Subgroups — #41003197

Order 26944.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212354	C₂	1	1	yes	maximal, characteristic
2	#27212289	C₂	1	2	yes	maximal, complemented
3	#27212286	C₂	1	2	yes	maximal
4	C₄×#16809890	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
8	C₂×#16809890	C₂²	1	2	yes	—
9	#20269748	—	2	2	no	—
10	C₄⋊C₄×C₄₂₁	—	2	2	no	—
11	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
12	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄₂₁	—	2	1	no	—
14	Dic₈₄₂	—	2	2	no	—
15	D₁₆₈₄	—	2	2	no	—
16	C₂×Dic₄₂₁	—	2	1	no	—
17	#16809890	—	2	2	no	—
18	C₂×C₄×C₄₂₁	—	2	2	no	—
19	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
20	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
22	C₂×D₄₂₁	—	4	1	no	—
23	C₂×D₄₂₁	—	4	2	no	—
24	C₄×C₄₂₁	—	2	2	no	—
25	C₄×C₄₂₁	—	4	2	no	—
26	C₄×C₄₂₁	#41	1	2	yes	—
27	Dic₄₂₁	—	4	1	no	—
28	C₂×C₄₂₁	#124	1	1	yes	characteristic
29	C₂×C₄₂₁	#135	1	2	yes	—
30	D₄₂₁	—	8	1	no	—
31	C₄₂₁	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	421	1	no	maximal, Sylow
33	#125	D₄₂₁	1	1	yes	characteristic, complemented
34	#124	—	421	1	no	—
35	#103	—	421	2	no	—
36	#102	—	421	2	no	—
37	C₄×C₈	—	421	1	no	—
38	C₂×Q₈	—	421	1	no	—
39	C₂×D₄	—	421	1	no	—
40	C₂×C₈	—	421	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
43	C₂²⋊C₄	—	842	2	no	—
44	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
45	C₂³	—	842	1	no	—
46	Q₈	—	842	2	no	—
47	D₄	—	842	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	842	1	no	—
50	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
51	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
52	C₈	—	842	2	no	—
53	C₂²	—	1684	1	no	—
54	C₂²	—	1684	2	no	—
55	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	1684	1	no	—
59	C₄	#20269772	1	2	yes	—
60	C₂	—	3368	1	no	—
61	C₂	#27212355	1	1	yes	characteristic
62	C₂	#27212386	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#125×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	#125	D₄₂₁
hypocenter	C₄₂₁	#488
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #27212289	C₂	semidirect
4. C₄×#16809890	C₂	semidirect
5. #125×C₄₂₁	C₂	semidirect
31. C₄₂₁	#488	semidirect
33. #125	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212354	2
#27212289	2
#27212286	2
C₄×#16809890	2
#125×C₄₂₁	2
#488	421