Subgroups — #22574977

Order 8576.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864231	C₂	1	2	yes	maximal, complemented
2	#16864213	C₂	1	1	yes	maximal, characteristic, complemented
3	#16864210	C₂	1	3	yes	maximal, complemented
4	#16864207	C₂	1	2	yes	maximal, complemented
5	#16864206	C₂	1	6	yes	maximal, complemented
6	#16864197	C₂	1	3	yes	maximal, complemented
7	#16864192	C₂	1	6	yes	maximal, complemented
8	C₂×#14524062	C₂	1	3	yes	maximal, complemented
9	#135×D₆₇	C₂	1	3	yes	maximal, complemented
10	#131×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#576×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#14524090	—	2	2	no	—
13	#14524090	C₂²	1	2	yes	complemented
14	#14524090	C₂²	1	6	yes	complemented
15	#14524088	—	2	6	no	—
16	#14524088	C₂²	1	6	yes	complemented
17	C₂×#11761627	—	2	6	no	—
18	C₂×#11761627	C₂²	1	6	yes	complemented
19	#41×D₆₇	—	2	2	no	—
20	#41×D₆₇	C₂²	1	1	yes	characteristic
21	#41×D₆₇	C₂²	1	6	yes	complemented
22	#14524076	C₂²	1	6	yes	complemented
23	#14524073	C₂²	1	6	yes	complemented
24	#14524072	C₂²	1	12	yes	complemented
25	#14524069	C₂²	1	6	yes	complemented
26	#14524067	C₂²	1	12	yes	complemented
27	#14524066	C₂²	1	1	yes	characteristic, complemented
28	#14524065	C₂²	1	3	yes	complemented
29	#14524064	C₂²	1	6	yes	complemented
30	#14524063	C₂²	1	6	yes	complemented
31	#14524062	C₂²	1	3	yes	complemented
32	#14524062	C₂²	1	6	yes	complemented
33	#14524062	C₂²	1	6	yes	complemented
34	#14524062	C₂²	1	6	yes	complemented
35	C₂×Dic₂₆₈	C₂²	1	3	yes	complemented
36	Q₁₆×D₆₇	C₂²	1	3	yes	complemented
37	C₂×#11761616	C₂²	1	6	yes	complemented
38	SD₁₆×D₆₇	C₂²	1	6	yes	complemented
39	D₈×D₆₇	C₂²	1	3	yes	complemented
40	#14524051	C₂²	1	3	yes	complemented
41	#14524050	C₂²	1	1	yes	characteristic, complemented
42	#14524049	C₂²	1	3	yes	complemented
43	M₁₆×D₆₇	C₂²	1	3	yes	complemented
44	C₂×C₈×D₆₇	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₆₇	C₂²	1	6	yes	complemented
47	#135×C₆₇	C₂²	1	3	yes	complemented
48	C₂×D₈×C₆₇	C₂²	1	3	yes	complemented
49	#131×C₆₇	C₂²	1	1	yes	characteristic, complemented
50	#11761629	D₄	1	1	yes	characteristic
51	#11761629	C₂³	1	2	yes	—
52	#11761628	—	2	6	no	—
53	#11761628	—	2	6	no	—
