Subgroups — #27213326

Order 13504.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270022	C₂	1	1	yes	maximal, characteristic
2	#20270010	C₂	1	2	yes	maximal, complemented
3	#20269964	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269903	C₂	1	2	yes	maximal, complemented
5	#100×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	2	yes	—
7	#16809961	—	2	2	no	—
8	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₂²⋊D₂₁₁	—	2	2	no	—
10	D₄×D₂₁₁	—	2	2	no	—
11	Dic₈₄₄	—	2	2	no	—
12	#16809951	—	2	2	no	—
13	C₂²⋊Dic₂₁₁	—	2	2	no	—
14	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	M₁₆×C₂₁₁	C₂²	1	2	yes	complemented
17	C₂²×D₂₁₁	—	2	2	no	—
18	Dic₄₂₂	—	2	2	no	—
19	Dic₄₂₂	D₄	1	2	yes	—
20	D₈₄₄	D₄	1	2	yes	—
21	C₂²⋊D₂₁₁	—	2	4	no	—
22	C₂²⋊D₂₁₁	—	2	2	no	—
23	C₂×Dic₂₁₁	—	2	2	no	—
24	C₂×Dic₂₁₁	—	2	1	no	—
25	C₂×Dic₂₁₁	—	2	1	no	—
26	C₄×D₂₁₁	—	2	2	no	—
27	C₂³×C₂₁₁	D₄	1	2	yes	—
28	D₄×C₂₁₁	—	2	2	no	—
29	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₈×C₂₁₁	—	2	2	no	—
31	C₂²×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	—	4	2	no	—
33	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₁₁	—	2	2	no	—
35	C₂×D₂₁₁	—	8	2	no	—
36	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
37	Dic₂₁₁	—	2	2	no	—
38	Dic₂₁₁	—	4	1	no	—
39	Dic₂₁₁	—	4	1	no	—
40	C₂×C₂₁₁	—	2	1	no	—
41	C₂×C₂₁₁	—	4	2	no	—
42	C₂×C₂₁₁	#120	1	1	yes	characteristic
43	D₂₁₁	—	4	2	no	—
44	C₂₁₁	#455	1	1	yes	characteristic, complemented, Sylow
45	#455	—	211	1	no	maximal, Sylow
46	H(5,𝔽₂)	—	211	1	no	—
47	#137	—	211	2	no	—
48	#124	—	211	1	no	—
49	#104	—	211	2	no	—
50	#100	D₂₁₁	1	1	yes	characteristic, complemented
51	#41	—	422	2	no	—
52	#41	—	211	2	no	—
53	C₂×Q₈	—	211	1	no	—
54	C₂×D₄	—	422	2	no	—
55	C₂×D₄	—	422	2	no	—
56	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
57	Q₁₆	—	422	2	no	—
58	SD₁₆	—	422	2	no	—
59	M₁₆	C₂×D₂₁₁	1	2	yes	complemented
60	C₂²⋊C₄	—	422	2	no	—
61	C₄²	—	211	1	no	—
62	C₂³	—	422	2	no	—
63	C₂³	D₈₄₄	1	2	yes	—
64	Q₈	—	422	2	no	—
65	Q₈	—	211	2	no	—
66	D₄	—	2	2	no	—
67	D₄	—	422	4	no	—
68	D₄	—	422	2	no	—
69	D₄	—	211	2	no	—
70	C₂×C₄	—	422	1	no	—
71	C₂×C₄	—	422	1	no	—
72	C₂×C₄	—	422	2	no	—
73	C₂×C₄	—	422	2	no	—
74	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	—	1688	2	no	—
79	C₂²	—	422	2	no	—
80	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
81	C₄	—	844	1	no	—
82	C₄	—	844	1	no	—
83	C₄	—	422	2	no	—
84	C₄	D₄×D₂₁₁	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	844	2	no	—
88	C₂	#20269947	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269947
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#100×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
prime radical	#20270022	C₂
hypercenter	#100	D₂₁₁
hypocenter	C₂₁₁	#455
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. #20270010	C₂	semidirect
3. #20269964	C₂	semidirect
4. #20269903	C₂	semidirect
5. #100×C₂₁₁	C₂	semidirect
8. C₂×Dic₄₂₂	C₂²	semidirect
14. C₄×Dic₂₁₁	C₂²	semidirect
16. M₁₆×C₂₁₁	C₂²	semidirect
44. C₂₁₁	#455	semidirect
50. #100	D₂₁₁	semidirect
59. M₁₆	C₂×D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270022	2
#20270010	2
#20269964	2
#20269903	2
#100×C₂₁₁	2
#455	211