Subgroups — #27213323

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	#20270008	C₂	1	2	yes	maximal, complemented
3	#20269905	C₂	1	2	yes	maximal, complemented
4	#20269894	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	2	yes	—
7	#16809961	—	2	2	no	—
8	C₂×C₂²⋊D₂₁₁	—	2	2	no	—
9	D₄×D₂₁₁	—	2	2	no	—
10	#16809952	—	2	2	no	—
11	#16809949	—	2	2	no	—
12	#16809946	C₂²	1	2	yes	complemented
13	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
16	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₂₁₁	—	2	2	no	—
18	Dic₄₂₂	D₄	1	2	yes	—
19	D₈₄₄	D₄	1	2	yes	—
20	C₂²⋊D₂₁₁	—	2	4	no	—
21	C₂²⋊D₂₁₁	—	2	2	no	—
22	C₂×Dic₂₁₁	—	2	2	no	—
23	C₄×D₂₁₁	—	2	2	no	—
24	C₂₁₁⋊C₈	—	2	2	no	—
25	C₂³×C₂₁₁	D₄	1	2	yes	—
26	Q₈×C₂₁₁	—	2	2	no	—
27	D₄×C₂₁₁	—	2	2	no	—
28	C₂×C₄×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	—	2	1	no	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
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	—	4	1	no	—
37	C₄×C₂₁₁	—	4	1	no	—
38	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
39	Dic₂₁₁	—	2	2	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	D₂₁₁	1	1	yes	characteristic, complemented
49	#104	—	211	2	no	—
50	#100	—	211	1	no	—
51	#41	—	422	2	no	—
52	#41	—	211	2	no	—
53	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic, complemented
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	—	211	2	no	—
60	C₂²⋊C₄	—	2	2	no	—
61	C₄²	C₂×D₂₁₁	1	1	yes	characteristic, complemented
62	C₂³	—	422	2	no	—
63	C₂³	C₂²⋊D₂₁₁	1	2	yes	—
64	Q₈	—	2	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	—	2	1	no	—
71	C₂×C₄	—	2	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	—	422	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×C₂²⋊D₂₁₁	1	1	yes	characteristic
81	C₄	—	4	1	no	—
82	C₄	—	4	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	#20269945	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269945
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#124×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
prime radical	#20270022	C₂
hypercenter	#124	D₂₁₁
hypocenter	C₂₁₁	#455
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #124

Complements

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

N	H ≅ G/N	type
2. #20270008	C₂	semidirect
3. #20269905	C₂	semidirect
4. #20269894	C₂	semidirect
5. #124×C₂₁₁	C₂	semidirect
12. #16809946	C₂²	semidirect
13. C₂×Q₈×C₂₁₁	C₂²	semidirect
16. C₄²×C₂₁₁	C₂²	semidirect
44. C₂₁₁	#455	semidirect
48. #124	D₂₁₁	semidirect
53. C₂×Q₈	C₂×D₂₁₁	semidirect
61. C₄²	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
#20270008	2
#20269905	2
#20269894	2
#124×C₂₁₁	2
#455	211