Subgroups — C₂×#22560256 #30855480

Order 17152.

1314 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524049	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863378	C₂	1	2	yes	maximal
4	#22560256	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₆₇⋊#338	C₂	1	2	yes	maximal, complemented
6	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761628	C₂²	1	1	yes	characteristic
8	C₂×#14524049	—	2	2	no	—
9	C₂×#14524049	—	2	2	no	—
10	C₂×#14524049	—	2	2	no	—
11	C₂×#14524049	—	2	2	no	—
12	C₂×#14524049	C₄	1	4	yes	—
13	C₂×#14524049	C₂²	1	4	yes	—
14	C₂²×#11761612	C₂²	1	2	yes	—
15	C₂²×#11761612	C₂²	1	1	yes	characteristic
16	C₂²×#11761611	C₄	1	2	yes	—
17	C₂²×C₈×D₆₇	C₄	1	2	yes	—
18	C₂×#14523979	—	2	2	no	—
19	C₂×#14523979	—	2	2	no	—
20	C₂×#14523979	C₂²	1	4	yes	complemented
21	#16863378	—	2	4	no	—
22	#16863378	C₂²	1	8	yes	—
23	C₂×#14523961	—	2	2	no	—
24	C₂×#14523959	—	2	2	no	—
25	C₆₇⋊#338	C₂²	1	8	yes	complemented
26	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
27	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#108×C₆₇	—	2	2	no	—
29	C₂×#108×C₆₇	—	2	2	no	—
30	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
31	C₂×#11761628	—	2	1	no	—
32	C₂×#11761628	—	2	1	no	—
33	C₂×#11761628	—	2	1	no	—
34	C₂×#11761628	—	2	1	no	—
35	C₂×#11761628	—	2	1	no	—
36	C₂×#11761628	—	2	1	no	—
37	C₂×#11761628	—	2	1	no	—
38	C₂×#11761628	—	2	1	no	—
39	C₂×#11761628	C₂×C₄	1	2	yes	—
40	C₂×#11761628	D₄	1	2	yes	—
41	C₂×#11761628	Q₈	1	2	yes	—
42	C₂×#11761628	C₂³	1	2	yes	—
43	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
44	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
45	C₂²×C₂²⋊D₆₇	Q₈	1	2	yes	—
46	C₂²×C₄×D₆₇	C₂×C₄	1	2	yes	—
47	#14524049	—	2	8	no	—
48	#14524049	—	2	8	no	—
49	#14524049	—	2	8	no	—
50	#14524049	—	2	8	no	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761612	—	2	1	no	—
53	C₂×#11761612	—	2	1	no	—
54	C₂×#11761612	—	2	1	no	—
55	C₂×#11761612	—	2	2	no	—
56	C₂×#11761612	—	2	2	no	—
57	C₂×#11761612	—	2	2	no	—
58	C₂×#11761612	—	2	2	no	—
59	C₂×#11761612	—	2	2	no	—
60	C₂×#11761612	C₂×C₄	1	2	yes	—
61	C₂×#11761612	C₂³	1	4	yes	—
62	C₂×#11761612	C₂³	1	2	yes	—
63	C₂×#11761611	—	2	4	no	—
64	C₂×#11761611	—	2	4	no	—
65	C₂×#11761611	C₂×C₄	1	8	yes	—
66	C₂²×C₆₇⋊C₈	—	2	2	no	—
67	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
68	C₂×C₈×D₆₇	—	2	4	no	—
69	C₂×C₈×D₆₇	—	2	4	no	—
70	C₂×C₈×D₆₇	C₂×C₄	1	8	yes	—
71	#14523979	—	2	8	no	—
72	#14523961	—	2	8	no	—
73	#14523959	—	2	8	no	—
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₆₇	—	2	2	no	—
76	C₂×M₁₆×C₆₇	—	2	2	no	—
77	C₂×M₁₆×C₆₇	—	2	2	no	—
78	C₂×M₁₆×C₆₇	—	2	2	no	—
79	C₂×M₁₆×C₆₇	—	2	2	no	—
80	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
81	C₂²×C₈×C₆₇	—	2	2	no	—
82	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
85	C₂²×C₈×C₆₇	D₄	1	2	yes	—
86	C₂²×C₈×C₆₇	D₄	1	2	yes	—
87	C₂²×C₈×C₆₇	D₄	1	2	yes	—
88	C₂²×C₈×C₆₇	D₄	1	2	yes	—
89	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
90	#108×C₆₇	—	2	8	no	—
91	C₂³×D₆₇	C₄⋊C₄	1	2	yes	—
