Subgroups — #22558313

Order 8576.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523979	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11761612	—	2	2	no	—
4	#14523979	—	2	2	no	—
5	#14523979	C₄	1	4	yes	—
6	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₆₇	—	2	2	no	—
8	C₂×C₁₆×C₆₇	C₄	1	2	yes	—
9	#11761612	—	2	4	no	—
10	#11761612	—	4	2	no	—
11	C₂×C₆₇⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₆₇	D₄	1	2	yes	—
16	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₆₇	—	2	2	no	—
18	C₂³×C₆₇	Q₁₆	1	1	yes	characteristic
19	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
23	C₈×C₆₇	C₄²	1	1	yes	characteristic
24	C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
26	C₆₇⋊C₈	—	4	4	no	—
27	C₂²×C₆₇	—	4	1	no	—
28	C₂²×C₆₇	#103	1	2	yes	—
29	C₂²×C₆₇	#107	1	1	yes	characteristic
30	C₄×C₆₇	#95	1	1	yes	characteristic
31	C₄×C₆₇	#102	1	2	yes	—
32	C₄×C₆₇	#106	1	1	yes	characteristic
33	C₂×C₆₇	—	2	2	no	—
34	C₂×C₆₇	—	2	1	no	—
35	C₂×C₆₇	#340	1	1	yes	characteristic
36	#1385	—	67	1	no	maximal, Sylow
37	C₆₇	#1385	1	1	yes	characteristic, complemented, Sylow
38	C₂×M₃₂	D₆₇	1	1	yes	characteristic
39	C₂×#108	—	67	2	no	—
40	C₂×M₁₆	—	134	2	no	—
41	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	Dic₆₇	1	2	yes	—
44	#108	—	67	4	no	—
45	#108	—	134	2	no	—
46	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
47	M₁₆	—	268	2	no	—
48	M₁₆	—	134	4	no	—
49	C₂×C₈	—	268	2	no	—
50	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
51	C₂×C₈	C₄×D₆₇	1	2	yes	—
52	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₂₆₈	1	1	yes	characteristic
56	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
58	C₂×C₄	D₅₃₆	1	1	yes	characteristic
59	C₂×C₄	#11761617	1	2	yes	—
60	C₈	—	268	4	no	—
61	C₈	C₄×Dic₆₇	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
63	C₈	Dic₆₇⋊C₄	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#14523968	1	2	yes	—
66	C₂²	#14523977	1	1	yes	characteristic
67	C₄	C₆₇⋊#95	1	1	yes	characteristic
68	C₄	#14523965	1	2	yes	—
69	C₄	#14523975	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#16863159	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14523975
commutator	C₈×C₆₇	C₄²
Frattini	C₂²×C₈	C₂×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂×C₆₇	#340
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#1385
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
37. C₆₇	#1385	semidirect