Subgroups — #41003721

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269780	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269780	C₂	1	1	yes	maximal, characteristic
3	#27212322	C₂	1	1	yes	maximal, characteristic
4	#20269780	—	2	1	no	—
5	#20269780	—	2	1	no	—
6	#20269780	—	2	1	no	—
7	#20269780	C₄	1	2	yes	complemented
8	C₂×#16809897	—	2	1	no	—
9	C₂×#16809897	C₄	1	2	yes	complemented
10	C₂²×Dic₄₂₁	C₂²	1	1	yes	characteristic
11	#20269750	—	2	1	no	—
12	C₂²⋊Dic₄₂₁	C₄	1	1	yes	characteristic
13	C₄×Dic₄₂₁	—	2	1	no	—
14	C₂²⋊C₄×C₄₂₁	C₄	1	1	yes	characteristic
15	#16809897	—	2	2	no	—
16	#16809897	—	2	2	no	—
17	C₂×Dic₄₂₁	—	2	1	no	—
18	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
21	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
22	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
23	C₂×Dic₄₂₁	Q₈	1	1	yes	characteristic
24	C₂³×C₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	—	2	1	no	—
26	C₂²×C₄₂₁	—	4	1	no	—
27	C₂²×C₄₂₁	C₄²	1	1	yes	characteristic
28	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄₂₁	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₄₂₁	—	4	1	no	—
31	Dic₄₂₁	—	4	1	no	—
32	Dic₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₄₂₁	C₄⋊C₄	1	1	yes	characteristic
35	Dic₄₂₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄₂₁	—	2	1	no	—
37	C₂×C₄₂₁	—	2	1	no	—
38	C₂×C₄₂₁	—	2	1	no	—
39	C₂×C₄₂₁	#95	1	1	yes	characteristic
40	C₄₂₁	#344	1	1	yes	characteristic, complemented, Sylow
41	#344	—	421	1	no	maximal, Sylow
42	C₂×M₁₆	—	421	1	no	—
43	C₂×M₁₆	—	421	1	no	—
44	#117	—	421	1	no	—
45	C₂²×C₄	—	421	1	no	—
46	M₁₆	—	842	1	no	—
47	M₁₆	—	842	1	no	—
48	M₁₆	—	842	1	no	—
49	M₁₆	—	421	2	no	—
50	C₂×C₈	—	842	1	no	—
51	C₂×C₈	—	421	2	no	—
52	C₄⋊C₄	—	842	1	no	—
53	C₂²⋊C₄	—	421	1	no	—
54	C₂²⋊C₄	#13132997	1	1	yes	characteristic
55	C₄²	—	842	1	no	—
56	C₂³	C₂×#13132997	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	842	1	no	—
59	C₂×C₄	—	421	1	no	—
60	C₂×C₄	—	421	1	no	—
61	C₂×C₄	—	421	1	no	—
62	C₂×C₄	—	421	1	no	—
63	C₂×C₄	—	421	1	no	—
64	C₂×C₄	—	421	1	no	—
65	C₈	—	842	2	no	—
66	C₈	—	842	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	C₄×#13132997	1	1	yes	characteristic
69	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
70	C₂²	C₄⋊#13132997	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	1684	1	no	—
73	C₄	—	421	1	no	—
74	C₄	—	421	1	no	—
75	C₄	—	421	1	no	—
76	C₄	—	421	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	(C₂×#13132997)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#13132997)⋊C₄
commutator	C₂²×C₄₂₁	C₄²
Frattini	C₂³	C₂×#13132997
Fitting	C₂²⋊C₄×C₄₂₁	C₄
socle	C₂×C₄₂₁	#95
prime radical	C₂³×C₄₂₁	C₂×C₄
cosocle	C₂²×Dic₄₂₁	C₂²
hypercenter	C₂²⋊C₄	#13132997
hypocenter	C₄₂₁	#344
LCS	G ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
7. #20269780	C₄	semidirect
9. C₂×#16809897	C₄	semidirect
40. C₄₂₁	#344	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
#27212322	2
#344	421