92	#11761628	—	2	2	no	—
93	#11761628	—	2	2	no	—
94	#11761628	—	2	2	no	—
95	#11761628	—	2	2	no	—
96	#11761628	—	2	2	no	—
97	#11761628	—	2	2	no	—
98	#11761628	—	2	2	no	—
99	#11761628	—	2	2	no	—
100	#11761628	—	2	2	no	—
101	#11761628	—	2	2	no	—
102	#11761628	—	2	2	no	—
103	#11761628	—	2	2	no	—
104	#11761628	—	2	2	no	—
105	#11761628	—	2	2	no	—
106	#11761628	—	2	2	no	—
107	#11761628	—	2	2	no	—
108	C₂×Dic₁₃₄	—	2	1	no	—
109	C₂×Dic₁₃₄	—	2	1	no	—
110	C₂×Dic₁₃₄	—	2	1	no	—
111	C₂×Dic₁₃₄	—	2	1	no	—
112	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
113	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
114	C₂×D₂₆₈	—	2	1	no	—
115	C₂×D₂₆₈	—	2	1	no	—
116	C₂×D₂₆₈	—	2	1	no	—
117	C₂×D₂₆₈	—	2	1	no	—
118	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
119	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
120	C₂×C₂²⋊D₆₇	—	2	2	no	—
121	C₂×C₂²⋊D₆₇	—	2	2	no	—
122	C₂×C₂²⋊D₆₇	—	2	2	no	—
123	C₂×C₂²⋊D₆₇	—	2	2	no	—
124	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	4	yes	—
125	C₂×C₂²⋊D₆₇	C₂×Q₈	1	4	yes	—
126	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
127	C₂×C₄×D₆₇	—	2	2	no	—
128	C₂×C₄×D₆₇	—	2	2	no	—
129	C₂×C₄×D₆₇	—	2	2	no	—
130	C₂×C₄×D₆₇	—	2	2	no	—
131	C₂×C₄×D₆₇	C₄⋊C₄	1	4	yes	—
132	C₂×C₄×D₆₇	C₂²×C₄	1	4	yes	—
133	#11761612	—	2	2	no	—
134	#11761612	—	2	2	no	—
135	#11761612	—	2	2	no	—
136	#11761612	—	2	2	no	—
137	#11761612	—	2	4	no	—
138	#11761612	—	2	4	no	—
139	#11761612	—	4	2	no	—
140	#11761612	—	4	2	no	—
141	#11761611	—	2	16	no	—
142	C₂×C₆₇⋊C₈	—	2	2	no	—
143	C₂×C₆₇⋊C₈	—	2	2	no	—
144	C₂×C₆₇⋊C₈	—	2	4	no	—
145	C₂×C₆₇⋊C₈	—	4	2	no	—
146	C₂×C₆₇⋊C₈	—	4	2	no	—
147	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
148	C₈×D₆₇	—	2	16	no	—
149	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
164	M₁₆×C₆₇	—	2	4	no	—
165	M₁₆×C₆₇	—	2	4	no	—
166	M₁₆×C₆₇	—	4	2	no	—
167	M₁₆×C₆₇	—	4	2	no	—
168	C₂×C₈×C₆₇	—	2	4	no	—
169	C₂×C₈×C₆₇	—	4	2	no	—
170	C₂×C₈×C₆₇	—	4	2	no	—
171	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
172	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
173	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
174	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
175	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
176	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
177	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
178	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
179	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
180	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
181	C₂³×C₆₇	—	4	2	no	—
182	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
185	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
186	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
187	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
188	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
189	C₂²×D₆₇	—	2	2	no	—
190	C₂²×D₆₇	—	2	2	no	—
191	C₂²×D₆₇	—	4	4	no	—
192	C₂²×D₆₇	C₂×C₄⋊C₄	1	4	yes	—
