Subgroups — #41002703

Order 26944.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Dic₄₂₁	C₂	1	2	yes	maximal
2	#27212273	C₂	1	4	yes	maximal, complemented
3	C₂³×C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₄₂₁	C₂²	1	4	yes	—
5	C₂²⋊Dic₄₂₁	—	2	4	no	—
6	C₂⁴×C₄₂₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₄₂₁	—	2	4	no	—
8	C₂²×C₄×C₄₂₁	C₄	1	4	yes	complemented
9	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
10	C₂×Dic₄₂₁	—	2	4	no	—
11	C₂×Dic₄₂₁	—	2	8	no	—
12	C₂³×C₄₂₁	—	2	4	no	—
13	C₂³×C₄₂₁	C₂×C₄	1	2	yes	—
14	C₂³×C₄₂₁	D₄	1	4	yes	—
15	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₄₂₁	—	2	8	no	—
17	C₂×C₄×C₄₂₁	—	2	4	no	—
18	C₂×C₄×C₄₂₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄₂₁	—	2	4	no	—
20	C₂²×C₄₂₁	—	2	8	no	—
21	C₂²×C₄₂₁	C₂²⋊C₄	1	4	yes	—
22	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
24	C₂²×C₄₂₁	#41	1	4	yes	—
25	C₄×C₄₂₁	—	2	4	no	—
26	Dic₄₂₁	—	4	4	no	—
27	C₂×C₄₂₁	—	2	4	no	—
28	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄₂₁	#117	1	2	yes	—
30	C₂×C₄₂₁	#123	1	4	yes	—
31	C₄₂₁	#381	1	1	yes	characteristic, complemented, Sylow
32	#381	—	421	1	no	maximal, Sylow
33	C₂³×C₄	D₄₂₁	1	1	yes	characteristic
34	C₂×C₂²⋊C₄	—	421	2	no	—
35	#95	—	421	4	no	—
36	C₂⁴	C₂×D₄₂₁	1	1	yes	characteristic
37	C₂²×C₄	—	2	4	no	—
38	C₂²×C₄	—	421	4	no	—
39	C₂²×C₄	Dic₄₂₁	1	4	yes	complemented
40	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
41	C₂²⋊C₄	—	842	4	no	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×Dic₄₂₁	1	2	yes	—
44	C₂³	C₂²⋊D₄₂₁	1	4	yes	—
45	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
46	C₂×C₄	—	2	8	no	—
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	842	4	no	—
49	C₂×C₄	—	842	8	no	—
50	C₂×C₄	C₂×Dic₄₂₁	1	4	yes	—
51	C₂²	—	2	4	no	—
52	C₂²	—	2	8	no	—
53	C₂²	C₂²⋊Dic₄₂₁	1	4	yes	—
54	C₂²	C₂²×Dic₄₂₁	1	1	yes	characteristic
55	C₂²	C₂×C₂²⋊D₄₂₁	1	2	yes	—
56	C₂²	#20269771	1	4	yes	—
57	C₄	—	2	4	no	—
58	C₄	—	1684	4	no	—
59	C₂	—	2	4	no	—
60	C₂	C₂×C₂²⋊Dic₄₂₁	1	1	yes	characteristic
61	C₂	#27212325	1	2	yes	—
62	C₂	#27212349	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₂₁
commutator	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₂³×C₄×C₄₂₁	C₂
socle, cosocle	C₂³×C₄₂₁	C₂³
prime radical	C₂⁴×C₄₂₁	C₂²
hypercenter	C₂³×C₄	D₄₂₁
hypocenter	C₄₂₁	#381
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. #27212273	C₂	semidirect
8. C₂²×C₄×C₄₂₁	C₄	semidirect
31. C₄₂₁	#381	semidirect
39. C₂²×C₄	Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Dic₄₂₁	2
#27212273	2
C₂³×C₄×C₄₂₁	2
#381	421