Group #10695156 — subgroups

Subgroups — #10695156

Order 768.

406 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₃×#3160	C₂	1	1	yes	maximal, characteristic
2	#71826	C₂	1	4	yes	maximal
3	#69792	C₂	1	1	yes	maximal, characteristic
4	C₂×#4455	C₂	1	1	yes	maximal, characteristic
5	#9861	—	3	1	no	maximal, Sylow
6	C₃×#533	C₂²	1	2	yes	—
7	C₃×#500	C₄	1	2	yes	—
8	C₃×#499	—	2	2	no	—
9	C₃×#498	C₄	1	2	yes	—
10	C₃×#427	—	2	2	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#819	—	2	2	no	—
13	#4924	C₂²	1	2	yes	—
14	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
15	#4535	—	2	8	no	—
16	#4455	—	2	2	no	—
17	#3160	D₃	1	1	yes	characteristic
18	#1848	—	3	1	no	—
19	#1493	—	3	4	no	—
20	C₂×#322	—	3	1	no	—
21	C₂²×C₃×C₈	D₄	1	2	yes	—
22	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	C₃×#125	—	2	2	no	—
24	C₃×C₂²⋊Q₈	—	2	4	no	—
25	C₃×C₄×Q₈	—	2	4	no	—
26	C₃×#117	—	2	2	no	—
27	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₂×#208	—	2	2	no	—
29	C₂²×C₃⋊C₈	—	2	2	no	—
30	C₃×#107	—	2	4	no	—
31	C₃×#106	—	2	4	no	—
32	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×C₈	D₄	1	2	yes	—
35	#819	—	4	4	no	—
36	C₄⋊(C₃⋊C₈)	—	2	4	no	—
37	C₄×C₃⋊C₈	—	2	2	no	—
38	C₄×C₃⋊C₈	D₄	1	4	yes	—
39	#533	C₂×D₃	1	2	yes	—
40	#500	Dic₃	1	2	yes	—
41	#499	—	2	2	no	—
42	#498	Dic₃	1	2	yes	—
43	C₂×#108	—	6	2	no	—
44	#427	—	2	2	no	—
45	#423	—	3	2	no	—
46	C₂×C₄×C₈	—	3	2	no	—
47	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
48	Q₈⋊C₈	—	6	8	no	—
49	#322	—	6	2	no	—
50	C₂×C₃×Q₈	—	2	4	no	—
51	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄	#41	1	2	yes	—
53	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₈	SD₁₆	1	4	yes	—
56	C₂×C₃×C₈	Q₁₆	1	4	yes	—
57	C₃×C₄⋊C₄	—	2	4	no	—
58	C₃×C₄⋊C₄	—	2	4	no	—
59	C₃×C₄⋊C₄	—	4	4	no	—
60	C₃×C₂²⋊C₄	—	4	2	no	—
61	C₃×C₄²	—	4	2	no	—
62	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₄²	C₂×D₄	1	2	yes	—
64	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
65	#208	—	4	4	no	—
66	C₂×C₃⋊C₈	—	2	4	no	—
67	C₂×C₃⋊C₈	—	4	4	no	—
68	C₂×C₃⋊C₈	—	4	2	no	—
69	C₂×M₁₆	—	6	2	no	—
70	C₂²×C₈	—	6	2	no	—
71	C₂²×C₈	#67	1	2	yes	—
72	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
73	#125	—	2	2	no	—
74	C₂²⋊Q₈	—	2	4	no	—
75	C₄×Q₈	—	2	4	no	—
76	#117	—	2	2	no	—
77	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
78	#108	—	12	4	no	—
79	#107	—	2	4	no	—
80	#106	—	2	4	no	—
81	C₄⋊C₈	—	6	4	no	—
82	C₄×C₈	—	3	4	no	—
83	C₄×C₈	—	6	2	no	—
84	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
86	C₄×C₈	#67	1	2	yes	—
87	C₂³×C₃	#124	1	1	yes	characteristic
88	C₃×Q₈	—	4	8	no	—
89	C₂×C₃×C₄	—	2	4	no	—
90	C₂×C₃×C₄	—	4	4	no	—
91	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
93	C₂×C₃×C₄	#121	1	2	yes	—
94	C₂×C₃×C₄	#127	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
96	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
97	C₃×C₈	#103	1	4	yes	—
98	C₃×C₈	#103	1	4	yes	—
99	C₃⋊C₈	—	4	4	no	—
100	C₃⋊C₈	—	8	2	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂²×C₄	#237	1	2	yes	—
103	C₂²×C₄	C₂×#67	1	1	yes	characteristic
104	M₁₆	—	12	4	no	—
105	C₂×C₈	—	6	4	no	—
106	C₂×C₈	—	12	4	no	—
107	C₂×C₈	—	12	2	no	—
108	C₂×C₈	#215	1	4	yes	—
109	C₂×C₈	#216	1	4	yes	—
110	C₂×C₈	#217	1	2	yes	—
111	C₂×C₈	#217	1	2	yes	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	4	4	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₄²	—	4	2	no	—
117	C₄²	D₃×D₄	1	2	yes	—
118	C₄²	C₂²×Dic₃	1	1	yes	characteristic
119	C₄²	C₂×#67	1	1	yes	characteristic
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	#390	1	1	yes	characteristic
122	C₂²×C₃	#487	1	2	yes	—
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	—	8	4	no	—
125	C₃×C₄	C₂×#103	1	2	yes	—
126	C₃×C₄	#444	1	4	yes	—
127	C₃×C₄	#492	1	1	yes	characteristic
128	C₃×C₄	#494	1	1	yes	characteristic
129	C₂³	#936	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	D₄×Dic₃	1	2	yes	—
134	C₂×C₄	#938	1	2	yes	—
135	C₂×C₄	#940	1	1	yes	characteristic
136	C₂×C₄	C₂×#215	1	2	yes	—
137	C₂×C₄	C₂×#216	1	2	yes	—
138	C₂×C₄	C₂×#217	1	1	yes	characteristic
139	C₈	—	12	4	no	—
140	C₈	—	24	2	no	—
141	C₈	#835	1	4	yes	—
142	C₈	#835	1	4	yes	—
143	C₂×C₃	—	2	2	no	—
144	C₂×C₃	#1956	1	1	yes	characteristic
145	C₂×C₃	#1975	1	2	yes	—
146	C₂²	—	2	2	no	—
147	C₂²	#5079	1	2	yes	—
148	C₂²	#5221	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	8	4	no	—
151	C₄	#5085	1	1	yes	characteristic
152	C₄	#5092	1	1	yes	characteristic
153	C₄	#5197	1	4	yes	—
154	C₄	C₂×#835	1	2	yes	—
155	C₃	#9861	1	1	yes	characteristic, complemented, Sylow
156	C₂	—	2	2	no	—
157	C₂	#71785	1	1	yes	characteristic
158	C₂	#72085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#3160

