Subgroups — #41003588

Order 26944.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212354	C₂	1	2	yes	maximal, complemented
2	#27212351	C₂	1	4	yes	maximal, complemented
3	#27212344	C₂	1	4	yes	maximal, complemented
4	#27212341	C₂	1	4	yes	maximal, complemented
5	#125×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	2	yes	—
7	C₂×D₁₆₈₄	C₂²	1	4	yes	—
8	C₂×D₁₆₈₄	C₂²	1	2	yes	—
9	C₂×C₄×D₄₂₁	C₂²	1	4	yes	—
10	#20269750	C₂²	1	4	yes	—
11	#20269748	C₂²	1	4	yes	—
12	#20269748	C₂²	1	4	yes	—
13	#20269748	C₂²	1	4	yes	—
14	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
15	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
16	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₂²×D₄₂₁	C₂³	1	4	yes	—
18	Dic₈₄₂	—	4	2	no	—
19	D₁₆₈₄	—	4	4	no	—
20	D₁₆₈₄	—	4	2	no	—
21	C₂×Dic₄₂₁	C₂³	1	4	yes	—
22	C₄×D₄₂₁	—	4	4	no	—
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
25	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
26	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	4	no	—
28	C₂×D₄₂₁	—	2	4	no	—
29	C₂×D₄₂₁	—	2	4	no	—
30	C₄×C₄₂₁	—	2	4	no	—
31	C₄×C₄₂₁	—	2	2	no	—
32	C₄×C₄₂₁	—	2	1	no	—
33	Dic₄₂₁	—	2	4	no	—
34	C₂×C₄₂₁	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₄₂₁	—	4	4	no	—
37	C₄₂₁	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	421	1	no	maximal, Sylow
39	#125	D₄₂₁	1	1	yes	characteristic, complemented
40	#124	—	421	2	no	—
41	#123	—	421	4	no	—
42	C₂²⋊Q₈	—	421	4	no	—
43	#121	—	421	4	no	—
44	C₂×Q₈	—	421	2	no	—
45	C₂×D₄	—	421	4	no	—
46	C₂×D₄	—	421	2	no	—
47	C₂²×C₄	—	421	4	no	—
48	C₄⋊C₄	—	421	4	no	—
49	C₄⋊C₄	C₂×D₄₂₁	1	4	yes	—
50	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
51	C₂²⋊C₄	—	421	4	no	—
52	C₂²⋊C₄	—	421	4	no	—
53	C₂²⋊C₄	—	421	4	no	—
54	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
55	C₂³	—	421	4	no	—
56	Q₈	—	1684	2	no	—
57	D₄	—	1684	4	no	—
58	D₄	—	1684	2	no	—
59	C₂×C₄	—	1684	4	no	—
60	C₂×C₄	—	421	4	no	—
61	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
63	C₂×C₄	C₂²×D₄₂₁	1	4	yes	—
64	C₂²	—	842	4	no	—
65	C₂²	—	842	4	no	—
66	C₂²	—	842	4	no	—
67	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	842	4	no	—
72	C₂	—	1684	4	no	—
73	C₂	#27212412	1	2	yes	—
74	C₂	#27212414	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#125×C₄₂₁	C₂
hypercenter	#125	D₄₂₁
hypocenter	C₄₂₁	#562
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #27212354	C₂	semidirect
2. #27212351	C₂	semidirect
3. #27212344	C₂	semidirect
4. #27212341	C₂	semidirect
5. #125×C₄₂₁	C₂	semidirect
37. C₄₂₁	#562	semidirect
39. #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
#27212351	2
#27212344	2
#27212341	2
#125×C₄₂₁	2
#562	421