Subgroups — #27213324

Order 13504.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270007	C₂	1	2	yes	maximal, complemented
2	#20269964	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269904	C₂	1	2	yes	maximal, complemented
4	#20269894	C₂	1	1	yes	maximal, characteristic, complemented
5	H(5,𝔽₂)×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic, complemented
7	#16809952	—	2	2	no	—
8	#16809950	—	2	2	no	—
9	#16809946	C₂²	1	2	yes	complemented
10	C₂²⋊Dic₂₁₁	—	2	2	no	—
11	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic, complemented
12	#41×C₂₁₁	—	2	2	no	—
13	#41×C₂₁₁	C₂²	1	2	yes	—
14	C₂×D₄×C₂₁₁	—	2	2	no	—
15	C₂×D₄×C₂₁₁	—	2	2	no	—
16	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₂₁₁	—	2	1	no	—
19	C₂×Dic₂₁₁	—	2	1	no	—
20	C₂₁₁⋊C₈	—	2	2	no	—
21	C₂³×C₂₁₁	—	2	2	no	—
22	C₂³×C₂₁₁	D₄	1	2	yes	—
23	Q₈×C₂₁₁	D₄	1	2	yes	—
24	D₄×C₂₁₁	—	2	4	no	—
25	D₄×C₂₁₁	—	2	2	no	—
26	D₄×C₂₁₁	—	2	2	no	—
27	D₄×C₂₁₁	D₄	1	2	yes	—
28	C₂×C₄×C₂₁₁	—	2	2	no	—
29	C₂×C₄×C₂₁₁	—	2	2	no	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	—	2	2	no	—
33	C₂²×C₂₁₁	—	4	2	no	—
34	C₂²×C₂₁₁	—	8	2	no	—
35	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₁₁	—	2	2	no	—
37	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
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	—	4	2	no	—
43	C₂×C₂₁₁	#120	1	1	yes	characteristic
44	C₂₁₁	#455	1	1	yes	characteristic, complemented, Sylow
45	#455	—	211	1	no	maximal, Sylow
46	H(5,𝔽₂)	D₂₁₁	1	1	yes	characteristic
47	#137	—	211	2	no	—
48	#124	—	211	1	no	—
49	#104	—	211	2	no	—
50	#100	—	211	1	no	—
51	#41	—	2	2	no	—
52	#41	C₂×D₂₁₁	1	2	yes	—
53	C₂×Q₈	—	211	1	no	—
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	—	2	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	—	422	2	no	—
61	C₄²	—	211	1	no	—
62	C₂³	—	2	2	no	—
63	C₂³	C₂²⋊D₂₁₁	1	2	yes	—
64	Q₈	—	422	2	no	—
65	Q₈	C₂²⋊D₂₁₁	1	2	yes	—
66	D₄	—	2	4	no	—
67	D₄	—	2	2	no	—
68	D₄	—	2	2	no	—
69	D₄	C₂²⋊D₂₁₁	1	2	yes	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	422	1	no	—
73	C₂×C₄	—	422	1	no	—
74	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	C₈	—	422	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	8	2	no	—
80	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	844	1	no	—
83	C₄	—	844	1	no	—
84	C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	4	2	no	—
88	C₂	#20269946	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269946
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting, prime radical	H(5,𝔽₂)×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
hypercenter	H(5,𝔽₂)	D₂₁₁
hypocenter	C₂₁₁	#455
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20270007	C₂	semidirect
2. #20269964	C₂	semidirect
3. #20269904	C₂	semidirect
4. #20269894	C₂	semidirect
6. C₂×Dic₄₂₂	C₂²	semidirect
9. #16809946	C₂²	semidirect
11. C₄×Dic₂₁₁	C₂²	semidirect
44. C₂₁₁	#455	semidirect

Maximal subgroups

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

subgroup	index
#20270007	2
#20269964	2
#20269904	2
#20269894	2
H(5,𝔽₂)×C₂₁₁	2
#455	211