Group #11752011 — subgroups

Subgroups — #11752011

Order 960.

228 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 82 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×#6762	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#170	C₂	1	1	yes	maximal, characteristic
3	#91528	C₂	1	1	yes	maximal, characteristic
4	#91177	C₂	1	2	yes	maximal, complemented
5	Dic₃⋊#170	C₂	1	2	yes	maximal, complemented
6	#65696	—	3	1	no	maximal, Hall
7	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
8	#6762	—	2	2	no	—
9	C₂×C₃⋊#170	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊#170	C₂²	1	1	yes	characteristic
11	C₂×C₃×#170	—	2	2	no	—
12	C₂×C₃×#170	C₂²	1	2	yes	—
13	#6719	C₄	1	1	yes	characteristic
14	C₅×#217	C₄	1	1	yes	characteristic
15	#6668	C₄	1	2	yes	—
16	Dic₃×Dic₅	C₂²	1	2	yes	—
17	#5102	—	5	1	no	maximal, Hall
18	C₂×#704	—	3	1	no	—
19	C₂²×#170	D₃	1	1	yes	characteristic
20	#4023	—	3	1	no	—
21	#4004	—	3	2	no	—
22	C₄⋊#170	—	3	2	no	—
23	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₅	C₂×C₄	1	2	yes	—
25	C₂×C₅×Dic₃	C₂×C₄	1	2	yes	—
26	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×Dic₅	D₄	1	2	yes	—
28	C₂×C₃×Dic₅	Q₈	1	2	yes	—
29	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
30	C₃⋊#170	—	2	1	no	—
31	C₃⋊#170	—	2	1	no	—
32	C₃×#170	—	2	2	no	—
33	C₂²×C₃×C₈	—	5	1	no	—
34	#926	—	5	1	no	—
35	C₂×#208	—	5	1	no	—
36	Dic₃⋊C₈	—	5	2	no	—
37	Dic₃⋊C₈	—	5	2	no	—
38	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
39	#704	—	6	2	no	—
40	C₂×#170	—	2	2	no	—
41	C₂×#170	—	3	1	no	—
42	C₂×#170	—	3	1	no	—
43	C₂×#170	C₂×D₃	1	2	yes	—
44	C₅×C₂²⋊C₄	—	3	1	no	—
45	C₂²⋊Dic₅	—	3	1	no	—
46	#683	—	3	2	no	—
47	C₄×Dic₅	—	3	2	no	—
48	#424	—	15	1	no	Sylow
49	C₂²×C₃×C₅	—	2	2	no	—
50	C₂²×C₃×C₅	C₄⋊C₄	1	2	yes	—
51	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
52	Dic₁₅	—	2	2	no	—
53	C₃×Dic₅	C₄⋊C₄	1	2	yes	—
54	C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
55	C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
56	C₅×Dic₃	—	2	2	no	—
57	C₂²×C₃×C₄	—	5	1	no	—
58	C₂×C₃×C₈	—	5	2	no	—
59	C₂×C₃×C₈	—	10	2	no	—
60	#217	—	5	1	no	—
61	#217	AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₄⋊Dic₃	—	5	2	no	—
63	C₄×Dic₃	—	5	2	no	—
64	#208	—	10	2	no	—
65	C₂×C₃⋊C₈	—	5	1	no	—
66	C₂×C₃⋊C₈	—	5	1	no	—
67	C₂³×C₅	C₄×D₃	1	1	yes	characteristic
68	C₂×C₄×C₅	—	3	2	no	—
69	C₂×Dic₅	—	3	2	no	—
70	C₂×Dic₅	Dic₆	1	2	yes	—
