Subgroups — #22573871

Order 8576.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864065	C₂	1	2	yes	maximal, complemented
3	#16864053	C₂	1	2	yes	maximal, complemented
4	#16864040	C₂	1	6	yes	maximal, complemented
5	D₄×C₂²⋊D₆₇	C₂	1	6	yes	maximal, complemented
6	#16863974	C₂	1	3	yes	maximal, complemented
7	#16863963	C₂	1	3	yes	maximal, complemented
8	#16863903	C₂	1	3	yes	maximal, complemented
9	C₂×#14524014	C₂	1	3	yes	maximal, complemented
10	#16863879	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761629	C₂²	1	1	yes	characteristic, complemented
13	C₂×#11761628	C₂²	1	3	yes	complemented
14	C₂×#11761627	C₂²	1	3	yes	complemented
15	#41×D₆₇	—	2	8	no	—
16	C₂×Q₈×D₆₇	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
18	C₂×D₄×D₆₇	C₂²	1	3	yes	complemented
19	C₂²×C₄×D₆₇	C₂²	1	3	yes	complemented
20	#14524039	C₂²	1	6	yes	complemented
21	#14524030	C₂²	1	6	yes	complemented
22	#14524024	C₂²	1	6	yes	complemented
23	#14524022	C₂²	1	6	yes	complemented
24	#14524019	C₂²	1	1	yes	characteristic, complemented
25	#14524019	C₂²	1	3	yes	complemented
26	#14524017	C₂²	1	6	yes	complemented
27	#14524015	C₂²	1	12	yes	complemented
28	#14524014	C₂²	1	6	yes	complemented
29	#14524014	C₂²	1	6	yes	complemented
30	#14524014	C₂²	1	3	yes	complemented
31	#14524014	C₂²	1	6	yes	complemented
32	#14524014	C₂²	1	3	yes	complemented
33	#14524012	C₂²	1	12	yes	complemented
34	#14524011	C₂²	1	6	yes	complemented
35	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
36	D₄×Dic₆₇	C₂²	1	6	yes	complemented
37	C₄×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
38	C₄×C₂²⋊D₆₇	C₂²	1	6	yes	complemented
39	#14524002	C₂²	1	3	yes	complemented
40	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	complemented
41	C₂×C₂²⋊Dic₆₇	C₂²	1	6	yes	complemented
42	H(5,𝔽₂)×C₆₇	—	2	8	no	—
43	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#41×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#41×C₆₇	C₂²	1	3	yes	complemented
46	C₂²×D₄×C₆₇	C₂²	1	6	yes	complemented
47	C₂²×D₄×C₆₇	C₂²	1	3	yes	complemented
48	C₂³×D₆₇	C₂³	1	3	yes	—
49	#11761629	—	2	4	no	—
50	#11761628	—	2	12	no	—
51	#11761627	—	2	12	no	—
52	C₂×Dic₁₃₄	C₂³	1	3	yes	—
53	Q₈×D₆₇	—	2	4	no	—
54	C₂×D₂₆₈	C₂³	1	3	yes	—
55	C₂×C₂²⋊D₆₇	—	4	12	no	—
