Subgroups — #27213416

Order 13504.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270002	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269893	C₂	1	2	yes	maximal, complemented
5	#131×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₄₄	C₂²	1	2	yes	—
7	#16809949	—	2	2	no	—
8	D₁₆₈₈	—	2	1	no	—
9	D₁₆₈₈	—	2	1	no	—
10	#16809947	—	2	2	no	—
11	#16809946	C₂²	1	2	yes	—
12	#41×C₂₁₁	C₂²	1	1	yes	characteristic
13	M₁₆×C₂₁₁	—	2	1	no	—
14	M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₂₁₁	—	2	1	no	—
16	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂²×D₂₁₁	—	2	2	no	—
18	D₈₄₄	—	2	2	no	—
19	D₈₄₄	—	2	2	no	—
20	C₂₁₁⋊C₈	—	2	2	no	—
21	Q₈×C₂₁₁	D₄	1	1	yes	characteristic
22	D₄×C₂₁₁	—	2	1	no	—
23	D₄×C₂₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	—	2	1	no	—
25	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₈×C₂₁₁	—	2	1	no	—
27	C₈×C₂₁₁	D₄	1	1	yes	characteristic
28	C₈×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₂₁₁	—	2	1	no	—
30	C₂²×C₂₁₁	#41	1	1	yes	characteristic
31	C₂×D₂₁₁	—	4	2	no	—
32	C₂×D₂₁₁	—	8	2	no	—
33	C₄×C₂₁₁	—	2	1	no	—
34	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₂₁₁	—	2	1	no	—
37	C₂×C₂₁₁	—	4	1	no	—
38	C₂×C₂₁₁	#121	1	1	yes	characteristic
39	D₂₁₁	—	8	2	no	—
40	C₂₁₁	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	211	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	211	2	no	—
43	C₂×D₈	—	211	1	no	—
44	#131	D₂₁₁	1	1	yes	characteristic, complemented
45	#108	—	211	1	no	—
46	#100	—	211	2	no	—
47	#41	C₂×D₂₁₁	1	1	yes	characteristic
48	C₂×D₄	—	211	2	no	—
49	SD₁₆	—	422	2	no	—
50	D₈	—	422	2	no	—
51	D₈	—	422	1	no	—
52	D₈	—	422	1	no	—
53	M₁₆	—	2	1	no	—
54	M₁₆	—	211	2	no	—
55	M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
58	C₂³	—	422	2	no	—
59	Q₈	D₈₄₄	1	1	yes	characteristic
60	D₄	—	2	1	no	—
61	D₄	—	422	2	no	—
62	D₄	—	422	2	no	—
63	D₄	D₈₄₄	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	422	2	no	—
68	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
69	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	844	2	no	—
72	C₂²	—	1688	2	no	—
73	C₂²	#16809962	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
76	C₄	C₂×D₈₄₄	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	1688	2	no	—
80	C₂	#20269951	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269951
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#131×C₂₁₁	C₂
socle	C₂×C₂₁₁	#121
hypercenter	#131	D₂₁₁
hypocenter	C₂₁₁	#472
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20270002	C₂	semidirect
2. C₂×D₁₆₈₈	C₂	semidirect
3. #20269913	C₂	semidirect
4. #20269893	C₂	semidirect
5. #131×C₂₁₁	C₂	semidirect
40. C₂₁₁	#472	semidirect
44. #131	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270002	2
C₂×D₁₆₈₈	2
#20269913	2
#20269893	2
#131×C₂₁₁	2
#472	211