Group #30698962 — subgroups

Subgroups — C₆₇×#19711 #30698962

Order 17152.

636 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 230 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	C₆₇×#3231	C₂	1	4	yes	maximal
2	C₆₇×#3170	C₂	1	2	yes	maximal
3	C₆₇×#1723	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1718	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
6	#533×C₆₇	C₂²	1	4	yes	—
7	#500×C₆₇	C₂²	1	2	yes	—
8	#500×C₆₇	C₂²	1	4	yes	—
9	#498×C₆₇	C₂²	1	4	yes	—
10	#494×C₆₇	C₂²	1	2	yes	—
11	#487×C₆₇	C₂²	1	4	yes	—
12	#487×C₆₇	C₂²	1	4	yes	—
13	#487×C₆₇	C₂²	1	4	yes	—
14	#485×C₆₇	—	2	8	no	—
15	#479×C₆₇	—	2	8	no	—
16	#467×C₆₇	—	2	4	no	—
17	C₄×Q₁₆×C₆₇	—	2	4	no	—
18	#431×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#322×C₆₇	C₂²	1	4	yes	—
21	C₂×Q₁₆×C₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	D₄	1	2	yes	—
23	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
24	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
25	#125×C₆₇	—	2	4	no	—
26	C₂²⋊Q₈×C₆₇	—	2	4	no	—
27	C₂²⋊Q₈×C₆₇	—	2	4	no	—
28	C₄×Q₈×C₆₇	—	2	4	no	—
29	C₄×Q₈×C₆₇	—	2	4	no	—
30	#117×C₆₇	—	2	4	no	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	#107×C₆₇	—	2	4	no	—
33	#107×C₆₇	—	2	8	no	—
34	#106×C₆₇	—	2	8	no	—
35	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
36	#103×C₆₇	—	2	8	no	—
37	#103×C₆₇	—	2	8	no	—
38	#103×C₆₇	—	2	8	no	—
39	C₂²⋊C₈×C₆₇	D₄	1	4	yes	—
40	C₄×C₈×C₆₇	D₄	1	2	yes	—
41	C₄×C₈×C₆₇	C₂³	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	4	yes	—
43	C₂×Q₈×C₆₇	—	2	4	no	—
44	C₂×Q₈×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₆₇	—	4	8	no	—
48	C₂×C₈×C₆₇	—	4	2	no	—
49	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
51	C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₄⋊C₄×C₆₇	—	4	4	no	—
56	C₄⋊C₄×C₆₇	—	4	4	no	—
57	C₂²⋊C₄×C₆₇	—	4	4	no	—
58	C₄²×C₆₇	—	4	4	no	—
59	C₄²×C₆₇	C₂×D₄	1	2	yes	—
60	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂³×C₆₇	#127	1	1	yes	characteristic
63	Q₈×C₆₇	—	4	8	no	—
64	Q₈×C₆₇	—	4	8	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	#127	1	2	yes	—
70	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
73	C₈×C₆₇	—	2	4	no	—
74	C₈×C₆₇	—	2	8	no	—
75	C₂²×C₆₇	—	2	1	no	—
76	C₂²×C₆₇	—	2	2	no	—
77	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
78	C₄×C₆₇	—	2	1	no	—
79	C₄×C₆₇	—	2	2	no	—
80	C₄×C₆₇	—	8	4	no	—
81	C₄×C₆₇	—	8	4	no	—
82	C₄×C₆₇	#575	1	2	yes	—
83	C₄×C₆₇	#578	1	4	yes	—
84	#19711	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₆₇	—	4	1	no	—
86	C₂×C₆₇	#3150	1	1	yes	characteristic
87	C₂×C₆₇	#3156	1	2	yes	—
88	#3231	C₂×C₆₇	1	4	yes	—
89	#3170	C₂×C₆₇	1	2	yes	—
90	#1723	C₂×C₆₇	1	4	yes	complemented
91	#1718	C₂×C₆₇	1	4	yes	complemented
92	#1452	C₂×C₆₇	1	1	yes	characteristic
93	C₆₇	#19711	1	1	yes	characteristic, direct product factor, Sylow
94	#533	C₂²×C₆₇	1	4	yes	—
95	#500	C₂²×C₆₇	1	2	yes	—
96	#500	C₂²×C₆₇	1	4	yes	—
97	#498	C₂²×C₆₇	1	4	yes	—
98	#494	C₂²×C₆₇	1	2	yes	—
99	#487	C₂²×C₆₇	1	4	yes	—
100	#487	C₂²×C₆₇	1	4	yes	—
101	#487	C₂²×C₆₇	1	4	yes	—
102	#485	—	2	8	no	—
103	#479	—	2	8	no	—
104	#467	—	2	4	no	—
105	C₄×Q₁₆	—	2	4	no	—
106	#431	C₂²×C₆₇	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
108	#322	C₂²×C₆₇	1	4	yes	—
109	C₂×Q₁₆	—	2	4	no	—
110	C₂²×C₈	D₄×C₆₇	1	2	yes	—
111	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
112	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
113	#125	—	2	4	no	—
114	C₂²⋊Q₈	—	2	4	no	—
115	C₂²⋊Q₈	—	2	4	no	—
116	C₄×Q₈	—	2	4	no	—
117	C₄×Q₈	—	2	4	no	—
118	#117	—	2	4	no	—
119	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
120	#107	—	2	4	no	—
121	#107	—	2	8	no	—
122	#106	—	2	8	no	—
123	C₄⋊C₈	D₄×C₆₇	1	4	yes	—
124	#103	—	2	8	no	—
125	#103	—	2	8	no	—
126	#103	—	2	8	no	—
127	C₂²⋊C₈	D₄×C₆₇	1	4	yes	—
128	C₄×C₈	D₄×C₆₇	1	2	yes	—
129	C₄×C₈	C₂³×C₆₇	1	2	yes	—
130	C₄×C₈	C₂³×C₆₇	1	4	yes	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	4	no	—
133	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
135	Q₁₆	—	4	8	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
138	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₂²⋊C₄	—	4	4	no	—
146	C₄²	—	4	4	no	—
147	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
148	C₄²	C₂×D₄×C₆₇	1	2	yes	—
149	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
150	C₂³	#127×C₆₇	1	1	yes	characteristic
151	Q₈	—	4	8	no	—
152	Q₈	—	4	8	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	#127×C₆₇	1	2	yes	—
158	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
161	C₈	—	2	4	no	—
162	C₈	—	2	8	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	2	no	—
165	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	8	4	no	—
169	C₄	—	8	4	no	—
170	C₄	#575×C₆₇	1	2	yes	—
171	C₄	#578×C₆₇	1	4	yes	—
172	C₂	—	4	1	no	—
173	C₂	C₆₇×#3150	1	1	yes	characteristic
174	C₂	C₆₇×#3156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#3231

