Subgroups — #41002971

Order 26944.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16809890	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#16809890	C₂	1	2	yes	maximal
3	#27212277	C₂	1	2	yes	maximal
4	C₄×#16809890	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809890	C₂²	1	1	yes	characteristic
7	C₂×#16809890	C₂²	1	2	yes	—
8	C₂×#16809890	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₄₂₁	C₄	1	1	yes	characteristic
10	C₄⋊C₄×C₄₂₁	C₄	1	2	yes	—
11	C₄⋊C₄×C₄₂₁	C₄	1	1	yes	characteristic
12	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄²×C₄₂₁	C₂²	1	2	yes	—
14	#16809890	—	2	1	no	—
15	#16809890	—	2	2	no	—
16	#16809890	Q₈	1	2	yes	complemented
17	Q₈×C₄₂₁	—	2	2	no	—
18	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₂₁	—	2	2	no	—
25	C₄×C₄₂₁	—	2	1	no	—
26	C₄×C₄₂₁	—	2	2	no	—
27	C₄×C₄₂₁	M₁₆	1	2	yes	—
28	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₄₂₁	#41	1	1	yes	characteristic
30	C₂×C₄₂₁	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₄₂₁	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₄₂₁	#131	1	1	yes	characteristic
33	C₄₂₁	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	421	1	no	maximal, Sylow
35	C₄×Q₈	D₄₂₁	1	1	yes	characteristic
36	C₄⋊C₈	—	421	1	no	—
37	C₄⋊C₈	—	421	2	no	—
38	#97	—	421	2	no	—
39	C₄×C₈	—	421	1	no	—
40	C₂×Q₈	Dic₄₂₁	1	1	yes	characteristic
41	C₂×C₈	—	421	1	no	—
42	C₂×C₈	—	421	2	no	—
43	C₂×C₈	—	421	1	no	—
44	C₄⋊C₄	Dic₄₂₁	1	2	yes	—
45	C₄⋊C₄	Dic₄₂₁	1	1	yes	characteristic
46	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
47	C₄²	C₂×D₄₂₁	1	2	yes	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	C₂×Dic₄₂₁	1	1	yes	characteristic
50	C₂×C₄	C₂×Dic₄₂₁	1	2	yes	—
51	C₂×C₄	C₂×Dic₄₂₁	1	2	yes	—
52	C₂×C₄	C₂×Dic₄₂₁	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
54	C₈	—	842	1	no	—
55	C₈	—	842	2	no	—
56	C₈	—	421	2	no	—
57	C₂²	C₂²×Dic₄₂₁	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	—	2	2	no	—
61	C₄	#20269755	1	2	yes	—
62	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
63	C₄	#20269772	1	1	yes	characteristic
64	C₂	Q₈×Dic₄₂₁	1	1	yes	characteristic
65	C₂	C₂×#20269755	1	1	yes	characteristic
66	C₂	#27212373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₂₁
commutator, socle, prime radical	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₄×Q₈×C₄₂₁	C₂
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₄×Q₈	D₄₂₁
hypocenter	C₄₂₁	#446
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. #16809890	Q₈	semidirect
33. C₄₂₁	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#16809890	2
#27212277	2
C₄×#16809890	2
C₄×Q₈×C₄₂₁	2
#446	421