56	C₂×C₂²⋊D₆₇	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₆₇	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₆₇	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₆₇	C₂³	1	12	yes	—
60	D₄×D₆₇	—	2	12	no	—
61	C₂²×Dic₆₇	C₂³	1	3	yes	—
62	C₂²×Dic₆₇	C₂³	1	6	yes	—
63	C₂×C₄×D₆₇	—	2	12	no	—
64	C₂×C₄×D₆₇	C₂³	1	3	yes	—
65	C₂×C₄×D₆₇	C₂³	1	3	yes	—
66	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₆₇	C₂³	1	3	yes	—
68	C₄⋊Dic₆₇	C₂³	1	3	yes	—
69	C₄⋊Dic₆₇	C₂³	1	6	yes	—
70	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
71	Dic₆₇⋊C₄	C₂³	1	2	yes	—
72	Dic₆₇⋊C₄	C₂³	1	6	yes	—
73	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
74	C₂²⋊Dic₆₇	C₂³	1	12	yes	—
75	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
76	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
77	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
78	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
79	C₄×Dic₆₇	C₂³	1	6	yes	—
80	C₂⁴×C₆₇	C₂³	1	6	yes	—
81	#41×C₆₇	—	2	8	no	—
82	#41×C₆₇	—	2	12	no	—
83	#41×C₆₇	D₄	1	8	yes	complemented
84	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₆₇	—	2	24	no	—
87	C₂×D₄×C₆₇	—	2	12	no	—
88	C₂×D₄×C₆₇	C₂³	1	6	yes	—
89	C₂×D₄×C₆₇	C₂³	1	3	yes	—
90	C₂×D₄×C₆₇	C₂³	1	6	yes	—
91	C₂×D₄×C₆₇	C₂³	1	3	yes	—
92	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
93	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
94	C₂³×C₆₇	—	2	12	no	—
95	C₂³×C₆₇	—	4	6	no	—
96	C₂³×C₆₇	—	4	6	no	—
97	C₂³×C₆₇	C₂⁴	1	6	yes	—
98	C₂³×C₆₇	C₂⁴	1	3	yes	—
99	C₂²×D₆₇	—	2	6	no	—
100	C₂²×D₆₇	—	2	6	no	—
101	C₂²×D₆₇	—	4	3	no	—
102	C₂²×D₆₇	C₂⁴	1	3	yes	—
103	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
104	Q₈×C₆₇	—	2	2	no	—
105	Q₈×C₆₇	C₂×D₄	1	4	yes	complemented
106	D₄×C₆₇	—	2	12	no	—
107	D₄×C₆₇	—	2	12	no	—
108	D₄×C₆₇	—	2	6	no	—
109	D₄×C₆₇	C₂×D₄	1	12	yes	complemented
110	C₂×C₄×C₆₇	—	2	12	no	—
111	C₂×C₄×C₆₇	C₂×D₄	1	12	yes	complemented
112	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
113	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
114	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₆₇	—	2	12	no	—
116	C₂²⋊D₆₇	—	2	12	no	—
117	C₂²⋊D₆₇	—	4	12	no	—
118	C₂²⋊D₆₇	—	4	6	no	—
119	C₂×Dic₆₇	—	2	6	no	—
120	C₂×Dic₆₇	—	4	6	no	—
121	C₂×Dic₆₇	C₂⁴	1	2	yes	—
122	C₂×Dic₆₇	C₂⁴	1	6	yes	—
123	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₆₇	C₂⁴	1	3	yes	—
125	D₂₆₈	—	2	6	no	—
126	C₄×D₆₇	—	2	2	no	—
127	C₄×D₆₇	—	2	6	no	—
128	C₄×D₆₇	—	2	6	no	—
