Subgroups — #65268

Order 320.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#683	C₂	1	2	yes	maximal
4	#3963	C₂	1	2	yes	maximal
5	#3963	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅	C₂²	1	2	yes	—
10	C₂²×Dic₅	C₂²	1	2	yes	—
11	C₂×Dic₁₀	—	2	2	no	—
12	C₂×Dic₁₀	—	2	4	no	—
13	C₅×C₄⋊C₄	—	2	2	no	—
14	#683	—	2	4	no	—
15	#395	—	5	1	no	maximal, Sylow
16	C₂³×C₅	C₂³	1	1	yes	characteristic
17	C₂×C₄×C₅	—	2	1	no	—
18	C₂×C₄×C₅	—	2	2	no	—
19	C₂×C₄×C₅	—	2	1	no	—
20	C₂×C₄×C₅	—	2	2	no	—
21	C₂×C₄×C₅	—	2	1	no	—
22	C₂×C₄×C₅	D₄	1	2	yes	—
23	C₂×C₄×C₅	Q₈	1	2	yes	complemented
24	C₂×Dic₅	—	2	2	no	—
25	C₂×Dic₅	—	2	2	no	—
26	C₂×Dic₅	—	2	2	no	—
27	C₂×Dic₅	—	2	2	no	—
28	C₂×Dic₅	D₄	1	4	yes	—
29	C₂×Dic₅	Q₈	1	4	yes	complemented
30	Dic₁₀	—	2	8	no	—
31	C₂²×Q₈	—	5	1	no	—
32	C₂×C₄⋊C₄	—	5	2	no	—
33	C₂×C₄⋊C₄	D₅	1	1	yes	characteristic
34	#95	—	5	1	no	—
35	#95	—	5	2	no	—
36	C₂²×C₅	C₂×D₄	1	2	yes	—
37	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₅	C₂×Q₈	1	2	yes	—
40	C₂²×C₅	#41	1	1	yes	characteristic
41	C₄×C₅	—	2	2	no	—
42	C₄×C₅	—	2	2	no	—
43	C₄×C₅	—	4	1	no	—
44	Dic₅	—	2	4	no	—
45	Dic₅	—	2	4	no	—
46	C₂×Q₈	—	10	4	no	—
47	C₂×Q₈	—	10	2	no	—
48	C₂²×C₄	—	5	2	no	—
49	C₂²×C₄	—	5	2	no	—
50	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	10	4	no	—
55	C₂×C₅	#120	1	1	yes	characteristic
56	C₂×C₅	C₂²⋊Q₈	1	2	yes	—
57	C₂×C₅	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₅	C₄⋊Q₈	1	2	yes	—
59	C₂×C₅	C₄⋊Q₈	1	1	yes	characteristic
60	C₂³	C₂²×D₅	1	1	yes	characteristic
61	Q₈	—	10	8	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	5	4	no	—
68	C₂×C₄	—	5	4	no	—
69	C₂×C₄	—	10	2	no	—
70	C₂×C₄	—	10	2	no	—
71	C₂×C₄	—	10	2	no	—
72	C₂×C₄	—	10	2	no	—
73	C₂×C₄	Dic₁₀	1	2	yes	complemented
74	C₂×C₄	C₂²⋊D₅	1	2	yes	—
75	C₅	#395	1	1	yes	characteristic, complemented, Sylow
76	C₂²	C₂×Dic₁₀	1	1	yes	characteristic
77	C₂²	#709	1	1	yes	characteristic
78	C₂²	D₄×D₅	1	2	yes	—
79	C₂²	Q₈×D₅	1	2	yes	—
80	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	—	10	4	no	—
85	C₄	—	10	4	no	—
86	C₂	#4034	1	2	yes	—
87	C₂	#4042	1	2	yes	—
88	C₂	#4070	1	1	yes	characteristic
89	C₂	#4083	1	1	yes	characteristic
90	C₂	#4090	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂³
Fitting	C₂×C₅×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	D₅
hypocenter	C₅	#395
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. C₂×C₄×C₅	Q₈	semidirect
29. C₂×Dic₅	Q₈	semidirect
73. C₂×C₄	Dic₁₀	semidirect
75. C₅	#395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀	2
C₂×C₅×C₄⋊C₄	2
C₂×#683	2
#3963	2
#395	5