Group #11751829 — subgroups

Subgroups — #11751829

Order 960.

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#6758	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#700	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#91177	C₂	1	4	yes	maximal, complemented
5	#65682	—	3	1	no	maximal, Hall
6	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂×D₅×Dic₃	C₄	1	2	yes	—
8	C₂×C₃⋊#170	C₂²	1	2	yes	—
9	#6758	—	2	2	no	—
10	C₂×C₃×#170	C₂²	1	2	yes	—
11	C₃×#700	—	2	2	no	—
12	C₄×Dic₁₅	C₄	1	1	yes	characteristic
13	C₄×C₅×Dic₃	C₄	1	1	yes	characteristic
14	Dic₃×Dic₅	C₂²	1	2	yes	—
15	#5118	—	5	1	no	maximal, Hall
16	C₂×#700	—	3	1	no	—
17	C₂×#700	D₃	1	1	yes	characteristic
18	C₄²×D₅	—	3	1	no	—
19	#4004	—	3	4	no	—
20	C₂²×C₃×D₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₅	C₂×C₄	1	2	yes	—
23	C₂×C₅×Dic₃	C₂×C₄	1	2	yes	—
24	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
25	C₃×C₄×D₅	D₄	1	2	yes	—
26	C₃×C₄×D₅	Q₈	1	2	yes	—
27	D₅×Dic₃	—	2	4	no	—
28	C₃⋊#170	—	2	2	no	—
29	C₃×#170	—	2	2	no	—
30	C₂×C₃×M₁₆	—	5	1	no	—
31	C₂×C₄×Dic₃	—	5	1	no	—
32	C₂×#208	—	5	1	no	—
33	Dic₃⋊C₈	—	5	4	no	—
34	C₂×C₄×D₅	—	3	2	no	—
35	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
36	C₂×#170	—	3	2	no	—
37	C₂×#170	C₂×D₃	1	2	yes	—
38	#700	—	2	2	no	—
39	#700	—	6	2	no	—
40	C₄²×C₅	—	3	1	no	—
41	C₄×Dic₅	—	3	1	no	—
42	C₄×Dic₅	—	3	2	no	—
43	#423	—	15	1	no	Sylow
44	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×D₅	—	2	2	no	—
46	C₂×C₃×D₅	C₄⋊C₄	1	2	yes	—
47	C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	Dic₁₅	M₁₆	1	4	yes	complemented
49	C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
50	C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
51	C₅×Dic₃	—	2	2	no	—
52	C₂²×C₃×C₄	—	5	1	no	—
53	C₂²×Dic₃	—	5	2	no	—
54	C₃×M₁₆	—	10	2	no	—
55	C₂×C₃×C₈	—	5	2	no	—
56	C₄×Dic₃	—	5	1	no	—
57	C₄×Dic₃	—	5	2	no	—
58	C₄×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
59	#208	—	10	2	no	—
60	C₂×C₃⋊C₈	—	5	2	no	—
61	C₂²×D₅	C₄×D₃	1	1	yes	characteristic
62	C₂×C₄×C₅	—	3	2	no	—
63	C₂×C₄×C₅	C₄×D₃	1	1	yes	characteristic
64	C₂×Dic₅	—	3	2	no	—
65	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
66	C₄×D₅	—	6	4	no	—
67	C₄×D₅	Dic₆	1	2	yes	—
