Group #10698310 — subgroups

Subgroups — #10698310

Order 768.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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	#73310	C₂	1	1	yes	maximal, characteristic
2	C₃×#1994	C₂	1	1	yes	maximal, characteristic
3	#71838	C₂	1	2	yes	maximal
4	#67713	C₂	1	2	yes	maximal
5	#67611	C₂	1	1	yes	maximal, characteristic
6	#10369	—	3	1	no	maximal, Sylow
7	C₃×#533	C₂²	1	1	yes	characteristic
8	C₃×#423	C₂²	1	1	yes	characteristic
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	#4936	C₂²	1	2	yes	—
11	#4933	—	2	4	no	—
12	#4928	—	2	2	no	—
13	#4925	C₂²	1	2	yes	—
14	C₃×#339	—	2	2	no	—
15	#4538	—	2	4	no	—
16	#2864	—	3	1	no	—
17	#1994	D₃	1	1	yes	characteristic
18	#1518	—	3	2	no	—
19	#1286	—	3	2	no	—
20	#1276	—	3	1	no	—
21	C₂×C₃×M₁₆	—	2	1	no	—
22	C₃×C₄⋊Q₈	Q₈	1	2	yes	—
23	C₃×#125	—	2	1	no	—
24	C₃×C₂²⋊Q₈	—	2	2	no	—
25	C₃×C₄×Q₈	—	2	2	no	—
26	C₃×#117	—	2	1	no	—
27	C₂×C₃×C₄²	—	2	2	no	—
28	C₂×C₃×C₄²	D₄	1	2	yes	—
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×Dic₃⋊C₄	—	2	4	no	—
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃⋊#95	—	2	4	no	—
34	C₃⋊#95	—	2	2	no	—
35	C₃⋊#95	D₄	1	4	yes	—
36	#830	—	2	2	no	—
37	C₄⋊(C₃⋊C₈)	—	2	2	no	—
38	C₄×C₃⋊C₈	—	2	2	no	—
39	#533	C₂×D₃	1	1	yes	characteristic
40	#431	—	3	2	no	—
41	#423	C₂×D₃	1	1	yes	characteristic
42	#389	—	6	2	no	—
43	#387	—	6	4	no	—
44	#382	—	3	2	no	—
45	C₄³	C₂×D₃	1	1	yes	characteristic
46	#339	—	2	2	no	—
47	#328	—	6	4	no	—
48	C₂×C₃×Q₈	—	2	2	no	—
49	C₂²×C₃×C₄	—	2	2	no	—
50	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
51	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₃	—	2	4	no	—
53	C₃×M₁₆	—	4	2	no	—
54	C₂×C₃×C₈	—	4	1	no	—
55	C₃×C₄⋊C₄	—	2	2	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	4	2	no	—
58	C₃×C₂²⋊C₄	—	4	1	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₃×C₄²	—	2	2	no	—
61	C₃×C₄²	—	4	4	no	—
62	C₃×C₄²	—	4	1	no	—
63	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
64	C₃×C₄²	#41	1	1	yes	characteristic
65	C₃×C₄²	#41	1	2	yes	—
66	C₄⋊Dic₃	—	4	2	no	—
67	C₄⋊Dic₃	—	4	2	no	—
68	Dic₃⋊C₄	—	4	8	no	—
69	C₂×C₃⋊C₈	—	4	2	no	—
70	C₂×M₁₆	—	2	1	no	—
71	C₄⋊Q₈	Dic₆	1	2	yes	—
72	#125	—	2	1	no	—
73	C₂²⋊Q₈	—	2	2	no	—
74	C₄×Q₈	—	2	2	no	—
75	#117	—	2	1	no	—
76	C₂×C₄⋊C₄	—	6	4	no	—
77	C₂×C₄⋊C₄	—	6	2	no	—
78	C₂×C₄²	—	2	2	no	—
79	C₂×C₄²	#67	1	2	yes	—
80	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	6	2	no	—
83	C₂²⋊C₈	—	6	2	no	—
84	C₄×C₈	—	6	2	no	—
85	#95	—	3	4	no	—
86	#95	—	6	4	no	—
87	#95	—	6	2	no	—
88	C₂³×C₃	#120	1	1	yes	characteristic
89	C₃×Q₈	—	8	2	no	—
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	4	2	no	—
92	C₂×C₃×C₄	—	4	2	no	—
93	C₂×C₃×C₄	—	4	2	no	—
94	C₂×C₃×C₄	—	4	2	no	—
95	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
96	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₃×C₄	#124	1	2	yes	—
98	C₂×C₃×C₄	#124	1	1	yes	characteristic
99	C₂×C₃×C₄	#135	1	2	yes	—
100	C₂×C₃×C₄	#137	1	2	yes	—
101	C₂×Dic₃	—	4	8	no	—
102	C₂×Dic₃	—	4	4	no	—
103	C₃×C₈	—	8	1	no	—
104	C₃⋊C₈	—	8	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	6	4	no	—
108	C₂²×C₄	D₃×D₄	1	2	yes	—
109	C₂²×C₄	C₂×#67	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	12	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	12	8	no	—
117	C₄⋊C₄	—	12	2	no	—
118	C₄⋊C₄	—	12	2	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	4	4	no	—
123	C₄²	—	4	1	no	—
124	C₄²	C₂×Dic₆	1	1	yes	characteristic
125	C₄²	#235	1	1	yes	characteristic
126	C₄²	#237	1	2	yes	—
127	C₂²×C₃	—	2	2	no	—
128	C₂²×C₃	#393	1	1	yes	characteristic
129	C₂²×C₃	#452	1	2	yes	—
130	C₃×C₄	—	2	2	no	—
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	4	4	no	—
133	C₃×C₄	—	8	2	no	—
134	C₃×C₄	#479	1	2	yes	—
135	C₃×C₄	#488	1	1	yes	characteristic
136	C₃×C₄	#491	1	1	yes	characteristic
137	Dic₃	—	8	4	no	—
138	C₂³	#937	1	1	yes	characteristic
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	12	8	no	—
146	C₂×C₄	—	12	4	no	—
147	C₂×C₄	#878	1	2	yes	—
148	C₂×C₄	#885	1	2	yes	—
149	C₂×C₄	#924	1	1	yes	characteristic
150	C₂×C₄	#936	1	1	yes	characteristic
151	C₂×C₄	#942	1	2	yes	—
152	C₂×C₄	#950	1	2	yes	—
153	C₈	—	8	1	no	—
154	C₈	—	24	2	no	—
155	C₂×C₃	—	2	2	no	—
156	C₂×C₃	#2026	1	1	yes	characteristic
157	C₂×C₃	#2031	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#4947	1	1	yes	characteristic
160	C₂²	#5047	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	2	2	no	—
163	C₄	—	4	4	no	—
164	C₄	—	8	2	no	—
165	C₄	—	24	4	no	—
166	C₄	#5022	1	2	yes	—
167	C₄	#5056	1	1	yes	characteristic
168	C₄	#5082	1	1	yes	characteristic
169	C₃	#10369	1	1	yes	characteristic, complemented, Sylow
170	C₂	—	2	2	no	—
171	C₂	#69669	1	1	yes	characteristic
172	C₂	#70368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#73310

