Subgroups — #22574084

Order 8576.

1232 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864128	C₂	1	4	yes	maximal, complemented
2	#16864066	C₂	1	4	yes	maximal, complemented
3	#16864033	C₂	1	8	yes	maximal, complemented
4	#16863999	C₂	1	4	yes	maximal, complemented
5	#16863960	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863949	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863941	C₂	1	4	yes	maximal, complemented
8	C₂×#14524039	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863871	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863870	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#535×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#11761627	C₂²	1	8	yes	complemented
14	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₆₇	C₂²	1	4	yes	—
16	#14524042	C₂²	1	2	yes	complemented
17	#14524039	C₂²	1	4	yes	—
18	#14524039	C₂²	1	2	yes	—
19	#14524039	C₂²	1	2	yes	—
20	#14524039	C₂²	1	2	yes	complemented
21	#14524034	C₂²	1	2	yes	complemented
22	#14524032	C₂²	1	2	yes	complemented
23	#14524029	C₂²	1	8	yes	complemented
24	#14524027	C₂²	1	8	yes	complemented
25	#14524025	C₂²	1	8	yes	complemented
26	#14524020	C₂²	1	4	yes	—
27	#14524017	C₂²	1	4	yes	complemented
28	#14524015	C₂²	1	8	yes	complemented
29	#14524015	C₂²	1	8	yes	complemented
30	#14524014	C₂²	1	8	yes	complemented
31	#14524011	C₂²	1	4	yes	—
32	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
33	C₄×Dic₁₃₄	C₂²	1	4	yes	complemented
34	C₄×D₂₆₈	C₂²	1	4	yes	complemented
35	D₄×Dic₆₇	C₂²	1	4	yes	—
36	D₄×Dic₆₇	C₂²	1	2	yes	complemented
37	D₄×Dic₆₇	C₂²	1	4	yes	complemented
38	D₄×Dic₆₇	C₂²	1	4	yes	complemented
39	#14524005	C₂²	1	4	yes	complemented
40	(C₂²⋊D₆₇)⋊C₄	C₂²	1	8	yes	complemented
41	#14524002	C₂²	1	1	yes	characteristic
42	#14524002	C₂²	1	1	yes	characteristic
43	#14524002	C₂²	1	2	yes	complemented
44	C₂×C₂²⋊Dic₆₇	C₂²	1	4	yes	—
45	C₂²⋊C₄×D₆₇	C₂²	1	4	yes	—
46	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄×Dic₆₇	C₂²	1	2	yes	complemented
48	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
50	#127×C₆₇	C₂²	1	2	yes	complemented
51	#124×C₆₇	C₂²	1	2	yes	complemented
52	#121×C₆₇	C₂²	1	4	yes	complemented
53	#120×C₆₇	C₂²	1	4	yes	—
54	#117×C₆₇	C₂²	1	1	yes	characteristic
55	C₂³×D₆₇	C₂³	1	2	yes	—
56	#11761627	—	4	16	no	—
57	C₂×Dic₁₃₄	C₂³	1	4	yes	—
58	C₂×D₂₆₈	—	2	4	no	—
59	C₂×D₂₆₈	C₂³	1	2	yes	—
60	C₂×D₂₆₈	C₂³	1	2	yes	—
61	C₂×C₂²⋊D₆₇	—	4	8	no	—
62	C₂×C₂²⋊D₆₇	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₆₇	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₆₇	C₂³	1	8	yes	—
65	C₂²×Dic₆₇	C₂³	1	2	yes	—
66	C₂²×Dic₆₇	C₂³	1	4	yes	—
67	C₂²×Dic₆₇	C₂³	1	8	yes	—
68	C₂×C₄×D₆₇	C₂³	1	8	yes	—
69	C₄⋊Dic₆₇	C₂³	1	2	yes	—
