Subgroups — #27213076

Order 13504.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁₁⋊C₄	C₂	1	2	yes	maximal
3	C₂₁₁⋊#95	C₂	1	2	yes	maximal
4	C₂₁₁⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₁	C₂²	1	2	yes	—
7	C₂²×Dic₂₁₁	C₂²	1	2	yes	—
8	C₄⋊Dic₂₁₁	—	2	2	no	—
9	Dic₂₁₁⋊C₄	—	2	4	no	—
10	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₁₁	—	2	2	no	—
14	C₂×Dic₂₁₁	—	2	2	no	—
15	C₂×Dic₂₁₁	—	2	2	no	—
16	C₂×Dic₂₁₁	—	2	2	no	—
17	C₂×Dic₂₁₁	—	2	2	no	—
18	C₂×Dic₂₁₁	—	2	2	no	—
19	C₂×Dic₂₁₁	Q₈	1	4	yes	—
20	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₂₁₁	—	2	1	no	—
22	C₂×C₄×C₂₁₁	—	2	2	no	—
23	C₂×C₄×C₂₁₁	—	2	1	no	—
24	C₂×C₄×C₂₁₁	—	2	1	no	—
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
27	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
28	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₂₁₁	C₂×Q₈	1	2	yes	—
31	C₂²×C₂₁₁	#41	1	2	yes	—
32	C₂²×C₂₁₁	#41	1	1	yes	characteristic
33	C₄×C₂₁₁	—	2	2	no	—
34	C₄×C₂₁₁	—	2	2	no	—
35	C₄×C₂₁₁	—	4	1	no	—
36	Dic₂₁₁	—	2	4	no	—
37	Dic₂₁₁	—	4	2	no	—
38	C₂×C₂₁₁	#121	1	1	yes	characteristic
39	C₂×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
40	C₂×C₂₁₁	#123	1	1	yes	characteristic
41	C₂×C₂₁₁	#125	1	2	yes	—
42	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
43	C₂₁₁	#398	1	1	yes	characteristic, complemented, Sylow
44	#398	—	211	1	no	maximal, Sylow
45	C₂×C₄⋊C₄	—	211	2	no	—
46	C₂×C₄⋊C₄	—	211	1	no	—
47	C₂×C₄⋊C₄	D₂₁₁	1	1	yes	characteristic
48	#95	—	211	2	no	—
49	#95	—	211	1	no	—
50	C₂²×C₄	—	211	2	no	—
51	C₂²×C₄	—	211	2	no	—
52	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
53	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	422	2	no	—
57	C₄⋊C₄	—	422	4	no	—
58	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	422	2	no	—
65	C₂×C₄	—	422	2	no	—
66	C₂×C₄	—	422	2	no	—
67	C₂×C₄	—	422	2	no	—
68	C₂×C₄	—	422	2	no	—
69	C₂×C₄	—	211	4	no	—
70	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
71	C₂×C₄	D₈₄₄	1	2	yes	—
72	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
73	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
74	C₂²	Q₈×D₂₁₁	1	2	yes	—
75	C₂²	#16809961	1	2	yes	—
76	C₂²	#16809962	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	4	1	no	—
80	C₄	—	844	2	no	—
81	C₄	—	422	4	no	—
82	C₂	#20269951	1	1	yes	characteristic
83	C₂	#20269957	1	2	yes	—
84	C₂	#20269958	1	1	yes	characteristic
85	C₂	#20269967	1	2	yes	—
86	C₂	#20269975	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁₁
commutator, socle, prime radical, cosocle	C₂³×C₂₁₁	C₂³
Fitting	C₂×C₄⋊C₄×C₂₁₁	C₂
hypercenter	C₂×C₄⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#398
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
43. C₂₁₁	#398	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₁₁	2
C₂×Dic₂₁₁⋊C₄	2
C₂₁₁⋊#95	2
C₂×C₄⋊C₄×C₂₁₁	2
#398	211