54	#11761628	C₂³	1	3	yes	—
55	#11761628	C₂³	1	6	yes	—
56	#11761627	—	2	6	no	—
57	#11761627	—	2	12	no	—
58	#11761627	—	2	6	no	—
59	#11761627	—	2	6	no	—
60	#11761627	—	2	6	no	—
61	#11761627	—	2	6	no	—
62	#11761627	D₄	1	3	yes	—
63	#11761627	C₂³	1	6	yes	—
64	#11761627	C₂³	1	12	yes	—
65	C₂×Dic₁₃₄	—	2	6	no	—
66	C₂×Dic₁₃₄	C₂³	1	6	yes	—
67	Q₈×D₆₇	—	2	2	no	—
68	Q₈×D₆₇	D₄	1	1	yes	characteristic
69	Q₈×D₆₇	C₂³	1	6	yes	—
70	C₂×C₂²⋊D₆₇	—	2	6	no	—
71	C₂×C₂²⋊D₆₇	—	2	12	no	—
72	D₄×D₆₇	—	2	6	no	—
73	D₄×D₆₇	—	2	6	no	—
74	D₄×D₆₇	D₄	1	3	yes	—
75	D₄×D₆₇	C₂³	1	6	yes	—
76	C₂²×Dic₆₇	—	2	6	no	—
77	C₂×C₄×D₆₇	—	2	6	no	—
78	C₂×C₄×D₆₇	C₂³	1	3	yes	—
79	Dic₂₆₈	C₂³	1	3	yes	—
80	Dic₂₆₈	C₂³	1	6	yes	—
81	#11761618	C₂³	1	6	yes	—
82	#11761617	C₂³	1	6	yes	—
83	#11761616	C₂³	1	6	yes	—
84	#11761616	C₂³	1	12	yes	—
85	#11761615	C₂³	1	2	yes	—
86	D₅₃₆	C₂³	1	1	yes	characteristic
87	#11761613	C₂³	1	6	yes	—
88	#11761612	C₂³	1	3	yes	—
89	#11761611	C₂³	1	6	yes	—
90	C₂×C₆₇⋊C₈	C₂³	1	3	yes	—
91	C₈×D₆₇	C₂³	1	1	yes	characteristic
92	C₈×D₆₇	C₂³	1	3	yes	—
93	C₈×D₆₇	C₂³	1	3	yes	—
94	C₈×D₆₇	C₂³	1	3	yes	—
95	#41×C₆₇	—	2	2	no	—
96	#41×C₆₇	C₂³	1	6	yes	—
97	#41×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₆₇	—	2	6	no	—
99	C₂×D₄×C₆₇	C₂³	1	6	yes	—
100	Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	SD₁₆×C₆₇	C₂³	1	6	yes	—
102	D₈×C₆₇	C₂³	1	3	yes	—
103	D₈×C₆₇	C₂³	1	6	yes	—
104	M₁₆×C₆₇	C₂³	1	3	yes	—
105	C₂×C₈×C₆₇	C₂³	1	3	yes	—
106	C₂³×C₆₇	—	2	6	no	—
107	C₂²×D₆₇	—	2	6	no	—
108	C₂²×D₆₇	C₂×D₄	1	3	yes	—
109	Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₆₇	C₂⁴	1	2	yes	—
111	D₄×C₆₇	—	2	6	no	—
112	D₄×C₆₇	—	2	6	no	—
113	D₄×C₆₇	C₂×D₄	1	3	yes	—
114	D₄×C₆₇	C₂⁴	1	6	yes	—
115	C₂×C₄×C₆₇	—	2	6	no	—
116	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
117	C₂²⋊D₆₇	—	2	6	no	—
118	C₂²⋊D₆₇	—	2	12	no	—
119	C₂²⋊D₆₇	—	2	6	no	—
120	C₂²⋊D₆₇	—	2	6	no	—
121	C₂²⋊D₆₇	C₂×D₄	1	6	yes	—
122	C₂×Dic₆₇	—	2	6	no	—
123	C₂×Dic₆₇	—	2	6	no	—
124	C₂×Dic₆₇	—	2	12	no	—
125	C₂×Dic₆₇	C₂×D₄	1	3	yes	—
126	D₂₆₈	—	2	2	no	—
127	D₂₆₈	C₂×D₄	1	3	yes	—
128	D₂₆₈	C₂⁴	1	2	yes	—
129	C₄×D₆₇	—	2	6	no	—
130	C₄×D₆₇	—	2	6	no	—
131	C₄×D₆₇	—	2	2	no	—
132	C₄×D₆₇	C₂×D₄	1	3	yes	—
133	C₄×D₆₇	C₂×D₄	1	3	yes	—
134	C₄×D₆₇	C₂⁴	1	1	yes	characteristic