193	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
200	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
201	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
202	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
203	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
204	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
205	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
206	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
207	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
208	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
209	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
210	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
221	C₂²⋊D₆₇	—	2	4	no	—
222	C₂²⋊D₆₇	—	2	4	no	—
223	C₂²⋊D₆₇	—	2	4	no	—
224	C₂²⋊D₆₇	—	2	4	no	—
225	C₂×Dic₆₇	—	2	2	no	—
226	C₂×Dic₆₇	—	2	2	no	—
227	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
228	D₂₆₈	—	2	2	no	—
229	D₂₆₈	—	2	2	no	—
230	D₂₆₈	—	2	2	no	—
231	D₂₆₈	—	2	2	no	—
232	C₄×D₆₇	—	2	4	no	—
233	C₄×D₆₇	—	2	4	no	—
234	C₄×D₆₇	—	2	4	no	—
235	C₄×D₆₇	—	2	4	no	—
236	Dic₁₃₄	—	2	2	no	—
237	Dic₁₃₄	—	2	2	no	—
238	Dic₁₃₄	—	2	2	no	—
239	Dic₁₃₄	—	2	2	no	—
240	C₈×C₆₇	—	4	4	no	—
241	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
242	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
243	C₆₇⋊C₈	—	2	4	no	—
244	C₆₇⋊C₈	—	4	4	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	2	no	—
248	C₂²×C₆₇	—	4	1	no	—
249	C₂²×C₆₇	—	4	1	no	—
250	C₂²×C₆₇	—	4	1	no	—
251	C₂²×C₆₇	—	4	1	no	—
252	C₂²×C₆₇	#380	1	1	yes	characteristic
253	C₂²×C₆₇	#380	1	1	yes	characteristic
254	C₂²×C₆₇	#380	1	1	yes	characteristic
255	C₂²×C₆₇	#380	1	1	yes	characteristic
256	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
257	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
259	C₂×D₆₇	—	2	4	no	—
260	C₂×D₆₇	—	4	4	no	—
261	C₂×D₆₇	—	4	4	no	—
262	C₂×D₆₇	—	4	4	no	—
263	C₄×C₆₇	#380	1	1	yes	characteristic
264	C₄×C₆₇	#380	1	1	yes	characteristic
265	C₄×C₆₇	#380	1	1	yes	characteristic
266	C₄×C₆₇	#380	1	1	yes	characteristic
267	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
268	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
269	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
270	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
271	Dic₆₇	—	2	4	no	—
272	C₂×#1813	—	67	1	no	maximal, Sylow
273	C₂×C₆₇	—	2	1	no	—
274	C₂×C₆₇	—	2	1	no	—
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	2	1	no	—
277	C₂×C₆₇	—	2	1	no	—
278	C₂×C₆₇	—	2	1	no	—
279	C₂×C₆₇	#1813	1	2	yes	complemented
280	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
281	D₆₇	—	4	4	no	—
282	C₂²×#131	—	67	1	no	—
283	C₂²×#108	—	67	1	no	—
284	C₂²×#108	D₆₇	1	1	yes	characteristic, complemented
285	C₂×#413	—	67	2	no	—
286	#1813	—	67	8	no	—
287	C₂×#338	—	67	2	no	—
288	C₆₇	C₂×#1813	1	1	yes	characteristic, complemented, Sylow
289	C₂²×#41	—	67	1	no	—
290	C₂×#131	—	67	4	no	—
291	C₂×#131	—	67	4	no	—
292	C₂×#131	—	134	2	no	—
293	C₂×#131	—	134	2	no	—
294	C₂×#131	—	134	2	no	—
