Subgroups — #14581377

Order 2624.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#4223	C₂	1	2	yes	maximal, complemented
2	#13050232	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050226	C₂	1	2	yes	maximal, complemented
4	#13050218	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#4223	—	2	2	no	—
7	D₄×D₄₁	C₂²	1	2	yes	—
8	C₂²⋊#4223	—	2	2	no	—
9	C₄⋊#4223	C₂²	1	2	yes	—
10	C₄×#4223	C₂²	1	1	yes	characteristic
11	#10620296	C₂²	1	1	yes	characteristic
12	D₈×C₄₁	C₄	1	1	yes	characteristic, complemented
13	#10620282	C₄	1	2	yes	complemented
14	D₃₂₈	C₄	1	1	yes	characteristic, complemented
15	C₈×D₄₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄₁	—	2	2	no	—
17	C₂×#4223	—	2	2	no	—
18	C₂×#4223	—	4	4	no	—
19	C₂×#4223	D₄	1	2	yes	—
20	D₄×C₄₁	C₂×C₄	1	2	yes	complemented
21	C₂²⋊D₄₁	—	2	2	no	—
22	D₁₆₄	C₂×C₄	1	2	yes	complemented
23	C₄×D₄₁	C₂³	1	1	yes	characteristic
24	#66493	—	2	1	no	—
25	C₈×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
26	#66491	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₄₁	—	2	2	no	—
28	C₂×D₄₁	—	2	2	no	—
29	C₂×D₄₁	—	4	2	no	—
30	C₂×D₄₁	—	4	2	no	—
31	C₂×D₄₁	C₂×D₄	1	1	yes	characteristic
32	#4223	—	2	2	no	—
33	#4223	—	4	2	no	—
34	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
35	Dic₄₁	#41	1	1	yes	characteristic
36	C₂×C₄₁	—	4	2	no	—
37	C₂×C₄₁	C₄×D₄	1	1	yes	characteristic
38	D₄₁	—	4	2	no	—
39	D₄₁	AGL(1,ℤ/8)	1	2	yes	—
40	#442	—	41	1	no	maximal, Sylow
41	C₄₁	#442	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₈	—	41	1	no	—
43	C₄×D₄	—	41	2	no	—
44	#106	—	41	1	no	—
45	#102	—	41	2	no	—
46	#97	—	41	1	no	—
47	C₂×D₄	—	41	2	no	—
48	C₂²×C₄	—	82	2	no	—
49	D₈	—	41	2	no	—
50	D₈	—	41	1	no	—
51	D₈	#4223	1	1	yes	characteristic, complemented
52	C₂×C₈	—	41	1	no	—
53	C₂×C₈	—	41	1	no	—
54	C₄⋊C₄	—	41	2	no	—
55	C₂²⋊C₄	—	82	2	no	—
56	C₄²	—	41	1	no	—
57	C₂³	—	82	2	no	—
58	D₄	—	41	2	no	—
59	D₄	—	82	2	no	—
60	D₄	C₂×#4223	1	2	yes	complemented
61	C₂×C₄	—	41	1	no	—
62	C₂×C₄	—	41	2	no	—
63	C₂×C₄	—	82	2	no	—
64	C₂×C₄	—	164	4	no	—
65	C₈	—	41	1	no	—
66	C₈	—	82	1	no	—
67	C₈	C₂×#4223	1	1	yes	characteristic, complemented
68	C₂²	—	2	2	no	—
69	C₂²	—	41	1	no	—
70	C₂²	—	82	2	no	—
71	C₂²	—	164	2	no	—
72	C₂²	—	164	2	no	—
73	C₄	—	41	1	no	—
74	C₄	—	82	2	no	—
75	C₄	—	164	2	no	—
76	C₄	C₂²×#4223	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	41	2	no	—
79	C₂	—	164	2	no	—
80	C₂	D₄×#4223	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#4223
commutator	C₄×C₄₁	C₂²×C₄
Frattini	C₄	C₂²×#4223
Fitting	D₈×C₄₁	C₄
socle	C₂×C₄₁	C₄×D₄
prime radical	D₈×D₄₁	C₂
cosocle	C₄×D₄₁	C₂³
hypercenter	D₈	#4223
hypocenter	C₄₁	#442
LCS	G ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#4223	C₂	semidirect
2. #13050232	C₂	semidirect
3. #13050226	C₂	semidirect
4. #13050218	C₂	semidirect
12. D₈×C₄₁	C₄	semidirect
13. #10620282	C₄	semidirect
14. D₃₂₈	C₄	semidirect
20. D₄×C₄₁	C₂×C₄	semidirect
22. D₁₆₄	C₂×C₄	semidirect
25. C₈×C₄₁	C₂×C₄	semidirect
26. #66491	C₂×C₄	semidirect
41. C₄₁	#442	semidirect
51. D₈	#4223	semidirect
60. D₄	C₂×#4223	semidirect
67. C₈	C₂×#4223	semidirect

Maximal subgroups

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

subgroup	index
D₄×#4223	2
#13050232	2
#13050226	2
#13050218	2
D₈×D₄₁	2
#442	41