135	C₄×D₆₇	C₂⁴	1	3	yes	—
136	Dic₁₃₄	—	2	6	no	—
137	Dic₁₃₄	—	2	6	no	—
138	Dic₁₃₄	C₂×D₄	1	3	yes	—
139	Dic₁₃₄	C₂⁴	1	6	yes	—
140	C₈×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₈×C₆₇	C₂⁴	1	3	yes	—
142	C₆₇⋊C₈	C₂⁴	1	1	yes	characteristic
143	C₆₇⋊C₈	C₂⁴	1	3	yes	—
144	C₂²×C₆₇	—	2	6	no	—
145	C₂²×C₆₇	—	8	6	no	—
146	C₂²×C₆₇	C₂²×D₄	1	3	yes	—
147	C₂×D₆₇	—	2	2	no	—
148	C₂×D₆₇	—	4	3	no	—
149	C₂×D₆₇	—	8	6	no	—
150	C₂×D₆₇	C₂²×D₄	1	3	yes	—
151	C₂×D₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	C₂²×D₄	1	3	yes	—
154	C₄×C₆₇	C₂⁵	1	1	yes	characteristic
155	Dic₆₇	—	2	6	no	—
156	Dic₆₇	C₂²×D₄	1	3	yes	—
157	Dic₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₆₇	—	2	3	no	—
159	C₂×C₆₇	—	4	6	no	—
160	C₂×C₆₇	C₂³×D₄	1	1	yes	characteristic
161	D₆₇	—	2	1	no	—
162	D₆₇	—	2	3	no	—
163	D₆₇	—	4	2	no	—
164	#3584	—	67	1	no	maximal, Sylow
165	C₆₇	#3584	1	1	yes	characteristic, complemented, Sylow
166	#585	—	67	2	no	—
167	#578	—	67	3	no	—
168	#578	—	67	1	no	—
169	#577	—	67	2	no	—
170	#577	—	67	6	no	—
171	#576	—	67	3	no	—
172	#576	D₆₇	1	1	yes	characteristic, complemented
173	#575	—	67	6	no	—
174	C₂×#135	—	67	3	no	—
175	C₂×#135	—	67	3	no	—
176	C₂×#131	—	67	1	no	—
177	2₋¹⁺⁴	—	67	6	no	—
178	2₋¹⁺⁴	—	67	2	no	—
179	2₋¹⁺⁴	—	134	2	no	—
180	H(5,𝔽₂)	—	67	6	no	—
181	H(5,𝔽₂)	—	134	6	no	—
182	H(5,𝔽₂)	C₂×D₆₇	1	2	yes	complemented
183	C₂×#41	—	67	1	no	—
184	C₂×#41	—	67	6	no	—
185	C₂×#41	—	67	6	no	—
186	C₂×#41	—	134	6	no	—
187	C₂×#41	—	134	2	no	—
188	#137	—	67	6	no	—
189	#137	—	67	12	no	—
190	#137	—	67	6	no	—
191	AGL(1,ℤ/8)	—	67	6	no	—
192	AGL(1,ℤ/8)	—	67	12	no	—
193	AGL(1,ℤ/8)	C₂×D₆₇	1	6	yes	complemented
194	#135	—	67	6	no	—
195	#135	—	67	1	no	—
196	#135	—	67	6	no	—
197	#135	—	67	3	no	—
198	#135	—	67	6	no	—
199	#135	—	67	6	no	—
200	#135	—	67	6	no	—
201	#135	—	67	3	no	—
202	#135	C₂×D₆₇	1	3	yes	complemented
203	C₂×Q₁₆	—	67	3	no	—
204	C₂×Q₁₆	—	67	3	no	—
205	C₂×SD₁₆	—	67	6	no	—
206	C₂×SD₁₆	—	67	6	no	—
207	C₂×D₈	—	67	3	no	—
208	C₂×D₈	C₂×D₆₇	1	3	yes	complemented
209	#131	—	67	1	no	—
210	#131	—	67	3	no	—
211	#131	—	67	3	no	—
212	#131	C₂×D₆₇	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	67	3	no	—
214	C₂²×C₈	—	67	3	no	—
215	#41	—	2	2	no	—
