Group #10615059 — subgroups

Subgroups — #10615059

Order 640.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	#65313	C₂	1	4	yes	maximal, complemented
3	Q₈×#168	C₂	1	4	yes	maximal, complemented
4	#65288	C₂	1	2	yes	maximal
5	#65286	C₂	1	2	yes	maximal
6	#65213	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×#168	C₂	1	1	yes	maximal, characteristic
8	C₅×C₄⋊Q₈	C₄	1	2	yes	—
9	C₅×#125	C₄	1	2	yes	—
10	C₅×C₂²⋊Q₈	C₄	1	4	yes	—
11	C₄×C₅×Q₈	C₂²	1	4	yes	—
12	C₅×#117	C₂²	1	2	yes	—
13	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×#681	C₂²	1	2	yes	—
15	C₂²×#168	C₂²	1	1	yes	characteristic
16	C₂²×#168	C₂²	1	1	yes	characteristic
17	C₄⋊#168	C₂²	1	8	yes	—
18	C₄⋊#168	C₂²	1	4	yes	—
19	#3935	C₂²	1	4	yes	—
20	C₄×#168	C₂²	1	4	yes	—
21	C₄×#168	C₂²	1	2	yes	—
22	C₄×#168	C₂²	1	2	yes	—
23	#2960	—	5	1	no	maximal, Sylow
24	C₂×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
28	C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
29	C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
31	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₅	C₂³	1	2	yes	—
34	C₄²×C₅	C₂³	1	4	yes	—
35	#681	—	2	4	no	—
36	C₂×#168	—	2	2	no	—
37	C₂×#168	Q₈	1	4	yes	complemented
38	C₂×#168	C₂³	1	2	yes	—
39	C₂×#168	C₂³	1	4	yes	—
40	C₂×#168	C₂³	1	1	yes	characteristic
41	C₂×#168	C₂³	1	1	yes	characteristic
42	#533	D₅	1	1	yes	characteristic
43	#446	—	5	4	no	—
44	C₈×Q₈	—	5	4	no	—
45	#424	—	5	2	no	—
46	#423	—	5	1	no	—
47	#405	—	5	2	no	—
48	C₂×C₄×C₈	—	5	1	no	—
49	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₅×Q₈	—	2	8	no	—
51	C₂×C₄×C₅	—	2	4	no	—
52	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₅	C₂²×C₄	1	8	yes	—
54	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₅	#41	1	2	yes	—
58	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
59	#168	—	2	2	no	—
60	#168	—	2	4	no	—
61	#168	C₂×Q₈	1	2	yes	—
62	#168	C₂×Q₈	1	2	yes	—
63	C₂×M₁₆	—	5	2	no	—
64	C₂²×C₈	—	5	1	no	—
65	C₂²×C₈	—	5	1	no	—
66	C₄⋊Q₈	Dic₅	1	2	yes	—
67	#125	Dic₅	1	2	yes	—
68	C₂²⋊Q₈	Dic₅	1	4	yes	—
69	C₄×Q₈	C₂×D₅	1	4	yes	—
70	#117	C₂×D₅	1	2	yes	—
71	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₄⋊C₈	—	5	4	no	—
73	C₄⋊C₈	—	5	8	no	—
74	#97	—	5	4	no	—
75	C₄×C₈	—	5	2	no	—
76	C₄×C₈	—	5	2	no	—
77	C₄×C₈	—	5	4	no	—
78	C₂²×C₅	—	2	2	no	—
79	C₂²×C₅	C₄×Q₈	1	2	yes	—
80	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
81	C₄×C₅	—	2	2	no	—
82	C₄×C₅	—	2	8	no	—
83	C₄×C₅	C₄×Q₈	1	2	yes	—
84	C₄×C₅	#131	1	4	yes	—
85	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₅	C₂×#41	1	1	yes	characteristic
87	C₂×Q₈	C₂×Dic₅	1	4	yes	—
88	C₂²×C₄	C₂×Dic₅	1	2	yes	—
89	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
90	M₁₆	—	10	4	no	—
91	C₂×C₈	—	5	4	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	5	1	no	—
94	C₂×C₈	—	5	2	no	—
95	C₂×C₈	—	5	4	no	—
96	C₂×C₈	—	10	2	no	—
97	C₄⋊C₄	C₂×Dic₅	1	4	yes	—
98	C₄⋊C₄	C₂×Dic₅	1	8	yes	—
99	C₄⋊C₄	C₂×Dic₅	1	4	yes	—
100	C₂²⋊C₄	C₂×Dic₅	1	4	yes	—
101	C₄²	C₂×Dic₅	1	1	yes	characteristic
102	C₄²	C₂×Dic₅	1	1	yes	characteristic
103	C₄²	C₂²×D₅	1	2	yes	—
104	C₄²	C₂²×D₅	1	4	yes	—
105	C₂×C₅	—	2	2	no	—
106	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂×C₅	C₂×#131	1	2	yes	—
108	C₂³	C₂²×Dic₅	1	1	yes	characteristic
109	Q₈	—	2	8	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	Q₈×D₅	1	2	yes	—
112	C₂×C₄	#713	1	2	yes	—
113	C₂×C₄	C₂²×Dic₅	1	2	yes	—
114	C₂×C₄	C₂²×Dic₅	1	8	yes	—
115	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₅	1	2	yes	—
117	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
118	C₈	—	5	2	no	—
119	C₈	—	5	2	no	—
120	C₈	—	10	2	no	—
121	C₈	—	10	4	no	—
122	C₅	#2960	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₅	1	2	yes	—
125	C₂²	C₂³×Dic₅	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₅	1	2	yes	—
129	C₄	#4094	1	4	yes	—
130	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
131	C₄	C₂×#713	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₅	1	1	yes	characteristic
134	C₂	C₂×#4094	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#533

