Group #10694674 — subgroups

Subgroups — #10694674

Order 768.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₃×#498	C₂	1	1	yes	maximal, characteristic
2	#71775	C₂	1	2	yes	maximal
3	#71774	C₂	1	2	yes	maximal
4	#69791	C₂	1	1	yes	maximal, characteristic
5	#69726	C₂	1	1	yes	maximal, characteristic
6	#9776	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#498	—	2	2	no	—
9	C₃×#498	C₄	1	4	yes	—
10	C₂×C₃×#106	—	2	2	no	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#4925	C₂²	1	2	yes	—
13	#4924	C₂²	1	1	yes	characteristic
14	#4924	C₂²	1	1	yes	characteristic
15	#4552	—	2	2	no	—
16	#4547	—	2	4	no	—
17	#4546	—	2	4	no	—
18	#4454	—	2	2	no	—
19	C₂×#498	D₃	1	1	yes	characteristic
20	#1837	—	3	1	no	—
21	#1549	—	3	2	no	—
22	#1548	—	3	2	no	—
23	#1452	—	3	1	no	—
24	C₂²×C₃×Q₈	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₃×C₄⋊Q₈	—	2	2	no	—
27	C₃×C₄⋊Q₈	—	2	2	no	—
28	C₃×C₄⋊Q₈	—	2	2	no	—
29	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂×#208	—	2	1	no	—
34	C₂×#208	—	2	1	no	—
35	C₃×#106	—	2	8	no	—
36	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₃×C₄×C₈	D₄	1	2	yes	—
38	#830	—	2	2	no	—
39	C₄⋊(C₃⋊C₈)	—	2	2	no	—
40	C₄⋊(C₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
42	C₄×C₃⋊C₈	—	2	2	no	—
43	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
44	#498	—	2	2	no	—
45	#498	Dic₃	1	4	yes	—
46	#431	—	3	2	no	—
47	C₂×#106	—	2	2	no	—
48	#423	—	3	1	no	—
49	#423	—	3	1	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	#341	—	6	2	no	—
52	#333	—	6	4	no	—
53	#332	—	6	4	no	—
54	#322	—	6	2	no	—
55	C₂×C₃×Q₈	—	2	4	no	—
56	C₂×C₃×Q₈	—	4	2	no	—
57	C₂×C₃×Q₈	—	8	4	no	—
58	C₂²×C₃×C₄	—	4	2	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	#41	1	2	yes	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
64	C₃×C₄⋊C₄	—	2	4	no	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₄⋊C₄	—	4	2	no	—
67	C₃×C₄⋊C₄	—	4	2	no	—
68	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄²	C₂×D₄	1	2	yes	—
70	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
71	#208	—	4	2	no	—
72	#208	—	4	2	no	—
73	C₂×C₃⋊C₈	—	2	4	no	—
74	C₂×C₃⋊C₈	—	4	1	no	—
75	C₂×C₃⋊C₈	—	4	1	no	—
76	C₂²×Q₈	—	2	2	no	—
77	C₂×M₁₆	—	6	1	no	—
78	C₂×M₁₆	—	6	1	no	—
79	C₂²×C₈	#67	1	2	yes	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	C₂×Dic₃	1	4	yes	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
87	#106	—	2	8	no	—
88	C₄⋊C₈	—	3	4	no	—
89	C₄⋊C₈	—	6	2	no	—
90	C₄⋊C₈	—	6	2	no	—
91	C₂²⋊C₈	—	6	2	no	—
92	C₄×C₈	—	6	2	no	—
93	C₄×C₈	C₂×Dic₃	1	2	yes	—
94	C₄×C₈	#67	1	2	yes	—
95	C₂³×C₃	#124	1	1	yes	characteristic
96	C₃×Q₈	—	8	4	no	—
97	C₃×Q₈	—	8	4	no	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	4	4	no	—
100	C₂×C₃×C₄	—	8	4	no	—
101	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
103	C₂×C₃×C₄	#121	1	2	yes	—
104	C₂×C₃×C₄	#127	1	1	yes	characteristic
105	C₂×C₃×C₄	#137	1	2	yes	—
106	C₂×C₃×C₄	#137	1	2	yes	—
107	C₃×C₈	—	2	4	no	—
108	C₃⋊C₈	—	4	4	no	—
109	C₃⋊C₈	—	8	1	no	—
110	C₃⋊C₈	—	8	1	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	8	4	no	—
114	C₂²×C₄	—	4	2	no	—
115	C₂²×C₄	#237	1	2	yes	—
116	C₂²×C₄	C₂×#67	1	1	yes	characteristic
117	M₁₆	—	12	2	no	—
118	M₁₆	—	12	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	6	4	no	—
122	C₂×C₈	—	12	1	no	—
123	C₂×C₈	—	12	1	no	—
124	C₂×C₈	#217	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄²	D₃×D₄	1	2	yes	—
130	C₄²	C₂²×Dic₃	1	1	yes	characteristic
131	C₄²	C₂×#67	1	1	yes	characteristic
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	#390	1	1	yes	characteristic
134	C₂²×C₃	#491	1	2	yes	—
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	8	4	no	—
137	C₃×C₄	#419	1	2	yes	—
138	C₃×C₄	#471	1	4	yes	—
139	C₃×C₄	#497	1	1	yes	characteristic
140	C₃×C₄	#497	1	1	yes	characteristic
141	C₂³	#936	1	1	yes	characteristic
142	Q₈	—	8	4	no	—
143	Q₈	—	8	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	D₄×Dic₃	1	2	yes	—
148	C₂×C₄	#938	1	2	yes	—
149	C₂×C₄	#940	1	1	yes	characteristic
150	C₂×C₄	#942	1	2	yes	—
151	C₂×C₄	#942	1	2	yes	—
152	C₂×C₄	C₂×#217	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	12	4	no	—
155	C₈	—	24	1	no	—
156	C₈	—	24	1	no	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	#1960	1	1	yes	characteristic
159	C₂×C₃	#1977	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#5082	1	2	yes	—
162	C₂²	#5221	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	8	4	no	—
165	C₄	#5086	1	1	yes	characteristic
166	C₄	#5086	1	1	yes	characteristic
167	C₄	#5177	1	4	yes	—
168	C₄	#5225	1	2	yes	—
169	C₃	#9776	1	1	yes	characteristic, complemented, Sylow
170	C₂	—	2	2	no	—
171	C₂	#71786	1	1	yes	characteristic
172	C₂	#72052	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#498

