Subgroups — #41002747

Order 26944.

186 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	C₂²×Q₈×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	—	2	2	no	—
9	C₂×Q₈×C₄₂₁	—	2	2	no	—
10	C₂×Q₈×C₄₂₁	—	2	2	no	—
11	C₂×Q₈×C₄₂₁	C₄	1	4	yes	complemented
12	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₂₁	—	2	2	no	—
15	C₂×Dic₄₂₁	—	2	2	no	—
16	C₂×Dic₄₂₁	—	2	2	no	—
17	C₂×Dic₄₂₁	—	2	4	no	—
18	C₂×Dic₄₂₁	Q₈	1	4	yes	complemented
19	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
20	Q₈×C₄₂₁	—	2	8	no	—
21	C₂×C₄×C₄₂₁	—	2	2	no	—
22	C₂×C₄×C₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
26	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
27	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄₂₁	C₂×Q₈	1	2	yes	—
31	C₂²×C₄₂₁	#41	1	2	yes	—
32	C₄×C₄₂₁	—	2	4	no	—
33	C₄×C₄₂₁	C₂²⋊C₄	1	4	yes	—
34	Dic₄₂₁	—	2	4	no	—
35	Dic₄₂₁	—	4	2	no	—
36	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄₂₁	C₄×Q₈	1	2	yes	—
38	C₂×C₄₂₁	C₂²⋊Q₈	1	2	yes	—
39	C₂×C₄₂₁	#124	1	1	yes	characteristic
40	C₂×C₄₂₁	C₄⋊Q₈	1	1	yes	characteristic
41	C₄₂₁	#391	1	1	yes	characteristic, complemented, Sylow
42	#391	—	421	1	no	maximal, Sylow
43	C₂²×Q₈	D₄₂₁	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	421	1	no	—
45	C₂×C₄²	—	421	1	no	—
46	#95	—	421	4	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	Dic₄₂₁	1	4	yes	complemented
50	C₂²×C₄	—	421	2	no	—
51	C₂²×C₄	—	421	2	no	—
52	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
53	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
54	C₄⋊C₄	—	842	2	no	—
55	C₄²	—	842	2	no	—
56	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
57	Q₈	—	2	8	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	842	2	no	—
61	C₂×C₄	—	842	2	no	—
62	C₂×C₄	—	842	2	no	—
63	C₂×C₄	—	842	4	no	—
64	C₂×C₄	—	421	4	no	—
65	C₂×C₄	C₂×Dic₄₂₁	1	4	yes	—
66	C₂×C₄	C₂×Dic₄₂₁	1	2	yes	—
67	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
68	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
69	C₂²	C₂²×Dic₄₂₁	1	1	yes	characteristic
70	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
71	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
72	C₂²	Q₈×D₄₂₁	1	2	yes	—
73	C₂²	#20269772	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	842	4	no	—
76	C₄	—	1684	2	no	—
77	C₄	C₂²⋊Dic₄₂₁	1	4	yes	—
78	C₂	C₂×C₂²⋊Dic₄₂₁	1	1	yes	characteristic
79	C₂	Q₈×Dic₄₂₁	1	2	yes	—
80	C₂	#27212342	1	2	yes	—
81	C₂	#27212355	1	1	yes	characteristic
82	C₂	#27212364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₂₁
commutator	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₂²×Q₈×C₄₂₁	C₂
socle, prime radical, cosocle	C₂³×C₄₂₁	C₂³
hypercenter	C₂²×Q₈	D₄₂₁
hypocenter	C₄₂₁	#391
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
11. C₂×Q₈×C₄₂₁	C₄	semidirect
18. C₂×Dic₄₂₁	Q₈	semidirect
41. C₄₂₁	#391	semidirect
49. C₂×Q₈	Dic₄₂₁	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
C₂²×Q₈×C₄₂₁	2
#391	421