71	C₂×Dic₅	C₄×D₃	1	1	yes	characteristic
72	C₂×Dic₅	#67	1	2	yes	—
73	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
74	#170	—	2	2	no	—
75	#170	—	6	1	no	—
76	#170	—	6	1	no	—
77	C₂×M₁₆	—	15	1	no	—
78	C₂²×C₈	—	5	1	no	—
79	#117	—	15	1	no	—
80	C₄⋊C₈	—	15	2	no	—
81	C₄⋊C₈	—	15	2	no	—
82	C₂×C₃×C₅	—	2	2	no	—
83	C₂×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₅	#131	1	2	yes	—
85	C₂³×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₂×C₃×C₄	—	5	2	no	—
87	C₂×C₃×C₄	—	5	2	no	—
88	C₂×C₃×C₄	—	5	1	no	—
89	C₂×C₃×C₄	—	5	1	no	—
90	C₂×Dic₃	—	5	2	no	—
91	C₂×Dic₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
92	C₃×C₈	—	10	2	no	—
93	C₃⋊C₈	—	10	1	no	—
94	C₃⋊C₈	—	10	1	no	—
95	C₂²×C₅	—	2	2	no	—
96	C₂²×C₅	Dic₃⋊C₄	1	2	yes	—
97	C₂²×C₅	C₂×C₄×D₃	1	1	yes	characteristic
98	C₄×C₅	—	6	2	no	—
99	Dic₅	—	6	2	no	—
100	Dic₅	Dic₃⋊C₄	1	2	yes	—
101	Dic₅	C₂×Dic₆	1	1	yes	characteristic
102	Dic₅	C₂×#67	1	1	yes	characteristic
103	C₂²×C₄	—	5	1	no	—
104	M₁₆	—	30	2	no	—
105	C₂×C₈	—	5	2	no	—
106	C₂×C₈	—	10	2	no	—
107	C₂×C₈	—	15	1	no	—
108	C₂×C₈	—	15	1	no	—
109	C₄⋊C₄	—	15	2	no	—
110	C₂²⋊C₄	—	3	1	no	—
111	C₂²⋊C₄	—	15	1	no	—
112	C₄²	—	15	2	no	—
113	C₃×C₅	#424	1	1	yes	characteristic, complemented, Hall
114	C₂²×C₃	—	2	2	no	—
115	C₂²×C₃	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
116	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₃×C₄	—	5	2	no	—
118	C₃×C₄	—	5	1	no	—
119	C₃×C₄	—	5	1	no	—
120	Dic₃	—	2	2	no	—
121	Dic₃	—	10	2	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	#901	1	2	yes	—
124	C₂×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
125	C₂³	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₂×C₄	—	3	2	no	—
127	C₂×C₄	—	5	2	no	—
128	C₂×C₄	—	5	2	no	—
129	C₂×C₄	—	5	1	no	—
130	C₂×C₄	—	5	1	no	—
131	C₂×C₄	—	15	2	no	—
132	C₈	—	10	2	no	—
133	C₈	—	30	1	no	—
134	C₈	—	30	1	no	—
135	C₂×C₃	—	2	2	no	—
136	C₂×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₂×C₃	#4131	1	2	yes	—
138	C₅	#5102	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	2	2	no	—
140	C₂²	Dic₃⋊AGL(1,𝔽₅)	1	2	yes	—
141	C₂²	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₄	—	5	1	no	—
143	C₄	—	5	2	no	—
144	C₄	—	5	1	no	—
145	C₄	—	6	2	no	—
146	C₄	—	30	2	no	—
147	C₃	#65696	1	1	yes	characteristic, complemented, Sylow
148	C₂	—	2	2	no	—
149	C₂	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂	#91927	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6762

