Subgroups — #41002848

Order 26944.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269755	C₂	1	1	yes	maximal, characteristic
3	#27212289	C₂	1	4	yes	maximal, complemented
4	#117×C₄₂₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₄₂	—	2	1	no	—
6	C₂×Dic₈₄₂	—	2	2	no	—
7	C₂×Dic₈₄₂	C₄	1	4	yes	complemented
8	C₂×Dic₈₄₂	C₂²	1	2	yes	—
9	C₂²×Dic₄₂₁	—	2	1	no	—
10	#20269755	—	2	2	no	—
11	C₂×#16809890	C₂²	1	2	yes	—
12	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₄₂₁	—	2	1	no	—
15	C₄²×C₄₂₁	—	2	1	no	—
16	Dic₈₄₂	—	2	4	no	—
17	Dic₈₄₂	—	2	2	no	—
18	Dic₈₄₂	C₂×C₄	1	4	yes	—
19	C₂×Dic₄₂₁	—	2	4	no	—
20	C₂×Dic₄₂₁	—	2	2	no	—
21	#16809890	—	2	2	no	—
22	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	—	2	2	no	—
24	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
27	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄₂₁	—	2	2	no	—
29	C₂²×C₄₂₁	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₂₁	—	4	2	no	—
32	C₄×C₄₂₁	C₂²⋊C₄	1	2	yes	—
33	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
35	Dic₄₂₁	—	2	4	no	—
36	C₂×C₄₂₁	—	2	2	no	—
37	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄₂₁	#137	1	2	yes	—
39	C₄₂₁	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	421	1	no	maximal, Sylow
41	C₂²×Q₈	—	421	1	no	—
42	C₂×M₁₆	—	421	1	no	—
43	#117	D₄₂₁	1	1	yes	characteristic
44	#103	—	421	4	no	—
45	C₂×Q₈	—	842	1	no	—
46	C₂×Q₈	—	842	2	no	—
47	C₂×Q₈	—	421	4	no	—
48	C₂×Q₈	—	421	2	no	—
49	C₂²×C₄	—	842	1	no	—
50	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
51	M₁₆	—	842	2	no	—
52	C₂×C₈	—	421	2	no	—
53	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₂³	C₂²⋊D₄₂₁	1	1	yes	characteristic
57	Q₈	—	842	4	no	—
58	Q₈	—	842	2	no	—
59	Q₈	—	421	4	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	842	4	no	—
62	C₂×C₄	—	842	2	no	—
63	C₂×C₄	C₄×D₄₂₁	1	2	yes	—
64	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
65	C₂×C₄	D₁₆₈₄	1	2	yes	—
66	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
67	C₈	—	842	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	#20269748	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	842	4	no	—
73	C₄	#20269748	1	2	yes	—
74	C₄	C₂×C₄×D₄₂₁	1	1	yes	characteristic
75	C₄	C₂×D₁₆₈₄	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×#20269748	1	1	yes	characteristic
78	C₂	#27212396	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#117×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂³×C₄₂₁	D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	#117	D₄₂₁
hypocenter	C₄₂₁	#419
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #27212289	C₂	semidirect
7. C₂×Dic₈₄₂	C₄	semidirect
39. C₄₂₁	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₈₄₂	2
C₂×#20269755	2
#27212289	2
#117×C₄₂₁	2
#419	421