Group #30692709 — subgroups

Subgroups — C₆₇×#13458 #30692709

Order 17152.

324 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 122 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×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#365×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1945	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1386	C₂	1	2	yes	maximal
5	C₆₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₆₇	—	2	1	no	—
7	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
8	#499×C₆₇	—	2	4	no	—
9	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
10	#427×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#107×C₆₇	—	2	1	no	—
12	C₂×#106×C₆₇	C₂²	1	2	yes	—
13	#424×C₆₇	—	2	1	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#365×C₆₇	—	2	4	no	—
16	#339×C₆₇	—	2	1	no	—
17	C₂×M₁₆×C₆₇	—	2	1	no	—
18	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	#125×C₆₇	—	2	4	no	—
21	#125×C₆₇	—	4	2	no	—
22	#117×C₆₇	—	2	1	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
25	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
26	M₃₂×C₆₇	—	2	2	no	—
27	M₃₂×C₆₇	Q₈	1	4	yes	—
28	C₂×C₁₆×C₆₇	—	2	2	no	—
29	#108×C₆₇	—	2	2	no	—
30	#107×C₆₇	—	2	4	no	—
31	#107×C₆₇	Q₈	1	2	yes	—
32	#106×C₆₇	—	2	2	no	—
33	#106×C₆₇	—	2	1	no	—
34	#106×C₆₇	D₄	1	4	yes	—
35	C₄⋊C₈×C₆₇	—	4	2	no	—
36	C₄×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₄×C₆₇	—	2	1	no	—
38	C₂²×C₄×C₆₇	—	4	2	no	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₆₇	—	4	1	no	—
41	M₁₆×C₆₇	—	4	1	no	—
42	C₂×C₈×C₆₇	—	2	1	no	—
43	C₂×C₈×C₆₇	—	2	1	no	—
44	C₂×C₈×C₆₇	—	4	1	no	—
45	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₆₇	—	2	4	no	—
51	C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₄⋊C₄×C₆₇	—	4	8	no	—
53	C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₂²⋊C₄×C₆₇	—	4	1	no	—
55	C₄²×C₆₇	—	2	1	no	—
56	C₄²×C₆₇	—	4	1	no	—
57	C₄²×C₆₇	SD₁₆	1	2	yes	—
58	C₁₆×C₆₇	—	2	4	no	—
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	—	4	2	no	—
64	C₂×C₄×C₆₇	—	8	4	no	—
65	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
71	C₈×C₆₇	—	2	2	no	—
72	C₈×C₆₇	—	8	1	no	—
73	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
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	—	8	2	no	—
82	C₄×C₆₇	#395	1	1	yes	characteristic
83	C₄×C₆₇	#448	1	1	yes	characteristic
84	C₄×C₆₇	#475	1	1	yes	characteristic
85	C₄×C₆₇	#498	1	1	yes	characteristic
86	#13458	C₆₇	1	1	yes	maximal, characteristic, direct product factor, 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	C₂×C₆₇	1	1	yes	characteristic
92	C₂×#365	C₂×C₆₇	1	2	yes	—
93	#1945	C₂×C₆₇	1	1	yes	characteristic
94	#1386	C₂×C₆₇	1	2	yes	—
95	#1376	C₂×C₆₇	1	1	yes	characteristic
96	C₆₇	#13458	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×#125	—	2	1	no	—
98	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
99	#499	—	2	4	no	—
100	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
101	#427	C₂²×C₆₇	1	1	yes	characteristic
102	C₂×#107	—	2	1	no	—
103	C₂×#106	C₂²×C₆₇	1	2	yes	—
104	#424	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
106	#365	—	2	4	no	—
107	#339	—	2	1	no	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
111	#125	—	2	4	no	—
112	#125	—	4	2	no	—
113	#117	—	2	1	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	2	no	—
116	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	Q₈×C₆₇	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	#108	—	2	2	no	—
121	#107	—	2	4	no	—
122	#107	Q₈×C₆₇	1	2	yes	—
123	#106	—	2	2	no	—
124	#106	—	2	1	no	—
125	#106	D₄×C₆₇	1	4	yes	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
131	M₁₆	—	4	1	no	—
132	M₁₆	—	4	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	SD₁₆×C₆₇	1	2	yes	—
137	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
138	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
139	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
140	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₂²⋊C₄	—	4	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	SD₁₆×C₆₇	1	2	yes	—
149	C₁₆	—	2	4	no	—
150	C₂³	AGL(1,ℤ/8)×C₆₇	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	—	4	2	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
157	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
158	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
161	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	8	1	no	—
164	C₈	C₂²⋊Q₈×C₆₇	1	2	yes	—
165	C₈	C₄⋊Q₈×C₆₇	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#450×C₆₇	1	1	yes	characteristic
168	C₂²	#476×C₆₇	1	1	yes	characteristic
169	C₂²	#501×C₆₇	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	8	2	no	—
173	C₄	#395×C₆₇	1	1	yes	characteristic
174	C₄	#448×C₆₇	1	1	yes	characteristic
175	C₄	#475×C₆₇	1	1	yes	characteristic
176	C₄	#498×C₆₇	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₆₇×#2058	1	1	yes	characteristic
180	C₂	C₆₇×#2231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#499×C₆₇

