Subgroups — #41003918

Order 26944.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212453	C₂	1	8	yes	maximal, complemented
2	D₄₂₁⋊M₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×#20269780	C₂	1	4	yes	maximal, complemented
4	C₂×#20269770	C₂	1	1	yes	maximal, characteristic
5	#20269781	C₂²	1	4	yes	complemented
6	#20269780	C₂²	1	4	yes	—
7	#20269780	C₂²	1	8	yes	complemented
8	D₄₂₁⋊C₈	C₂²	1	4	yes	complemented
9	C₂×#16809897	C₂²	1	8	yes	complemented
10	#20269770	C₄	1	4	yes	—
11	#20269770	C₂²	1	4	yes	—
12	C₂×Dic₈₄₂	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₂₁	C₄	1	2	yes	—
14	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₄₂₁	C₄	1	1	yes	characteristic
17	#16809897	C₂³	1	8	yes	—
18	C₂²×D₄₂₁	C₂×C₄	1	1	yes	characteristic
19	Dic₈₄₂	C₂×C₄	1	4	yes	—
20	C₂²⋊D₄₂₁	C₂×C₄	1	8	yes	—
21	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
22	C₂×Dic₄₂₁	C₂×C₄	1	4	yes	—
23	C₂×Dic₄₂₁	C₂³	1	1	yes	characteristic
24	C₂×Dic₄₂₁	C₂³	1	4	yes	—
25	C₄×D₄₂₁	C₂×C₄	1	2	yes	—
26	C₄×D₄₂₁	C₂³	1	2	yes	—
27	C₂³×C₄₂₁	C₂×C₄	1	2	yes	—
28	D₄×C₄₂₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄₂₁	—	4	2	no	—
31	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄₂₁	C₂²×C₄	1	4	yes	—
33	C₂×D₄₂₁	—	4	1	no	—
34	C₂×D₄₂₁	C₂²×C₄	1	2	yes	—
35	C₄×C₄₂₁	C₂²×C₄	1	2	yes	—
36	Dic₄₂₁	C₂²×C₄	1	1	yes	characteristic
37	Dic₄₂₁	C₂²×C₄	1	4	yes	—
38	Dic₄₂₁	C₂⁴	1	1	yes	characteristic
39	C₂×C₄₂₁	—	2	1	no	—
40	C₂×C₄₂₁	—	2	4	no	—
41	C₂×C₄₂₁	C₂³×C₄	1	1	yes	characteristic
42	D₄₂₁	—	2	2	no	—
43	C₄₂₁	#568	1	1	yes	characteristic, complemented, Sylow
44	#568	—	421	1	no	maximal, Sylow
45	C₂×#41	—	421	1	no	—
46	#131	—	421	8	no	—
47	C₂×M₁₆	—	421	4	no	—
48	C₂×M₁₆	—	421	2	no	—
49	#41	—	421	4	no	—
50	#41	—	421	4	no	—
51	C₂×Q₈	—	421	1	no	—
52	C₂×D₄	—	421	2	no	—
53	C₂×D₄	#13132997	1	1	yes	characteristic
54	C₂²×C₄	—	421	2	no	—
55	C₂²×C₄	—	421	1	no	—
56	M₁₆	—	421	4	no	—
57	M₁₆	—	421	8	no	—
58	M₁₆	—	421	4	no	—
59	C₂×C₈	—	421	8	no	—
60	C₂×C₈	—	421	4	no	—
61	C₂³	—	421	1	no	—
62	C₂³	C₂×#13132997	1	2	yes	—
63	Q₈	—	421	4	no	—
64	D₄	—	421	8	no	—
65	D₄	C₂×#13132997	1	4	yes	—
66	C₂×C₄	—	421	4	no	—
67	C₂×C₄	—	421	1	no	—
68	C₂×C₄	—	421	2	no	—
69	C₂×C₄	—	421	4	no	—
70	C₂×C₄	—	421	2	no	—
71	C₂×C₄	—	421	2	no	—
72	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
73	C₈	—	421	8	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	1684	1	no	—
76	C₂²	—	421	2	no	—
77	C₂²	C₂²×#13132997	1	1	yes	characteristic
78	C₂²	C₂²×#13132997	1	4	yes	—
79	C₄	—	421	1	no	—
80	C₄	—	421	1	no	—
81	C₄	—	421	4	no	—
82	C₄	C₂²×#13132997	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	2	4	no	—
85	C₂	—	842	2	no	—
86	C₂	C₂³×#13132997	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#13132997
commutator, socle	C₂×C₄₂₁	C₂³×C₄
Fitting	C₂×D₄×C₄₂₁	C₄
prime radical	C₂×#20269770	C₂
cosocle	Dic₄₂₁	C₂⁴
hypercenter	C₂×D₄	#13132997
hypocenter	C₄₂₁	#568
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #27212453	C₂	semidirect
2. D₄₂₁⋊M₁₆	C₂	semidirect
3. C₂×#20269780	C₂	semidirect
5. #20269781	C₂²	semidirect
7. #20269780	C₂²	semidirect
8. D₄₂₁⋊C₈	C₂²	semidirect
9. C₂×#16809897	C₂²	semidirect
43. C₄₂₁	#568	semidirect

Maximal subgroups

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

subgroup	index
#27212453	2
D₄₂₁⋊M₁₆	2
C₂×#20269780	2
C₂×#20269770	2
#568	421