Subgroups — #41003209

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212366	C₂	1	1	yes	maximal, characteristic
2	#27212358	C₂	1	1	yes	maximal, characteristic
3	#27212291	C₂	1	4	yes	maximal
4	#97×C₄₂₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₄₂	C₂²	1	2	yes	—
6	C₄⋊Dic₄₂₁	—	2	2	no	—
7	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
8	#20269750	—	2	2	no	—
9	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
10	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
11	Dic₈₄₂	—	2	4	no	—
12	C₂×Dic₄₂₁	—	2	2	no	—
13	C₂×Dic₄₂₁	—	2	2	no	—
14	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
15	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
16	C₈×C₄₂₁	—	2	2	no	—
17	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
18	C₄×C₄₂₁	—	2	2	no	—
19	C₄×C₄₂₁	#41	1	2	yes	—
20	Dic₄₂₁	—	4	2	no	—
21	Dic₄₂₁	—	4	2	no	—
22	C₂×C₄₂₁	#124	1	1	yes	characteristic
23	C₂×C₄₂₁	#137	1	2	yes	—
24	C₄₂₁	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	421	1	no	maximal, Sylow
26	C₄⋊Q₈	—	421	1	no	—
27	#125	—	421	1	no	—
28	#103	—	421	4	no	—
29	#97	D₄₂₁	1	1	yes	characteristic
30	C₂×Q₈	—	421	2	no	—
31	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
32	C₄⋊C₄	—	842	2	no	—
33	C₄⋊C₄	—	842	2	no	—
34	C₄⋊C₄	—	421	2	no	—
35	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
36	Q₈	—	842	4	no	—
37	C₂×C₄	—	842	2	no	—
38	C₂×C₄	—	842	2	no	—
39	C₂×C₄	D₁₆₈₄	1	2	yes	—
40	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	1684	2	no	—
45	C₄	—	1684	2	no	—
46	C₄	#20269771	1	2	yes	—
47	C₂	#27212354	1	1	yes	characteristic
48	C₂	#27212399	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#97×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	#97	D₄₂₁
hypocenter	C₄₂₁	#491
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
24. C₄₂₁	#491	semidirect

Maximal subgroups

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

subgroup	index
#27212366	2
#27212358	2
#27212291	2
#97×C₄₂₁	2
#491	421