Subgroups — #16864201

Order 4288.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14524089	C₂	1	2	yes	maximal, complemented
2	#14524071	C₂	1	6	yes	maximal, complemented
3	#14524065	C₂	1	3	yes	maximal, complemented
4	C₂×D₅₃₆	C₂	1	3	yes	maximal, complemented
5	#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#11761629	—	2	2	no	—
7	#11761628	C₂²	1	6	yes	complemented
8	C₂×D₂₆₈	C₂²	1	6	yes	complemented
9	D₄×D₆₇	—	2	6	no	—
10	Dic₂₆₈	C₂²	1	1	yes	characteristic, complemented
11	#11761617	C₂²	1	6	yes	complemented
12	D₅₃₆	C₂²	1	3	yes	complemented
13	D₅₃₆	C₂²	1	6	yes	complemented
14	#41×C₆₇	C₂²	1	1	yes	characteristic
15	M₁₆×C₆₇	C₂²	1	3	yes	complemented
16	C₂×C₈×C₆₇	C₂²	1	3	yes	complemented
17	C₂²×D₆₇	—	2	6	no	—
18	Q₈×C₆₇	D₄	1	1	yes	characteristic
19	D₄×C₆₇	D₄	1	3	yes	—
20	C₂×C₄×C₆₇	C₂³	1	3	yes	—
21	C₂²⋊D₆₇	—	2	6	no	—
22	D₂₆₈	—	2	6	no	—
23	D₂₆₈	C₂³	1	6	yes	—
24	C₄×D₆₇	—	2	6	no	—
25	Dic₁₃₄	C₂³	1	2	yes	—
26	C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₈×C₆₇	C₂³	1	3	yes	—
28	C₂²×C₆₇	C₂×D₄	1	3	yes	—
29	C₂×D₆₇	—	2	6	no	—
30	C₂×D₆₇	—	8	6	no	—
31	C₄×C₆₇	C₂×D₄	1	3	yes	—
32	C₄×C₆₇	C₂⁴	1	1	yes	characteristic
33	Dic₆₇	—	2	2	no	—
34	C₂×C₆₇	—	2	3	no	—
35	C₂×C₆₇	C₂²×D₄	1	1	yes	characteristic
36	D₆₇	—	4	6	no	—
37	C₆₇	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	67	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	67	2	no	—
40	AGL(1,ℤ/8)	—	67	6	no	—
41	#135	—	67	3	no	—
42	C₂×D₈	—	67	3	no	—
43	#131	D₆₇	1	1	yes	characteristic, complemented
44	#41	—	67	6	no	—
45	#41	—	134	2	no	—
46	#41	C₂×D₆₇	1	1	yes	characteristic
47	C₂×D₄	—	67	6	no	—
48	C₂×D₄	—	134	6	no	—
49	Q₁₆	—	67	1	no	—
50	SD₁₆	—	67	6	no	—
51	D₈	—	67	3	no	—
52	D₈	—	67	6	no	—
53	M₁₆	C₂×D₆₇	1	3	yes	complemented
54	C₂×C₈	C₂×D₆₇	1	3	yes	complemented
55	C₂³	—	134	6	no	—
56	Q₈	—	67	2	no	—
57	Q₈	D₂₆₈	1	1	yes	characteristic
58	D₄	—	67	6	no	—
59	D₄	—	134	6	no	—
60	D₄	—	134	6	no	—
61	D₄	D₂₆₈	1	3	yes	—
62	C₂×C₄	—	134	6	no	—
63	C₂×C₄	C₂²×D₆₇	1	3	yes	—
64	C₈	C₂²×D₆₇	1	1	yes	characteristic
65	C₈	C₂²×D₆₇	1	3	yes	—
66	C₂²	—	536	6	no	—
67	C₂²	—	134	6	no	—
68	C₂²	C₂×D₂₆₈	1	3	yes	—
69	C₄	—	134	2	no	—
70	C₄	C₂×D₂₆₈	1	3	yes	—
71	C₄	C₂³×D₆₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	268	6	no	—
74	C₂	C₂²×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₆₈
commutator, cosocle	C₄×C₆₇	C₂⁴
Frattini	C₄	C₂³×D₆₇
Fitting	#131×C₆₇	C₂
socle	C₂×C₆₇	C₂²×D₄
hypercenter	#131	D₆₇
hypocenter	C₆₇	#576
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #14524089	C₂	semidirect
2. #14524071	C₂	semidirect
3. #14524065	C₂	semidirect
4. C₂×D₅₃₆	C₂	semidirect
5. #131×C₆₇	C₂	semidirect
7. #11761628	C₂²	semidirect
8. C₂×D₂₆₈	C₂²	semidirect
10. Dic₂₆₈	C₂²	semidirect
11. #11761617	C₂²	semidirect
12. D₅₃₆	C₂²	semidirect
13. D₅₃₆	C₂²	semidirect
15. M₁₆×C₆₇	C₂²	semidirect
16. C₂×C₈×C₆₇	C₂²	semidirect
37. C₆₇	#576	semidirect
43. #131	D₆₇	semidirect
53. M₁₆	C₂×D₆₇	semidirect
54. C₂×C₈	C₂×D₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14524089	2
#14524071	2
#14524065	2
C₂×D₅₃₆	2
#131×C₆₇	2
#576	67