C₂

yes

maximal, characteristic

#65313

C₂

yes

maximal, complemented

Q₈×#168

C₂

yes

maximal, complemented

#65288

C₂

yes

maximal

#65286

C₂

yes

maximal

#65213

C₂

yes

maximal, characteristic

C₂×C₄×#168

C₂

yes

maximal, characteristic

C₅×C₄⋊Q₈

C₄

yes

—

C₅×#125

C₄

yes

—

C₅×C₂²⋊Q₈

C₄

yes

—

C₄×C₅×Q₈

C₂²

yes

—

C₅×#117

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×#681

C₂²

yes

—

C₂²×#168

C₂²

yes

characteristic

C₂²×#168

C₂²

yes

characteristic

C₄⋊#168

C₂²

yes

—

C₄⋊#168

C₂²

yes

—

#3935

C₂²

yes

—

C₄×#168

C₂²

yes

—

C₄×#168

C₂²

yes

—

C₄×#168

C₂²

yes

—

#2960

—

maximal, Sylow

C₂×C₅×Q₈

C₂×C₄

yes

—

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₅×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

C₂³

yes

—

C₄²×C₅

C₂³

yes

—

#681

—

C₂×#168

—

C₂×#168

Q₈

yes

complemented

C₂×#168

C₂³

yes

—

C₂×#168

C₂³

yes

—

C₂×#168

C₂³

yes

characteristic

C₂×#168

C₂³

yes

characteristic

#533

D₅

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×C₄×C₅

C₂⁴

yes

—

—

C₂×Q₈

yes

—

#168

C₂×Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₅

yes

—

#125

Dic₅

yes

—

C₂²⋊Q₈

Dic₅

yes

—

C₄×Q₈

C₂×D₅

yes

—

#117

C₂×D₅

yes

—

C₂×C₄²

C₂×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

C₄×Q₈

yes

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

C₄×C₅

#131

yes

—

C₄×C₅

C₂²×Q₈

yes

characteristic

C₄×C₅

C₂×#41

yes

characteristic

C₂×Q₈

C₂×Dic₅

yes

—

C₂²×C₄

C₂×Dic₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅

yes

—

C₄⋊C₄

C₂×Dic₅

yes

—

C₄⋊C₄

C₂×Dic₅

yes

—

100

C₂²⋊C₄

C₂×Dic₅

yes

—

101

C₄²

C₂×Dic₅

yes

characteristic

102

C₄²

C₂×Dic₅

yes

characteristic

103

C₄²

C₂²×D₅

yes

—

104

C₄²

C₂²×D₅

yes

—

105

C₂×C₅

—

106

C₂×C₅

C₂×C₄×Q₈

yes

characteristic

107

C₂×C₅

C₂×#131

yes

—

108

C₂³

C₂²×Dic₅

yes

characteristic

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

Q₈×D₅

yes

—

112

C₂×C₄

#713

yes

—

113

C₂×C₄

C₂²×Dic₅

yes

—

114

C₂×C₄

C₂²×Dic₅

yes

—

115

C₂×C₄

C₂²×Dic₅

yes

characteristic

116

C₂×C₄

C₂²×Dic₅

yes

—

117

C₂×C₄

C₂³×D₅

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₅

#2960

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

Q₈×Dic₅

yes

—

125

C₂²

C₂³×Dic₅

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₅

yes

—

129

C₄

#4094

yes

—

130

C₄

C₂×Q₈×D₅

yes

characteristic

131

C₄

C₂×#713

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₅

yes

characteristic

134

C₂

C₂×#4094

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#533	C₂
prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#533	D₅
hypocenter	C₅	#2960
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅

commutator, socle

C₂²×C₅

C₂³×C₄

Fitting

C₅×#533

C₂

prime radical

C₂³×C₅

C₂²×C₄

cosocle

C₂×C₄×C₅

C₂⁴

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
2. #65313	C₂	semidirect
3. Q₈×#168	C₂	semidirect
37. C₂×#168	Q₈	semidirect
122. C₅	#2960	semidirect

H ≅ G/N

type

2. #65313

C₂

semidirect

3. Q₈×#168

C₂

semidirect

37. C₂×#168

Q₈

semidirect

122. C₅

#2960

semidirect

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

subgroup	index
C₅×#533	2
#65313	2
Q₈×#168	2
#65288	2
#65286	2
#65213	2
C₂×C₄×#168	2
#2960	5

subgroup

index

C₂×C₄×#168

Subgroups — #10615059

Conjugacy classes of subgroups

Complements

Maximal subgroups