Group #30775481 — subgroups

Subgroups — #30775481

Order 17152.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16863218	C₂	1	2	yes	maximal
2	#22560957	C₂	1	1	yes	maximal, characteristic
3	#22558316	C₂	1	2	yes	maximal
4	#22558291	C₂	1	1	yes	maximal, characteristic
5	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523979	C₂²	1	1	yes	characteristic
8	#16863434	C₂²	1	1	yes	characteristic
9	#16863422	—	2	1	no	—
10	#16863218	—	2	4	no	—
11	#16863156	—	2	1	no	—
12	C₂×#125×C₆₇	—	2	1	no	—
13	#499×C₆₇	—	2	4	no	—
14	C₂×#107×C₆₇	—	2	1	no	—
15	C₂×#106×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#11761612	—	2	1	no	—
18	#14524002	—	2	1	no	—
19	#14523983	—	2	2	no	—
20	#14523983	Q₈	1	4	yes	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	#14523979	—	2	2	no	—
23	#14523977	Q₈	1	2	yes	—
24	#14523975	—	2	1	no	—
25	Dic₆₇⋊C₈	—	4	2	no	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	#125×C₆₇	—	2	4	no	—
29	#125×C₆₇	—	4	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	#107×C₆₇	—	2	4	no	—
34	#106×C₆₇	—	2	2	no	—
35	#106×C₆₇	D₄	1	4	yes	—
36	C₄×C₈×C₆₇	—	2	2	no	—
37	#11761612	—	4	1	no	—
38	#11761612	—	4	1	no	—
39	C₂×C₆₇⋊C₈	—	4	1	no	—
40	C₄⋊Dic₆₇	—	2	2	no	—
41	C₂²⋊Dic₆₇	—	4	1	no	—
42	C₄×Dic₆₇	—	4	1	no	—
43	C₆₇⋊C₁₆	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₄⋊C₄×C₆₇	—	4	8	no	—
56	C₄⋊C₄×C₆₇	—	4	2	no	—
57	C₄²×C₆₇	—	2	1	no	—
58	C₄²×C₆₇	SD₁₆	1	2	yes	—
59	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	4	2	no	—
62	C₂×C₄×C₆₇	—	4	4	no	—
63	C₂×C₄×C₆₇	—	8	4	no	—
64	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
70	C₂×Dic₆₇	—	4	2	no	—
71	C₈×C₆₇	—	2	2	no	—
72	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
74	C₆₇⋊C₈	—	8	1	no	—
75	C₂²×C₆₇	—	2	2	no	—
76	C₂²×C₆₇	#450	1	1	yes	characteristic
77	C₂²×C₆₇	#476	1	1	yes	characteristic
78	C₂²×C₆₇	#501	1	1	yes	characteristic
79	C₄×C₆₇	—	4	2	no	—
80	C₄×C₆₇	—	8	4	no	—
81	C₄×C₆₇	#395	1	1	yes	characteristic
82	C₄×C₆₇	#448	1	1	yes	characteristic
83	C₄×C₆₇	#475	1	1	yes	characteristic
84	C₄×C₆₇	#498	1	1	yes	characteristic
85	Dic₆₇	—	8	2	no	—
86	#13458	—	67	1	no	maximal, Sylow
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	—	2	1	no	—
89	C₂×C₆₇	#2058	1	1	yes	characteristic
90	C₂×C₆₇	#2231	1	2	yes	—
91	C₂×#499	D₆₇	1	1	yes	characteristic
92	C₂×#365	—	67	2	no	—
93	#1945	—	67	1	no	—
94	#1386	—	67	2	no	—
95	#1376	—	67	1	no	—
96	C₆₇	#13458	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	2	1	no	—
98	C₂×M₃₂	—	67	2	no	—
99	#499	—	2	4	no	—
100	C₂×#108	—	67	1	no	—
101	#427	—	67	1	no	—
102	C₂×#107	—	2	1	no	—
103	C₂×#106	C₂×D₆₇	1	2	yes	—
104	#424	—	134	1	no	—
105	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
106	#365	—	134	4	no	—
107	#339	—	134	1	no	—
108	C₂×M₁₆	—	134	1	no	—
109	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
111	#125	—	2	4	no	—
112	#125	—	4	2	no	—
113	#117	—	134	1	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
117	M₃₂	—	67	4	no	—
118	M₃₂	—	134	2	no	—
119	C₂×C₁₆	—	134	2	no	—
120	#108	—	134	2	no	—
121	#107	—	2	4	no	—
122	#107	—	67	2	no	—
123	#106	—	2	2	no	—
124	#106	—	134	1	no	—
125	#106	C₂²⋊D₆₇	1	4	yes	—
126	C₄⋊C₈	—	268	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	M₁₆	—	268	1	no	—
132	M₁₆	—	268	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	268	1	no	—
136	C₂×C₈	#11761615	1	2	yes	—
137	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
138	C₂×C₈	Q₈×D₆₇	1	2	yes	—
139	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
140	C₂×C₈	#11761629	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	134	2	no	—
145	C₂²⋊C₄	—	268	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	268	1	no	—
148	C₄²	#11761616	1	2	yes	—
149	C₁₆	—	134	4	no	—
150	C₂³	#14524069	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	268	2	no	—
156	C₂×C₄	#14524012	1	1	yes	characteristic
157	C₂×C₄	#14524019	1	1	yes	characteristic
158	C₂×C₄	#14524041	1	1	yes	characteristic
159	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
160	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
161	C₂×C₄	#14524074	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	536	1	no	—
164	C₈	#14524019	1	2	yes	—
165	C₈	#14524041	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#16863547	1	1	yes	characteristic
168	C₂²	#16863683	1	1	yes	characteristic
169	C₂²	#16863802	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	536	2	no	—
173	C₄	#16863315	1	1	yes	characteristic
174	C₄	#16863535	1	1	yes	characteristic
175	C₄	#16863678	1	1	yes	characteristic
176	C₄	#16863792	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#22561539	1	1	yes	characteristic
180	C₂	#22562382	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863218

