Subgroups — #41003886

Order 26944.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212454	C₂	1	2	yes	maximal, complemented
2	#27212436	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212431	C₂	1	2	yes	maximal, complemented
4	#27212420	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212389	C₂	1	1	yes	maximal, characteristic
6	#20269781	—	2	2	no	—
7	#20269781	C₂²	1	2	yes	—
8	D₄₂₁⋊C₈	—	2	2	no	—
9	D₄₂₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×#13132997	C₂²	1	1	yes	characteristic
11	#20269772	C₂²	1	2	yes	—
12	#20269758	C₄	1	2	yes	complemented
13	D₃₃₆₈	C₄	1	1	yes	characteristic, complemented
14	C₈×D₄₂₁	C₂²	1	1	yes	characteristic
15	Q₁₆×C₄₂₁	C₄	1	1	yes	characteristic, complemented
16	C₂×#13132997	—	2	1	no	—
17	#16809897	—	2	2	no	—
18	#16809897	D₄	1	2	yes	—
19	D₁₆₈₄	—	2	2	no	—
20	D₁₆₈₄	C₂×C₄	1	2	yes	—
21	C₄×D₄₂₁	—	2	2	no	—
22	C₄×D₄₂₁	C₂³	1	1	yes	characteristic
23	#16809890	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₄₂₁	C₂×C₄	1	2	yes	—
25	C₈×C₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄₂₁	—	2	2	no	—
27	C₂×D₄₂₁	#41	1	1	yes	characteristic
28	#13132997	—	4	1	no	—
29	C₄×C₄₂₁	—	2	2	no	—
30	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
31	Dic₄₂₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
33	D₄₂₁	—	2	1	no	—
34	D₄₂₁	—	4	2	no	—
35	C₄₂₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	421	1	no	maximal, Sylow
37	#135	—	421	1	no	—
38	#131	—	421	2	no	—
39	#108	—	421	1	no	—
40	#104	—	421	2	no	—
41	#97	—	421	1	no	—
42	#41	—	421	2	no	—
43	Q₁₆	#13132997	1	1	yes	characteristic, complemented
44	SD₁₆	—	421	2	no	—
45	D₈	—	421	1	no	—
46	M₁₆	—	842	2	no	—
47	M₁₆	—	421	2	no	—
48	C₂×C₈	—	842	2	no	—
49	C₂×C₈	—	421	1	no	—
50	C₂×C₈	—	421	1	no	—
51	C₄²	—	421	1	no	—
52	Q₈	C₂×#13132997	1	2	yes	—
53	D₄	—	842	2	no	—
54	D₄	—	421	2	no	—
55	C₂×C₄	—	842	2	no	—
56	C₂×C₄	—	842	1	no	—
57	C₂×C₄	—	421	1	no	—
58	C₈	—	842	2	no	—
59	C₈	—	421	2	no	—
60	C₈	—	421	1	no	—
61	C₈	C₂×#13132997	1	1	yes	characteristic
62	C₂²	—	842	2	no	—
63	C₂²	—	421	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	1684	1	no	—
66	C₄	—	421	1	no	—
67	C₄	C₂²×#13132997	1	1	yes	characteristic
68	C₂	—	842	1	no	—
69	C₂	—	1684	2	no	—
70	C₂	D₄×#13132997	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13132997
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₄	C₂²×#13132997
Fitting	Q₁₆×C₄₂₁	C₄
socle	C₂×C₄₂₁	C₄×D₄
prime radical	#27212389	C₂
cosocle	C₄×D₄₂₁	C₂³
hypercenter	Q₁₆	#13132997
hypocenter	C₄₂₁	#444
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #27212454	C₂	semidirect
2. #27212436	C₂	semidirect
3. #27212431	C₂	semidirect
4. #27212420	C₂	semidirect
12. #20269758	C₄	semidirect
13. D₃₃₆₈	C₄	semidirect
15. Q₁₆×C₄₂₁	C₄	semidirect
35. C₄₂₁	#444	semidirect
43. Q₁₆	#13132997	semidirect

Maximal subgroups

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

subgroup	index
#27212454	2
#27212436	2
#27212431	2
#27212420	2
#27212389	2
#444	421