Subgroups — #27213507

Order 13504.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269976	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269966	C₂	1	1	yes	maximal, characteristic
3	#20269901	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269899	C₂	1	1	yes	maximal, characteristic
5	#20269897	C₂	1	1	yes	maximal, characteristic
6	#20269887	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₂₂	—	2	1	no	—
9	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
12	Dic₂₁₁⋊C₄	—	2	1	no	—
13	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
14	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
15	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂²×D₂₁₁	—	2	1	no	—
20	Dic₄₂₂	—	4	2	no	—
21	D₈₄₄	—	2	1	no	—
22	D₈₄₄	—	2	1	no	—
23	C₂×Dic₂₁₁	—	2	1	no	—
24	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
25	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
26	C₂₁₁⋊C₈	—	2	1	no	—
27	Q₈×C₂₁₁	—	2	1	no	—
28	Q₈×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	—	2	1	no	—
30	C₂×C₄×C₂₁₁	—	2	1	no	—
31	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₈×C₂₁₁	—	2	1	no	—
33	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₁₁	—	4	1	no	—
35	C₂×D₂₁₁	—	4	1	no	—
36	C₂×D₂₁₁	—	4	1	no	—
37	C₄×C₂₁₁	—	4	1	no	—
38	C₄×C₂₁₁	—	4	1	no	—
39	C₄×C₂₁₁	#41	1	1	yes	characteristic
40	C₄×C₂₁₁	#41	1	1	yes	characteristic
41	Dic₂₁₁	—	2	1	no	—
42	Dic₂₁₁	—	2	1	no	—
43	Dic₂₁₁	—	4	1	no	—
44	C₂×C₂₁₁	#124	1	1	yes	characteristic
45	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
46	C₂×C₂₁₁	#137	1	1	yes	characteristic
47	D₂₁₁	—	8	1	no	—
48	C₂₁₁	#489	1	1	yes	characteristic, complemented, Sylow
49	#489	—	211	1	no	maximal, Sylow
50	C₄⋊Q₈	—	211	1	no	—
51	#124	—	211	1	no	—
52	#103	—	211	1	no	—
53	#103	D₂₁₁	1	1	yes	characteristic, complemented
54	#102	—	211	1	no	—
55	#102	—	211	1	no	—
56	#97	—	211	1	no	—
57	C₂×Q₈	—	422	1	no	—
58	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
59	C₂×D₄	—	211	1	no	—
60	C₂×C₈	—	211	1	no	—
61	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
62	C₄⋊C₄	—	422	1	no	—
63	C₄⋊C₄	—	211	1	no	—
64	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
65	C₂²⋊C₄	—	422	1	no	—
66	C₂²⋊C₄	—	422	1	no	—
67	C₄²	—	211	1	no	—
68	C₂³	—	422	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	844	2	no	—
72	D₄	—	422	1	no	—
73	D₄	—	422	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	422	1	no	—
77	C₂×C₄	—	211	1	no	—
78	C₂×C₄	—	211	1	no	—
79	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	422	1	no	—
82	C₂²	—	844	1	no	—
83	C₂²	—	844	1	no	—
84	C₂²	—	844	1	no	—
85	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	844	1	no	—
89	C₄	—	422	1	no	—
90	C₄	—	422	1	no	—
91	C₄	#16809961	1	1	yes	characteristic
92	C₄	#16809962	1	1	yes	characteristic
93	C₂	—	1688	1	no	—
94	C₂	#20269963	1	1	yes	characteristic
95	C₂	#20270004	1	1	yes	characteristic
96	C₂	#20270009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#103×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₈₄₄	C₂²
hypercenter	#103	D₂₁₁
hypocenter	C₂₁₁	#489
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
1. #20269976	C₂	semidirect
3. #20269901	C₂	semidirect
6. #20269887	C₂	semidirect
7. #103×C₂₁₁	C₂	semidirect
48. C₂₁₁	#489	semidirect
53. #103	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
#20269966	2
#20269901	2
#20269899	2
#20269897	2
#20269887	2
#103×C₂₁₁	2
#489	211