Subgroups — #27213963

Order 13504.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270024	C₂	1	2	yes	maximal, complemented
2	#20270010	C₂	1	6	yes	maximal, complemented
3	#20269999	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₈₄₄	C₂	1	3	yes	maximal, complemented
5	#131×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	6	yes	—
7	#16809961	—	2	6	no	—
8	C₂×Dic₄₂₂	C₂²	1	6	yes	—
9	Q₈×D₂₁₁	—	2	2	no	—
10	Dic₈₄₄	C₂²	1	3	yes	—
11	Dic₈₄₄	C₂²	1	6	yes	—
12	#16809951	C₂²	1	6	yes	—
13	D₁₆₈₈	C₂²	1	1	yes	characteristic
14	#41×C₂₁₁	C₂²	1	1	yes	characteristic
15	M₁₆×C₂₁₁	C₂²	1	3	yes	—
16	C₂×C₈×C₂₁₁	C₂²	1	3	yes	—
17	Dic₄₂₂	—	2	6	no	—
18	Dic₄₂₂	C₂³	1	6	yes	—
19	D₈₄₄	C₂³	1	2	yes	—
20	C₂²⋊D₂₁₁	—	2	6	no	—
21	C₂×Dic₂₁₁	—	2	6	no	—
22	C₄×D₂₁₁	—	2	6	no	—
23	Q₈×C₂₁₁	D₄	1	1	yes	characteristic
24	D₄×C₂₁₁	D₄	1	3	yes	—
25	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
26	C₈×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₈×C₂₁₁	C₂³	1	3	yes	—
28	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
29	C₂×D₂₁₁	—	2	2	no	—
30	C₄×C₂₁₁	C₂×D₄	1	3	yes	—
31	C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
32	Dic₂₁₁	—	2	6	no	—
33	C₂×C₂₁₁	—	2	3	no	—
34	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
35	D₂₁₁	—	4	2	no	—
36	C₂₁₁	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	211	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	211	2	no	—
39	#137	—	211	6	no	—
40	#135	—	211	3	no	—
41	C₂×Q₁₆	—	211	3	no	—
42	#131	D₂₁₁	1	1	yes	characteristic, complemented
43	#41	—	422	6	no	—
44	#41	—	211	6	no	—
45	#41	C₂×D₂₁₁	1	1	yes	characteristic
46	C₂×Q₈	—	422	2	no	—
47	C₂×Q₈	—	211	6	no	—
48	Q₁₆	—	211	3	no	—
49	Q₁₆	—	211	6	no	—
50	SD₁₆	—	211	6	no	—
51	D₈	—	211	1	no	—
52	M₁₆	C₂×D₂₁₁	1	3	yes	—
53	C₂×C₈	C₂×D₂₁₁	1	3	yes	—
54	Q₈	—	422	6	no	—
55	Q₈	—	211	6	no	—
56	Q₈	D₈₄₄	1	1	yes	characteristic
57	D₄	—	422	6	no	—
58	D₄	—	211	2	no	—
59	D₄	D₈₄₄	1	3	yes	—
60	C₂×C₄	—	422	6	no	—
61	C₂×C₄	—	422	6	no	—
62	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
63	C₈	C₂²×D₂₁₁	1	1	yes	characteristic
64	C₈	C₂²×D₂₁₁	1	3	yes	—
65	C₂²	—	422	2	no	—
66	C₂²	C₂×D₈₄₄	1	3	yes	—
67	C₄	—	422	6	no	—
68	C₄	C₂×D₈₄₄	1	3	yes	—
69	C₄	C₂³×D₂₁₁	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	844	2	no	—
72	C₂	C₂²×D₈₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₈₄₄
commutator, cosocle	C₄×C₂₁₁	C₂⁴
Frattini	C₄	C₂³×D₂₁₁
Fitting	#131×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂²×D₄
hypercenter	#131	D₂₁₁
hypocenter	C₂₁₁	#578
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20270024	C₂	semidirect
2. #20270010	C₂	semidirect
3. #20269999	C₂	semidirect
4. C₂×Dic₈₄₄	C₂	semidirect
5. #131×C₂₁₁	C₂	semidirect
36. C₂₁₁	#578	semidirect
42. #131	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270024	2
#20270010	2
#20269999	2
C₂×Dic₈₄₄	2
#131×C₂₁₁	2
#578	211