Subgroups — #41003740

Order 26944.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212426	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212424	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212335	C₂	1	1	yes	maximal, characteristic
4	#20269780	—	2	1	no	—
5	C₂²⋊#13132997	—	2	1	no	—
6	C₂×C₂²⋊D₄₂₁	—	2	1	no	—
7	C₂×C₂²⋊D₄₂₁	C₂²	1	1	yes	characteristic
8	C₂²⋊Dic₄₂₁	—	2	1	no	—
9	C₂²⋊Dic₄₂₁	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₄₂₁	C₄	1	1	yes	characteristic, complemented
11	C₂×#13132997	—	4	1	no	—
12	#16809897	—	4	1	no	—
13	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
14	C₂²⋊D₄₂₁	—	4	1	no	—
15	C₂²⋊D₄₂₁	—	4	2	no	—
16	C₂×Dic₄₂₁	—	2	1	no	—
17	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
18	C₂³×C₄₂₁	—	2	1	no	—
19	C₂³×C₄₂₁	—	4	2	no	—
20	C₂³×C₄₂₁	—	4	1	no	—
21	C₂³×C₄₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₂₁	—	2	2	no	—
23	C₂²×C₄₂₁	—	2	1	no	—
24	C₂²×C₄₂₁	—	4	1	no	—
25	C₂²×C₄₂₁	—	4	1	no	—
26	C₂²×C₄₂₁	—	4	2	no	—
27	C₂²×C₄₂₁	—	4	2	no	—
28	C₂²×C₄₂₁	—	4	1	no	—
29	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₄₂₁	—	2	1	no	—
31	C₂×D₄₂₁	—	4	1	no	—
32	#13132997	—	8	1	no	—
33	Dic₄₂₁	—	2	1	no	—
34	Dic₄₂₁	—	4	1	no	—
35	C₂×C₄₂₁	—	2	1	no	—
36	C₂×C₄₂₁	—	4	2	no	—
37	C₂×C₄₂₁	—	4	1	no	—
38	C₂×C₄₂₁	#99	1	1	yes	characteristic
39	D₄₂₁	—	4	1	no	—
40	C₄₂₁	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	421	1	no	maximal, Sylow
42	#120	—	421	1	no	—
43	#100	—	421	1	no	—
44	#99	—	421	1	no	—
45	C₂⁴	#13132997	1	1	yes	characteristic, complemented
46	C₂×D₄	—	842	1	no	—
47	C₂×D₄	—	421	1	no	—
48	M₁₆	—	842	1	no	—
49	C₂²⋊C₄	—	842	1	no	—
50	C₂²⋊C₄	—	842	1	no	—
51	C₂²⋊C₄	—	421	1	no	—
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	—	421	1	no	—
56	C₂³	C₂×#13132997	1	1	yes	characteristic
57	D₄	—	1684	1	no	—
58	D₄	—	1684	2	no	—
59	C₂×C₄	—	842	1	no	—
60	C₂×C₄	—	1684	1	no	—
61	C₂×C₄	—	421	1	no	—
62	C₈	—	1684	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	842	1	no	—
71	C₂²	—	1684	1	no	—
72	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
73	C₄	—	3368	1	no	—
74	C₄	—	842	1	no	—
75	C₄	—	1684	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	—	1684	1	no	—
80	C₂	#27212424	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212424
commutator	C₂³×C₄₂₁	C₂×C₄
Frattini	C₂³	C₂×#13132997
Fitting	C₂⁴×C₄₂₁	C₄
socle	C₂×C₄₂₁	#99
prime radical	#27212335	C₂
cosocle	C₂×C₂²⋊D₄₂₁	C₂²
hypercenter	C₂⁴	#13132997
hypocenter	C₄₂₁	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₄₂₁ ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. #27212426	C₂	semidirect
2. #27212424	C₂	semidirect
9. C₂²⋊Dic₄₂₁	C₄	semidirect
10. C₂⁴×C₄₂₁	C₄	semidirect
40. C₄₂₁	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	#13132997	semidirect

Maximal subgroups

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

subgroup	index
#27212426	2
#27212424	2
#27212335	2
AΣL(1,𝔽₁₆)	421