Subgroups — #22562436

Order 8576.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863801	C₂	1	1	yes	maximal, characteristic
2	#16863200	C₂	1	4	yes	maximal
3	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#494×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#14524043	—	2	1	no	—
6	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	#14523977	—	2	1	no	—
8	#14523977	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₆₇	—	2	1	no	—
10	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₆₇	—	2	1	no	—
12	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₃₄	—	4	1	no	—
14	C₄⋊Dic₆₇	—	2	2	no	—
15	C₄⋊Dic₆₇	—	4	1	no	—
16	C₆₇⋊C₁₆	—	2	2	no	—
17	C₂×Q₈×C₆₇	—	2	2	no	—
18	Q₁₆×C₆₇	—	2	4	no	—
19	Q₁₆×C₆₇	—	4	2	no	—
20	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₆₇	—	4	1	no	—
23	C₄⋊C₄×C₆₇	—	4	1	no	—
24	C₄²×C₆₇	D₄	1	1	yes	characteristic
25	Q₈×C₆₇	—	4	2	no	—
26	Q₈×C₆₇	—	4	2	no	—
27	C₂×C₄×C₆₇	—	4	2	no	—
28	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₆₇	—	4	2	no	—
32	Dic₁₃₄	—	8	2	no	—
33	C₈×C₆₇	SD₁₆	1	2	yes	—
34	C₈×C₆₇	#41	1	2	yes	—
35	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₆₇	—	8	2	no	—
37	C₄×C₆₇	SD₃₂	1	2	yes	—
38	C₄×C₆₇	Q₃₂	1	2	yes	—
39	C₄×C₆₇	#124	1	1	yes	characteristic
40	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₆₇	—	8	2	no	—
42	C₂×C₆₇	#486	1	1	yes	characteristic
43	C₂×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
45	#2241	—	67	1	no	maximal, Sylow
46	C₆₇	#2241	1	1	yes	characteristic, complemented, Sylow
47	#500	—	67	1	no	—
48	#494	D₆₇	1	1	yes	characteristic
49	#358	—	67	4	no	—
50	C₄×C₁₆	—	67	1	no	—
51	C₂×Q₁₆	—	2	1	no	—
52	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	134	1	no	—
55	C₂×C₁₆	—	67	2	no	—
56	#107	—	67	2	no	—
57	#107	—	134	1	no	—
58	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	268	1	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	268	1	no	—
68	C₄⋊C₄	—	134	2	no	—
69	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
70	C₁₆	—	134	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	536	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	268	2	no	—
76	C₂×C₄	#11761613	1	1	yes	characteristic
77	C₂×C₄	#11761613	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
79	C₈	#11761616	1	2	yes	—
80	C₈	#11761627	1	2	yes	—
81	C₂²	C₂×#11761613	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	536	2	no	—
84	C₄	#14523986	1	2	yes	—
85	C₄	#14523989	1	2	yes	—
86	C₄	#14524030	1	1	yes	characteristic
87	C₄	C₂×#11761616	1	1	yes	characteristic
88	C₂	#16863742	1	1	yes	characteristic
89	C₂	C₂×#14523986	1	1	yes	characteristic
90	C₂	C₂×#14523989	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#494	D₆₇
hypocenter	C₆₇	#2241
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
46. C₆₇	#2241	semidirect