Group #30692531 — subgroups

Subgroups — C₆₇×#13280 #30692531

Order 17152.

334 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 150 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	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1387	C₂	1	4	yes	maximal
5	C₂×#125×C₆₇	—	2	1	no	—
6	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
7	#499×C₆₇	—	2	2	no	—
8	#499×C₆₇	C₄	1	4	yes	—
9	C₂×#108×C₆₇	C₂²	1	2	yes	—
10	C₂×#107×C₆₇	C₂²	1	2	yes	—
11	C₂×#106×C₆₇	—	2	1	no	—
12	C₄×M₁₆×C₆₇	—	2	1	no	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₆₇	—	2	2	no	—
15	#341×C₆₇	—	2	1	no	—
16	C₂×M₁₆×C₆₇	—	2	2	no	—
17	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
19	#125×C₆₇	—	2	4	no	—
20	#125×C₆₇	—	4	1	no	—
21	#125×C₆₇	—	4	1	no	—
22	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
24	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
25	M₃₂×C₆₇	—	2	4	no	—
26	C₂×C₁₆×C₆₇	—	2	2	no	—
27	#108×C₆₇	—	2	2	no	—
28	#108×C₆₇	Q₈	1	4	yes	—
29	#107×C₆₇	—	2	2	no	—
30	#107×C₆₇	C₂×C₄	1	4	yes	—
31	#106×C₆₇	—	2	4	no	—
32	#97×C₆₇	—	4	1	no	—
33	C₄×C₈×C₆₇	—	4	1	no	—
34	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₂²×C₄×C₆₇	—	4	2	no	—
37	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₆₇	—	2	4	no	—
40	M₁₆×C₆₇	—	4	2	no	—
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	—	4	2	no	—
43	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₄⋊C₄×C₆₇	—	4	4	no	—
50	C₄⋊C₄×C₆₇	—	4	4	no	—
51	C₄⋊C₄×C₆₇	—	4	2	no	—
52	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₆₇	D₈	1	1	yes	characteristic
54	C₄²×C₆₇	D₈	1	1	yes	characteristic
55	C₁₆×C₆₇	—	4	2	no	—
56	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	4	2	no	—
58	C₂×C₄×C₆₇	—	4	4	no	—
59	C₂×C₄×C₆₇	—	8	4	no	—
60	C₂×C₄×C₆₇	#102	1	2	yes	—
61	C₂×C₄×C₆₇	#102	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#137	1	2	yes	—
66	C₈×C₆₇	—	2	2	no	—
67	C₈×C₆₇	—	4	4	no	—
68	C₈×C₆₇	C₄×Q₈	1	2	yes	—
69	C₈×C₆₇	#124	1	1	yes	characteristic
70	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
71	C₂²×C₆₇	—	2	2	no	—
72	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
73	C₂²×C₆₇	#475	1	2	yes	—
74	C₄×C₆₇	—	2	2	no	—
75	C₄×C₆₇	—	2	2	no	—
76	C₄×C₆₇	—	8	4	no	—
77	C₄×C₆₇	#391	1	1	yes	characteristic
78	C₄×C₆₇	#419	1	1	yes	characteristic
79	C₄×C₆₇	#474	1	2	yes	—
80	#13280	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₆₇	—	2	2	no	—
82	C₂×C₆₇	#1988	1	1	yes	characteristic
83	C₂×C₆₇	#2148	1	2	yes	—
84	C₂×#499	C₂×C₆₇	1	1	yes	characteristic
85	#2150	C₂×C₆₇	1	1	yes	characteristic
86	C₄×#108	C₂×C₆₇	1	1	yes	characteristic
87	#1387	C₂×C₆₇	1	4	yes	—
88	C₆₇	#13280	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×#125	—	2	1	no	—
90	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
91	#499	—	2	2	no	—
92	#499	C₄×C₆₇	1	4	yes	—
93	C₂×#108	C₂²×C₆₇	1	2	yes	—
94	C₂×#107	C₂²×C₆₇	1	2	yes	—
95	C₂×#106	—	2	1	no	—
96	C₄×M₁₆	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
98	C₄⋊C₁₆	—	2	2	no	—
99	#341	—	2	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	1	no	—
105	#125	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
109	M₃₂	—	2	4	no	—
110	C₂×C₁₆	—	2	2	no	—
111	#108	—	2	2	no	—
112	#108	Q₈×C₆₇	1	4	yes	—
113	#107	—	2	2	no	—
114	#107	C₂×C₄×C₆₇	1	4	yes	—
115	#106	—	2	4	no	—
116	#97	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
119	C₄×C₈	D₄×C₆₇	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
122	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
128	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
129	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
130	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
131	C₂×C₈	#41×C₆₇	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
137	C₄²	D₈×C₆₇	1	1	yes	characteristic
138	C₄²	D₈×C₆₇	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#102×C₆₇	1	2	yes	—
145	C₂×C₄	#102×C₆₇	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
147	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
148	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
149	C₂×C₄	#137×C₆₇	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	C₄×Q₈×C₆₇	1	2	yes	—
153	C₈	#124×C₆₇	1	1	yes	characteristic
154	C₈	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
157	C₂²	#475×C₆₇	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#391×C₆₇	1	1	yes	characteristic
162	C₄	#419×C₆₇	1	1	yes	characteristic
163	C₄	#474×C₆₇	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₆₇×#1988	1	1	yes	characteristic
166	C₂	C₆₇×#2148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#499×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#2150