216	#41	—	67	1	no	—
217	#41	—	67	3	no	—
218	#41	—	67	3	no	—
219	#41	—	67	2	no	—
220	#41	—	67	6	no	—
221	#41	—	67	12	no	—
222	#41	—	67	6	no	—
223	#41	—	134	6	no	—
224	#41	—	134	6	no	—
225	#41	—	134	6	no	—
226	#41	—	134	6	no	—
227	#41	—	134	6	no	—
228	#41	—	134	12	no	—
229	#41	—	134	6	no	—
230	#41	—	134	6	no	—
231	#41	C₂²×D₆₇	1	6	yes	—
232	#41	C₂²×D₆₇	1	1	yes	characteristic
233	C₂×Q₈	—	67	1	no	—
234	C₂×Q₈	—	67	6	no	—
235	C₂×Q₈	—	67	6	no	—
236	C₂×Q₈	—	134	6	no	—
237	C₂×Q₈	—	134	2	no	—
238	C₂×D₄	—	2	6	no	—
239	C₂×D₄	—	67	3	no	—
240	C₂×D₄	—	67	6	no	—
241	C₂×D₄	—	134	6	no	—
242	C₂×D₄	—	134	6	no	—
243	C₂×D₄	—	134	12	no	—
244	C₂×D₄	—	134	6	no	—
245	C₂×D₄	C₂²×D₆₇	1	6	yes	—
246	C₂²×C₄	—	67	3	no	—
247	C₂²×C₄	—	134	6	no	—
248	C₂²×C₄	—	134	6	no	—
249	Q₁₆	—	67	6	no	—
250	Q₁₆	—	67	3	no	—
251	Q₁₆	—	67	6	no	—
252	Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
253	SD₁₆	—	67	2	no	—
254	SD₁₆	—	67	6	no	—
255	SD₁₆	—	67	12	no	—
256	SD₁₆	—	67	6	no	—
257	SD₁₆	C₂²×D₆₇	1	6	yes	—
258	D₈	—	67	6	no	—
259	D₈	—	67	1	no	—
260	D₈	C₂²×D₆₇	1	3	yes	—
261	D₈	C₂²×D₆₇	1	6	yes	—
262	M₁₆	—	67	3	no	—
263	M₁₆	—	67	6	no	—
264	M₁₆	C₂²×D₆₇	1	3	yes	—
265	C₂×C₈	—	67	3	no	—
266	C₂×C₈	—	67	1	no	—
267	C₂×C₈	—	67	3	no	—
268	C₂×C₈	—	67	3	no	—
269	C₂×C₈	—	67	3	no	—
270	C₂×C₈	C₂²×D₆₇	1	3	yes	—
271	C₂³	—	2	6	no	—
272	C₂³	—	67	3	no	—
273	C₂³	—	134	6	no	—
274	Q₈	—	67	3	no	—
275	Q₈	—	67	6	no	—
276	Q₈	—	134	6	no	—
277	Q₈	—	134	6	no	—
278	Q₈	D₄×D₆₇	1	1	yes	characteristic
279	Q₈	C₂³×D₆₇	1	2	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	2	6	no	—
282	D₄	—	67	6	no	—
283	D₄	—	67	3	no	—
284	D₄	—	67	2	no	—
285	D₄	—	134	2	no	—
286	D₄	—	134	6	no	—
287	D₄	—	134	6	no	—
288	D₄	—	134	12	no	—
289	D₄	—	134	6	no	—
290	D₄	D₄×D₆₇	1	3	yes	—
291	D₄	C₂³×D₆₇	1	6	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	67	3	no	—
294	C₂×C₄	—	67	3	no	—
295	C₂×C₄	—	67	3	no	—
296	C₂×C₄	—	67	1	no	—
297	C₂×C₄	—	67	3	no	—
298	C₂×C₄	—	134	2	no	—
299	C₂×C₄	—	134	6	no	—
300	C₂×C₄	—	134	6	no	—
301	C₂×C₄	—	134	12	no	—
302	C₂×C₄	—	134	6	no	—
303	C₂×C₄	—	134	6	no	—
304	C₂×C₄	C₂³×D₆₇	1	3	yes	—
305	C₈	—	67	1	no	—
306	C₈	—	67	3	no	—
307	C₈	C₂³×D₆₇	1	1	yes	characteristic
308	C₈	C₂³×D₆₇	1	3	yes	—
