Subgroups — #22562515

Order 8576.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863825	C₂	1	2	yes	maximal, complemented
2	Dic₁₀₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863234	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524062	—	2	2	no	—
7	Dic₅₃₆	C₂²	1	2	yes	—
8	#14523987	—	2	2	no	—
9	C₁₆×D₆₇	C₂²	1	1	yes	characteristic
10	C₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	D₁₆×C₆₇	C₂²	1	2	yes	—
12	C₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	#11761627	—	4	2	no	—
14	Dic₂₆₈	—	2	2	no	—
15	#11761616	—	4	2	no	—
16	C₈×D₆₇	D₄	1	1	yes	characteristic
17	C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
18	D₈×C₆₇	—	2	2	no	—
19	C₁₆×C₆₇	C₂³	1	1	yes	characteristic
20	D₄×C₆₇	—	4	2	no	—
21	C₂²⋊D₆₇	—	8	2	no	—
22	C₂×Dic₆₇	—	8	2	no	—
23	C₄×D₆₇	D₈	1	1	yes	characteristic
24	Dic₁₃₄	—	4	2	no	—
25	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
26	C₆₇⋊C₈	D₈	1	1	yes	characteristic
27	C₂²×C₆₇	—	8	2	no	—
28	C₂×D₆₇	D₁₆	1	1	yes	characteristic
29	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
30	Dic₆₇	—	8	2	no	—
31	Dic₆₇	D₁₆	1	1	yes	characteristic
32	C₂×C₆₇	—	16	2	no	—
33	C₂×C₆₇	C₂×D₁₆	1	1	yes	characteristic
34	D₆₇	—	2	1	no	—
35	#2262	—	67	1	no	maximal, Sylow
36	C₆₇	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	67	2	no	—
38	Q₆₄	—	67	1	no	—
39	SD₆₄	—	67	2	no	—
40	D₃₂	D₆₇	1	1	yes	characteristic, complemented
41	C₂×C₃₂	—	67	1	no	—
42	#135	—	134	2	no	—
43	Q₃₂	—	67	2	no	—
44	SD₃₂	—	134	2	no	—
45	D₁₆	C₂×D₆₇	1	2	yes	—
46	C₂×C₁₆	—	67	1	no	—
47	C₃₂	—	67	1	no	—
48	C₃₂	C₂×D₆₇	1	1	yes	characteristic
49	#41	—	268	2	no	—
50	Q₁₆	—	134	2	no	—
51	SD₁₆	—	268	2	no	—
52	D₈	—	2	2	no	—
53	C₂×C₈	—	67	1	no	—
54	C₁₆	—	67	1	no	—
55	C₁₆	C₂²×D₆₇	1	1	yes	characteristic
56	Q₈	—	268	2	no	—
57	D₄	—	4	2	no	—
58	D₄	—	536	2	no	—
59	C₂×C₄	—	536	2	no	—
60	C₂×C₄	—	67	1	no	—
61	C₈	—	67	1	no	—
62	C₈	D₄×D₆₇	1	1	yes	characteristic
63	C₂²	—	8	2	no	—
64	C₂²	—	67	1	no	—
65	C₄	—	536	2	no	—
66	C₄	—	67	1	no	—
67	C₄	D₈×D₆₇	1	1	yes	characteristic
68	C₂	—	16	2	no	—
69	C₂	—	134	1	no	—
70	C₂	D₁₆×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₆₇
commutator	C₁₆×C₆₇	C₂³
Frattini	C₁₆	C₂²×D₆₇
Fitting	D₃₂×C₆₇	C₂
socle	C₂×C₆₇	C₂×D₁₆
hypercenter	D₃₂	D₆₇
hypocenter	C₆₇	#2262
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₃₂

Complements

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

N	H ≅ G/N	type
1. #16863825	C₂	semidirect
2. Dic₁₀₇₂	C₂	semidirect
3. #16863234	C₂	semidirect
4. C₃₂×D₆₇	C₂	semidirect
5. D₃₂×C₆₇	C₂	semidirect
36. C₆₇	#2262	semidirect
40. D₃₂	D₆₇	semidirect