Subgroups — #10601310

Order 640.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#423	C₂	1	1	yes	maximal, characteristic
2	#65213	C₂	1	1	yes	maximal, characteristic
3	C₂×#3935	C₂	1	1	yes	maximal, characteristic
4	#64778	C₂	1	4	yes	maximal
5	C₂×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂×#681	C₂²	1	2	yes	—
8	C₂²×#168	C₂²	1	2	yes	—
9	C₅×C₄⋊C₈	—	2	2	no	—
10	C₄⋊#168	—	2	2	no	—
11	#3935	C₄	1	4	yes	—
12	#1838	—	5	1	no	maximal, Sylow
13	C₂²×C₄×C₅	Q₈	1	2	yes	—
14	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
15	C₅×M₁₆	—	2	4	no	—
16	C₂×C₅×C₈	—	2	2	no	—
17	C₄²×C₅	C₂×C₄	1	2	yes	—
18	C₄²×C₅	D₄	1	2	yes	—
19	#681	—	2	4	no	—
20	C₂×#168	—	2	2	no	—
21	C₂×#168	—	2	2	no	—
22	C₂×#168	—	2	2	no	—
23	C₂×#168	C₂×C₄	1	4	yes	—
24	#423	—	5	1	no	—
25	#423	D₅	1	1	yes	characteristic
26	C₂×#97	—	5	1	no	—
27	#341	—	5	4	no	—
28	C₂³×C₅	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄×C₅	—	4	2	no	—
30	C₂×C₄×C₅	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅	#41	1	2	yes	—
34	C₂×C₄×C₅	#41	1	2	yes	—
35	C₅×C₈	—	4	2	no	—
36	#168	—	2	4	no	—
37	#168	—	4	2	no	—
38	C₂×M₁₆	—	5	2	no	—
39	C₂×M₁₆	C₂×D₅	1	2	yes	—
40	C₂²×C₈	—	5	2	no	—
41	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	10	2	no	—
44	#97	—	5	4	no	—
45	C₂²×C₅	—	2	2	no	—
46	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₅	#125	1	2	yes	—
48	C₄×C₅	—	2	4	no	—
49	C₄×C₅	#117	1	2	yes	—
50	C₄×C₅	#124	1	1	yes	characteristic
51	C₄×C₅	#124	1	1	yes	characteristic
52	C₂²×C₄	Dic₁₀	1	2	yes	—
53	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
54	M₁₆	—	2	4	no	—
55	M₁₆	—	10	4	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	5	4	no	—
58	C₂×C₈	—	10	2	no	—
59	C₂×C₈	—	10	2	no	—
60	C₂×C₈	—	10	2	no	—
61	C₄²	C₄×D₅	1	2	yes	—
62	C₄²	C₂²⋊D₅	1	2	yes	—
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	#382	1	1	yes	characteristic
65	C₂×C₅	#430	1	2	yes	—
66	C₂³	C₂×Dic₁₀	1	1	yes	characteristic
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	#683	1	4	yes	—
69	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
70	C₂×C₄	#711	1	2	yes	—
71	C₂×C₄	#713	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
73	C₈	—	4	2	no	—
74	C₈	—	10	4	no	—
75	C₈	—	20	2	no	—
76	C₅	#1838	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	2	2	no	—
78	C₂²	#4036	1	2	yes	—
79	C₂²	C₂×#683	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	#4038	1	2	yes	—
82	C₄	#4082	1	1	yes	characteristic
83	C₄	#4093	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#65266	1	1	yes	characteristic
86	C₂	#65413	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#423	C₂
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#423	D₅
hypocenter	C₅	#1838
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
76. C₅	#1838	semidirect

Maximal subgroups

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

subgroup	index
C₅×#423	2
#65213	2
C₂×#3935	2
#64778	2
#1838	5