129	C₄×D₆₇	—	2	6	no	—
130	Dic₁₃₄	—	2	6	no	—
131	C₂²×C₆₇	—	2	3	no	—
132	C₂²×C₆₇	—	2	3	no	—
133	C₂²×C₆₇	—	2	6	no	—
134	C₂²×C₆₇	—	2	6	no	—
135	C₂²×C₆₇	—	2	6	no	—
136	C₂²×C₆₇	—	8	12	no	—
137	C₂²×C₆₇	C₂²×D₄	1	6	yes	—
138	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
139	C₂×D₆₇	—	2	1	no	—
140	C₂×D₆₇	—	2	1	no	—
141	C₂×D₆₇	—	2	3	no	—
142	C₂×D₆₇	—	2	3	no	—
143	C₂×D₆₇	—	2	3	no	—
144	C₂×D₆₇	—	4	12	no	—
145	C₂×D₆₇	H(5,𝔽₂)	1	2	yes	complemented
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	C₂²×D₄	1	6	yes	—
148	C₄×C₆₇	C₂²×D₄	1	2	yes	—
149	Dic₆₇	—	2	3	no	—
150	Dic₆₇	—	2	1	no	—
151	Dic₆₇	—	2	6	no	—
152	Dic₆₇	—	2	2	no	—
153	C₂×C₆₇	—	2	6	no	—
154	C₂×C₆₇	—	4	6	no	—
155	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
156	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₆₇	#585	1	1	yes	characteristic
158	D₆₇	—	2	2	no	—
159	D₆₇	—	4	3	no	—
160	#3488	—	67	1	no	maximal, Sylow
161	C₆₇	#3488	1	1	yes	characteristic, complemented, Sylow
162	C₂×H(5,𝔽₂)	D₆₇	1	1	yes	characteristic, complemented
163	C₂²×#41	—	67	1	no	—
164	#550	—	67	2	no	—
165	#548	—	67	2	no	—
166	#546	—	67	6	no	—
167	D₄²	—	67	6	no	—
168	#537	—	67	3	no	—
169	#535	—	67	3	no	—
170	#525	—	67	3	no	—
171	C₂×#121	—	67	3	no	—
172	#520	—	67	1	no	—
173	H(5,𝔽₂)	—	2	8	no	—
174	C₂×#41	—	67	1	no	—
175	C₂×#41	—	67	3	no	—
176	C₂×#41	—	67	3	no	—
177	C₂×#41	—	134	8	no	—
178	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
179	C₂×#41	C₂×D₆₇	1	3	yes	complemented
180	C₂×#41	C₂×D₆₇	1	2	yes	complemented
181	C₂²×Q₈	—	67	1	no	—
182	C₂²×D₄	—	67	6	no	—
183	C₂²×D₄	—	67	3	no	—
184	C₂²×D₄	C₂×D₆₇	1	3	yes	complemented
185	C₂²×D₄	C₂×D₆₇	1	6	yes	complemented
186	C₂³×C₄	—	67	3	no	—
187	#127	—	67	6	no	—
188	#124	—	67	6	no	—
189	#123	—	67	6	no	—
190	C₂²⋊Q₈	—	67	1	no	—
191	C₂²⋊Q₈	—	67	3	no	—
192	C₂²⋊Q₈	—	67	6	no	—
193	#121	—	67	3	no	—
194	#121	—	67	6	no	—
195	#121	—	67	6	no	—
196	#121	—	67	6	no	—
197	#121	—	67	3	no	—
198	#121	—	67	12	no	—
199	#121	—	67	6	no	—
200	#120	—	67	12	no	—
201	#120	—	67	6	no	—
202	C₄×Q₈	—	67	2	no	—
203	C₄×D₄	—	67	6	no	—
204	C₄×D₄	—	67	6	no	—
205	C₄×D₄	—	67	6	no	—
206	#117	—	67	3	no	—
207	C₂×C₄⋊C₄	—	67	3	no	—
208	C₂×C₂²⋊C₄	—	67	6	no	—
209	C₂⁴	—	67	3	no	—
210	C₂⁴	C₂²×D₆₇	1	6	yes	—
211	#41	—	2	8	no	—
212	#41	—	2	12	no	—
213	#41	—	134	4	no	—
214	#41	—	134	12	no	—
215	#41	—	134	12	no	—
