Subgroups — #22562117

Order 8576.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal
4	#16863171	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
7	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
9	Dic₆₇⋊C₈	C₄	1	2	yes	—
10	C₈×Dic₆₇	C₂²	1	2	yes	—
11	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₃₄	C₈	1	1	yes	characteristic
15	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
17	Dic₆₇⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
19	C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₆₇	—	2	1	no	—
24	C₁₆×C₆₇	Q₈	1	2	yes	complemented
25	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₆₇	C₂×C₈	1	2	yes	—
30	Dic₁₃₄	—	2	2	no	—
31	C₈×C₆₇	—	2	1	no	—
32	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₆₇	#41	1	1	yes	characteristic
34	C₆₇⋊C₈	—	2	2	no	—
35	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₆₇	—	2	1	no	—
37	C₄×C₆₇	M₃₂	1	2	yes	—
38	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₆₇	#131	1	1	yes	characteristic
40	Dic₆₇	—	2	2	no	—
41	Dic₆₇	—	2	2	no	—
42	C₂×C₆₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₆₇	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₆₇	#504	1	1	yes	characteristic
45	#2183	—	67	1	no	maximal, Sylow
46	C₆₇	#2183	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	67	1	no	—
48	C₄⋊C₁₆	—	67	2	no	—
49	C₄⋊C₁₆	—	67	1	no	—
50	#346	—	67	2	no	—
51	C₄×C₁₆	D₆₇	1	1	yes	characteristic
52	C₄×Q₈	—	67	1	no	—
53	C₂×C₁₆	—	67	2	no	—
54	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
55	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
56	C₄⋊C₈	—	67	2	no	—
57	C₄⋊C₈	—	67	1	no	—
58	C₄×C₈	—	67	2	no	—
59	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
60	C₂×Q₈	—	67	1	no	—
61	C₂×C₈	—	67	2	no	—
62	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
64	C₄⋊C₄	—	67	2	no	—
65	C₄⋊C₄	—	67	1	no	—
66	C₄²	—	67	2	no	—
67	C₄²	C₄×D₆₇	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	134	2	no	—
70	C₁₆	Dic₁₃₄	1	2	yes	complemented
71	Q₈	—	134	2	no	—
72	C₂×C₄	—	67	2	no	—
73	C₂×C₄	—	67	2	no	—
74	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
75	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	134	2	no	—
79	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
80	C₈	#11761628	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₆₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	134	2	no	—
84	C₄	—	134	2	no	—
85	C₄	#14523982	1	2	yes	—
86	C₄	C₄×Dic₁₃₄	1	1	yes	characteristic
87	C₄	#14524049	1	1	yes	characteristic
88	C₂	C₈×Dic₁₃₄	1	1	yes	characteristic
89	C₂	C₂×#14523982	1	1	yes	characteristic
90	C₂	#16863812	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₈
Fitting	C₄×C₁₆×C₆₇	C₂
hypercenter	C₄×C₁₆	D₆₇
hypocenter	C₆₇	#2183
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₆₇	Q₈	semidirect
46. C₆₇	#2183	semidirect
70. C₁₆	Dic₁₃₄	semidirect