Subgroups — #22570733

Order 8576.

348 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863953	C₂	1	2	yes	maximal
2	#16863805	C₂	1	8	yes	maximal, complemented
3	#16863434	C₂	1	4	yes	maximal
4	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#14524043	C₂²	1	4	yes	—
6	#14524035	C₂²	1	4	yes	—
7	#14524022	—	2	4	no	—
8	C₄×Dic₁₃₄	—	2	4	no	—
9	#14524002	C₂²	1	4	yes	—
10	#14523977	C₂²	1	8	yes	—
11	#14523975	C₂²	1	8	yes	—
12	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#97×C₆₇	C₂²	1	4	yes	—
15	C₂×Dic₁₃₄	—	2	4	no	—
16	C₄⋊Dic₆₇	—	2	4	no	—
17	C₄⋊Dic₆₇	C₂³	1	8	yes	—
18	Dic₆₇⋊C₄	—	2	8	no	—
19	C₂²⋊Dic₆₇	—	2	4	no	—
20	C₄×Dic₆₇	—	2	4	no	—
21	C₂²×C₄×C₆₇	D₄	1	2	yes	—
22	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₆₇	Q₈	1	8	yes	complemented
24	C₂×C₈×C₆₇	C₂³	1	4	yes	—
25	C₄²×C₆₇	D₄	1	2	yes	—
26	C₄²×C₆₇	C₂³	1	2	yes	—
27	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₆₇	—	4	2	no	—
29	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
30	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
32	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
33	C₂×Dic₆₇	—	2	8	no	—
34	Dic₁₃₄	—	4	8	no	—
35	C₈×C₆₇	C₂×Q₈	1	8	yes	—
36	C₂²×C₆₇	—	2	2	no	—
37	C₂²×C₆₇	C₄⋊Q₈	1	2	yes	—
38	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
39	C₄×C₆₇	—	2	4	no	—
40	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
41	C₄×C₆₇	C₂²×Q₈	1	2	yes	—
42	Dic₆₇	—	4	8	no	—
43	C₂×C₆₇	—	2	2	no	—
44	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₆₇	#575	1	2	yes	—
46	#3162	—	67	1	no	maximal, Sylow
47	C₆₇	#3162	1	1	yes	characteristic, complemented, Sylow
48	#533	—	67	2	no	—
49	#501	—	67	8	no	—
50	#427	—	67	4	no	—
51	C₂×#97	D₆₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₆₇	1	2	yes	—
53	C₄⋊Q₈	—	67	4	no	—
54	#125	—	67	4	no	—
55	C₂²⋊Q₈	—	134	4	no	—
56	C₄×Q₈	—	134	4	no	—
57	#117	—	67	4	no	—
58	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
59	#107	—	67	8	no	—
60	#106	—	67	8	no	—
61	#97	C₂×D₆₇	1	4	yes	—
62	C₂×Q₈	—	134	4	no	—
63	C₂²×C₄	D₂₆₈	1	2	yes	—
64	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
65	C₂×C₈	Dic₁₃₄	1	8	yes	complemented
66	C₂×C₈	C₂²×D₆₇	1	4	yes	—
67	C₄⋊C₄	—	67	8	no	—
68	C₄⋊C₄	—	134	8	no	—
69	C₄⋊C₄	—	134	4	no	—
70	C₂²⋊C₄	—	134	4	no	—
71	C₄²	—	134	4	no	—
72	C₄²	D₂₆₈	1	2	yes	—
73	C₄²	C₂²×D₆₇	1	2	yes	—
74	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
75	Q₈	—	268	8	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	134	8	no	—
78	C₂×C₄	C₂×D₂₆₈	1	4	yes	—
79	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
81	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
82	C₈	C₂×Dic₁₃₄	1	8	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	#14524043	1	2	yes	—
85	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	—	268	8	no	—
88	C₄	#14524043	1	2	yes	—
89	C₄	C₂²×Dic₁₃₄	1	2	yes	—
90	C₂	—	2	2	no	—
91	C₂	C₂×#14524043	1	1	yes	characteristic
92	C₂	#16864196	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#97×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×#97	D₆₇
hypocenter	C₆₇	#3162
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #16863805	C₂	semidirect
23. C₂×C₈×C₆₇	Q₈	semidirect
47. C₆₇	#3162	semidirect
65. C₂×C₈	Dic₁₃₄	semidirect