216	#41	C₂²⋊D₆₇	1	8	yes	complemented
217	C₂×Q₈	—	67	3	no	—
218	C₂×Q₈	—	134	4	no	—
219	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
220	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₂×D₄	—	2	24	no	—
222	C₂×D₄	—	2	12	no	—
223	C₂×D₄	—	268	12	no	—
224	C₂×D₄	—	67	6	no	—
225	C₂×D₄	—	67	12	no	—
226	C₂×D₄	—	67	6	no	—
227	C₂×D₄	—	67	6	no	—
228	C₂×D₄	—	67	3	no	—
229	C₂×D₄	—	134	12	no	—
230	C₂×D₄	C₂²×D₆₇	1	6	yes	—
231	C₂×D₄	C₂²×D₆₇	1	3	yes	—
232	C₂×D₄	C₂²×D₆₇	1	6	yes	—
233	C₂×D₄	C₂²×D₆₇	1	3	yes	—
234	C₂²×C₄	—	67	6	no	—
235	C₂²×C₄	—	67	3	no	—
236	C₂²×C₄	—	67	3	no	—
237	C₂²×C₄	—	67	3	no	—
238	C₂²×C₄	—	67	1	no	—
239	C₂²×C₄	—	67	3	no	—
240	C₂²×C₄	—	134	12	no	—
241	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
242	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
243	C₄⋊C₄	—	67	3	no	—
244	C₄⋊C₄	—	67	6	no	—
245	C₄⋊C₄	—	67	1	no	—
246	C₄⋊C₄	—	67	2	no	—
247	C₄⋊C₄	—	67	6	no	—
248	C₂²⋊C₄	—	67	6	no	—
249	C₂²⋊C₄	—	67	12	no	—
250	C₂²⋊C₄	—	67	6	no	—
251	C₂²⋊C₄	—	67	6	no	—
252	C₂²⋊C₄	—	67	2	no	—
253	C₂²⋊C₄	—	67	6	no	—
254	C₄²	—	67	6	no	—
255	C₂³	—	2	12	no	—
256	C₂³	—	4	6	no	—
257	C₂³	—	4	6	no	—
258	C₂³	—	268	3	no	—
259	C₂³	—	67	1	no	—
260	C₂³	—	67	3	no	—
261	C₂³	—	134	6	no	—
262	C₂³	—	134	6	no	—
263	C₂³	C₂³×D₆₇	1	6	yes	—
264	C₂³	C₂³×D₆₇	1	3	yes	—
265	Q₈	—	2	2	no	—
266	Q₈	—	134	6	no	—
267	Q₈	C₂×C₂²⋊D₆₇	1	4	yes	complemented
268	D₄	—	2	12	no	—
269	D₄	—	2	12	no	—
270	D₄	—	2	6	no	—
271	D₄	—	268	12	no	—
272	D₄	—	268	6	no	—
273	D₄	—	134	12	no	—
274	D₄	—	134	12	no	—
275	D₄	—	134	6	no	—
276	D₄	C₂×C₂²⋊D₆₇	1	12	yes	complemented
277	C₂×C₄	—	2	12	no	—
278	C₂×C₄	—	268	6	no	—
279	C₂×C₄	—	67	2	no	—
280	C₂×C₄	—	67	6	no	—
281	C₂×C₄	—	67	1	no	—
282	C₂×C₄	—	67	3	no	—
283	C₂×C₄	—	134	6	no	—
284	C₂×C₄	—	134	2	no	—
285	C₂×C₄	—	134	6	no	—
286	C₂×C₄	—	134	6	no	—
287	C₂×C₄	—	134	6	no	—
288	C₂×C₄	C₂×C₂²⋊D₆₇	1	12	yes	complemented
289	C₂×C₄	C₂³×D₆₇	1	2	yes	—
290	C₂×C₄	C₂³×D₆₇	1	3	yes	—
291	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
292	C₂²	—	2	3	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	6	no	—
295	C₂²	—	2	6	no	—
296	C₂²	—	2	6	no	—
297	C₂²	—	8	12	no	—
298	C₂²	—	268	12	no	—
299	C₂²	—	67	2	no	—
300	C₂²	—	134	3	no	—
301	C₂²	—	134	3	no	—
302	C₂²	—	134	3	no	—
303	C₂²	—	134	1	no	—
304	C₂²	—	134	1	no	—
305	C₂²	C₂²×C₂²⋊D₆₇	1	6	yes	—