309	C₂²	—	2	6	no	—
310	C₂²	—	8	6	no	—
311	C₂²	—	268	3	no	—
312	C₂²	—	536	6	no	—
313	C₂²	—	67	1	no	—
314	C₂²	—	67	3	no	—
315	C₂²	—	134	2	no	—
316	C₂²	C₂×D₄×D₆₇	1	3	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	67	1	no	—
319	C₄	—	67	3	no	—
320	C₄	—	134	6	no	—
321	C₄	C₂×D₄×D₆₇	1	3	yes	—
322	C₄	C₂⁴×D₆₇	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	6	no	—
325	C₂	—	268	2	no	—
326	C₂	—	134	1	no	—
327	C₂	—	134	3	no	—
328	C₂	C₂²×D₄×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₆₇
commutator	C₄×C₆₇	C₂⁵
Frattini	C₄	C₂⁴×D₆₇
Fitting	#576×C₆₇	C₂
socle	C₂×C₆₇	C₂³×D₄
hypercenter	#576	D₆₇
hypocenter	C₆₇	#3584
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. #16864231	C₂	semidirect
2. #16864213	C₂	semidirect
3. #16864210	C₂	semidirect
4. #16864207	C₂	semidirect
5. #16864206	C₂	semidirect
6. #16864197	C₂	semidirect
7. #16864192	C₂	semidirect
8. C₂×#14524062	C₂	semidirect
9. #135×D₆₇	C₂	semidirect
10. #131×D₆₇	C₂	semidirect
11. #576×C₆₇	C₂	semidirect
13. #14524090	C₂²	semidirect
14. #14524090	C₂²	semidirect
16. #14524088	C₂²	semidirect
18. C₂×#11761627	C₂²	semidirect
21. #41×D₆₇	C₂²	semidirect
22. #14524076	C₂²	semidirect
23. #14524073	C₂²	semidirect
24. #14524072	C₂²	semidirect
25. #14524069	C₂²	semidirect
26. #14524067	C₂²	semidirect
27. #14524066	C₂²	semidirect
28. #14524065	C₂²	semidirect
29. #14524064	C₂²	semidirect
30. #14524063	C₂²	semidirect
31. #14524062	C₂²	semidirect
32. #14524062	C₂²	semidirect
33. #14524062	C₂²	semidirect
34. #14524062	C₂²	semidirect
35. C₂×Dic₂₆₈	C₂²	semidirect
36. Q₁₆×D₆₇	C₂²	semidirect
37. C₂×#11761616	C₂²	semidirect
38. SD₁₆×D₆₇	C₂²	semidirect
39. D₈×D₆₇	C₂²	semidirect
40. #14524051	C₂²	semidirect
41. #14524050	C₂²	semidirect
42. #14524049	C₂²	semidirect
43. M₁₆×D₆₇	C₂²	semidirect
44. C₂×C₈×D₆₇	C₂²	semidirect
45. H(5,𝔽₂)×C₆₇	C₂²	semidirect
46. AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
47. #135×C₆₇	C₂²	semidirect
48. C₂×D₈×C₆₇	C₂²	semidirect
49. #131×C₆₇	C₂²	semidirect
165. C₆₇	#3584	semidirect
172. #576	D₆₇	semidirect
182. H(5,𝔽₂)	C₂×D₆₇	semidirect
193. AGL(1,ℤ/8)	C₂×D₆₇	semidirect
202. #135	C₂×D₆₇	semidirect
208. C₂×D₈	C₂×D₆₇	semidirect
212. #131	C₂×D₆₇	semidirect