Subgroups — #41002705

Order 26944.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
3	#27212273	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
6	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
7	C₄⋊Dic₄₂₁	—	2	2	no	—
8	C₄×Dic₄₂₁	C₄	1	4	yes	complemented
9	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
10	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₄₂₁	—	2	2	no	—
12	C₂×Dic₄₂₁	—	2	2	no	—
13	C₂×Dic₄₂₁	—	2	2	no	—
14	C₂×Dic₄₂₁	—	2	2	no	—
15	C₂×Dic₄₂₁	—	2	4	no	—
16	C₂×Dic₄₂₁	C₂×C₄	1	4	yes	—
17	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₄₂₁	—	2	2	no	—
19	C₂×C₄×C₄₂₁	—	2	4	no	—
20	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
22	C₂×C₄×C₄₂₁	Q₈	1	2	yes	—
23	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₄₂₁	#41	1	2	yes	—
27	C₂²×C₄₂₁	#41	1	2	yes	—
28	C₄×C₄₂₁	—	4	2	no	—
29	C₄×C₄₂₁	C₄⋊C₄	1	4	yes	—
30	Dic₄₂₁	—	2	4	no	—
31	Dic₄₂₁	—	4	2	no	—
32	C₂×C₄₂₁	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄₂₁	#117	1	2	yes	—
34	C₂×C₄₂₁	#124	1	1	yes	characteristic
35	C₂×C₄₂₁	#124	1	1	yes	characteristic
36	C₂×C₄₂₁	#125	1	2	yes	—
37	C₄₂₁	#382	1	1	yes	characteristic, complemented, Sylow
38	#382	—	421	1	no	maximal, Sylow
39	C₂×C₄⋊C₄	—	421	1	no	—
40	C₂×C₄⋊C₄	D₄₂₁	1	1	yes	characteristic
41	C₂×C₄²	—	421	1	no	—
42	#95	—	421	4	no	—
43	C₂²×C₄	—	421	2	no	—
44	C₂²×C₄	—	421	2	no	—
45	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
46	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	842	2	no	—
49	C₄²	—	421	4	no	—
50	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	842	2	no	—
54	C₂×C₄	—	842	2	no	—
55	C₂×C₄	—	842	2	no	—
56	C₂×C₄	—	842	4	no	—
57	C₂×C₄	—	421	4	no	—
58	C₂×C₄	C₄×D₄₂₁	1	2	yes	—
59	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
60	C₂×C₄	Dic₈₄₂	1	2	yes	—
61	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
62	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
63	C₂²	C₂×Dic₈₄₂	1	1	yes	characteristic
64	C₂²	#20269770	1	2	yes	—
65	C₂²	#20269772	1	2	yes	—
66	C₄	—	4	2	no	—
67	C₄	—	842	4	no	—
68	C₄	—	1684	2	no	—
69	C₄	#20269750	1	4	yes	—
70	C₂	C₂×#20269750	1	1	yes	characteristic
71	C₂	#27212324	1	2	yes	—
72	C₂	#27212353	1	1	yes	characteristic
73	C₂	#27212355	1	1	yes	characteristic
74	C₂	#27212357	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₂₁
commutator	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₄₂₁	C₂
socle, prime radical, cosocle	C₂³×C₄₂₁	C₂³
hypercenter	C₂×C₄⋊C₄	D₄₂₁
hypocenter	C₄₂₁	#382
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. C₄×Dic₄₂₁	C₄	semidirect
37. C₄₂₁	#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₄₂₁	2
C₂×C₄×Dic₄₂₁	2
#27212273	2
C₂×C₄⋊C₄×C₄₂₁	2
#382	421