Subgroups — #41003690

Order 26944.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16809897	C₂	1	1	yes	maximal, characteristic
2	C₄×#16809897	C₂	1	1	yes	maximal, characteristic
3	#27212331	C₂	1	1	yes	maximal, characteristic
4	C₂×#16809897	—	2	1	no	—
5	C₂×#16809897	—	2	1	no	—
6	C₂×Dic₈₄₂	C₄	1	1	yes	characteristic
7	#20269750	—	2	1	no	—
8	C₄×Dic₄₂₁	—	2	1	no	—
9	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₄₂₁	C₄	1	1	yes	characteristic
11	#16809897	—	2	2	no	—
12	#16809897	—	4	1	no	—
13	Dic₈₄₂	—	2	1	no	—
14	Dic₈₄₂	C₈	1	2	yes	complemented
15	C₂×Dic₄₂₁	—	2	1	no	—
16	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
17	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₄₂₁	—	2	1	no	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₄₂₁	—	4	1	no	—
22	C₄×C₄₂₁	C₂×C₈	1	1	yes	characteristic
23	C₄×C₄₂₁	M₁₆	1	1	yes	characteristic
24	Dic₄₂₁	—	2	2	no	—
25	Dic₄₂₁	—	2	1	no	—
26	Dic₄₂₁	SD₁₆	1	1	yes	characteristic
27	Dic₄₂₁	Q₁₆	1	1	yes	characteristic
28	C₂×C₄₂₁	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₄₂₁	#103	1	1	yes	characteristic
30	C₂×C₄₂₁	#104	1	1	yes	characteristic
31	C₄₂₁	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	421	1	no	maximal, Sylow
33	C₄×Q₈	—	421	1	no	—
34	C₄⋊C₈	—	421	1	no	—
35	C₄×C₈	—	421	1	no	—
36	C₂×Q₈	—	421	1	no	—
37	C₂×C₈	—	842	1	no	—
38	C₂×C₈	—	842	1	no	—
39	C₄⋊C₄	—	842	1	no	—
40	C₄⋊C₄	#13132997	1	1	yes	characteristic
41	C₄²	—	842	1	no	—
42	C₄²	—	421	1	no	—
43	Q₈	—	842	1	no	—
44	Q₈	—	421	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	842	1	no	—
47	C₂×C₄	—	421	1	no	—
48	C₂×C₄	—	421	1	no	—
49	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
50	C₈	—	842	2	no	—
51	C₈	—	1684	1	no	—
52	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
53	C₄	—	4	1	no	—
54	C₄	—	842	2	no	—
55	C₄	—	842	1	no	—
56	C₄	—	421	1	no	—
57	C₄	—	421	1	no	—
58	C₄	D₄₂₁⋊C₈	1	1	yes	characteristic
59	C₄	#20269781	1	1	yes	characteristic
60	C₂	#27212423	1	1	yes	characteristic
61	C₂	#27212429	1	1	yes	characteristic
62	C₂	#27212430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13132997
commutator	C₄×C₄₂₁	C₂×C₈
Frattini	C₂×C₄	C₂×#13132997
Fitting	C₄⋊C₄×C₄₂₁	C₄
socle, prime radical	C₂²×C₄₂₁	C₂²⋊C₄
cosocle	C₄×Dic₄₂₁	C₂²
hypercenter	C₄⋊C₄	#13132997
hypocenter	C₄₂₁	Q₈⋊C₈
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Dic₈₄₂	C₈	semidirect
31. C₄₂₁	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#16809897	2
C₄×#16809897	2
#27212331	2
Q₈⋊C₈	421