Group #30748826 — subgroups

Subgroups — #30748826

Order 17152.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22569139	C₂	1	1	yes	maximal, characteristic
2	#22558970	C₂	1	2	yes	maximal
3	#22558846	C₂	1	2	yes	maximal
4	#22558120	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1307	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863523	—	2	2	no	—
8	#16863520	—	2	2	no	—
9	#16863453	C₂²	1	2	yes	—
10	#16863424	C₄	1	2	yes	—
11	#16863354	C₄	1	2	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#16863148	—	2	2	no	—
14	#16863146	—	2	2	no	—
15	#16863137	—	2	4	no	—
16	#432×C₆₇	C₂²	1	2	yes	—
17	#423×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	#14524043	Q₈	1	2	yes	—
21	#14524035	D₄	1	2	yes	—
22	#14524022	—	2	2	no	—
23	C₄×Dic₁₃₄	—	2	2	no	—
24	#14524002	C₂×C₄	1	2	yes	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	Dic₆₇⋊C₈	—	2	4	no	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
28	#14523954	—	2	2	no	—
29	#14523953	—	2	2	no	—
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
32	C₈×Dic₆₇	—	2	2	no	—
33	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄⋊C₈×C₆₇	—	2	2	no	—
37	C₂²⋊C₈×C₆₇	—	2	2	no	—
38	#97×C₆₇	—	2	2	no	—
39	C₂×Dic₁₃₄	—	2	2	no	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	4	1	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊Dic₆₇	—	2	2	no	—
47	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
48	Dic₆₇⋊C₄	—	2	4	no	—
49	C₂²⋊Dic₆₇	—	2	2	no	—
50	C₄×Dic₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	#41	1	2	yes	—
53	M₁₆×C₆₇	—	4	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	C₂×D₄	1	2	yes	—
57	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₆₇	#117	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	1	no	—
61	C₂×C₄×C₆₇	—	2	1	no	—
62	C₂×C₄×C₆₇	—	4	2	no	—
63	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#121	1	2	yes	—
66	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
67	C₂×Dic₆₇	—	2	4	no	—
68	Dic₁₃₄	—	4	4	no	—
69	C₈×C₆₇	—	4	2	no	—
70	C₈×C₆₇	—	8	2	no	—
71	C₆₇⋊C₈	—	2	1	no	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	—	8	2	no	—
74	C₆₇⋊C₈	#101	1	2	yes	—
75	C₂²×C₆₇	—	2	1	no	—
76	C₂²×C₆₇	—	2	2	no	—
77	C₂²×C₆₇	#380	1	1	yes	characteristic
78	C₄×C₆₇	—	2	1	no	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
82	C₄×C₆₇	#413	1	1	yes	characteristic
83	Dic₆₇	—	4	4	no	—
84	#8631	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	—	4	1	no	—
86	C₂×C₆₇	#1801	1	1	yes	characteristic
87	C₂×C₆₇	#1814	1	2	yes	—
88	#2960	—	67	1	no	—
89	#1568	—	67	2	no	—
90	#1537	—	67	2	no	—
91	#1307	—	67	1	no	—
92	#1307	D₆₇	1	1	yes	characteristic
93	C₆₇	#8631	1	1	yes	characteristic, complemented, Sylow
94	#533	—	67	1	no	—
95	#446	—	134	2	no	—
96	C₈×Q₈	—	134	2	no	—
97	#432	—	67	2	no	—
98	#432	C₂×D₆₇	1	2	yes	—
99	#424	—	67	2	no	—
100	#423	C₂×D₆₇	1	1	yes	characteristic
101	#405	—	67	2	no	—
102	C₂×C₄×C₈	—	67	1	no	—
103	#335	—	134	2	no	—
104	#334	—	134	2	no	—
105	#330	—	134	4	no	—
106	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
107	C₂²×C₈	—	67	1	no	—
108	C₂²×C₈	—	67	1	no	—
109	C₄⋊Q₈	—	67	2	no	—
110	#125	—	67	2	no	—
111	C₂²⋊Q₈	—	134	2	no	—
112	C₄×Q₈	—	134	2	no	—
113	#117	—	67	2	no	—
114	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	134	4	no	—
118	C₄⋊C₈	—	134	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	134	2	no	—
121	#97	—	2	2	no	—
122	#97	—	134	2	no	—
123	#97	—	134	2	no	—
124	C₄×C₈	—	67	2	no	—
125	C₄×C₈	—	67	2	no	—
126	C₄×C₈	—	134	2	no	—
127	C₂×Q₈	—	134	2	no	—
128	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	#11761628	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	268	2	no	—
134	C₂×C₈	—	268	1	no	—
135	C₂×C₈	—	67	1	no	—
136	C₂×C₈	—	67	1	no	—
137	C₂×C₈	—	67	2	no	—
138	C₂×C₈	—	134	2	no	—
139	C₄⋊C₄	—	67	4	no	—
140	C₄⋊C₄	—	134	4	no	—
141	C₄⋊C₄	—	134	2	no	—
142	C₂²⋊C₄	—	134	2	no	—
143	C₄²	—	134	2	no	—
144	C₄²	D₄×D₆₇	1	2	yes	—
145	C₄²	D₄×D₆₇	1	1	yes	characteristic
146	C₄²	Q₈×D₆₇	1	1	yes	characteristic
147	C₂³	#14524000	1	1	yes	characteristic
148	Q₈	—	268	4	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	134	4	no	—
153	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
155	C₂×C₄	#14524016	1	2	yes	—
156	C₂×C₄	#14524021	1	2	yes	—
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	536	2	no	—
160	C₈	—	67	2	no	—
161	C₈	—	134	1	no	—
162	C₈	—	134	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#16863255	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	268	4	no	—
170	C₄	#101×D₆₇	1	1	yes	characteristic
171	C₄	#16863377	1	1	yes	characteristic
172	C₂	—	4	1	no	—
173	C₂	#22560204	1	1	yes	characteristic
174	C₂	#22560262	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569139