295	C₂×#131	—	134	2	no	—
296	C₂²×M₁₆	—	67	1	no	—
297	C₂²×M₁₆	—	67	2	no	—
298	C₂²×M₁₆	—	67	2	no	—
299	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
300	C₂³×C₈	—	67	2	no	—
301	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
302	C₂×#108	—	2	2	no	—
303	C₂×#108	—	2	2	no	—
304	C₂×#108	—	67	4	no	—
305	C₂×#108	—	134	2	no	—
306	C₂×#108	—	134	2	no	—
307	C₂×#108	C₂×D₆₇	1	4	yes	complemented
308	#413	—	67	8	no	—
309	#413	—	134	4	no	—
310	C₂×#101	—	134	2	no	—
311	C₂×#100	—	134	2	no	—
312	#338	—	67	8	no	—
313	C₂×#41	—	67	2	no	—
314	C₂×#41	—	67	2	no	—
315	C₂×#41	—	67	2	no	—
316	C₂×#41	—	67	2	no	—
317	C₂×#41	—	134	1	no	—
318	C₂×#41	—	134	1	no	—
319	C₂×#41	—	134	1	no	—
320	C₂×#41	—	134	1	no	—
321	C₂×#41	—	134	1	no	—
322	C₂×#41	—	134	1	no	—
323	C₂×#41	—	134	1	no	—
324	C₂×#41	—	134	1	no	—
325	C₂²×Q₈	—	67	1	no	—
326	C₂²×D₄	—	67	2	no	—
327	C₂²×D₄	—	67	1	no	—
328	C₂³×C₄	—	67	2	no	—
329	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
330	#131	—	134	8	no	—
331	#131	—	134	8	no	—
332	#131	—	134	8	no	—
333	#131	—	134	8	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	—	2	2	no	—
336	C₂×M₁₆	—	2	2	no	—
337	C₂×M₁₆	—	2	2	no	—
338	C₂×M₁₆	—	2	2	no	—
339	C₂×M₁₆	—	67	2	no	—
340	C₂×M₁₆	—	67	2	no	—
341	C₂×M₁₆	—	67	8	no	—
342	C₂×M₁₆	—	67	4	no	—
343	C₂×M₁₆	—	134	2	no	—
344	C₂×M₁₆	—	134	2	no	—
345	C₂×M₁₆	—	134	2	no	—
346	C₂×M₁₆	—	134	2	no	—
347	C₂×M₁₆	—	134	2	no	—
348	C₂×M₁₆	—	134	1	no	—
349	C₂×M₁₆	—	134	1	no	—
350	C₂×M₁₆	—	134	1	no	—
351	C₂×M₁₆	—	134	1	no	—
352	C₂×M₁₆	—	134	4	no	—
353	C₂×M₁₆	—	134	4	no	—
354	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
355	C₂²×C₈	—	2	2	no	—
356	C₂²×C₈	—	67	2	no	—
357	C₂²×C₈	—	67	8	no	—
358	C₂²×C₈	—	134	2	no	—
359	C₂²×C₈	—	134	4	no	—
360	C₂²×C₈	—	134	4	no	—
361	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
362	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
363	C₂²×C₈	C₄×D₆₇	1	2	yes	—
364	C₂²×C₈	D₂₆₈	1	2	yes	—
365	C₂²×C₈	D₂₆₈	1	2	yes	—
366	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
367	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
368	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
369	#108	—	2	8	no	—
370	#108	—	134	8	no	—
371	#101	—	134	8	no	—
372	#100	—	134	8	no	—
373	C₂⁴	—	67	2	no	—
374	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
375	#41	—	134	2	no	—
376	#41	—	134	2	no	—
377	#41	—	134	2	no	—
378	#41	—	134	2	no	—
379	#41	—	134	2	no	—
380	#41	—	134	2	no	—
381	#41	—	134	2	no	—
382	#41	—	134	2	no	—
383	#41	—	134	2	no	—
384	#41	—	134	2	no	—
385	#41	—	134	2	no	—
386	#41	—	134	2	no	—
387	#41	—	134	2	no	—
388	#41	—	134	2	no	—
389	#41	—	134	2	no	—
390	#41	—	134	2	no	—
391	C₂×Q₈	—	67	2	no	—
392	C₂×Q₈	—	67	2	no	—
393	C₂×Q₈	—	134	1	no	—
394	C₂×Q₈	—	134	1	no	—
395	C₂×Q₈	—	134	1	no	—
396	C₂×Q₈	—	134	1	no	—
397	C₂×D₄	—	67	4	no	—
398	C₂×D₄	—	67	4	no	—
399	C₂×D₄	—	67	2	no	—
400	C₂×D₄	—	67	2	no	—
401	C₂×D₄	—	134	2	no	—
402	C₂×D₄	—	134	2	no	—
403	C₂×D₄	—	134	2	no	—
404	C₂×D₄	—	134	2	no	—
405	C₂×D₄	—	134	1	no	—
