Group #30748825 — subgroups

Subgroups — #30748825

Order 17152.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22569144	C₂	1	1	yes	maximal, characteristic
2	#22558970	C₂	1	2	yes	maximal
3	#22558847	C₂	1	1	yes	maximal, characteristic
4	#22558119	C₂	1	2	yes	maximal
5	C₆₇×#1537	C₂	1	1	yes	maximal, characteristic
6	#16863527	—	2	2	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863453	C₂²	1	2	yes	—
9	#16863423	C₄	1	2	yes	—
10	#16863420	C₂²	1	1	yes	characteristic
11	#16863352	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	#16863138	—	2	2	no	—
16	#533×C₆₇	C₂²	1	1	yes	characteristic
17	#432×C₆₇	C₂²	1	2	yes	—
18	#330×C₆₇	—	2	2	no	—
19	C₂×#11761612	C₂×C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523954	—	2	2	no	—
27	#14523954	—	2	2	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	—
32	#125×C₆₇	D₄	1	2	yes	—
33	C₂²⋊Q₈×C₆₇	—	2	2	no	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	#117×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	—	2	2	no	—
38	C₂²⋊C₈×C₆₇	—	2	2	no	—
39	#97×C₆₇	—	2	2	no	—
40	#11761612	—	4	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	4	1	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	#41	1	2	yes	—
51	C₂×C₈×C₆₇	—	4	2	no	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
55	C₂²⋊C₄×C₆₇	—	2	2	no	—
56	C₄²×C₆₇	—	2	2	no	—
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₆₇	#117	1	1	yes	characteristic
61	Q₈×C₆₇	—	4	4	no	—
62	C₂×C₄×C₆₇	—	2	1	no	—
63	C₂×C₄×C₆₇	—	2	1	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#121	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₆₇	—	8	2	no	—
71	C₆₇⋊C₈	—	2	1	no	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₆₇⋊C₈	—	4	2	no	—
74	C₆₇⋊C₈	—	8	2	no	—
75	C₆₇⋊C₈	#101	1	2	yes	—
76	C₂²×C₆₇	—	2	1	no	—
77	C₂²×C₆₇	—	2	2	no	—
78	C₂²×C₆₇	#380	1	1	yes	characteristic
79	C₄×C₆₇	—	2	1	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	—	4	4	no	—
83	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
84	C₄×C₆₇	#413	1	1	yes	characteristic
85	#8631	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	4	1	no	—
87	C₂×C₆₇	#1801	1	1	yes	characteristic
88	C₂×C₆₇	#1814	1	2	yes	—
89	#2960	—	67	1	no	—
90	#1568	—	67	2	no	—
91	#1537	—	67	1	no	—
92	#1537	D₆₇	1	1	yes	characteristic
93	#1307	—	67	2	no	—
94	C₆₇	#8631	1	1	yes	characteristic, complemented, Sylow
95	#533	C₂×D₆₇	1	1	yes	characteristic
96	#446	—	134	2	no	—
97	C₈×Q₈	—	134	2	no	—
98	#432	—	67	2	no	—
99	#432	C₂×D₆₇	1	2	yes	—
100	#424	—	67	2	no	—
101	#423	—	67	1	no	—
102	#405	—	67	2	no	—
103	C₂×C₄×C₈	—	67	1	no	—
104	#335	—	134	2	no	—
105	#334	—	134	2	no	—
106	#330	—	2	2	no	—
107	#330	—	134	2	no	—
108	C₂×M₁₆	—	67	2	no	—
109	C₂²×C₈	—	67	1	no	—
110	C₂²×C₈	—	67	1	no	—
111	C₄⋊Q₈	Dic₁₃₄	1	2	yes	—
112	#125	C₂²⋊D₆₇	1	2	yes	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	C₄×D₆₇	1	2	yes	—
116	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	134	4	no	—
119	C₄⋊C₈	—	134	2	no	—
120	C₄⋊C₈	—	134	2	no	—
121	C₂²⋊C₈	—	2	2	no	—
122	C₂²⋊C₈	—	134	2	no	—
123	#97	—	2	2	no	—
124	#97	—	134	2	no	—
125	#97	—	134	2	no	—
126	C₄×C₈	—	67	2	no	—
127	C₄×C₈	—	67	2	no	—
128	C₄×C₈	—	134	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
131	C₂²×C₄	#11761627	1	2	yes	—
132	M₁₆	—	268	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	268	2	no	—
135	C₂×C₈	—	268	1	no	—
136	C₂×C₈	—	67	1	no	—
137	C₂×C₈	—	67	1	no	—
138	C₂×C₈	—	67	2	no	—
139	C₂×C₈	—	134	2	no	—
140	C₂×C₈	—	134	2	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	Dic₆₇⋊C₄	1	4	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	D₄×D₆₇	1	2	yes	—
147	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
148	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
149	C₂³	#14523999	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
157	C₂×C₄	#14524015	1	2	yes	—
158	C₂×C₄	#14524020	1	2	yes	—
159	C₈	—	8	2	no	—
160	C₈	—	268	2	no	—
161	C₈	—	536	2	no	—
162	C₈	—	67	2	no	—
163	C₈	—	134	1	no	—
164	C₈	—	134	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#16863254	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	4	no	—
172	C₄	#101×D₆₇	1	1	yes	characteristic
173	C₄	#16863379	1	1	yes	characteristic
174	C₂	—	4	1	no	—
175	C₂	#22560203	1	1	yes	characteristic
176	C₂	#22560261	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22569144