68	C₄×D₅	#67	1	2	yes	—
69	#170	—	2	2	no	—
70	#170	—	6	2	no	—
71	C₂×M₁₆	—	5	1	no	—
72	C₂×M₁₆	—	15	1	no	—
73	C₂×C₄²	—	15	1	no	—
74	C₄⋊C₈	—	15	4	no	—
75	C₂×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₅	C₂×M₁₆	1	2	yes	—
77	C₃×D₅	—	2	2	no	—
78	C₂³×C₃	—	5	1	no	—
79	C₂×C₃×C₄	—	5	1	no	—
80	C₂×C₃×C₄	—	5	2	no	—
81	C₂×C₃×C₄	—	5	2	no	—
82	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×Dic₃	—	5	2	no	—
84	C₂×Dic₃	—	10	4	no	—
85	C₂×Dic₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
86	C₃×C₈	—	10	2	no	—
87	C₃⋊C₈	—	10	2	no	—
88	C₂²×C₅	C₂×C₄×D₃	1	1	yes	characteristic
89	C₂×D₅	—	2	2	no	—
90	C₂×D₅	Dic₃⋊C₄	1	2	yes	—
91	C₄×C₅	—	6	2	no	—
92	C₄×C₅	Dic₃⋊C₄	1	2	yes	—
93	Dic₅	—	3	4	no	—
94	Dic₅	C₂×Dic₆	1	1	yes	characteristic
95	Dic₅	C₂×#67	1	1	yes	characteristic
96	C₂²×C₄	—	5	1	no	—
97	C₂²×C₄	—	15	2	no	—
98	M₁₆	—	10	2	no	—
99	M₁₆	—	30	2	no	—
100	C₂×C₈	—	5	2	no	—
101	C₂×C₈	—	15	2	no	—
102	C₄²	—	3	1	no	—
103	C₄²	—	15	2	no	—
104	C₄²	—	15	1	no	—
105	C₃×C₅	#423	1	1	yes	characteristic, complemented, Hall
106	C₂²×C₃	—	5	2	no	—
107	C₂²×C₃	—	10	2	no	—
108	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
109	C₃×C₄	—	5	1	no	—
110	C₃×C₄	—	5	1	no	—
111	C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
112	Dic₃	—	2	2	no	—
113	Dic₃	—	5	4	no	—
114	C₂×C₅	D₃×M₁₆	1	2	yes	—
115	C₂×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
116	D₅	—	2	2	no	—
117	C₂³	—	5	1	no	—
118	C₂×C₄	—	3	2	no	—
119	C₂×C₄	—	5	1	no	—
120	C₂×C₄	—	5	2	no	—
121	C₂×C₄	—	5	2	no	—
122	C₂×C₄	—	15	2	no	—
123	C₂×C₄	—	30	4	no	—
124	C₂×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₈	—	10	2	no	—
126	C₈	—	30	2	no	—
127	C₂×C₃	—	10	2	no	—
128	C₂×C₃	D₅⋊M₁₆	1	2	yes	—
129	C₂×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₅	#5118	1	1	yes	characteristic, complemented, Sylow
131	C₂²	—	5	2	no	—
132	C₂²	—	10	2	no	—
133	C₂²	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₄	—	5	1	no	—
135	C₄	—	5	1	no	—
136	C₄	—	6	2	no	—
137	C₄	—	15	4	no	—
138	C₄	Dic₃⋊AGL(1,𝔽₅)	1	2	yes	—
139	C₃	#65682	1	1	yes	characteristic, complemented, Sylow
140	C₂	—	10	2	no	—
141	C₂	#91915	1	2	yes	—
142	C₂	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6758

