Subgroups — #41003223

Order 26944.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₆₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20269759	C₂	1	2	yes	maximal, complemented
3	#27212353	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	C₂²	1	2	yes	—
7	Dic₁₆₈₄	—	2	2	no	—
8	#20269759	—	2	4	no	—
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₄₂₁	—	2	4	no	—
11	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
13	D₈×C₄₂₁	—	2	2	no	—
14	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
15	Dic₈₄₂	—	2	2	no	—
16	Dic₈₄₂	—	2	2	no	—
17	C₂×Dic₄₂₁	—	2	2	no	—
18	C₂×Dic₄₂₁	D₄	1	2	yes	—
19	#16809890	D₄	1	2	yes	—
20	C₂³×C₄₂₁	—	2	2	no	—
21	D₄×C₄₂₁	—	2	2	no	—
22	D₄×C₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₈×C₄₂₁	D₄	1	2	yes	—
25	C₂²×C₄₂₁	—	4	2	no	—
26	C₂²×C₄₂₁	—	4	4	no	—
27	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
29	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	Dic₄₂₁	—	2	2	no	—
31	Dic₄₂₁	—	4	2	no	—
32	C₂×C₄₂₁	—	8	2	no	—
33	C₂×C₄₂₁	#127	1	1	yes	characteristic
34	C₂×C₄₂₁	#135	1	2	yes	—
35	C₄₂₁	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	421	1	no	maximal, Sylow
37	C₂×Q₁₆	—	421	1	no	—
38	C₂×SD₁₆	—	421	2	no	—
39	C₂×D₈	D₄₂₁	1	1	yes	characteristic
40	#124	—	421	2	no	—
41	C₄×C₈	—	421	1	no	—
42	C₂×Q₈	—	421	2	no	—
43	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
44	Q₁₆	—	842	2	no	—
45	SD₁₆	—	842	4	no	—
46	D₈	—	2	2	no	—
47	C₂×C₈	—	421	1	no	—
48	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
49	C₂²⋊C₄	—	842	4	no	—
50	C₄²	—	421	1	no	—
51	C₂³	—	2	2	no	—
52	Q₈	—	842	2	no	—
53	Q₈	—	842	2	no	—
54	D₄	—	2	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	842	2	no	—
57	C₂×C₄	—	421	2	no	—
58	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
59	C₈	—	421	2	no	—
60	C₈	C₂²⋊D₄₂₁	1	2	yes	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	4	no	—
63	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
64	C₄	—	842	2	no	—
65	C₄	—	1684	2	no	—
66	C₄	D₄×D₄₂₁	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
68	C₂	—	8	2	no	—
69	C₂	#27212362	1	1	yes	characteristic
70	C₂	#27212385	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting, prime radical	C₂×D₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂×D₈	D₄₂₁
hypocenter	C₄₂₁	#495
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₆₈₄	C₂	semidirect
2. C₂×#20269759	C₂	semidirect
3. #27212353	C₂	semidirect
4. C₈×Dic₄₂₁	C₂	semidirect
35. C₄₂₁	#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₆₈₄	2
C₂×#20269759	2
#27212353	2
C₈×Dic₄₂₁	2
C₂×D₈×C₄₂₁	2
#495	421