C₂

yes

maximal

#22560957

C₂

yes

maximal, characteristic

#22558316

C₂

yes

maximal

#22558291

C₂

yes

maximal, characteristic

C₂×#499×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523983

C₂²

yes

—

C₂×#14523979

C₂²

yes

characteristic

#16863434

C₂²

yes

—

—

—

C₂×#125×C₆₇

—

#499×C₆₇

—

C₂×#107×C₆₇

—

C₂×#106×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

—

—

Q₈

yes

—

C₂×C₆₇⋊C₁₆

—

#14523979

—

#14523977

Q₈

yes

—

#14523975

—

Dic₆₇⋊C₈

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#125×C₆₇

—

#125×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#107×C₆₇

—

#106×C₆₇

—

#106×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#137

yes

characteristic

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#13458

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2058

yes

characteristic

C₂×C₆₇

#2231

yes

—

C₂×#499

D₆₇

yes

—

—

—

—

C₆₇

#13458

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×M₃₂

—

#499

—

100

C₂×#108

—

101

#427

—

102

C₂×#107

—

103

C₂×#106

C₂×D₆₇

yes

—

104

#424

—

134

—

105

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

106

#365

—

134

—

107

#339

—

134

—

108

C₂×M₁₆

—

134

—

109

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

110

C₂²×C₈

C₂²×D₆₇

yes

characteristic

111

#125

—

112

#125

—

113

#117

—

134

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

117

M₃₂

—

118

M₃₂

—

134

—

119

C₂×C₁₆

—

134

—

120

#108

—

134

—

121

#107

—

122

#107

—

123

#106

—

124

#106

—

134

—

125

#106

C₂²⋊D₆₇

yes

—

126

C₄⋊C₈

—

268

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

131

M₁₆

—

268

—

132

M₁₆

—

268

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

268

—

136

C₂×C₈

#11761615

yes

—

137

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

138

C₂×C₈

Q₈×D₆₇

yes

—

139

C₂×C₈

Q₈×D₆₇

yes

characteristic

140

C₂×C₈

#11761629

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

134

—

145

C₂²⋊C₄

—

268

—

146

C₄²

—

147

C₄²

—

268

—

148

C₄²

#11761616

yes

—

149

C₁₆

—

134

—

150

C₂³

#14524069

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

268

—

156

C₂×C₄

#14524012

yes

characteristic

157

C₂×C₄

#14524019

yes

characteristic

158

C₂×C₄

#14524041

yes

characteristic

159

C₂×C₄

C₂×#11761615

yes

characteristic

160

C₂×C₄

C₂×#11761616

yes

characteristic

161

C₂×C₄

#14524074

yes

characteristic

162

C₈

—

163

C₈

—

536

—

164

C₈

#14524019

yes

—

165

C₈

#14524041

yes

—

166

C₂²

—

167

C₂²

#16863547

yes

characteristic

168

C₂²

#16863683

yes

characteristic

169

C₂²

#16863802

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

536

—

173

C₄

#16863315

yes

characteristic

174

C₄

#16863535

yes

characteristic

175

C₄

#16863678

yes

characteristic

176

C₄

#16863792

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#22561539

yes

characteristic

180

C₂

#22562382

yes

—

	subgroup	G/H
center	C₂²	#16863683
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle	C₂²×C₆₇	#476
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	#13458
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

#16863683

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#499×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
96. C₆₇	#13458	semidirect

H ≅ G/N

type

96. C₆₇

#13458

semidirect

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

subgroup	index
C₂×#16863218	2
#22560957	2
#22558316	2
#22558291	2
C₂×#499×C₆₇	2
#13458	67

subgroup

index

C₂×#499×C₆₇

Subgroups — #30775481

Conjugacy classes of subgroups

Complements

Maximal subgroups