Subgroups — #65418

Order 320.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#3952	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊D₅	C₄	1	2	yes	—
7	#709	—	2	2	no	—
8	#709	—	2	2	no	—
9	C₂×D₂₀	C₄	1	1	yes	characteristic
10	C₂×C₄×D₅	C₂²	1	2	yes	—
11	C₂×Dic₁₀	C₄	1	1	yes	characteristic
12	C₅×M₁₆	—	2	2	no	—
13	C₂×C₅×C₈	C₂²	1	2	yes	—
14	C₂×#168	—	2	2	no	—
15	C₂×#168	C₂²	1	2	yes	—
16	#408	—	5	1	no	maximal, Sylow
17	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×D₅	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅	D₄	1	2	yes	—
21	C₂×C₄×C₅	D₄	1	2	yes	—
22	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²⋊D₅	—	2	4	no	—
24	C₂×Dic₅	C₂×C₄	1	2	yes	—
25	D₂₀	—	2	2	no	—
26	C₄×D₅	—	2	2	no	—
27	C₄×D₅	—	2	2	no	—
28	Dic₁₀	—	2	2	no	—
29	C₅×C₈	—	2	2	no	—
30	#168	—	2	2	no	—
31	C₂×#41	—	5	1	no	—
32	C₂×M₁₆	D₅	1	1	yes	characteristic, complemented
33	C₂²×C₈	—	5	1	no	—
34	C₂²⋊C₈	—	5	4	no	—
35	C₂²×C₅	—	2	2	no	—
36	C₂²×C₅	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₂×D₅	—	2	2	no	—
39	C₂×D₅	—	2	4	no	—
40	C₄×C₅	C₂²⋊C₄	1	2	yes	—
41	C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	Dic₅	—	2	2	no	—
44	#41	—	10	2	no	—
45	#41	—	10	2	no	—
46	C₂×Q₈	—	5	1	no	—
47	C₂×D₄	—	5	2	no	—
48	C₂×D₄	—	5	1	no	—
49	C₂²×C₄	—	5	2	no	—
50	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	—	5	2	no	—
53	C₂×C₈	—	10	2	no	—
54	C₂×C₈	C₂×D₅	1	2	yes	—
55	C₂×C₅	—	2	2	no	—
56	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₅	#131	1	2	yes	—
58	D₅	—	4	2	no	—
59	C₂³	—	5	2	no	—
60	C₂³	C₄×D₅	1	1	yes	characteristic
61	Q₈	—	10	2	no	—
62	D₄	—	10	4	no	—
63	D₄	—	10	2	no	—
64	C₂×C₄	—	5	2	no	—
65	C₂×C₄	—	10	2	no	—
66	C₂×C₄	—	10	2	no	—
67	C₂×C₄	C₄×D₅	1	1	yes	characteristic
68	C₂×C₄	D₂₀	1	2	yes	—
69	C₂×C₄	C₂²⋊D₅	1	2	yes	—
70	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	10	2	no	—
73	C₅	#408	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	2	no	—
75	C₂²	—	10	4	no	—
76	C₂²	—	10	2	no	—
77	C₂²	#685	1	2	yes	—
78	C₂²	C₂×C₄×D₅	1	1	yes	characteristic
79	C₄	—	10	2	no	—
80	C₄	#685	1	2	yes	—
81	C₄	C₂×D₂₀	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	20	2	no	—
85	C₂	#4053	1	2	yes	—
86	C₂	C₂×#685	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₅×M₁₆	C₂
prime radical	C₂×#709	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#408
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×M₁₆	C₂	semidirect
3. C₂²×#168	C₂	semidirect
4. #3952	C₂	semidirect
32. C₂×M₁₆	D₅	semidirect
73. C₅	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#709	2
C₂×C₅×M₁₆	2
C₂²×#168	2
#3952	2
#408	5