no	—
352	C₂⁴	—	134	2	no	—
353	C₂⁴	#11761629	1	1	yes	characteristic
354	#41	—	67	8	no	—
355	#41	—	67	16	no	—
356	#41	—	67	8	no	—
357	#41	—	134	8	no	—
358	#41	—	134	8	no	—
359	#41	—	134	8	no	—
360	#41	—	134	8	no	—
361	#41	—	134	4	no	—
362	#41	—	134	4	no	—
363	#41	—	134	8	no	—
364	C₂×Q₈	—	67	4	no	—
365	C₂×Q₈	—	67	8	no	—
366	C₂×Q₈	—	134	2	no	—
367	C₂×Q₈	—	134	4	no	—
368	C₂×D₄	—	67	4	no	—
369	C₂×D₄	—	67	8	no	—
370	C₂×D₄	—	134	4	no	—
371	C₂×D₄	—	134	8	no	—
372	C₂×D₄	—	134	4	no	—
373	C₂×D₄	—	134	8	no	—
374	C₂×D₄	—	134	2	no	—
375	C₂×D₄	—	134	4	no	—
376	C₂²×C₄	—	134	2	no	—
377	C₂²×C₄	—	134	2	no	—
378	C₂²×C₄	—	134	4	no	—
379	C₂²×C₄	—	134	8	no	—
380	C₂²×C₄	—	134	8	no	—
381	C₂²×C₄	—	134	4	no	—
382	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
385	C₂²×C₄	D₄×D₆₇	1	2	yes	—
386	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
387	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
388	C₂²×C₄	#11761629	1	1	yes	characteristic
389	C₂²×C₄	#11761629	1	2	yes	—
390	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
391	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
392	Q₁₆	—	67	8	no	—
393	Q₁₆	—	67	8	no	—
394	SD₁₆	—	67	16	no	—
395	SD₁₆	—	67	16	no	—
396	D₈	—	67	8	no	—
397	D₈	—	67	8	no	—
398	M₁₆	—	2	8	no	—
399	M₁₆	—	2	4	no	—
400	M₁₆	—	67	8	no	—
401	M₁₆	—	67	8	no	—
402	M₁₆	—	134	32	no	—
403	M₁₆	C₂³×D₆₇	1	8	yes	—
404	C₂×C₈	—	2	8	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	67	4	no	—
407	C₂×C₈	—	67	4	no	—
408	C₂×C₈	—	67	4	no	—
409	C₂×C₈	—	134	32	no	—
410	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
411	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
412	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
413	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
414	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
415	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
416	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
417	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
418	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
419	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
420	C₂×C₈	C₂³×D₆₇	1	2	yes	—
421	C₂×C₈	C₂³×D₆₇	1	2	yes	—
422	C₄⋊C₄	—	134	4	no	—
423	C₂²⋊C₄	—	134	4	no	—
424	C₄²	—	67	4	no	—
425	C₄²	—	134	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	536	4	no	—
428	C₂³	—	134	4	no	—
429	C₂³	—	134	8	no	—
430	C₂³	#14524005	1	1	yes	characteristic
431	C₂³	#14524005	1	1	yes	characteristic
432	C₂³	#14524005	1	2	yes	—
433	C₂³	C₂×#11761629	1	1	yes	characteristic
434	C₂³	C₂×#11761629	1	2	yes	—
435	Q₈	—	67	8	no	—
436	Q₈	—	134	8	no	—
437	Q₈	—	134	4	no	—
438	D₄	—	67	8	no	—
439	D₄	—	134	16	no	—
440	D₄	—	134	8	no	—
441	D₄	—	134	8	no	—
442	D₄	—	134	8	no	—
443	D₄	—	134	4	no	—
444	C₂×C₄	—	268	4	no	—
445	C₂×C₄	—	134	4	no	—
446	C₂×C₄	—	134	8	no	—
447	C₂×C₄	—	134	4	no	—
448	C₂×C₄	—	134	16	no	—
449	C₂×C₄	—	134	8	no	—
450	C₂×C₄	—	134	8	no	—
451	C₂×C₄	#14524005	1	1	yes	characteristic
452	C₂×C₄	#14524005	1	2	yes	—
453	C₂×C₄	#14524005	1	2	yes	—
454	C₂×C₄	#14524005	1	1	yes	characteristic
455	C₂×C₄	#14524005	1	2	yes	—
456	C₂×C₄	#14524005	1	2	yes	—
457	C₂×C₄	#14524005	1	2	yes	—
458	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
