Group #10615073 — subgroups

Subgroups — #10615073

Order 640.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×#4016	C₂	1	1	yes	maximal, characteristic
3	#65369	C₂	1	4	yes	maximal, complemented
4	#65353	C₂	1	4	yes	maximal, complemented
5	#65284	C₂	1	4	yes	maximal
6	C₄×#681	C₂	1	1	yes	maximal, characteristic
7	C₅×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₅×#125	C₂²	1	2	yes	—
9	C₅×C₂²⋊Q₈	—	2	2	no	—
10	C₄×C₅×Q₈	—	2	2	no	—
11	C₅×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×#683	—	2	2	no	—
15	C₂×#681	C₂²	1	2	yes	—
16	#4016	—	2	2	no	—
17	#4016	C₂²	1	2	yes	—
18	#4016	C₂²	1	2	yes	—
19	#3940	C₂²	1	8	yes	—
20	#3939	C₂²	1	8	yes	—
21	#3935	C₂²	1	2	yes	—
22	C₄×#168	C₂²	1	2	yes	—
23	#3166	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₈	—	2	2	no	—
25	C₂²×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅	—	2	2	no	—
28	C₅×C₄⋊C₄	—	2	4	no	—
29	C₅×C₄⋊C₄	—	2	2	no	—
30	C₅×C₄⋊C₄	C₂³	1	4	yes	—
31	C₅×C₂²⋊C₄	—	2	2	no	—
32	C₄²×C₅	—	2	2	no	—
33	C₄²×C₅	D₄	1	2	yes	—
34	C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₅	—	2	2	no	—
37	C₄⋊Dic₅	C₂³	1	4	yes	—
38	#683	—	2	4	no	—
39	#683	—	2	4	no	—
40	#681	Q₈	1	8	yes	complemented
41	C₂×#168	C₂³	1	4	yes	—
42	#533	D₅	1	1	yes	characteristic
43	C₂×#125	—	5	1	no	—
44	#501	—	5	4	no	—
45	#499	—	5	4	no	—
46	#428	—	5	4	no	—
47	C₄×M₁₆	—	5	1	no	—
48	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
49	C₅×Q₈	—	4	4	no	—
50	C₂×C₄×C₅	—	2	4	no	—
51	C₂×C₄×C₅	—	4	2	no	—
52	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅	—	2	4	no	—
58	C₂×Dic₅	—	4	4	no	—
59	#168	C₂×Q₈	1	8	yes	—
60	C₂×M₁₆	—	5	2	no	—
61	C₄⋊Q₈	C₂×D₅	1	2	yes	—
62	#125	—	5	2	no	—
63	#125	—	5	2	no	—
64	#125	—	10	2	no	—
65	#125	C₂×D₅	1	2	yes	—
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₅	1	2	yes	—
69	C₂×C₄⋊C₄	—	5	2	no	—
70	C₂×C₄⋊C₄	—	10	2	no	—
71	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
72	#107	—	5	8	no	—
73	#106	—	5	8	no	—
74	#97	—	5	2	no	—
75	C₄×C₈	—	5	2	no	—
76	C₂²×C₅	—	2	2	no	—
77	C₂²×C₅	C₄⋊Q₈	1	2	yes	—
78	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	—	2	2	no	—
81	C₄×C₅	—	4	4	no	—
82	C₄×C₅	C₄⋊Q₈	1	2	yes	—
83	C₄×C₅	C₂²×Q₈	1	2	yes	—
84	Dic₅	—	4	4	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂²×C₄	—	10	2	no	—
87	C₂²×C₄	C₂²⋊D₅	1	2	yes	—
88	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
89	M₁₆	—	5	8	no	—
90	C₂×C₈	—	5	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	5	4	no	—
94	C₄⋊C₄	—	10	2	no	—
95	C₄⋊C₄	—	10	4	no	—
96	C₄⋊C₄	—	10	4	no	—
97	C₄⋊C₄	C₂²×D₅	1	4	yes	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₂²⋊D₅	1	2	yes	—
101	C₄²	C₂²×D₅	1	1	yes	characteristic
102	C₄²	C₂²×D₅	1	1	yes	characteristic
103	C₂×C₅	—	2	2	no	—
104	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₅	#575	1	2	yes	—
106	C₂³	C₂×C₂²⋊D₅	1	1	yes	characteristic
107	Q₈	—	4	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	10	4	no	—
111	C₂×C₄	—	20	4	no	—
112	C₂×C₄	Q₈×D₅	1	4	yes	—
113	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
117	C₈	—	5	8	no	—
118	C₅	#3166	1	1	yes	characteristic, complemented, Sylow
119	C₂²	—	2	2	no	—
120	C₂²	#4090	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	20	4	no	—
126	C₄	#4090	1	2	yes	—
127	C₄	C₂×Q₈×D₅	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#4090	1	1	yes	characteristic
130	C₂	#66156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#533

