Subgroups — #27213417

Order 13504.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270005	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269983	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269893	C₂	1	2	yes	maximal, complemented
5	#108×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	1	yes	characteristic
7	C₂×D₈₄₄	C₂²	1	2	yes	—
8	#16809951	—	2	2	no	—
9	D₁₆₈₈	—	2	2	no	—
10	D₁₆₈₈	—	2	1	no	—
11	D₁₆₈₈	—	2	1	no	—
12	#16809946	C₂²	1	1	yes	characteristic
13	#16809945	—	2	1	no	—
14	C₈×D₂₁₁	—	2	1	no	—
15	M₁₆×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂²×D₂₁₁	—	2	2	no	—
18	Dic₄₂₂	D₄	1	1	yes	characteristic
19	D₈₄₄	—	2	2	no	—
20	D₈₄₄	—	2	2	no	—
21	D₈₄₄	D₄	1	1	yes	characteristic
22	C₂²⋊D₂₁₁	—	2	1	no	—
23	C₄×D₂₁₁	—	2	1	no	—
24	C₂₁₁⋊C₈	—	2	1	no	—
25	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₈×C₂₁₁	—	2	2	no	—
27	C₈×C₂₁₁	D₄	1	1	yes	characteristic
28	C₈×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₂₁₁	#41	1	1	yes	characteristic
30	C₂×D₂₁₁	—	2	1	no	—
31	C₂×D₂₁₁	—	4	2	no	—
32	C₂×D₂₁₁	—	8	2	no	—
33	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	Dic₂₁₁	—	2	1	no	—
36	C₂×C₂₁₁	—	2	1	no	—
37	C₂×C₂₁₁	#121	1	1	yes	characteristic
38	D₂₁₁	—	4	1	no	—
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	—	211	1	no	—
45	#108	D₂₁₁	1	1	yes	characteristic, complemented
46	#100	—	211	2	no	—
47	#41	—	211	1	no	—
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	—	422	1	no	—
54	M₁₆	—	211	1	no	—
55	M₁₆	C₂×D₂₁₁	1	2	yes	—
56	C₂×C₈	—	422	1	no	—
57	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
58	C₂³	—	422	2	no	—
59	Q₈	—	211	1	no	—
60	D₄	—	422	1	no	—
61	D₄	—	422	2	no	—
62	D₄	—	422	2	no	—
63	D₄	—	211	1	no	—
64	C₂×C₄	—	422	1	no	—
65	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	422	1	no	—
68	C₈	D₈₄₄	1	1	yes	characteristic
69	C₈	D₈₄₄	1	1	yes	characteristic
70	C₂²	—	844	2	no	—
71	C₂²	—	1688	2	no	—
72	C₂²	—	422	1	no	—
73	C₂²	#16809963	1	1	yes	characteristic
74	C₄	—	422	1	no	—
75	C₄	D₄×D₂₁₁	1	1	yes	characteristic
76	C₄	C₂×D₈₄₄	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	844	1	no	—
79	C₂	—	1688	2	no	—
80	C₂	#20269952	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269952
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#108×C₂₁₁	C₂
socle	C₂×C₂₁₁	#121
hypercenter	#108	D₂₁₁
hypocenter	C₂₁₁	#472
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
1. #20270005	C₂	semidirect
2. C₂×D₁₆₈₈	C₂	semidirect
3. #20269983	C₂	semidirect
4. #20269893	C₂	semidirect
5. #108×C₂₁₁	C₂	semidirect
40. C₂₁₁	#472	semidirect
45. #108	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270005	2
C₂×D₁₆₈₈	2
#20269983	2
#20269893	2
#108×C₂₁₁	2
#472	211