C₂

yes

maximal, characteristic

C₂×#365×C₆₇

C₂

yes

maximal

C₆₇×#1945

C₂

yes

maximal, characteristic

C₆₇×#1386

C₂

yes

maximal

C₆₇×#1376

C₂

yes

maximal, characteristic

C₂×#125×C₆₇

—

C₂×M₃₂×C₆₇

C₂²

yes

—

#499×C₆₇

—

C₂×#108×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

characteristic

C₂×#107×C₆₇

—

C₂×#106×C₆₇

C₂²

yes

—

#424×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#365×C₆₇

—

#339×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#125×C₆₇

—

#125×C₆₇

—

#117×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

Q₈

yes

—

C₂×C₁₆×C₆₇

—

#108×C₆₇

—

#107×C₆₇

—

#107×C₆₇

Q₈

yes

—

#106×C₆₇

—

#106×C₆₇

—

#106×C₆₇

D₄

yes

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

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

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

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

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2058

yes

characteristic

C₂×C₆₇

#2231

yes

—

C₂×#499

C₂×C₆₇

yes

characteristic

C₂×#365

C₂×C₆₇

yes

—

#1945

C₂×C₆₇

yes

characteristic

#1386

C₂×C₆₇

yes

—

#1376

C₂×C₆₇

yes

characteristic

C₆₇

#13458

yes

characteristic, direct product factor, Sylow

C₂×#125

—

C₂×M₃₂

C₂²×C₆₇

yes

—

#499

—

100

C₂×#108

C₂²×C₆₇

yes

characteristic

101

#427

C₂²×C₆₇

yes

characteristic

102

C₂×#107

—

103

C₂×#106

C₂²×C₆₇

yes

—

104

#424

—

105

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

106

#365

—

107

#339

—

108

C₂×M₁₆

—

109

C₂²×C₈

D₄×C₆₇

yes

characteristic

110

C₂²×C₈

C₂³×C₆₇

yes

characteristic

111

#125

—

112

#125

—

113

#117

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

D₄×C₆₇

yes

characteristic

117

M₃₂

—

118

M₃₂

Q₈×C₆₇

yes

—

119

C₂×C₁₆

—

120

#108

—

121

#107

—

122

#107

Q₈×C₆₇

yes

—

123

#106

—

124

#106

—

125

#106

D₄×C₆₇

yes

—

126

C₄⋊C₈

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

SD₁₆×C₆₇

yes

—

137

C₂×C₈

C₂×D₄×C₆₇

yes

—

138

C₂×C₈

C₂×Q₈×C₆₇

yes

—

139

C₂×C₈

C₂×Q₈×C₆₇

yes

characteristic

140

C₂×C₈

#41×C₆₇

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

SD₁₆×C₆₇

yes

—

149

C₁₆

—

150

C₂³

AGL(1,ℤ/8)×C₆₇

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#120×C₆₇

yes

characteristic

157

C₂×C₄

C₂²⋊Q₈×C₆₇

yes

characteristic

158

C₂×C₄

C₄⋊Q₈×C₆₇

yes

characteristic

159

C₂×C₄

C₂×SD₁₆×C₆₇

yes

characteristic

160

C₂×C₄

C₂×SD₁₆×C₆₇

yes

characteristic

161

C₂×C₄

#137×C₆₇

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

C₂²⋊Q₈×C₆₇

yes

—

165

C₈

C₄⋊Q₈×C₆₇

yes

—

166

C₂²

—

167

C₂²

#450×C₆₇

yes

characteristic

168

C₂²

#476×C₆₇

yes

characteristic

169

C₂²

#501×C₆₇

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#395×C₆₇

yes

characteristic

174

C₄

#448×C₆₇

yes

characteristic

175

C₄

#475×C₆₇

yes

characteristic

176

C₄

#498×C₆₇

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

C₆₇×#2058

yes

characteristic

180

C₂

C₆₇×#2231

yes

—

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

subgroup

G/H

center, socle

C₂²×C₆₇

#476

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

prime radical

C₂³×C₆₇

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
86. #13458	C₆₇	direct (N × H)
96. C₆₇	#13458	direct (N × H)

H ≅ G/N

type

86. #13458

C₆₇

direct (N × H)

96. C₆₇

#13458

direct (N × H)

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

subgroup	index
C₂×#499×C₆₇	2
C₂×#365×C₆₇	2
C₆₇×#1945	2
C₆₇×#1386	2
C₆₇×#1376	2
#13458	67

subgroup

index

C₂×#499×C₆₇

C₂×#365×C₆₇

C₆₇×#1945

C₆₇×#1386

C₆₇×#1376

#13458

Subgroups — C₆₇×#13458 #30692709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#13458 #30692709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13458 #30692709