406	C₂×D₄	—	134	1	no	—
407	C₂×D₄	—	134	1	no	—
408	C₂×D₄	—	134	1	no	—
409	C₂²×C₄	—	67	2	no	—
410	C₂²×C₄	—	67	4	no	—
411	C₂²×C₄	—	67	4	no	—
412	C₂²×C₄	—	134	2	no	—
413	C₂²×C₄	—	134	2	no	—
414	C₂²×C₄	—	134	2	no	—
415	C₂²×C₄	—	134	2	no	—
416	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
418	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
419	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
424	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
425	C₂²×C₄	#11761627	1	1	yes	characteristic
426	C₂²×C₄	#11761627	1	1	yes	characteristic
427	C₂²×C₄	#11761629	1	1	yes	characteristic
428	C₂²×C₄	#11761629	1	1	yes	characteristic
429	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
430	M₁₆	—	2	4	no	—
431	M₁₆	—	2	4	no	—
432	M₁₆	—	4	2	no	—
433	M₁₆	—	4	2	no	—
434	M₁₆	—	268	2	no	—
435	M₁₆	—	268	2	no	—
436	M₁₆	—	134	4	no	—
437	M₁₆	—	134	4	no	—
438	M₁₆	—	134	2	no	—
439	M₁₆	—	134	2	no	—
440	M₁₆	—	134	2	no	—
441	M₁₆	—	134	2	no	—
442	M₁₆	—	134	16	no	—
443	C₂×C₈	—	2	4	no	—
444	C₂×C₈	—	4	2	no	—
445	C₂×C₈	—	4	2	no	—
446	C₂×C₈	—	268	2	no	—
447	C₂×C₈	—	268	2	no	—
448	C₂×C₈	—	67	4	no	—
449	C₂×C₈	—	134	4	no	—
450	C₂×C₈	—	134	2	no	—
451	C₂×C₈	—	134	2	no	—
452	C₂×C₈	—	134	16	no	—
453	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
454	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
455	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
456	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
457	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
458	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
459	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
460	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
461	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
462	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
463	C₂³	—	4	2	no	—
464	C₂³	—	268	4	no	—
465	C₂³	—	67	4	no	—
466	C₂³	—	134	2	no	—
467	C₂³	—	134	2	no	—
468	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
469	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
470	C₂³	#14524019	1	1	yes	characteristic
471	C₂³	#14524019	1	1	yes	characteristic
472	C₂³	#14524023	1	1	yes	characteristic
473	C₂³	#14524023	1	1	yes	characteristic
474	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
475	Q₈	—	134	2	no	—
476	Q₈	—	134	2	no	—
477	Q₈	—	134	2	no	—
478	Q₈	—	134	2	no	—
479	D₄	—	134	4	no	—
480	D₄	—	134	4	no	—
481	D₄	—	134	4	no	—
482	D₄	—	134	4	no	—
483	D₄	—	134	2	no	—
484	D₄	—	134	2	no	—
485	D₄	—	134	2	no	—
486	D₄	—	134	2	no	—
487	C₂×C₄	—	67	4	no	—
488	C₂×C₄	—	134	2	no	—
489	C₂×C₄	—	134	2	no	—
490	C₂×C₄	—	134	4	no	—
491	C₂×C₄	—	134	4	no	—
492	C₂×C₄	—	134	4	no	—
493	C₂×C₄	—	134	4	no	—
494	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
495	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
496	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
497	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
498	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