C₂

yes

maximal, characteristic

#71775

C₂

yes

maximal

#71774

C₂

yes

maximal

#69791

C₂

yes

maximal, characteristic

#69726

C₂

yes

maximal, characteristic

#9776

—

maximal, Sylow

C₂×C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#498

—

C₃×#498

C₄

yes

—

C₂×C₃×#106

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4925

C₂²

yes

—

#4924

C₂²

yes

characteristic

#4924

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

—

C₂²×C₃×Q₈

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂³

yes

—

—

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

#498

—

#498

Dic₃

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Dic₃

yes

—

C₄×C₈

#67

yes

—

C₂³×C₃

#124

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

102

C₂×C₃×C₄

C₄×D₄

yes

—

103

C₂×C₃×C₄

#121

yes

—

104

C₂×C₃×C₄

#127

yes

characteristic

105

C₂×C₃×C₄

#137

yes

—

106

C₂×C₃×C₄

#137

yes

—

107

C₃×C₈

—

108

C₃⋊C₈

—

109

C₃⋊C₈

—

110

C₃⋊C₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂²×C₄

—

115

C₂²×C₄

#237

yes

—

116

C₂²×C₄

C₂×#67

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#217

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

D₃×D₄

yes

—

130

C₄²

C₂²×Dic₃

yes

characteristic

131

C₄²

C₂×#67

yes

characteristic

132

C₂²×C₃

—

133

C₂²×C₃

#390

yes

characteristic

134

C₂²×C₃

#491

yes

—

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

#419

yes

—

138

C₃×C₄

#471

yes

—

139

C₃×C₄

#497

yes

characteristic

140

C₃×C₄

#497

yes

characteristic

141

C₂³

#936

yes

characteristic

142

Q₈

—

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

D₄×Dic₃

yes

—

148

C₂×C₄

#938

yes

—

149

C₂×C₄

#940

yes

characteristic

150

C₂×C₄

#942

yes

—

151

C₂×C₄

#942

yes

—

152

C₂×C₄

C₂×#217

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₂×C₃

—

158

C₂×C₃

#1960

yes

characteristic

159

C₂×C₃

#1977

yes

—

160

C₂²

—

161

C₂²

#5082

yes

—

162

C₂²

#5221

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

#5086

yes

characteristic

166

C₄

#5086

yes

characteristic

167

C₄

#5177

yes

—

168

C₄

#5225

yes

—

169

C₃

#9776

yes

characteristic, complemented, Sylow

170

C₂

—

171

C₂

#71786

yes

characteristic

172

C₂

#72052

yes

—

	subgroup	G/H
center	C₂²	#5221
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#498	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#498	D₃
hypocenter	C₃	#9776
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#5221

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#498

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
169. C₃	#9776	semidirect

H ≅ G/N

type

169. C₃

#9776

semidirect

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

subgroup	index
C₂×C₃×#498	2
#71775	2
#71774	2
#69791	2
#69726	2
#9776	3

subgroup

index

C₂×C₃×#498

Subgroups — #10694674

Conjugacy classes of subgroups

Complements

Maximal subgroups