70	C₄⋊Dic₆₇	C₂³	1	2	yes	—
71	C₄⋊Dic₆₇	C₂³	1	4	yes	—
72	C₄⋊Dic₆₇	C₂³	1	4	yes	—
73	Dic₆₇⋊C₄	C₂³	1	4	yes	—
74	Dic₆₇⋊C₄	C₂³	1	4	yes	—
75	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
76	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
77	C₂²⋊Dic₆₇	C₂³	1	8	yes	—
78	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
79	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₆₇)⋊C₄	C₂³	1	8	yes	—
81	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
82	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₆₇	C₂³	1	2	yes	—
84	C₄×Dic₆₇	C₂³	1	2	yes	—
85	C₄×Dic₆₇	C₂³	1	4	yes	—
86	C₄×Dic₆₇	C₂³	1	2	yes	—
87	C₂⁴×C₆₇	C₂³	1	2	yes	—
88	#41×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	C₂³	1	2	yes	—
92	C₂×D₄×C₆₇	C₂³	1	2	yes	—
93	C₂×D₄×C₆₇	C₂³	1	2	yes	—
94	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₆₇	C₂³	1	8	yes	—
96	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
101	C₄²×C₆₇	C₂³	1	2	yes	—
102	C₂³×C₆₇	—	2	4	no	—
103	C₂³×C₆₇	—	4	4	no	—
104	C₂³×C₆₇	C₂⁴	1	4	yes	—
105	C₂³×C₆₇	C₂⁴	1	2	yes	—
106	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂²×D₆₇	—	2	4	no	—
108	C₂²×D₆₇	—	4	4	no	—
109	C₂²×D₆₇	C₂⁴	1	4	yes	—
110	Q₈×C₆₇	—	2	2	no	—
111	D₄×C₆₇	—	2	4	no	—
112	D₄×C₆₇	—	2	4	no	—
113	D₄×C₆₇	—	2	4	no	—
114	D₄×C₆₇	—	2	2	no	—
115	D₄×C₆₇	—	4	8	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	#41	1	4	yes	complemented
118	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
119	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
120	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₆₇	—	2	16	no	—
123	C₂²⋊D₆₇	—	4	8	no	—
124	C₂²⋊D₆₇	—	4	8	no	—
125	C₂×Dic₆₇	—	2	4	no	—
126	C₂×Dic₆₇	—	2	8	no	—
127	C₂×Dic₆₇	—	4	8	no	—
128	C₂×Dic₆₇	C₂⁴	1	4	yes	—
129	C₂×Dic₆₇	C₂⁴	1	2	yes	—
130	C₂×Dic₆₇	C₂⁴	1	4	yes	—
131	C₂×Dic₆₇	C₂⁴	1	2	yes	—
132	D₂₆₈	—	2	4	no	—
133	D₂₆₈	—	2	4	no	—
134	C₄×D₆₇	—	4	8	no	—
135	Dic₁₃₄	—	2	8	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	4	no	—
139	C₂²×C₆₇	—	2	4	no	—
140	C₂²×C₆₇	—	2	8	no	—
141	C₂²×C₆₇	—	8	4	no	—
142	C₂²×C₆₇	C₂×#41	1	2	yes	—
143	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
144	C₂×D₆₇	—	2	4	no	—
145	C₂×D₆₇	—	2	8	no	—
146	C₂×D₆₇	—	8	4	no	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	C₂×#41	1	2	yes	—
150	C₄×C₆₇	C₂×#41	1	2	yes	—
151	Dic₆₇	—	2	4	no	—
152	Dic₆₇	—	2	4	no	—
153	Dic₆₇	—	2	2	no	—
154	Dic₆₇	H(5,𝔽₂)	1	4	yes	—
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	—	4	2	no	—
157	C₂×C₆₇	—	4	4	no	—
158	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
159	C₂×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
160	D₆₇	—	4	4	no	—
161	#3508	—	67	1	no	maximal, Sylow
162	C₆₇	#3508	1	1	yes	characteristic, complemented, Sylow
163	#559	—	67	4	no	—