C₂

yes

maximal, characteristic

C₃×#1994

C₂

yes

maximal, characteristic

#71838

C₂

yes

maximal

#67713

C₂

yes

maximal

#67611

C₂

yes

maximal, characteristic

#10369

—

maximal, Sylow

C₃×#533

C₂²

yes

characteristic

C₃×#423

C₂²

yes

characteristic

C₃×C₄³

C₂²

yes

characteristic

#4936

C₂²

yes

—

#4933

—

#4928

—

#4925

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

C₂×C₃×M₁₆

—

C₃×C₄⋊Q₈

Q₈

yes

—

C₃×#125

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×Dic₃⋊C₄

—

C₃×C₄⋊C₈

—

C₃⋊#95

—

C₃⋊#95

—

C₃⋊#95

D₄

yes

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

#533

C₂×D₃

yes

characteristic

#431

—

#423

C₂×D₃

yes

—

—

—

C₄³

C₂×D₃

yes

characteristic

#339

—

#328

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×Q₈

yes

C₃×C₄²

yes

C₃×C₄²

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₆

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

#95

—

#95

—

C₂³×C₃

#120

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

C₂×C₃×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₃×C₄

#124

yes

—

C₂×C₃×C₄

#124

yes

characteristic

C₂×C₃×C₄

#135

yes

—

100

C₂×C₃×C₄

#137

yes

—

101

C₂×Dic₃

—

102

C₂×Dic₃

—

103

C₃×C₈

—

104

C₃⋊C₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₃×D₄

yes

—

109

C₂²×C₄

C₂×#67

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂×Dic₆

yes

characteristic

125

C₄²

#235

yes

characteristic

126

C₄²

#237

yes

—

127

C₂²×C₃

—

128

C₂²×C₃

#393

yes

characteristic

129

C₂²×C₃

#452

yes

—

130

C₃×C₄

—

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

#479

yes

—

135

C₃×C₄

#488

yes

characteristic

136

C₃×C₄

#491

yes

characteristic

137

Dic₃

—

138

C₂³

#937

yes

characteristic

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

#878

yes

—

148

C₂×C₄

#885

yes

—

149

C₂×C₄

#924

yes

characteristic

150

C₂×C₄

#936

yes

characteristic

151

C₂×C₄

#942

yes

—

152

C₂×C₄

#950

yes

—

153

C₈

—

154

C₈

—

155

C₂×C₃

—

156

C₂×C₃

#2026

yes

characteristic

157

C₂×C₃

#2031

yes

—

158

C₂²

—

159

C₂²

#4947

yes

characteristic

160

C₂²

#5047

yes

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#5022

yes

—

167

C₄

#5056

yes

characteristic

168

C₄

#5082

yes

characteristic

169

C₃

#10369

yes

characteristic, complemented, Sylow

170

C₂

—

171

C₂

#69669

yes

characteristic

172

C₂

#70368

yes

—

	subgroup	G/H
center	C₂²	#4947
commutator, cosocle	C₂×C₃×C₄²	C₂³
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1994	C₂
socle	C₂²×C₃	#393
prime radical	C₂³×C₃	#120
hypercenter	#1994	D₃
hypocenter	C₃	#10369
LCS	G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#4947

commutator, cosocle

C₂×C₃×C₄²

C₂³

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
169. C₃	#10369	semidirect

H ≅ G/N

type

169. C₃

#10369

semidirect

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

subgroup	index
#73310	2
C₃×#1994	2
#71838	2
#67713	2
#67611	2
#10369	3

subgroup

index

Subgroups — #10698310

Conjugacy classes of subgroups

Complements

Maximal subgroups