C₂

yes

maximal, characteristic

#22558970

C₂

yes

maximal

#22558847

C₂

yes

maximal, characteristic

#22558119

C₂

yes

maximal

C₆₇×#1537

C₂

yes

maximal, characteristic

#16863527

—

Q₈×C₆₇⋊C₈

—

#16863453

C₂²

yes

—

#16863423

C₄

yes

—

#16863420

C₂²

yes

characteristic

#16863352

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

—

—

#533×C₆₇

C₂²

yes

characteristic

#432×C₆₇

C₂²

yes

—

#330×C₆₇

—

C₂×#11761612

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄⋊Q₈×C₆₇

Q₈

yes

—

#125×C₆₇

D₄

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

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

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₄⋊C₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

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

—

C₈×C₆₇

—

C₆₇⋊C₈

—

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

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1801

yes

characteristic

C₂×C₆₇

#1814

yes

—

—

—

—

D₆₇

yes

characteristic

#1307

—

C₆₇

#8631

yes

characteristic, complemented, Sylow

#533

C₂×D₆₇

yes

characteristic

#446

—

134

—

C₈×Q₈

—

134

—

#432

—

#432

C₂×D₆₇

yes

—

100

#424

—

101

#423

—

102

#405

—

103

C₂×C₄×C₈

—

104

#335

—

134

—

105

#334

—

134

—

106

#330

—

107

#330

—

134

—

108

C₂×M₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₄⋊Q₈

Dic₁₃₄

yes

—

112

#125

C₂²⋊D₆₇

yes

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

C₄×D₆₇

yes

—

116

C₂×C₄²

C₂²×D₆₇

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

134

—

119

C₄⋊C₈

—

134

—

120

C₄⋊C₈

—

134

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

134

—

123

#97

—

124

#97

—

134

—

125

#97

—

134

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

134

—

129

C₂×Q₈

—

130

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

131

C₂²×C₄

#11761627

yes

—

132

M₁₆

—

268

—

133

C₂×C₈

—

134

C₂×C₈

—

268

—

135

C₂×C₈

—

268

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

134

—

140

C₂×C₈

—

134

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

Dic₆₇⋊C₄

yes

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

D₄×D₆₇

yes

—

147

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

148

C₄²

C₂×Dic₁₃₄

yes

characteristic

149

C₂³

#14523999

yes

characteristic

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊C₄×D₆₇

yes

characteristic

156

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

157

C₂×C₄

#14524015

yes

—

158

C₂×C₄

#14524020

yes

—

159

C₈

—

160

C₈

—

268

—

161

C₈

—

536

—

162

C₈

—

163

C₈

—

134

—

164

C₈

—

134

—

165

C₂²

—

166

C₂²

—

167

C₂²

#16863254

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#101×D₆₇

yes

characteristic

173

C₄

#16863379

yes

characteristic

174

C₂

—

175

C₂

#22560203

yes

characteristic

176

C₂

#22560261

yes

—

	subgroup	G/H
center	C₂²	#16863254
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1537	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1537	D₆₇
hypocenter	C₆₇	#8631
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

subgroup

G/H

center

C₂²

#16863254

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1537

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1537

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

H ≅ G/N

type

94. C₆₇

#8631

semidirect

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

subgroup	index
#22569144	2
#22558970	2
#22558847	2
#22558119	2
C₆₇×#1537	2
#8631	67

subgroup

index

C₆₇×#1537

Subgroups — #30748825

Conjugacy classes of subgroups

Complements

Maximal subgroups