Subgroups — #27213505

Order 13504.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269977	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269901	C₂	1	2	yes	maximal, complemented
3	#20269898	C₂	1	2	yes	maximal
4	#20269888	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	—	2	1	no	—
7	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₁₁	—	2	1	no	—
9	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
10	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
13	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
14	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	Dic₄₂₂	—	4	2	no	—
16	C₂×Dic₂₁₁	—	2	2	no	—
17	C₂₁₁⋊C₈	—	2	2	no	—
18	C₂³×C₂₁₁	—	2	1	no	—
19	Q₈×C₂₁₁	—	2	2	no	—
20	D₄×C₂₁₁	—	2	2	no	—
21	C₂×C₄×C₂₁₁	—	2	1	no	—
22	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₂₁₁	—	4	1	no	—
26	C₂²×C₂₁₁	—	4	1	no	—
27	C₂²×C₂₁₁	—	4	1	no	—
28	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₁₁	—	2	1	no	—
30	C₄×C₂₁₁	—	2	1	no	—
31	C₄×C₂₁₁	—	4	1	no	—
32	C₄×C₂₁₁	#41	1	2	yes	—
33	Dic₂₁₁	—	4	2	no	—
34	C₂×C₂₁₁	—	8	1	no	—
35	C₂×C₂₁₁	#124	1	1	yes	characteristic
36	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
37	C₂×C₂₁₁	#137	1	1	yes	characteristic
38	C₂₁₁	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	211	1	no	maximal, Sylow
40	C₄⋊Q₈	—	211	1	no	—
41	#124	D₂₁₁	1	1	yes	characteristic
42	#103	—	211	2	no	—
43	#102	—	211	2	no	—
44	#97	—	211	1	no	—
45	C₂×Q₈	—	422	1	no	—
46	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
47	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
48	C₂×C₈	—	211	2	no	—
49	C₄⋊C₄	—	422	1	no	—
50	C₄⋊C₄	—	211	2	no	—
51	C₂²⋊C₄	—	2	1	no	—
52	C₂²⋊C₄	—	2	1	no	—
53	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	844	2	no	—
57	D₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	422	2	no	—
60	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	C₈	—	422	2	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	1	no	—
71	C₄	—	844	2	no	—
72	C₄	#16809961	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	#20269964	1	1	yes	characteristic
75	C₂	#20270002	1	1	yes	characteristic
76	C₂	#20270007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#124×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₄×C₂₁₁	C₂²
hypercenter	#124	D₂₁₁
hypocenter	C₂₁₁	#489
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #20269977	C₂	semidirect
2. #20269901	C₂	semidirect
4. #20269888	C₂	semidirect
38. C₂₁₁	#489	semidirect

Maximal subgroups

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

subgroup	index
#20269977	2
#20269901	2
#20269898	2
#20269888	2
#124×C₂₁₁	2
#489	211