Subgroups — #41003693

Order 26944.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212423	C₂	1	1	yes	maximal, characteristic
2	(C₂×#13132997)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#27212346	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13132997	—	2	1	no	—
5	C₂×#16809897	—	2	1	no	—
6	C₂×Dic₈₄₂	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₄₂₁	—	2	1	no	—
9	#20269748	—	2	1	no	—
10	C₄⋊C₄×C₄₂₁	C₄	1	1	yes	characteristic, complemented
11	C₂×#13132997	—	2	2	no	—
12	C₂×#13132997	—	4	1	no	—
13	C₂×#13132997	—	4	1	no	—
14	#16809897	—	4	1	no	—
15	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
16	Dic₈₄₂	—	4	1	no	—
17	C₂×Dic₄₂₁	—	2	1	no	—
18	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
19	C₄×D₄₂₁	—	4	1	no	—
20	C₂×C₄×C₄₂₁	—	2	1	no	—
21	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₄₂₁	—	2	1	no	—
24	C₂×D₄₂₁	—	2	1	no	—
25	C₂×D₄₂₁	SD₁₆	1	1	yes	characteristic
26	C₂×D₄₂₁	Q₁₆	1	1	yes	characteristic
27	#13132997	—	4	2	no	—
28	C₄×C₄₂₁	—	2	1	no	—
29	C₄×C₄₂₁	—	4	1	no	—
30	Dic₄₂₁	—	2	1	no	—
31	Dic₄₂₁	—	4	1	no	—
32	C₂×C₄₂₁	#99	1	1	yes	characteristic
33	C₂×C₄₂₁	#103	1	1	yes	characteristic
34	C₂×C₄₂₁	#104	1	1	yes	characteristic
35	D₄₂₁	—	2	1	no	—
36	D₄₂₁	—	2	1	no	—
37	C₄₂₁	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	421	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	421	1	no	—
40	C₂²⋊C₈	—	421	1	no	—
41	#95	—	421	1	no	—
42	C₂×Q₈	—	421	1	no	—
43	C₂²×C₄	—	842	1	no	—
44	C₂²×C₄	—	421	1	no	—
45	C₂×C₈	—	842	1	no	—
46	C₄⋊C₄	—	842	1	no	—
47	C₄⋊C₄	#13132997	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	842	1	no	—
49	C₂³	—	421	1	no	—
50	Q₈	—	1684	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	842	2	no	—
53	C₂×C₄	—	842	1	no	—
54	C₂×C₄	—	1684	1	no	—
55	C₂×C₄	—	1684	1	no	—
56	C₂×C₄	—	1684	1	no	—
57	C₂×C₄	—	421	1	no	—
58	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
59	C₈	—	1684	1	no	—
60	C₂²	—	842	1	no	—
61	C₂²	—	842	1	no	—
62	C₂²	—	421	1	no	—
63	C₂²	—	421	1	no	—
64	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	842	1	no	—
68	C₄	—	1684	2	no	—
69	C₄	—	1684	1	no	—
70	C₂	—	842	1	no	—
71	C₂	—	842	1	no	—
72	C₂	#27212425	1	1	yes	characteristic
73	C₂	#27212429	1	1	yes	characteristic
74	C₂	#27212430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13132997
commutator	C₂×C₄×C₄₂₁	C₂×C₄
Frattini	C₂×C₄	C₂×#13132997
Fitting	C₄⋊C₄×C₄₂₁	C₄
socle	C₂²×C₄₂₁	C₂²⋊C₄
prime radical	C₂²×D₄₂₁	D₄
cosocle	C₂×C₄×D₄₂₁	C₂²
hypercenter	C₄⋊C₄	#13132997
hypocenter	C₄₂₁	#328
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₂×Dic₈₄₂	C₄	semidirect
10. C₄⋊C₄×C₄₂₁	C₄	semidirect
37. C₄₂₁	#328	semidirect
47. C₄⋊C₄	#13132997	semidirect

Maximal subgroups

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

subgroup	index
#27212423	2
(C₂×#13132997)⋊C₄	2
#27212346	2
#328	421