Group #30758945 — subgroups

Subgroups — #30758945

Order 17152.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22568596	C₂	1	1	yes	maximal, characteristic
2	#22561186	C₂	1	1	yes	maximal, characteristic
3	#22558774	C₂	1	2	yes	maximal
4	#22558068	C₂	1	2	yes	maximal
5	C₆₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863420	C₂²	1	1	yes	characteristic
8	#16863294	—	2	2	no	—
9	#16863280	—	2	4	no	—
10	#16863262	C₂²	1	2	yes	—
11	#16863156	—	2	2	no	—
12	#16863134	—	2	4	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761612	—	2	1	no	—
16	#14524043	Q₈	1	2	yes	—
17	#14524035	—	2	1	no	—
18	#14524022	—	2	2	no	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	#14524002	—	2	1	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×Dic₆₇⋊C₄	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₆₇⋊#95	—	2	4	no	—
25	C₆₇⋊#95	—	2	2	no	—
26	C₆₇⋊#95	D₄	1	4	yes	—
27	C₂×C₄²×C₆₇	—	2	2	no	—
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₇	—	2	2	no	—
31	C₂²⋊C₈×C₆₇	—	2	2	no	—
32	C₄×C₈×C₆₇	—	2	2	no	—
33	C₂×Dic₁₃₄	—	2	2	no	—
34	C₂²×Dic₆₇	—	2	4	no	—
35	#11761612	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	1	no	—
37	C₄⋊Dic₆₇	—	2	2	no	—
38	C₄⋊Dic₆₇	—	2	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	Dic₆₇⋊C₄	—	4	8	no	—
42	Dic₆₇⋊C₄	—	4	2	no	—
43	C₂²⋊Dic₆₇	—	4	1	no	—
44	C₄×Dic₆₇	—	4	1	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	—	4	2	no	—
49	C₄²×C₆₇	—	2	2	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	—	4	4	no	—
52	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₆₇	#41	1	1	yes	characteristic
54	C₄²×C₆₇	#41	1	2	yes	—
55	C₂³×C₆₇	#120	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	—	4	2	no	—
60	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#124	1	2	yes	—
63	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#135	1	2	yes	—
65	C₂×C₄×C₆₇	#137	1	2	yes	—
66	C₂×Dic₆₇	—	4	8	no	—
67	C₂×Dic₆₇	—	4	2	no	—
68	C₂×Dic₆₇	—	4	4	no	—
69	Dic₁₃₄	—	8	2	no	—
70	C₈×C₆₇	—	8	2	no	—
71	C₆₇⋊C₈	—	8	1	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	#393	1	1	yes	characteristic
74	C₂²×C₆₇	#452	1	2	yes	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	—	4	4	no	—
78	C₄×C₆₇	#479	1	2	yes	—
79	C₄×C₆₇	#488	1	1	yes	characteristic
80	C₄×C₆₇	#491	1	1	yes	characteristic
81	Dic₆₇	—	8	4	no	—
82	Dic₆₇	—	8	2	no	—
83	#10367	—	67	1	no	maximal, Sylow
84	C₂×C₆₇	—	2	2	no	—
85	C₂×C₆₇	#2026	1	1	yes	characteristic
86	C₂×C₆₇	#2031	1	2	yes	—
87	#2864	—	67	1	no	—
88	#1994	—	67	1	no	—
89	#1518	—	67	2	no	—
90	#1286	—	67	2	no	—
91	#1276	D₆₇	1	1	yes	characteristic
92	C₆₇	#10367	1	1	yes	characteristic, complemented, Sylow
93	#533	—	67	1	no	—
94	#431	C₂×D₆₇	1	2	yes	—
95	#423	—	67	1	no	—
96	#389	—	134	2	no	—
97	#387	—	134	4	no	—
98	#382	—	67	2	no	—
99	C₄³	C₂×D₆₇	1	1	yes	characteristic
100	#339	—	134	2	no	—
101	#328	—	134	4	no	—
102	C₂×M₁₆	—	134	1	no	—
103	C₄⋊Q₈	—	67	2	no	—
104	#125	—	134	1	no	—
105	C₂²⋊Q₈	—	134	2	no	—
106	C₄×Q₈	—	134	2	no	—
107	#117	—	134	1	no	—
108	C₂×C₄⋊C₄	—	134	4	no	—
109	C₂×C₄⋊C₄	—	134	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₂₆₈	1	2	yes	—
112	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	134	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	67	4	no	—
118	#95	—	134	4	no	—
119	#95	—	134	2	no	—
120	C₂×Q₈	—	134	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	134	4	no	—
123	C₂²×C₄	D₄×D₆₇	1	2	yes	—
124	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
125	M₁₆	—	268	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	268	1	no	—
128	C₄⋊C₄	—	268	8	no	—
129	C₄⋊C₄	—	268	2	no	—
130	C₄⋊C₄	—	268	2	no	—
131	C₄⋊C₄	—	268	2	no	—
132	C₄⋊C₄	—	134	2	no	—
133	C₄⋊C₄	—	134	2	no	—
134	C₂²⋊C₄	—	268	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	268	1	no	—
139	C₄²	Q₈×D₆₇	1	1	yes	characteristic
140	C₄²	#11761627	1	1	yes	characteristic
141	C₄²	#11761628	1	2	yes	—
142	C₂³	#14524013	1	1	yes	characteristic
143	Q₈	—	536	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	268	8	no	—
149	C₂×C₄	—	268	4	no	—
150	C₂×C₄	—	268	2	no	—
151	C₂×C₄	#14524021	1	2	yes	—
152	C₂×C₄	#14524023	1	1	yes	characteristic
153	C₂×C₄	#14524029	1	2	yes	—
154	C₂×C₄	#14524031	1	1	yes	characteristic
155	C₂×C₄	#14524065	1	2	yes	—
156	C₂×C₄	#14524076	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	536	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#16863309	1	1	yes	characteristic
161	C₂²	#16863560	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	536	4	no	—
166	C₄	—	536	2	no	—
167	C₄	#16863704	1	2	yes	—
168	C₄	#16863751	1	1	yes	characteristic
169	C₄	#16863765	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#22561361	1	1	yes	characteristic
172	C₂	#22561382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22568596