C₂

yes

maximal, characteristic

C₂×C₃×#700

C₂

yes

maximal, characteristic

C₄×D₅×Dic₃

C₂

yes

maximal, characteristic

#91177

C₂

yes

maximal, complemented

#65682

—

maximal, Hall

C₂×C₃×C₄×D₅

C₂²

yes

characteristic

C₂×D₅×Dic₃

C₄

yes

—

C₂×C₃⋊#170

C₂²

yes

—

#6758

—

C₂×C₃×#170

C₂²

yes

—

C₃×#700

—

C₄×Dic₁₅

C₄

yes

characteristic

C₄×C₅×Dic₃

C₄

yes

characteristic

Dic₃×Dic₅

C₂²

yes

—

—

—

D₃

yes

characteristic

C₄²×D₅

—

#4004

—

C₂²×C₃×D₅

C₂×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×C₄

yes

characteristic

C₂×Dic₁₅

C₂×C₄

yes

—

C₂×C₅×Dic₃

C₂×C₄

yes

—

C₂×C₃×Dic₅

C₂³

yes

characteristic

C₃×C₄×D₅

D₄

yes

—

C₃×C₄×D₅

Q₈

yes

—

D₅×Dic₃

—

C₃⋊#170

—

C₃×#170

—

C₂×C₃×M₁₆

—

C₂×C₄×Dic₃

—

C₂×#208

—

Dic₃⋊C₈

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂×D₃

yes

characteristic

C₂×#170

—

C₂×#170

C₂×D₃

yes

—

#700

—

#700

—

C₄²×C₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#423

—

Sylow

C₂²×C₃×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

C₄⋊C₄

yes

—

C₃×C₄×C₅

C₄⋊C₄

yes

—

Dic₁₅

M₁₆

yes

complemented

C₃×Dic₅

C₂×D₄

yes

characteristic

C₃×Dic₅

C₂×Q₈

yes

characteristic

C₅×Dic₃

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

AGL(1,𝔽₅)

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂²×D₅

C₄×D₃

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₃

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂²×D₃

yes

characteristic

C₄×D₅

—

C₄×D₅

Dic₆

yes

—

C₄×D₅

#67

yes

—

#170

—

#170

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₂×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×C₅

C₂×M₁₆

yes

—

C₃×D₅

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×AGL(1,𝔽₅)

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₅

C₂×C₄×D₃

yes

characteristic

C₂×D₅

—

C₂×D₅

Dic₃⋊C₄

yes

—

C₄×C₅

—

C₄×C₅

Dic₃⋊C₄

yes

—

Dic₅

—

Dic₅

C₂×Dic₆

yes

characteristic

Dic₅

C₂×#67

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₃×C₅

#423

yes

characteristic, complemented, Hall

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

C₂²×AGL(1,𝔽₅)

yes

characteristic

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

112

Dic₃

—

113

Dic₃

—

114

C₂×C₅

D₃×M₁₆

yes

—

115

C₂×C₅

C₂×Dic₃⋊C₄

yes

characteristic

116

D₅

—

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₃×AGL(1,𝔽₅)

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂×C₃

—

128

C₂×C₃

D₅⋊M₁₆

yes

—

129

C₂×C₃

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

130

C₅

#5118

yes

characteristic, complemented, Sylow

131

C₂²

—

132

C₂²

—

133

C₂²

C₂×D₃×AGL(1,𝔽₅)

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

Dic₃⋊AGL(1,𝔽₅)

yes

—

139

C₃

#65682

yes

characteristic, complemented, Sylow

140

C₂

—

141

C₂

#91915

yes

—

142

C₂

C₂×Dic₃⋊AGL(1,𝔽₅)

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₄×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂×C₄
cosocle	C₂×C₃×Dic₅	C₂³
hypercenter	C₂×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#423
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₂²

C₂×D₃×AGL(1,𝔽₅)

commutator, socle

C₂²×C₃×C₅

C₂²×C₄

Fitting

C₂×C₃×C₄×C₅

C₂×C₄

prime radical

C₂²×C₃×D₅

C₂×C₄

cosocle

C₂×C₃×Dic₅

C₂³

C₂×C₄

C₃×C₅

G ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
4. #91177	C₂	semidirect
48. Dic₁₅	M₁₆	semidirect
105. C₃×C₅	#423	semidirect
130. C₅	#5118	semidirect
139. C₃	#65682	semidirect

H ≅ G/N

type

4. #91177

C₂

semidirect

48. Dic₁₅

M₁₆

semidirect

105. C₃×C₅

#423

semidirect

130. C₅

#5118

semidirect

139. C₃

#65682

semidirect

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

subgroup	index
C₂×#6758	2
C₂×C₃×#700	2
C₄×D₅×Dic₃	2
#91177	2
#65682	3
#5118	5

subgroup

index

C₂×#6758

C₂×C₃×#700

C₄×D₅×Dic₃

#91177

#65682

#5118

Subgroups — #11751829

Conjugacy classes of subgroups

Complements

Maximal subgroups