459	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
461	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
463	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
465	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
466	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
467	C₂×C₄	C₂×#11761629	1	2	yes	—
468	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
469	C₈	—	2	8	no	—
470	C₈	—	67	8	no	—
471	C₈	C₂²×C₄×D₆₇	1	2	yes	—
472	C₈	C₂²×C₄×D₆₇	1	2	yes	—
473	C₈	C₂²×C₄×D₆₇	1	2	yes	—
474	C₈	C₂²×C₄×D₆₇	1	2	yes	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	268	8	no	—
481	C₂²	—	536	8	no	—
482	C₂²	—	134	8	no	—
483	C₂²	C₂×#14524005	1	2	yes	—
484	C₂²	C₂×#14524005	1	2	yes	—
485	C₂²	C₂×#14524005	1	1	yes	characteristic
486	C₂²	C₂×#14524005	1	1	yes	characteristic
487	C₂²	C₂²×#11761629	1	1	yes	characteristic
488	C₄	—	268	4	no	—
489	C₄	—	134	8	no	—
490	C₄	C₂×#14524005	1	2	yes	—
491	C₄	C₂×#14524005	1	1	yes	characteristic
492	C₄	C₂×#14524005	1	2	yes	—
493	C₄	C₂×#14524005	1	1	yes	characteristic
494	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
495	C₄	C₂²×D₄×D₆₇	1	1	yes	characteristic
496	C₂	—	2	1	no	—
497	C₂	—	2	2	no	—
498	C₂	—	2	1	no	—
499	C₂	—	2	2	no	—
500	C₂	—	268	8	no	—
501	C₂	#22569125	1	2	yes	direct product factor
502	C₂	C₂²×#14524005	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×#14524065	C₂
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂×#2957
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
2. C₂×#16864162	C₂	semidirect
3. #22569125	C₂	direct (N × H)
4. C₂×#16863517	C₂	semidirect
5. C₂×#16863512	C₂	semidirect
6. C₂×#16863412	C₂	semidirect
7. C₂×#16863351	C₂	semidirect
8. C₂²×#108×C₆₇	C₂	semidirect
10. C₂×#14524065	C₄	semidirect
11. C₂×#14524065	C₄	semidirect
12. C₂×#14524065	C₄	semidirect
16. C₂²×Dic₂₆₈	C₄	semidirect
17. C₂²×#11761617	C₄	semidirect
18. C₂²×D₅₃₆	C₄	semidirect
20. #16864162	C₂²	semidirect
21. C₂×#14524051	C₂²	semidirect
26. #16863517	C₂²	semidirect
27. #16863512	C₂²	semidirect
28. #16863412	C₂²	semidirect
29. C₂×#14523969	C₂²	semidirect
30. #16863351	C₂²	semidirect
31. C₂×#14523953	C₂²	semidirect
32. C₂×C₈×Dic₆₇	C₂²	semidirect
35. C₂×#108×C₆₇	C₂²	semidirect
36. C₂×#108×C₆₇	C₂²	semidirect
37. C₂×#108×C₆₇	C₂²	semidirect
52. #14524065	C₂×C₄	semidirect
53. #14524065	C₂×C₄	semidirect
58. C₂×Dic₂₆₈	C₂×C₄	semidirect
61. C₂×#11761617	C₂×C₄	semidirect
64. C₂×D₅₃₆	C₂×C₄	semidirect
66. #14524051	C₂³	semidirect
81. #14523969	C₂³	semidirect
82. #14523953	C₂³	semidirect
83. C₈×Dic₆₇	C₂³	semidirect
98. #108×C₆₇	C₂³	semidirect
99. #108×C₆₇	C₂³	semidirect
249. C₂×C₆₇	#2957	semidirect
257. C₂²×#108	D₆₇	semidirect
260. C₆₇	C₂×#2957	semidirect
282. C₂×#108	C₂×D₆₇	semidirect
283. C₂×#108	C₂×D₆₇	semidirect
284. C₂×#108	C₂×D₆₇	semidirect
347. #108	C₂²×D₆₇	semidirect
348. #108	C₂²×D₆₇	semidirect
501. C₂	#22569125	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14524065	2
C₂×#16864162	2
#22569125	2
C₂×#16863517	2
C₂×#16863512	2
C₂×#16863412	2
C₂×#16863351	2
C₂²×#108×C₆₇	2
C₂×#2957	67

Subgroups — C2×#22569125 #31720151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569125 #31720151