C₂

yes

maximal, characteristic

#22558970

C₂

yes

maximal

#22558846

C₂

yes

maximal

#22558120

C₂

yes

maximal, characteristic

C₆₇×#1307

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

—

C₂²

yes

—

#16863424

C₄

yes

—

#16863354

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

#432×C₆₇

C₂²

yes

—

#423×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

#14524043

Q₈

yes

—

#14524035

D₄

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

Dic₆₇⋊C₈

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523953

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₄⋊C₄

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#101

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#380

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#8631

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1801

yes

characteristic

C₂×C₆₇

#1814

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8631

yes

characteristic, complemented, Sylow

#533

—

#446

—

134

—

C₈×Q₈

—

134

—

#432

—

#432

C₂×D₆₇

yes

—

#424

—

100

#423

C₂×D₆₇

yes

characteristic

101

#405

—

102

C₂×C₄×C₈

—

103

#335

—

134

—

104

#334

—

134

—

105

#330

—

134

—

106

C₂×M₁₆

C₄×D₆₇

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₄⋊Q₈

—

110

#125

—

111

C₂²⋊Q₈

—

134

—

112

C₄×Q₈

—

134

—

113

#117

—

114

C₂×C₄²

C₂²×D₆₇

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

134

—

118

C₄⋊C₈

—

134

—

119

C₂²⋊C₈

—

120

C₂²⋊C₈

—

134

—

121

#97

—

122

#97

—

134

—

123

#97

—

134

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

134

—

127

C₂×Q₈

—

134

—

128

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

129

C₂²×C₄

#11761628

yes

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

268

—

134

C₂×C₈

—

268

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

134

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

134

—

141

C₄⋊C₄

—

134

—

142

C₂²⋊C₄

—

134

—

143

C₄²

—

134

—

144

C₄²

D₄×D₆₇

yes

—

145

C₄²

D₄×D₆₇

yes

characteristic

146

C₄²

Q₈×D₆₇

yes

characteristic

147

C₂³

#14524000

yes

characteristic

148

Q₈

—

268

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

134

—

153

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

154

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

155

C₂×C₄

#14524016

yes

—

156

C₂×C₄

#14524021

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

536

—

160

C₈

—

161

C₈

—

134

—

162

C₈

—

134

—

163

C₂²

—

164

C₂²

—

165

C₂²

#16863255

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

268

—

170

C₄

#101×D₆₇

yes

characteristic

171

C₄

#16863377

yes

characteristic

172

C₂

—

173

C₂

#22560204

yes

characteristic

174

C₂

#22560262

yes

—

	subgroup	G/H
center	C₂²	#16863255
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1307	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1307	D₆₇
hypocenter	C₆₇	#8631
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#16863255

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1307

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
93. C₆₇	#8631	semidirect

H ≅ G/N

type

93. C₆₇

#8631

semidirect

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

subgroup	index
#22569139	2
#22558970	2
#22558846	2
#22558120	2
C₆₇×#1307	2
#8631	67

subgroup

index

C₆₇×#1307

Subgroups — #30748826

Conjugacy classes of subgroups

Complements

Maximal subgroups