C₂

yes

maximal, characteristic

#71826

C₂

yes

maximal

#69792

C₂

yes

maximal, characteristic

C₂×#4455

C₂

yes

maximal, characteristic

#9861

—

maximal, Sylow

C₃×#533

C₂²

yes

—

C₃×#500

C₄

yes

—

C₃×#499

—

C₃×#498

C₄

yes

—

C₃×#427

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#819

—

#4924

C₂²

yes

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

#4535

—

#4455

—

#3160

D₃

yes

—

—

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₃×#125

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×#208

—

C₂²×C₃⋊C₈

—

C₃×#107

—

C₃×#106

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

#819

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

D₄

yes

—

#533

C₂×D₃

yes

—

#500

Dic₃

yes

—

#499

—

#498

Dic₃

yes

—

C₂×#108

—

#427

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

Q₈⋊C₈

—

#322

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₂×C₃×C₈

Q₁₆

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

C₂×Dic₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×D₃

yes

—

—

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

characteristic

C₄×C₈

C₂×Dic₃

yes

characteristic

C₄×C₈

#67

yes

—

C₂³×C₃

#124

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄×D₄

yes

—

C₂×C₃×C₄

#121

yes

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×SD₁₆

yes

—

C₂×C₃×C₄

C₂×Q₁₆

yes

—

C₃×C₈

#103

yes

—

C₃×C₈

#103

yes

—

C₃⋊C₈

—

100

C₃⋊C₈

—

101

C₂×Q₈

—

102

C₂²×C₄

#237

yes

—

103

C₂²×C₄

C₂×#67

yes

characteristic

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#215

yes

—

109

C₂×C₈

#216

yes

—

110

C₂×C₈

#217

yes

—

111

C₂×C₈

#217

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

D₃×D₄

yes

—

118

C₄²

C₂²×Dic₃

yes

characteristic

119

C₄²

C₂×#67

yes

characteristic

120

C₂²×C₃

—

121

C₂²×C₃

#390

yes

characteristic

122

C₂²×C₃

#487

yes

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

C₂×#103

yes

—

126

C₃×C₄

#444

yes

—

127

C₃×C₄

#492

yes

characteristic

128

C₃×C₄

#494

yes

characteristic

129

C₂³

#936

yes

characteristic

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₄×Dic₃

yes

—

134

C₂×C₄

#938

yes

—

135

C₂×C₄

#940

yes

characteristic

136

C₂×C₄

C₂×#215

yes

—

137

C₂×C₄

C₂×#216

yes

—

138

C₂×C₄

C₂×#217

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

#835

yes

—

142

C₈

#835

yes

—

143

C₂×C₃

—

144

C₂×C₃

#1956

yes

characteristic

145

C₂×C₃

#1975

yes

—

146

C₂²

—

147

C₂²

#5079

yes

—

148

C₂²

#5221

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

#5085

yes

characteristic

152

C₄

#5092

yes

characteristic

153

C₄

#5197

yes

—

154

C₄

C₂×#835

yes

—

155

C₃

#9861

yes

characteristic, complemented, Sylow

156

C₂

—

157

C₂

#71785

yes

characteristic

158

C₂

#72085

yes

—

	subgroup	G/H
center	C₂×C₄	#940
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#3160	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#3160	D₃
hypocenter	C₃	#9861
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#940

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
155. C₃	#9861	semidirect

H ≅ G/N

type

155. C₃

#9861

semidirect

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

subgroup	index
C₃×#3160	2
#71826	2
#69792	2
C₂×#4455	2
#9861	3

subgroup

index

Subgroups — #10695156

Conjugacy classes of subgroups

Complements

Maximal subgroups