164	#550	—	67	4	no	—
165	#546	—	67	8	no	—
166	#540	—	67	4	no	—
167	#535	—	67	1	no	—
168	#535	D₆₇	1	1	yes	characteristic, complemented
169	#533	—	67	1	no	—
170	#532	—	67	4	no	—
171	C₂×#127	—	67	1	no	—
172	#518	—	67	1	no	—
173	#518	—	67	1	no	—
174	C₄×#41	—	67	1	no	—
175	C₂×#41	—	67	8	no	—
176	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
177	C₂²×D₄	—	67	4	no	—
178	C₂²×D₄	—	67	1	no	—
179	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	67	2	no	—
181	#127	—	67	2	no	—
182	#127	—	67	2	no	—
183	#127	—	67	4	no	—
184	#127	—	67	2	no	—
185	#127	C₂×D₆₇	1	2	yes	complemented
186	#125	—	67	2	no	—
187	#124	—	67	2	no	—
188	#124	—	67	8	no	—
189	#124	C₂×D₆₇	1	2	yes	complemented
190	#123	—	67	8	no	—
191	#123	—	67	8	no	—
192	C₂²⋊Q₈	—	67	4	no	—
193	#121	—	67	8	no	—
194	#121	—	67	8	no	—
195	#121	—	67	8	no	—
196	#121	—	67	4	no	—
197	#121	C₂×D₆₇	1	4	yes	complemented
198	#120	—	67	4	no	—
199	#120	C₂×D₆₇	1	4	yes	—
200	C₄×Q₈	—	67	2	no	—
201	C₄×Q₈	—	67	4	no	—
202	C₄×D₄	—	67	4	no	—
203	C₄×D₄	—	67	2	no	—
204	C₄×D₄	—	67	4	no	—
205	C₄×D₄	—	67	4	no	—
206	C₄×D₄	—	67	4	no	—
207	C₄×D₄	—	67	8	no	—
208	C₄×D₄	—	67	4	no	—
209	#117	—	67	1	no	—
210	#117	—	67	2	no	—
211	#117	—	67	1	no	—
212	#117	C₂×D₆₇	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	67	4	no	—
214	C₂×C₂²⋊C₄	—	67	4	no	—
215	C₂×C₄²	—	67	1	no	—
216	C₂×C₄²	—	67	2	no	—
217	C₂⁴	—	67	2	no	—
218	C₂⁴	C₂²×D₆₇	1	2	yes	—
219	#41	—	2	4	no	—
220	#41	—	268	16	no	—
221	C₂×Q₈	—	67	4	no	—
222	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	268	8	no	—
225	C₂×D₄	—	67	8	no	—
226	C₂×D₄	—	67	8	no	—
227	C₂×D₄	—	67	8	no	—
228	C₂×D₄	—	67	2	no	—
229	C₂×D₄	—	67	2	no	—
230	C₂×D₄	—	134	4	no	—
231	C₂×D₄	C₂²×D₆₇	1	2	yes	—
232	C₂×D₄	C₂²×D₆₇	1	2	yes	—
233	C₂×D₄	C₂²×D₆₇	1	2	yes	—
234	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₆₇	1	8	yes	—
236	C₂²×C₄	—	67	8	no	—
237	C₂²×C₄	—	67	8	no	—
238	C₂²×C₄	—	67	4	no	—
239	C₂²×C₄	—	67	2	no	—
240	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
241	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
242	C₄⋊C₄	—	67	2	no	—
243	C₄⋊C₄	—	67	2	no	—
244	C₄⋊C₄	—	67	4	no	—
245	C₄⋊C₄	—	67	4	no	—
246	C₄⋊C₄	—	67	4	no	—
247	C₄⋊C₄	—	67	4	no	—
248	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
249	C₂²⋊C₄	—	67	2	no	—
250	C₂²⋊C₄	—	67	2	no	—
251	C₂²⋊C₄	—	67	8	no	—
252	C₂²⋊C₄	—	67	4	no	—
253	C₂²⋊C₄	—	67	8	no	—
254	C₂²⋊C₄	—	67	8	no	—
255	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
256	C₂²⋊C₄	C₂²×D₆₇	1	8	yes	—
257	C₄²	—	67	1	no	—
258	C₄²	—	67	2	no	—
259	C₄²	—	67	1	no	—
260	C₄²	—	67	4	no	—
261	C₄²	—	67	2	no	—
262	C₄²	—	67	2	no	—
263	C₄²	C₂²×D₆₇	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	268	4	no	—
267	C₂³	—	67	4	no	—
