Subgroups — #27212938

Order 13504.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269973	C₂	1	1	yes	maximal, characteristic
2	#20269892	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₄₄	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₂₁₁	—	2	1	no	—
6	C₂×C₂²⋊D₂₁₁	—	2	1	no	—
7	C₂²⋊Dic₂₁₁	—	2	1	no	—
8	C₄×Dic₂₁₁	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
11	C₂²×D₂₁₁	—	2	1	no	—
12	D₈₄₄	—	4	1	no	—
13	C₂²⋊D₂₁₁	—	4	2	no	—
14	C₂²⋊D₂₁₁	—	4	2	no	—
15	C₂×Dic₂₁₁	—	2	1	no	—
16	C₂×Dic₂₁₁	—	4	1	no	—
17	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
18	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
19	D₄×C₂₁₁	—	4	1	no	—
20	C₂×C₄×C₂₁₁	—	4	1	no	—
21	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₂₁₁	—	2	1	no	—
23	C₂²×C₂₁₁	—	2	1	no	—
24	C₂²×C₂₁₁	—	4	1	no	—
25	C₂²×C₂₁₁	—	4	1	no	—
26	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₂₁₁	—	4	1	no	—
28	C₂×D₂₁₁	—	4	1	no	—
29	C₂×D₂₁₁	—	4	1	no	—
30	C₄×C₂₁₁	—	2	1	no	—
31	C₄×C₂₁₁	—	8	1	no	—
32	Dic₂₁₁	—	2	2	no	—
33	Dic₂₁₁	—	8	1	no	—
34	C₂×C₂₁₁	—	2	1	no	—
35	C₂×C₂₁₁	—	4	1	no	—
36	C₂×C₂₁₁	—	4	1	no	—
37	C₂×C₂₁₁	#99	1	1	yes	characteristic
38	D₂₁₁	—	8	1	no	—
39	C₂₁₁	#353	1	1	yes	characteristic, complemented, Sylow
40	#353	—	211	1	no	maximal, Sylow
41	#127	—	211	1	no	—
42	#99	—	211	1	no	—
43	#99	D₂₁₁	1	1	yes	characteristic, complemented
44	C₂×D₄	—	422	1	no	—
45	C₂×D₄	—	422	1	no	—
46	C₂×D₄	—	211	1	no	—
47	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	422	1	no	—
50	C₄²	—	211	1	no	—
51	C₂³	—	422	1	no	—
52	C₂³	C₂²⋊D₂₁₁	1	1	yes	characteristic
53	C₂³	D₈₄₄	1	1	yes	characteristic
54	D₄	—	4	1	no	—
55	D₄	—	844	1	no	—
56	D₄	—	844	2	no	—
57	D₄	—	844	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	844	1	no	—
60	C₂×C₄	—	422	1	no	—
61	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	844	1	no	—
67	C₂²	—	844	1	no	—
68	C₂²	—	844	1	no	—
69	C₂²	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	8	1	no	—
72	C₄	—	1688	1	no	—
73	C₄	—	422	2	no	—
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	1688	1	no	—
78	C₂	#20269891	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269891
commutator	C₂×C₄×C₂₁₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₁₁
Fitting	#99×C₂₁₁	C₂
socle	C₂×C₂₁₁	#99
prime radical	#20269973	C₂
cosocle	C₂×D₄×C₂₁₁	C₂²
hypercenter	#99	D₂₁₁
hypocenter	C₂₁₁	#353
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. #20269892	C₂	semidirect
3. #99×C₂₁₁	C₂	semidirect
4. C₂×D₈₄₄	C₄	semidirect
8. C₄×Dic₂₁₁	C₄	semidirect
39. C₂₁₁	#353	semidirect
43. #99	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269973	2
#20269892	2
#99×C₂₁₁	2
#353	211