Subgroups — #27213702

Order 13504.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269976	C₂	1	2	yes	maximal, complemented
2	#20269969	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269955	C₂	1	4	yes	maximal
4	#20269942	C₂	1	4	yes	maximal, complemented
5	#20269935	C₂	1	2	yes	maximal
6	C₄²×D₂₁₁	C₂	1	1	yes	maximal, characteristic
7	#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₂₂	C₂²	1	4	yes	—
9	C₂×C₄×D₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄×D₂₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
12	Dic₂₁₁⋊C₄	C₂²	1	4	yes	—
13	Dic₂₁₁⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₂₁₁)⋊C₄	C₂²	1	4	yes	—
15	C₄×Dic₂₁₁	C₂²	1	2	yes	—
16	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₁₁	C₂²	1	4	yes	—
18	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
20	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂²×D₂₁₁	C₂³	1	1	yes	characteristic
22	Dic₄₂₂	—	2	8	no	—
23	C₂×Dic₂₁₁	C₂³	1	4	yes	—
24	C₂×Dic₂₁₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₁₁	C₂³	1	2	yes	—
26	C₄×D₂₁₁	—	2	4	no	—
27	C₄×D₂₁₁	Q₈	1	4	yes	complemented
28	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
29	C₂×C₄×C₂₁₁	C₂³	1	4	yes	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
32	C₂×D₂₁₁	—	2	2	no	—
33	C₂×D₂₁₁	C₂×Q₈	1	2	yes	—
34	C₄×C₂₁₁	—	2	2	no	—
35	C₄×C₂₁₁	—	2	4	no	—
36	C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
37	Dic₂₁₁	—	2	4	no	—
38	Dic₂₁₁	C₂×Q₈	1	2	yes	—
39	Dic₂₁₁	#41	1	4	yes	—
40	C₂×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂×#41	1	2	yes	—
42	D₂₁₁	—	2	2	no	—
43	C₂₁₁	#533	1	1	yes	characteristic, complemented, Sylow
44	#533	—	211	1	no	maximal, Sylow
45	C₄⋊Q₈	—	211	2	no	—
46	#125	—	211	1	no	—
47	#125	D₂₁₁	1	1	yes	characteristic, complemented
48	C₂²⋊Q₈	—	211	4	no	—
49	C₄×Q₈	—	211	4	no	—
50	#117	—	211	2	no	—
51	C₂×C₄²	—	211	1	no	—
52	C₂×Q₈	—	211	4	no	—
53	C₂²×C₄	—	211	2	no	—
54	C₂²×C₄	—	211	1	no	—
55	C₄⋊C₄	—	211	2	no	—
56	C₄⋊C₄	—	211	4	no	—
57	C₄⋊C₄	—	211	4	no	—
58	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
59	C₄⋊C₄	C₂×D₂₁₁	1	4	yes	—
60	C₂²⋊C₄	—	211	4	no	—
61	C₄²	—	211	2	no	—
62	C₄²	—	211	1	no	—
63	C₄²	—	211	4	no	—
64	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
65	C₂³	—	211	1	no	—
66	Q₈	—	422	8	no	—
67	C₂×C₄	—	422	4	no	—
68	C₂×C₄	—	211	4	no	—
69	C₂×C₄	—	211	1	no	—
70	C₂×C₄	—	211	2	no	—
71	C₂×C₄	—	211	4	no	—
72	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
73	C₂×C₄	C₂²×D₂₁₁	1	4	yes	—
74	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	C₂²	—	422	2	no	—
76	C₂²	—	211	2	no	—
77	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	4	no	—
80	C₄	—	422	4	no	—
81	C₄	—	211	2	no	—
82	C₄	—	211	4	no	—
83	C₄	Q₈×D₂₁₁	1	2	yes	—
84	C₂	—	422	2	no	—
85	C₂	C₂×Q₈×D₂₁₁	1	1	yes	characteristic
86	C₂	#41×D₂₁₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	#125×C₂₁₁	C₂
prime radical	C₂²×D₂₁₁	C₂³
hypercenter	#125	D₂₁₁
hypocenter	C₂₁₁	#533
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #20269976	C₂	semidirect
2. #20269969	C₂	semidirect
4. #20269942	C₂	semidirect
7. #125×C₂₁₁	C₂	semidirect
27. C₄×D₂₁₁	Q₈	semidirect
43. C₂₁₁	#533	semidirect
47. #125	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269976	2
#20269969	2
#20269955	2
#20269942	2
#20269935	2
C₄²×D₂₁₁	2
#125×C₂₁₁	2
#533	211