306	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	134	1	no	—
309	C₄	—	134	3	no	—
310	C₄	—	134	6	no	—
311	C₄	—	134	2	no	—
312	C₄	C₂²×C₂²⋊D₆₇	1	6	yes	—
313	C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
314	C₂	—	2	6	no	—
315	C₂	—	4	6	no	—
316	C₂	—	268	3	no	—
317	C₂	—	134	2	no	—
318	C₂	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
319	C₂	H(5,𝔽₂)×D₆₇	1	1	yes	characteristic
320	C₂	#16864231	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₆₇	C₂
hypercenter	C₂×H(5,𝔽₂)	D₆₇
hypocenter	C₆₇	#3488
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₆₇	C₂	semidirect
2. #16864065	C₂	semidirect
3. #16864053	C₂	semidirect
4. #16864040	C₂	semidirect
5. D₄×C₂²⋊D₆₇	C₂	semidirect
6. #16863974	C₂	semidirect
7. #16863963	C₂	semidirect
8. #16863903	C₂	semidirect
9. C₂×#14524014	C₂	semidirect
10. #16863879	C₂	semidirect
11. C₂×H(5,𝔽₂)×C₆₇	C₂	semidirect
12. C₂×#11761629	C₂²	semidirect
13. C₂×#11761628	C₂²	semidirect
14. C₂×#11761627	C₂²	semidirect
16. C₂×Q₈×D₆₇	C₂²	semidirect
17. C₂²×C₂²⋊D₆₇	C₂²	semidirect
18. C₂×D₄×D₆₇	C₂²	semidirect
19. C₂²×C₄×D₆₇	C₂²	semidirect
20. #14524039	C₂²	semidirect
21. #14524030	C₂²	semidirect
22. #14524024	C₂²	semidirect
23. #14524022	C₂²	semidirect
24. #14524019	C₂²	semidirect
25. #14524019	C₂²	semidirect
26. #14524017	C₂²	semidirect
27. #14524015	C₂²	semidirect
28. #14524014	C₂²	semidirect
29. #14524014	C₂²	semidirect
30. #14524014	C₂²	semidirect
31. #14524014	C₂²	semidirect
32. #14524014	C₂²	semidirect
33. #14524012	C₂²	semidirect
34. #14524011	C₂²	semidirect
35. Q₈×Dic₆₇	C₂²	semidirect
36. D₄×Dic₆₇	C₂²	semidirect
37. C₄×C₂²⋊D₆₇	C₂²	semidirect
38. C₄×C₂²⋊D₆₇	C₂²	semidirect
39. #14524002	C₂²	semidirect
40. C₂×C₄⋊Dic₆₇	C₂²	semidirect
41. C₂×C₂²⋊Dic₆₇	C₂²	semidirect
44. C₂×#41×C₆₇	C₂²	semidirect
45. C₂×#41×C₆₇	C₂²	semidirect
46. C₂²×D₄×C₆₇	C₂²	semidirect
47. C₂²×D₄×C₆₇	C₂²	semidirect
83. #41×C₆₇	D₄	semidirect
105. Q₈×C₆₇	C₂×D₄	semidirect
109. D₄×C₆₇	C₂×D₄	semidirect
111. C₂×C₄×C₆₇	C₂×D₄	semidirect
145. C₂×D₆₇	H(5,𝔽₂)	semidirect
161. C₆₇	#3488	semidirect
162. C₂×H(5,𝔽₂)	D₆₇	semidirect
179. C₂×#41	C₂×D₆₇	semidirect
180. C₂×#41	C₂×D₆₇	semidirect
184. C₂²×D₄	C₂×D₆₇	semidirect
185. C₂²×D₄	C₂×D₆₇	semidirect
216. #41	C₂²⋊D₆₇	semidirect
267. Q₈	C₂×C₂²⋊D₆₇	semidirect
276. D₄	C₂×C₂²⋊D₆₇	semidirect
288. C₂×C₄	C₂×C₂²⋊D₆₇	semidirect