C₂

yes

maximal, characteristic

C₄×#108×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#1387

C₂

yes

maximal

C₂×#125×C₆₇

—

C₂×M₃₂×C₆₇

C₂²

yes

—

#499×C₆₇

—

#499×C₆₇

C₄

yes

—

C₂×#108×C₆₇

C₂²

yes

—

C₂×#107×C₆₇

C₂²

yes

—

C₂×#106×C₆₇

—

C₄×M₁₆×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

#341×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#125×C₆₇

—

#125×C₆₇

—

#125×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

#108×C₆₇

—

#108×C₆₇

Q₈

yes

—

#107×C₆₇

—

#107×C₆₇

C₂×C₄

yes

—

#106×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

D₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄×Q₈

yes

—

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#102

yes

characteristic

C₂²×C₆₇

#475

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#13280

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1988

yes

characteristic

C₂×C₆₇

#2148

yes

—

C₂×#499

C₂×C₆₇

yes

characteristic

#2150

C₂×C₆₇

yes

characteristic

C₄×#108

C₂×C₆₇

yes

characteristic

#1387

C₂×C₆₇

yes

—

C₆₇

#13280

yes

characteristic, direct product factor, Sylow

C₂×#125

—

C₂×M₃₂

C₂²×C₆₇

yes

—

#499

—

#499

C₄×C₆₇

yes

—

C₂×#108

C₂²×C₆₇

yes

—

C₂×#107

C₂²×C₆₇

yes

—

C₂×#106

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

C₄⋊C₁₆

—

#341

—

100

C₂×M₁₆

—

101

C₂²×C₈

D₄×C₆₇

yes

characteristic

102

C₂²×C₈

C₂³×C₆₇

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

D₄×C₆₇

yes

characteristic

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

112

#108

Q₈×C₆₇

yes

—

113

#107

—

114

#107

C₂×C₄×C₆₇

yes

—

115

#106

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

C₂×C₄×C₆₇

yes

—

119

C₄×C₈

D₄×C₆₇

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

SD₁₆×C₆₇

yes

—

122

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊C₄×C₆₇

yes

—

128

C₂×C₈

C₂²×C₄×C₆₇

yes

characteristic

129

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

130

C₂×C₈

C₂×Q₈×C₆₇

yes

—

131

C₂×C₈

#41×C₆₇

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂²⋊C₄×C₆₇

yes

—

137

C₄²

D₈×C₆₇

yes

characteristic

138

C₄²

D₈×C₆₇

yes

characteristic

139

C₁₆

—

140

C₂³

C₂×SD₁₆×C₆₇

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#102×C₆₇

yes

—

145

C₂×C₄

#102×C₆₇

yes

—

146

C₂×C₄

C₂×C₂²⋊C₄×C₆₇

yes

characteristic

147

C₂×C₄

C₂²⋊Q₈×C₆₇

yes

—

148

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

149

C₂×C₄

#137×C₆₇

yes

—

150

C₈

—

151

C₈

—

152

C₈

C₄×Q₈×C₆₇

yes

—

153

C₈

#124×C₆₇

yes

characteristic

154

C₈

C₄⋊Q₈×C₆₇

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#102×C₆₇

yes

characteristic

157

C₂²

#475×C₆₇

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#391×C₆₇

yes

characteristic

162

C₄

#419×C₆₇

yes

characteristic

163

C₄

#474×C₆₇

yes

—

164

C₂

—

165

C₂

C₆₇×#1988

yes

characteristic

166

C₂

C₆₇×#2148

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
prime radical	C₂³×C₆₇	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₆₇

Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

prime radical

C₂³×C₆₇

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
80. #13280	C₆₇	direct (N × H)
88. C₆₇	#13280	direct (N × H)

H ≅ G/N

type

80. #13280

C₆₇

direct (N × H)

88. C₆₇

#13280

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
C₆₇×#2150	2
C₄×#108×C₆₇	2
C₆₇×#1387	2
#13280	67

subgroup

index

C₂×#499×C₆₇

C₆₇×#2150

C₄×#108×C₆₇

C₆₇×#1387

#13280

Subgroups — C₆₇×#13280 #30692531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#13280 #30692531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13280 #30692531