Subgroups — #41003720

Order 26944.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269780	C₂	1	2	yes	maximal
2	C₂×#20269750	C₂	1	1	yes	maximal, characteristic
3	#20269780	—	2	2	no	—
4	#20269780	C₄	1	4	yes	complemented
5	C₂×#16809897	—	2	2	no	—
6	C₂²×Dic₄₂₁	C₄	1	1	yes	characteristic
7	C₂²×Dic₄₂₁	C₂²	1	1	yes	characteristic
8	#20269750	—	2	2	no	—
9	C₂²×C₄×C₄₂₁	C₄	1	1	yes	characteristic
10	#16809897	—	2	4	no	—
11	C₂×Dic₄₂₁	—	2	1	no	—
12	C₂×Dic₄₂₁	—	4	1	no	—
13	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
14	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
15	C₂×Dic₄₂₁	D₄	1	2	yes	—
16	C₂×Dic₄₂₁	Q₈	1	1	yes	characteristic
17	C₂³×C₄₂₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₄₂₁	—	2	1	no	—
19	C₂×C₄×C₄₂₁	—	4	1	no	—
20	C₂²×C₄₂₁	—	2	1	no	—
21	C₂²×C₄₂₁	—	2	1	no	—
22	C₂²×C₄₂₁	C₄²	1	1	yes	characteristic
23	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
25	C₄×C₄₂₁	—	4	1	no	—
26	Dic₄₂₁	—	4	1	no	—
27	Dic₄₂₁	C₂²⋊C₄	1	2	yes	—
28	Dic₄₂₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄₂₁	—	2	1	no	—
30	C₂×C₄₂₁	—	2	1	no	—
31	C₂×C₄₂₁	#95	1	1	yes	characteristic
32	C₂×C₄₂₁	#100	1	1	yes	characteristic
33	C₂×C₄₂₁	#101	1	1	yes	characteristic
34	C₄₂₁	#343	1	1	yes	characteristic, complemented, Sylow
35	#343	—	421	1	no	maximal, Sylow
36	C₂×M₁₆	—	421	2	no	—
37	C₂×C₄⋊C₄	—	421	1	no	—
38	C₂²×C₄	—	421	1	no	—
39	C₂²×C₄	—	421	1	no	—
40	C₂²×C₄	#13132997	1	1	yes	characteristic
41	M₁₆	—	842	2	no	—
42	M₁₆	—	421	4	no	—
43	C₂×C₈	—	842	2	no	—
44	C₄⋊C₄	—	842	2	no	—
45	C₂³	C₂×#13132997	1	1	yes	characteristic
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	842	1	no	—
49	C₂×C₄	—	1684	1	no	—
50	C₂×C₄	—	421	2	no	—
51	C₂×C₄	—	421	2	no	—
52	C₂×C₄	—	421	1	no	—
53	C₂×C₄	—	421	1	no	—
54	C₈	—	842	4	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	C₄×#13132997	1	1	yes	characteristic
58	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
59	C₂²	C₄⋊#13132997	1	1	yes	characteristic
60	C₄	—	4	1	no	—
61	C₄	—	1684	1	no	—
62	C₄	—	421	2	no	—
63	C₄	—	421	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	2	1	no	—
66	C₂	(C₂×#13132997)⋊C₄	1	1	yes	characteristic
67	C₂	#27212426	1	1	yes	characteristic
68	C₂	#27212427	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13132997
commutator	C₂²×C₄₂₁	C₄²
Frattini	C₂³	C₂×#13132997
Fitting	C₂²×C₄×C₄₂₁	C₄
socle	C₂²×C₄₂₁	C₂²⋊C₄
prime radical	C₂³×C₄₂₁	C₂×C₄
cosocle	C₂²×Dic₄₂₁	C₂²
hypercenter	C₂²×C₄	#13132997
hypocenter	C₄₂₁	#343
LCS	G ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #20269780	C₄	semidirect
34. C₄₂₁	#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269780	2
C₂×#20269750	2
#343	421