Subgroups — #27213225

Order 13504.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809945	C₂	1	2	yes	maximal, complemented
2	C₄×D₈₄₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₂₁₁)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₄₄	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₁₁	C₂²	1	2	yes	—
8	#16809945	—	2	4	no	—
9	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
10	C₄⋊Dic₂₁₁	C₄	1	1	yes	characteristic
11	(C₂×D₂₁₁)⋊C₄	C₄	1	2	yes	—
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	C₂×C₄	1	2	yes	—
16	D₈₄₄	—	2	2	no	—
17	C₂×Dic₂₁₁	C₂×C₄	1	2	yes	—
18	C₄×D₂₁₁	—	2	2	no	—
19	C₄×D₂₁₁	—	2	2	no	—
20	C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₈×C₂₁₁	—	2	1	no	—
25	C₈×C₂₁₁	D₄	1	2	yes	complemented
26	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₁₁	—	2	2	no	—
28	C₂×D₂₁₁	—	2	2	no	—
29	C₂×D₂₁₁	—	2	2	no	—
30	C₄×C₂₁₁	—	2	1	no	—
31	C₄×C₂₁₁	M₁₆	1	2	yes	complemented
32	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₁₁	#41	1	1	yes	characteristic
34	Dic₂₁₁	—	2	2	no	—
35	C₂×C₂₁₁	C₄×D₄	1	1	yes	characteristic
36	C₂×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₂₁₁	#131	1	1	yes	characteristic
38	D₂₁₁	—	4	2	no	—
39	C₂₁₁	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	211	1	no	maximal, Sylow
41	C₂×M₁₆	—	211	2	no	—
42	C₄×D₄	—	211	1	no	—
43	C₄⋊C₈	—	211	1	no	—
44	C₂²⋊C₈	—	211	2	no	—
45	C₄×C₈	D₂₁₁	1	1	yes	characteristic, complemented
46	C₂×D₄	—	211	1	no	—
47	C₂²×C₄	—	211	2	no	—
48	M₁₆	—	422	4	no	—
49	C₂×C₈	—	211	2	no	—
50	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
51	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
52	C₄⋊C₄	—	211	1	no	—
53	C₂²⋊C₄	—	211	2	no	—
54	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂³	—	211	2	no	—
56	D₄	—	422	2	no	—
57	C₂×C₄	—	422	2	no	—
58	C₂×C₄	—	422	2	no	—
59	C₂×C₄	—	211	2	no	—
60	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
61	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	422	2	no	—
65	C₈	D₈₄₄	1	2	yes	complemented
66	C₂²	—	422	2	no	—
67	C₂²	—	422	2	no	—
68	C₂²	—	422	2	no	—
69	C₂²	C₂×C₄×D₂₁₁	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	422	2	no	—
72	C₄	#16809945	1	2	yes	complemented
73	C₄	C₂×D₈₄₄	1	1	yes	characteristic
74	C₄	#16809962	1	1	yes	characteristic
75	C₂	—	844	2	no	—
76	C₂	C₄×D₈₄₄	1	1	yes	characteristic
77	C₂	C₂×#16809945	1	1	yes	characteristic
78	C₂	#20269983	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₁₁
commutator, socle	C₂²×C₂₁₁	C₂²×C₄
Fitting	C₄×C₈×C₂₁₁	C₂
prime radical	C₂×D₈₄₄	C₄
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#436
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16809945	C₂	semidirect
3. C₄⋊(C₂₁₁⋊C₈)	C₂	semidirect
4. (C₂×D₂₁₁)⋊C₈	C₂	semidirect
5. C₄×C₈×C₂₁₁	C₂	semidirect
25. C₈×C₂₁₁	D₄	semidirect
31. C₄×C₂₁₁	M₁₆	semidirect
39. C₂₁₁	#436	semidirect
45. C₄×C₈	D₂₁₁	semidirect
65. C₈	D₈₄₄	semidirect
72. C₄	#16809945	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809945	2
C₄×D₈₄₄	2
C₄⋊(C₂₁₁⋊C₈)	2
(C₂×D₂₁₁)⋊C₈	2
C₄×C₈×C₂₁₁	2
#436	211