Subgroups — #22562239

Order 8576.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863578	C₂	1	3	yes	maximal
2	#16863195	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14524091	C₂²	1	1	yes	characteristic
5	#14524074	—	2	3	no	—
6	#14523971	—	2	3	no	—
7	#14523962	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₆₇	—	2	3	no	—
9	C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
10	C₄×Q₈×C₆₇	—	2	3	no	—
11	#11761629	—	4	1	no	—
12	#11761628	—	2	3	no	—
13	Q₈×D₆₇	—	4	1	no	—
14	#11761618	—	4	3	no	—
15	#11761615	—	4	3	no	—
16	#11761612	—	2	3	no	—
17	C₂×Q₈×C₆₇	—	2	1	no	—
18	C₂×Q₈×C₆₇	—	2	1	no	—
19	C₂×Q₈×C₆₇	D₄	1	3	yes	—
20	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₆₇	—	2	3	no	—
22	C₄⋊C₄×C₆₇	—	2	6	no	—
23	C₄²×C₆₇	—	2	3	no	—
24	C₄²×C₆₇	D₄	1	3	yes	—
25	Q₈×C₆₇	—	2	4	no	—
26	Q₈×C₆₇	—	4	1	no	—
27	Q₈×C₆₇	—	4	3	no	—
28	Q₈×C₆₇	—	8	1	no	—
29	C₂×C₄×C₆₇	—	2	3	no	—
30	C₂×C₄×C₆₇	—	2	3	no	—
31	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
32	C₂²⋊D₆₇	—	4	1	no	—
33	D₂₆₈	—	2	3	no	—
34	C₄×D₆₇	—	4	3	no	—
35	Dic₁₃₄	—	2	3	no	—
36	C₆₇⋊C₈	—	4	3	no	—
37	C₂²×C₆₇	#120	1	1	yes	characteristic
38	C₂×D₆₇	—	4	1	no	—
39	C₄×C₆₇	—	2	6	no	—
40	C₄×C₆₇	—	2	3	no	—
41	C₄×C₆₇	—	4	3	no	—
42	Dic₆₇	—	4	1	no	—
43	C₂×C₆₇	—	2	1	no	—
44	C₂×C₆₇	B(4,𝔽₂)	1	1	yes	characteristic
45	D₆₇	—	8	1	no	—
46	#2205	—	67	1	no	maximal, Sylow
47	C₆₇	#2205	1	1	yes	characteristic, complemented, Sylow
48	Q₈²	D₆₇	1	1	yes	characteristic, complemented
49	#456	—	67	3	no	—
50	#356	—	67	3	no	—
51	2₋¹⁺⁴	—	67	1	no	—
52	#137	—	134	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₆₇	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	134	3	no	—
57	#101	—	67	3	no	—
58	#41	—	268	1	no	—
59	#41	—	134	3	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	268	1	no	—
63	C₂×Q₈	C₂²⋊D₆₇	1	3	yes	—
64	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
65	Q₁₆	—	268	3	no	—
66	SD₁₆	—	268	3	no	—
67	M₁₆	—	134	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₆₇	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	134	3	no	—
77	D₄	—	268	1	no	—
78	D₄	—	134	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	268	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
83	C₈	—	268	3	no	—
84	C₂²	—	268	1	no	—
85	C₂²	#14524012	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	268	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	536	1	no	—
92	C₂	#16863583	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16863583
commutator	C₂×Q₈×C₆₇	C₂³
Frattini	C₂×Q₈	C₂²×D₆₇
Fitting	Q₈²×C₆₇	C₂
socle	C₂×C₆₇	B(4,𝔽₂)
hypercenter	Q₈²	D₆₇
hypocenter	C₆₇	#2205
LCS	G ▷ C₂×Q₈×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #16863195	C₂	semidirect
3. Q₈²×C₆₇	C₂	semidirect
47. C₆₇	#2205	semidirect
48. Q₈²	D₆₇	semidirect