Subgroups — #27213333

Order 13504.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269945	C₂	1	3	yes	maximal, complemented
2	#20269892	C₂	1	3	yes	maximal, complemented
3	H(5,𝔽₂)×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₁₁	C₂²	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₂₁₁	—	2	3	no	—
6	C₂²⋊Dic₂₁₁	C₂²	1	3	yes	complemented
7	(C₂×D₂₁₁)⋊C₄	—	2	3	no	—
8	#41×C₂₁₁	—	2	3	no	—
9	C₂×D₄×C₂₁₁	—	2	3	no	—
10	C₂×D₄×C₂₁₁	C₂²	1	3	yes	—
11	C₂²×D₂₁₁	—	2	3	no	—
12	C₂²×D₂₁₁	—	4	2	no	—
13	C₂²⋊D₂₁₁	—	4	6	no	—
14	C₂×Dic₂₁₁	—	2	3	no	—
15	C₂³×C₂₁₁	—	2	1	no	—
16	C₂³×C₂₁₁	D₄	1	3	yes	complemented
17	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
18	Q₈×C₂₁₁	—	2	1	no	—
19	D₄×C₂₁₁	—	2	3	no	—
20	D₄×C₂₁₁	—	2	6	no	—
21	C₂×C₄×C₂₁₁	—	2	3	no	—
22	C₂×C₄×C₂₁₁	D₄	1	3	yes	complemented
23	C₂²×C₂₁₁	—	2	3	no	—
24	C₂²×C₂₁₁	—	4	1	no	—
25	C₂²×C₂₁₁	—	4	3	no	—
26	C₂²×C₂₁₁	—	8	1	no	—
27	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
28	C₂×D₂₁₁	—	2	6	no	—
29	C₂×D₂₁₁	—	4	3	no	—
30	C₂×D₂₁₁	—	4	1	no	—
31	C₄×C₂₁₁	—	2	3	no	—
32	Dic₂₁₁	—	4	3	no	—
33	C₂×C₂₁₁	—	2	3	no	—
34	C₂×C₂₁₁	—	4	3	no	—
35	C₂×C₂₁₁	#120	1	1	yes	characteristic
36	D₂₁₁	—	4	2	no	—
37	C₂₁₁	B(4,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
38	B(4,𝔽₂)	—	211	1	no	maximal, Sylow
39	H(5,𝔽₂)	D₂₁₁	1	1	yes	characteristic, complemented
40	#120	—	211	3	no	—
41	#99	—	211	3	no	—
42	C₂⁴	—	211	1	no	—
43	#41	—	2	3	no	—
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	—	422	3	no	—
46	C₂×D₄	C₂×D₂₁₁	1	3	yes	—
47	C₂²⋊C₄	—	422	3	no	—
48	C₂²⋊C₄	—	211	3	no	—
49	C₂³	—	2	1	no	—
50	C₂³	—	844	2	no	—
51	C₂³	—	422	3	no	—
52	C₂³	C₂²⋊D₂₁₁	1	3	yes	complemented
53	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
54	Q₈	—	2	1	no	—
55	D₄	—	2	3	no	—
56	D₄	—	2	6	no	—
57	D₄	—	844	6	no	—
58	C₂×C₄	—	2	3	no	—
59	C₂×C₄	—	422	3	no	—
60	C₂×C₄	C₂²⋊D₂₁₁	1	3	yes	complemented
61	C₂²	—	2	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	3	no	—
64	C₂²	—	8	1	no	—
65	C₂²	—	844	3	no	—
66	C₂²	—	844	1	no	—
67	C₂²	—	422	6	no	—
68	C₂²	C₂×C₂²⋊D₂₁₁	1	3	yes	—
69	C₄	—	2	3	no	—
70	C₄	—	844	3	no	—
71	C₂	—	2	3	no	—
72	C₂	—	4	3	no	—
73	C₂	—	844	2	no	—
74	C₂	#20269946	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269946
commutator, cosocle	C₂³×C₂₁₁	C₂³
Frattini	C₂³	C₂²×D₂₁₁
Fitting	H(5,𝔽₂)×C₂₁₁	C₂
socle	C₂×C₂₁₁	#120
hypercenter	H(5,𝔽₂)	D₂₁₁
hypocenter	C₂₁₁	B(4,𝔽₂)
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20269945	C₂	semidirect
2. #20269892	C₂	semidirect
3. H(5,𝔽₂)×C₂₁₁	C₂	semidirect
4. C₂³×D₂₁₁	C₂²	semidirect
6. C₂²⋊Dic₂₁₁	C₂²	semidirect
16. C₂³×C₂₁₁	D₄	semidirect
22. C₂×C₄×C₂₁₁	D₄	semidirect
37. C₂₁₁	B(4,𝔽₂)	semidirect
39. H(5,𝔽₂)	D₂₁₁	semidirect
52. C₂³	C₂²⋊D₂₁₁	semidirect
60. C₂×C₄	C₂²⋊D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269945	2
#20269892	2
H(5,𝔽₂)×C₂₁₁	2
B(4,𝔽₂)	211