268	C₂³	—	134	4	no	—
269	C₂³	C₂³×D₆₇	1	4	yes	—
270	C₂³	C₂³×D₆₇	1	2	yes	—
271	C₂³	C₂³×D₆₇	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	134	8	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	D₄	—	4	8	no	—
279	D₄	—	268	8	no	—
280	D₄	—	268	8	no	—
281	D₄	—	134	16	no	—
282	D₄	—	134	4	no	—
283	D₄	—	134	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	268	8	no	—
286	C₂×C₄	—	268	8	no	—
287	C₂×C₄	—	67	2	no	—
288	C₂×C₄	—	67	4	no	—
289	C₂×C₄	—	67	2	no	—
290	C₂×C₄	—	67	4	no	—
291	C₂×C₄	—	134	4	no	—
292	C₂×C₄	—	134	8	no	—
293	C₂×C₄	#11761627	1	4	yes	complemented
294	C₂×C₄	C₂³×D₆₇	1	2	yes	—
295	C₂×C₄	C₂³×D₆₇	1	4	yes	—
296	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	8	4	no	—
304	C₂²	—	536	4	no	—
305	C₂²	—	134	4	no	—
306	C₂²	—	134	8	no	—
307	C₂²	C₂×#11761627	1	2	yes	—
308	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	67	4	no	—
312	C₄	—	134	4	no	—
313	C₄	—	134	4	no	—
314	C₄	—	134	2	no	—
315	C₄	C₂×#11761627	1	2	yes	—
316	C₄	C₂×#11761627	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	4	2	no	—
320	C₂	—	268	4	no	—
321	C₂	C₂²×#11761627	1	1	yes	characteristic
322	C₂	H(5,𝔽₂)×D₆₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#535×C₆₇	C₂
hypercenter	#535	D₆₇
hypocenter	C₆₇	#3508
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #535

Complements

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

N	H ≅ G/N	type
1. #16864128	C₂	semidirect
2. #16864066	C₂	semidirect
3. #16864033	C₂	semidirect
4. #16863999	C₂	semidirect
5. #16863960	C₂	semidirect
6. #16863949	C₂	semidirect
7. #16863941	C₂	semidirect
8. C₂×#14524039	C₂	semidirect
9. #16863871	C₂	semidirect
10. #16863870	C₂	semidirect
11. #41×Dic₆₇	C₂	semidirect
12. #535×C₆₇	C₂	semidirect
13. C₂×#11761627	C₂²	semidirect
16. #14524042	C₂²	semidirect
20. #14524039	C₂²	semidirect
21. #14524034	C₂²	semidirect
22. #14524032	C₂²	semidirect
23. #14524029	C₂²	semidirect
24. #14524027	C₂²	semidirect
25. #14524025	C₂²	semidirect
27. #14524017	C₂²	semidirect
28. #14524015	C₂²	semidirect
29. #14524015	C₂²	semidirect
30. #14524014	C₂²	semidirect
32. Q₈×Dic₆₇	C₂²	semidirect
33. C₄×Dic₁₃₄	C₂²	semidirect
34. C₄×D₂₆₈	C₂²	semidirect
36. D₄×Dic₆₇	C₂²	semidirect
37. D₄×Dic₆₇	C₂²	semidirect
38. D₄×Dic₆₇	C₂²	semidirect
39. #14524005	C₂²	semidirect
40. (C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
43. #14524002	C₂²	semidirect
47. C₂×C₄×Dic₆₇	C₂²	semidirect
50. #127×C₆₇	C₂²	semidirect
51. #124×C₆₇	C₂²	semidirect
52. #121×C₆₇	C₂²	semidirect
117. C₂×C₄×C₆₇	#41	semidirect
162. C₆₇	#3508	semidirect
168. #535	D₆₇	semidirect
185. #127	C₂×D₆₇	semidirect
189. #124	C₂×D₆₇	semidirect
197. #121	C₂×D₆₇	semidirect
293. C₂×C₄	#11761627	semidirect