Subgroups — #41003215

Order 26944.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#20269756	C₂	1	2	yes	maximal, complemented
3	#27212362	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, complemented
6	C₂×D₁₆₈₄	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₄₂₁	—	2	4	no	—
8	D₃₃₆₈	—	2	2	no	—
9	#20269756	—	2	4	no	—
10	C₂×#16809890	C₂²	1	1	yes	characteristic
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	C₂²×D₄₂₁	—	2	2	no	—
16	D₁₆₈₄	—	2	2	no	—
17	D₁₆₈₄	—	2	2	no	—
18	C₂²⋊D₄₂₁	—	4	8	no	—
19	C₂×Dic₄₂₁	D₄	1	2	yes	—
20	#16809890	D₄	1	2	yes	complemented
21	C₂³×C₄₂₁	—	2	2	no	—
22	D₄×C₄₂₁	—	2	2	no	—
23	D₄×C₄₂₁	—	2	2	no	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	D₄	1	2	yes	complemented
26	C₂²×C₄₂₁	—	4	2	no	—
27	C₂²×C₄₂₁	—	4	4	no	—
28	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
29	C₂×D₄₂₁	—	4	2	no	—
30	C₂×D₄₂₁	—	4	4	no	—
31	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
33	Dic₄₂₁	D₈	1	4	yes	complemented
34	C₂×C₄₂₁	—	8	2	no	—
35	C₂×C₄₂₁	#127	1	1	yes	characteristic
36	C₂×C₄₂₁	C₂×D₈	1	2	yes	—
37	D₄₂₁	—	8	2	no	—
38	C₄₂₁	#493	1	1	yes	characteristic, complemented, Sylow
39	#493	—	421	1	no	maximal, Sylow
40	C₂×D₈	—	421	2	no	—
41	C₂×D₈	—	421	1	no	—
42	C₂×D₈	D₄₂₁	1	1	yes	characteristic, complemented
43	#127	—	421	2	no	—
44	C₄×C₈	—	421	1	no	—
45	C₂×D₄	—	842	4	no	—
46	C₂×D₄	—	421	2	no	—
47	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
48	D₈	—	2	2	no	—
49	D₈	—	842	4	no	—
50	D₈	—	842	2	no	—
51	C₂×C₈	—	421	1	no	—
52	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
53	C₄²	—	421	1	no	—
54	C₂³	—	2	2	no	—
55	C₂³	—	842	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	842	2	no	—
59	D₄	—	842	2	no	—
60	D₄	—	1684	8	no	—
61	C₂×C₄	—	421	2	no	—
62	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
63	C₈	—	421	2	no	—
64	C₈	C₂²⋊D₄₂₁	1	2	yes	complemented
65	C₂²	—	4	2	no	—
66	C₂²	—	4	4	no	—
67	C₂²	—	1684	2	no	—
68	C₂²	—	1684	4	no	—
69	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
70	C₄	—	421	4	no	—
71	C₄	D₄×D₄₂₁	1	1	yes	characteristic
72	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
73	C₂	—	8	2	no	—
74	C₂	—	3368	2	no	—
75	C₂	#27212362	1	1	yes	characteristic
76	C₂	D₈×D₄₂₁	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×D₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂×D₈	D₄₂₁
hypocenter	C₄₂₁	#493
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×D₃₃₆₈	C₂	semidirect
2. C₂×#20269756	C₂	semidirect
3. #27212362	C₂	semidirect
4. C₈×Dic₄₂₁	C₂	semidirect
5. C₂×D₈×C₄₂₁	C₂	semidirect
20. #16809890	D₄	semidirect
25. C₈×C₄₂₁	D₄	semidirect
33. Dic₄₂₁	D₈	semidirect
38. C₄₂₁	#493	semidirect
42. C₂×D₈	D₄₂₁	semidirect
64. C₈	C₂²⋊D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₃₆₈	2
C₂×#20269756	2
#27212362	2
C₈×Dic₄₂₁	2
C₂×D₈×C₄₂₁	2
#493	421