C₂

yes

maximal, characteristic

C₂×#4016

C₂

yes

maximal, characteristic

#65369

C₂

yes

maximal, complemented

#65353

C₂

yes

maximal, complemented

#65284

C₂

yes

maximal

C₄×#681

C₂

yes

maximal, characteristic

C₅×C₄⋊Q₈

C₂²

yes

—

C₅×#125

C₂²

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

—

C₂×#681

C₂²

yes

—

#4016

—

#4016

C₂²

yes

—

#4016

C₂²

yes

—

#3940

C₂²

yes

—

#3939

C₂²

yes

—

#3935

C₂²

yes

—

C₄×#168

C₂²

yes

—

#3166

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂³

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

D₄

yes

—

C₄²×C₅

C₂³

yes

characteristic

C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂³

yes

—

#683

—

#683

—

#681

Q₈

yes

complemented

C₂×#168

C₂³

yes

—

#533

D₅

yes

—

—

—

—

C₄×M₁₆

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

C₂⁴

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

#168

C₂×Q₈

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₅

yes

—

—

—

—

C₂×D₅

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅

yes

—

—

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

C₄⋊Q₈

yes

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄⋊Q₈

yes

—

C₄×C₅

C₂²×Q₈

yes

—

Dic₅

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂²⋊D₅

yes

—

101

C₄²

C₂²×D₅

yes

characteristic

102

C₄²

C₂²×D₅

yes

characteristic

103

C₂×C₅

—

104

C₂×C₅

C₂×C₄⋊Q₈

yes

characteristic

105

C₂×C₅

#575

yes

—

106

C₂³

C₂×C₂²⋊D₅

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

Q₈×D₅

yes

—

113

C₂×C₄

C₂×C₂²⋊D₅

yes

—

114

C₂×C₄

C₂×C₂²⋊D₅

yes

—

115

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

116

C₂×C₄

C₂³×D₅

yes

characteristic

117

C₈

—

118

C₅

#3166

yes

characteristic, complemented, Sylow

119

C₂²

—

120

C₂²

#4090

yes

—

121

C₂²

C₂²×C₂²⋊D₅

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#4090

yes

—

127

C₄

C₂×Q₈×D₅

yes

—

128

C₂

—

129

C₂

C₂×#4090

yes

characteristic

130

C₂

#66156

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₅
Fitting	C₅×#533	C₂
socle	C₂²×C₅	C₂²×D₄
prime radical	C₂³×C₅	C₂×D₄
hypercenter	#533	D₅
hypocenter	C₅	#3166
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅

commutator, cosocle

C₂×C₄×C₅

C₂⁴

Frattini

C₂×C₄

C₂³×D₅

Fitting

C₅×#533

C₂

socle

C₂²×C₅

C₂²×D₄

prime radical

C₂³×C₅

C₂×D₄

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #65369	C₂	semidirect
4. #65353	C₂	semidirect
40. #681	Q₈	semidirect
118. C₅	#3166	semidirect

H ≅ G/N

type

3. #65369

C₂

semidirect

4. #65353

C₂

semidirect

40. #681

Q₈

semidirect

118. C₅

#3166

semidirect

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

subgroup	index
C₅×#533	2
C₂×#4016	2
#65369	2
#65353	2
#65284	2
C₄×#681	2
#3166	5

subgroup

index

Subgroups — #10615073

Conjugacy classes of subgroups

Complements

Maximal subgroups