Subgroups — #27213530

Order 13504.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16809951	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₆₈₈	C₂	1	1	yes	maximal, characteristic
4	#20269965	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₂₂	C₂²	1	2	yes	—
7	C₂×D₈₄₄	C₂²	1	2	yes	—
8	Dic₈₄₄	—	2	2	no	—
9	#16809951	—	2	4	no	—
10	D₁₆₈₈	—	2	2	no	—
11	(C₂×D₂₁₁)⋊C₄	—	2	4	no	—
12	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₁₁	—	2	2	no	—
16	Dic₄₂₂	—	2	2	no	—
17	Dic₄₂₂	—	2	2	no	—
18	D₈₄₄	—	2	2	no	—
19	D₈₄₄	—	2	2	no	—
20	C₂×Dic₂₁₁	—	2	2	no	—
21	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
22	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₈×C₂₁₁	D₄	1	2	yes	—
24	C₈×C₂₁₁	D₄	1	2	yes	—
25	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₁₁	—	4	2	no	—
27	C₂×D₂₁₁	—	4	4	no	—
28	C₄×C₂₁₁	—	2	2	no	—
29	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
31	Dic₂₁₁	—	4	2	no	—
32	C₂×C₂₁₁	#127	1	1	yes	characteristic
33	C₂×C₂₁₁	#135	1	2	yes	—
34	D₂₁₁	—	8	2	no	—
35	C₂₁₁	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	211	1	no	maximal, Sylow
37	C₂×Q₁₆	—	211	1	no	—
38	C₂×SD₁₆	—	211	2	no	—
39	C₂×D₈	—	211	1	no	—
40	#124	—	211	2	no	—
41	C₄×C₈	D₂₁₁	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	211	2	no	—
43	C₂×D₄	—	211	2	no	—
44	Q₁₆	—	422	2	no	—
45	SD₁₆	—	422	4	no	—
46	D₈	—	422	2	no	—
47	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
48	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
49	C₂²⋊C₄	—	422	4	no	—
50	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
51	C₂³	—	422	2	no	—
52	Q₈	—	422	2	no	—
53	Q₈	—	422	2	no	—
54	D₄	—	422	2	no	—
55	D₄	—	422	2	no	—
56	C₂×C₄	—	422	2	no	—
57	C₂×C₄	D₈₄₄	1	2	yes	—
58	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
59	C₈	D₈₄₄	1	2	yes	—
60	C₈	D₈₄₄	1	2	yes	—
61	C₂²	—	844	2	no	—
62	C₂²	—	844	4	no	—
63	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	844	2	no	—
66	C₄	C₂×D₈₄₄	1	1	yes	characteristic
67	C₄	C₂×D₈₄₄	1	1	yes	characteristic
68	C₂	—	1688	2	no	—
69	C₂	#20269974	1	1	yes	characteristic
70	C₂	#20269999	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄₄
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄×C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₁₆₈₈	C₂
hypercenter	C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#495
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₈₄₄	C₂	semidirect
2. C₂×#16809951	C₂	semidirect
4. #20269965	C₂	semidirect
5. C₄×C₈×C₂₁₁	C₂	semidirect
35. C₂₁₁	#495	semidirect
41. C₄×C₈	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄₄	2
C₂×#16809951	2
C₂×D₁₆₈₈	2
#20269965	2
C₄×C₈×C₂₁₁	2
#495	211