C₂

yes

maximal, characteristic

C₂²×C₃×#170

C₂

yes

maximal, characteristic

#91528

C₂

yes

maximal, characteristic

#91177

C₂

yes

maximal, complemented

Dic₃⋊#170

C₂

yes

maximal, complemented

#65696

—

maximal, Hall

C₂²×C₃×Dic₅

C₂²

yes

characteristic

#6762

—

C₂×C₃⋊#170

C₂²

yes

characteristic

C₂×C₃⋊#170

C₂²

yes

characteristic

C₂×C₃×#170

—

C₂×C₃×#170

C₂²

yes

—

#6719

C₄

yes

characteristic

C₅×#217

C₄

yes

characteristic

#6668

C₄

yes

—

Dic₃×Dic₅

C₂²

yes

—

#5102

—

maximal, Hall

C₂×#704

—

C₂²×#170

D₃

yes

—

—

—

C₂³×C₃×C₅

C₂×C₄

yes

characteristic

C₂×Dic₁₅

C₂×C₄

yes

—

C₂×C₅×Dic₃

C₂×C₄

yes

—

C₂×C₃×Dic₅

C₂×C₄

yes

characteristic

C₂×C₃×Dic₅

D₄

yes

—

C₂×C₃×Dic₅

Q₈

yes

—

C₂×C₃×Dic₅

C₂³

yes

—

—

—

C₂²×C₃×C₈

—

#926

—

C₂×#208

—

Dic₃⋊C₈

—

Dic₃⋊C₈

—

C₂²×Dic₅

C₂×D₃

yes

—

—

—

—

C₂×D₃

yes

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

#683

—

C₄×Dic₅

—

#424

—

Sylow

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₄⋊C₄

yes

—

C₂²×C₃×C₅

C₂²×C₄

yes

characteristic

Dic₁₅

—

C₃×Dic₅

C₄⋊C₄

yes

—

C₃×Dic₅

C₂×D₄

yes

characteristic

C₃×Dic₅

C₂×Q₈

yes

characteristic

C₅×Dic₃

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

—

yes

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂³×C₅

C₄×D₃

yes

characteristic

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

Dic₆

yes

—

C₂×Dic₅

C₄×D₃

yes

characteristic

C₂×Dic₅

#67

yes

—

C₂×Dic₅

C₂²×D₃

yes

—

—

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂×C₃×C₅

—

C₂×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×C₅

#131

yes

—

C₂³×C₃

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

C₂×AGL(1,𝔽₅)

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₅

—

C₂²×C₅

Dic₃⋊C₄

yes

—

C₂²×C₅

C₂×C₄×D₃

yes

characteristic

C₄×C₅

—

Dic₅

—

100

Dic₅

Dic₃⋊C₄

yes

—

101

Dic₅

C₂×Dic₆

yes

characteristic

102

Dic₅

C₂×#67

yes

characteristic

103

C₂²×C₄

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₃×C₅

#424

yes

characteristic, complemented, Hall

114

C₂²×C₃

—

115

C₂²×C₃

C₄⋊AGL(1,𝔽₅)

yes

—

116

C₂²×C₃

C₂²×AGL(1,𝔽₅)

yes

characteristic

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

Dic₃

—

121

Dic₃

—

122

C₂×C₅

—

123

C₂×C₅

#901

yes

—

124

C₂×C₅

C₂×Dic₃⋊C₄

yes

characteristic

125

C₂³

D₃×AGL(1,𝔽₅)

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂×C₃

—

136

C₂×C₃

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

137

C₂×C₃

#4131

yes

—

138

C₅

#5102

yes

characteristic, complemented, Sylow

139

C₂²

—

140

C₂²

Dic₃⋊AGL(1,𝔽₅)

yes

—

141

C₂²

C₂×D₃×AGL(1,𝔽₅)

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₃

#65696

yes

characteristic, complemented, Sylow

148

C₂

—

149

C₂

C₂×Dic₃⋊AGL(1,𝔽₅)

yes

characteristic

150

C₂

#91927

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂²×C₄
Fitting, prime radical	C₂³×C₃×C₅	C₂×C₄
cosocle	C₂×C₃×Dic₅	C₂³
hypercenter	C₂³	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#424
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂³

subgroup

G/H

center, Frattini

C₂²

C₂×D₃×AGL(1,𝔽₅)

commutator, socle

C₂²×C₃×C₅

C₂²×C₄

Fitting, prime radical

C₂³×C₃×C₅

C₂×C₄

cosocle

C₂×C₃×Dic₅

C₂³

C₂³

C₃×C₅

G ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂³

N	H ≅ G/N	type
4. #91177	C₂	semidirect
5. Dic₃⋊#170	C₂	semidirect
113. C₃×C₅	#424	semidirect
138. C₅	#5102	semidirect
147. C₃	#65696	semidirect

H ≅ G/N

type

4. #91177

C₂

semidirect

5. Dic₃⋊#170

C₂

semidirect

113. C₃×C₅

#424

semidirect

138. C₅

#5102

semidirect

147. C₃

#65696

semidirect

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

subgroup	index
C₂×#6762	2
C₂²×C₃×#170	2
#91528	2
#91177	2
Dic₃⋊#170	2
#65696	3
#5102	5

subgroup

index

C₂²×C₃×#170

Subgroups — #11752011

Conjugacy classes of subgroups

Complements

Maximal subgroups