Subgroups — #41003397

Order 26944.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212366	C₂	1	2	yes	maximal, complemented
2	#27212358	C₂	1	2	yes	maximal, complemented
3	#27212345	C₂	1	4	yes	maximal
4	C₄×Dic₈₄₂	C₂	1	4	yes	maximal, complemented
5	#27212325	C₂	1	2	yes	maximal
6	C₂×C₄²×C₄₂₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₈₄₂	C₂²	1	4	yes	—
8	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
9	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
10	#20269750	C₂²	1	8	yes	—
11	C₂²⋊Dic₄₂₁	C₂²	1	4	yes	—
12	C₄×Dic₄₂₁	C₂²	1	4	yes	—
13	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
14	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₄²×C₄₂₁	C₂²	1	2	yes	—
16	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
18	Dic₈₄₂	—	2	8	no	—
19	C₂×Dic₄₂₁	C₂³	1	8	yes	—
20	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	—	2	4	no	—
22	C₂×C₄×C₄₂₁	Q₈	1	4	yes	complemented
23	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
26	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄₂₁	—	2	2	no	—
28	C₂²×C₄₂₁	C₂×Q₈	1	2	yes	—
29	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
30	C₄×C₄₂₁	—	2	2	no	—
31	C₄×C₄₂₁	C₂×Q₈	1	2	yes	—
32	C₄×C₄₂₁	C₂×Q₈	1	2	yes	—
33	C₄×C₄₂₁	#41	1	4	yes	—
34	Dic₄₂₁	—	2	8	no	—
35	C₂×C₄₂₁	—	2	2	no	—
36	C₂×C₄₂₁	C₂²×Q₈	1	1	yes	characteristic
37	C₂×C₄₂₁	C₂×#41	1	2	yes	—
38	C₄₂₁	#533	1	1	yes	characteristic, complemented, Sylow
39	#533	—	421	1	no	maximal, Sylow
40	C₄⋊Q₈	—	421	2	no	—
41	#125	—	421	2	no	—
42	C₂²⋊Q₈	—	421	4	no	—
43	C₄×Q₈	—	421	4	no	—
44	#117	—	421	2	no	—
45	C₂×C₄²	D₄₂₁	1	1	yes	characteristic
46	C₂×Q₈	—	421	4	no	—
47	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
48	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
49	C₄⋊C₄	—	421	4	no	—
50	C₄⋊C₄	—	421	8	no	—
51	C₄⋊C₄	—	421	4	no	—
52	C₂²⋊C₄	—	421	4	no	—
53	C₄²	—	421	4	no	—
54	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
55	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
56	C₄²	C₂×D₄₂₁	1	2	yes	—
57	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
58	Q₈	—	842	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	421	8	no	—
61	C₂×C₄	Dic₈₄₂	1	4	yes	complemented
62	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
63	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
65	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Dic₈₄₂	1	2	yes	—
68	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	842	8	no	—
71	C₄	C₂×Dic₈₄₂	1	2	yes	—
72	C₄	C₂×Dic₈₄₂	1	2	yes	—
73	C₄	#20269771	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₈₄₂	1	1	yes	characteristic
76	C₂	C₂×#20269771	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Frattini	C₂²	C₂³×D₄₂₁
Fitting	C₂×C₄²×C₄₂₁	C₂
prime radical	C₂³×C₄₂₁	C₂³
hypercenter	C₂×C₄²	D₄₂₁
hypocenter	C₄₂₁	#533
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #27212366	C₂	semidirect
2. #27212358	C₂	semidirect
4. C₄×Dic₈₄₂	C₂	semidirect
22. C₂×C₄×C₄₂₁	Q₈	semidirect
38. C₄₂₁	#533	semidirect
61. C₂×C₄	Dic₈₄₂	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
#27212345	2
C₄×Dic₈₄₂	2
#27212325	2
C₂×C₄²×C₄₂₁	2
#533	421