Subgroups — #41003202

Order 26944.

100 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	#27212277	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×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	1	no	—
10	#20269748	—	2	1	no	—
11	C₂×Q₈×C₄₂₁	—	2	1	no	—
12	C₄⋊C₄×C₄₂₁	—	2	1	no	—
13	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
14	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	—	2	1	no	—
16	Dic₈₄₂	—	2	2	no	—
17	D₁₆₈₄	—	2	2	no	—
18	C₂×Dic₄₂₁	—	2	1	no	—
19	#16809890	—	2	2	no	—
20	Q₈×C₄₂₁	—	4	2	no	—
21	C₂×C₄×C₄₂₁	—	2	2	no	—
22	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₄₂₁	—	4	1	no	—
27	C₂×D₄₂₁	—	4	1	no	—
28	C₂×D₄₂₁	—	4	1	no	—
29	C₄×C₄₂₁	—	2	1	no	—
30	C₄×C₄₂₁	—	2	1	no	—
31	C₄×C₄₂₁	—	4	2	no	—
32	C₄×C₄₂₁	#41	1	2	yes	—
33	Dic₄₂₁	—	4	1	no	—
34	C₂×C₄₂₁	#124	1	1	yes	characteristic
35	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₄₂₁	#137	1	1	yes	characteristic
37	D₄₂₁	—	8	1	no	—
38	C₄₂₁	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	421	1	no	maximal, Sylow
40	C₄⋊Q₈	D₄₂₁	1	1	yes	characteristic, complemented
41	#124	—	421	1	no	—
42	#103	—	421	2	no	—
43	#102	—	421	2	no	—
44	#97	—	421	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	421	1	no	—
47	C₂×D₄	—	421	1	no	—
48	C₂×C₈	—	421	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
51	C₂²⋊C₄	—	842	1	no	—
52	C₂²⋊C₄	—	842	1	no	—
53	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
54	C₂³	—	842	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	842	2	no	—
57	D₄	—	842	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	842	1	no	—
60	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
63	C₈	—	842	2	no	—
64	C₂²	—	1684	1	no	—
65	C₂²	—	1684	1	no	—
66	C₂²	—	1684	1	no	—
67	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	1684	1	no	—
72	C₄	#20269772	1	2	yes	—
73	C₂	—	3368	1	no	—
74	C₂	#27212355	1	1	yes	characteristic
75	C₂	#27212392	1	1	yes	characteristic
76	C₂	#27212397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₄⋊Q₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	C₄⋊Q₈	D₄₂₁
hypocenter	C₄₂₁	#489
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #27212289	C₂	semidirect
4. #27212277	C₂	semidirect
5. C₄⋊Q₈×C₄₂₁	C₂	semidirect
38. C₄₂₁	#489	semidirect
40. C₄⋊Q₈	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
#27212277	2
C₄⋊Q₈×C₄₂₁	2
#489	421