499	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
500	C₂×C₄	#14524001	1	1	yes	characteristic
501	C₂×C₄	#14524001	1	1	yes	characteristic
502	C₂×C₄	#14524001	1	1	yes	characteristic
503	C₂×C₄	#14524001	1	1	yes	characteristic
504	C₂×C₄	#14524014	1	1	yes	characteristic
505	C₂×C₄	#14524014	1	1	yes	characteristic
506	C₂×C₄	#14524014	1	1	yes	characteristic
507	C₂×C₄	#14524014	1	1	yes	characteristic
508	C₂×C₄	#14524018	1	1	yes	characteristic
509	C₂×C₄	#14524018	1	1	yes	characteristic
510	C₂×C₄	#14524018	1	1	yes	characteristic
511	C₂×C₄	#14524018	1	1	yes	characteristic
512	C₂×C₄	#14524019	1	1	yes	characteristic
513	C₂×C₄	#14524019	1	1	yes	characteristic
514	C₂×C₄	#14524023	1	1	yes	characteristic
515	C₂×C₄	#14524023	1	1	yes	characteristic
516	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
517	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
518	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
520	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
521	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
522	C₈	—	4	4	no	—
523	C₈	—	268	4	no	—
524	C₈	—	134	4	no	—
525	C₈	C₂×(C₂×D₆₇)⋊C₄	1	4	yes	—
526	C₈	C₂×(C₂×D₆₇)⋊C₄	1	4	yes	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	268	4	no	—
535	C₂²	—	268	4	no	—
536	C₂²	—	268	4	no	—
537	C₂²	—	134	4	no	—
538	C₂²	#16863253	1	1	yes	characteristic
539	C₂²	#16863253	1	1	yes	characteristic
540	C₂²	#16863253	1	1	yes	characteristic
541	C₂²	#16863253	1	1	yes	characteristic
542	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
543	C₂²	C₂×#14524019	1	1	yes	characteristic
544	C₂²	C₂×#14524023	1	1	yes	characteristic
545	C₄	—	134	4	no	—
546	C₄	#16863253	1	1	yes	characteristic
547	C₄	#16863253	1	1	yes	characteristic
548	C₄	#16863253	1	1	yes	characteristic
549	C₄	#16863253	1	1	yes	characteristic
550	C₄	C₂²×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
551	C₄	C₂×#14524001	1	1	yes	characteristic
552	C₄	C₂×#14524014	1	1	yes	characteristic
553	C₄	C₂×#14524018	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	2	1	no	—
557	C₂	—	2	1	no	—
558	C₂	—	2	1	no	—
559	C₂	—	2	1	no	—
560	C₂	—	268	4	no	—
561	C₂	#22560256	1	2	yes	direct product factor
562	C₂	C₂×#16863253	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×#11761628	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂×#1813
LCS	G ▷ C₂×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. C₂²×#14523979	C₂	semidirect
4. #22560256	C₂	direct (N × H)
5. C₂×C₆₇⋊#338	C₂	semidirect
6. C₂²×#108×C₆₇	C₂	semidirect
20. C₂×#14523979	C₂²	semidirect
25. C₆₇⋊#338	C₂²	semidirect
30. C₂×#108×C₆₇	C₂²	semidirect
279. C₂×C₆₇	#1813	semidirect
284. C₂²×#108	D₆₇	semidirect
288. C₆₇	C₂×#1813	semidirect
307. C₂×#108	C₂×D₆₇	semidirect
561. C₂	#22560256	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×#14524049	2
C₂²×#14523979	2
C₂×#16863378	2
#22560256	2
C₂×C₆₇⋊#338	2
C₂²×#108×C₆₇	2
C₂×#1813	67

Subgroups — C2×#22560256 #30855480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560256 #30855480