C₂

yes

maximal, characteristic

#22561186

C₂

yes

maximal, characteristic

#22558774

C₂

yes

maximal

#22558068

C₂

yes

maximal

C₆₇×#1276

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

characteristic

#16863420

C₂²

yes

—

—

C₂²

yes

—

#16863156

—

#16863134

—

#431×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

#14524043

Q₈

yes

—

#14524035

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

—

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

C₄⋊(C₆₇⋊C₈)

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

D₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

C₄²×C₆₇

yes

C₄²×C₆₇

yes

—

C₂³×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#393

yes

characteristic

C₂²×C₆₇

#452

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#479

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

#10367

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2026

yes

characteristic

C₂×C₆₇

#2031

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#10367

yes

characteristic, complemented, Sylow

#533

—

#431

C₂×D₆₇

yes

—

#423

—

#389

—

134

—

#387

—

134

—

#382

—

C₄³

C₂×D₆₇

yes

characteristic

100

#339

—

134

—

101

#328

—

134

—

102

C₂×M₁₆

—

134

—

103

C₄⋊Q₈

—

104

#125

—

134

—

105

C₂²⋊Q₈

—

134

—

106

C₄×Q₈

—

134

—

107

#117

—

134

—

108

C₂×C₄⋊C₄

—

134

—

109

C₂×C₄⋊C₄

—

134

—

110

C₂×C₄²

—

111

C₂×C₄²

D₂₆₈

yes

—

112

C₂×C₄²

C₂²×D₆₇

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

134

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

134

—

119

#95

—

134

—

120

C₂×Q₈

—

134

—

121

C₂²×C₄

—

122

C₂²×C₄

—

134

—

123

C₂²×C₄

D₄×D₆₇

yes

—

124

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

125

M₁₆

—

268

—

126

C₂×C₈

—

127

C₂×C₈

—

268

—

128

C₄⋊C₄

—

268

—

129

C₄⋊C₄

—

268

—

130

C₄⋊C₄

—

268

—

131

C₄⋊C₄

—

268

—

132

C₄⋊C₄

—

134

—

133

C₄⋊C₄

—

134

—

134

C₂²⋊C₄

—

268

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

268

—

139

C₄²

Q₈×D₆₇

yes

characteristic

140

C₄²

#11761627

yes

characteristic

141

C₄²

#11761628

yes

—

142

C₂³

#14524013

yes

characteristic

143

Q₈

—

536

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

268

—

149

C₂×C₄

—

268

—

150

C₂×C₄

—

268

—

151

C₂×C₄

#14524021

yes

—

152

C₂×C₄

#14524023

yes

characteristic

153

C₂×C₄

#14524029

yes

—

154

C₂×C₄

#14524031

yes

characteristic

155

C₂×C₄

#14524065

yes

—

156

C₂×C₄

#14524076

yes

—

157

C₈

—

158

C₈

—

536

—

159

C₂²

—

160

C₂²

#16863309

yes

characteristic

161

C₂²

#16863560

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

536

—

166

C₄

—

536

—

167

C₄

#16863704

yes

—

168

C₄

#16863751

yes

characteristic

169

C₄

#16863765

yes

characteristic

170

C₂

—

171

C₂

#22561361

yes

characteristic

172

C₂

#22561382

yes

—

	subgroup	G/H
center	C₂²	#16863309
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1276	C₂
socle	C₂²×C₆₇	#393
prime radical	C₂³×C₆₇	#120
hypercenter	#1276	D₆₇
hypocenter	C₆₇	#10367
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#16863309

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1276

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
92. C₆₇	#10367	semidirect

H ≅ G/N

type

92. C₆₇

#10367

semidirect

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

subgroup	index
#22568596	2
#22561186	2
#22558774	2
#22558068	2
C₆₇×#1276	2
#10367	67

subgroup

index

C₆₇×#1276

Subgroups — #30758945

Conjugacy classes of subgroups

Complements

Maximal subgroups