C₂

yes

maximal

C₆₇×#3170

C₂

yes

maximal

C₆₇×#1723

C₂

yes

maximal, complemented

C₆₇×#1718

C₂

yes

maximal, complemented

C₆₇×#1452

C₂

yes

maximal, characteristic

#533×C₆₇

C₂²

yes

—

#500×C₆₇

C₂²

yes

—

#500×C₆₇

C₂²

yes

—

#498×C₆₇

C₂²

yes

—

#494×C₆₇

C₂²

yes

—

#487×C₆₇

C₂²

yes

—

#487×C₆₇

C₂²

yes

—

#487×C₆₇

C₂²

yes

—

#485×C₆₇

—

#479×C₆₇

—

#467×C₆₇

—

C₄×Q₁₆×C₆₇

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

C₂³

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

#117×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

#107×C₆₇

—

#106×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

—

#103×C₆₇

—

#103×C₆₇

—

#103×C₆₇

—

C₂²⋊C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

—

C₄×C₈×C₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂⁴

yes

characteristic

C₂³×C₆₇

#127

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#127

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#575

yes

—

C₄×C₆₇

#578

yes

—

#19711

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#3150

yes

characteristic

C₂×C₆₇

#3156

yes

—

#3231

C₂×C₆₇

yes

—

#3170

C₂×C₆₇

yes

—

#1723

C₂×C₆₇

yes

complemented

#1718

C₂×C₆₇

yes

complemented

#1452

C₂×C₆₇

yes

characteristic

C₆₇

#19711

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₆₇

yes

—

#500

C₂²×C₆₇

yes

—

#500

C₂²×C₆₇

yes

—

#498

C₂²×C₆₇

yes

—

#494

C₂²×C₆₇

yes

—

#487

C₂²×C₆₇

yes

—

100

#487

C₂²×C₆₇

yes

—

101

#487

C₂²×C₆₇

yes

—

102

#485

—

103

#479

—

104

#467

—

105

C₄×Q₁₆

—

106

#431

C₂²×C₆₇

yes

—

107

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

108

#322

C₂²×C₆₇

yes

—

109

C₂×Q₁₆

—

110

C₂²×C₈

D₄×C₆₇

yes

—

111

C₄⋊Q₈

C₂³×C₆₇

yes

—

112

C₄⋊Q₈

C₂³×C₆₇

yes

—

113

#125

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

—

118

#117

—

119

C₂×C₄²

C₂³×C₆₇

yes

characteristic

120

#107

—

121

#107

—

122

#106

—

123

C₄⋊C₈

D₄×C₆₇

yes

—

124

#103

—

125

#103

—

126

#103

—

127

C₂²⋊C₈

D₄×C₆₇

yes

—

128

C₄×C₈

D₄×C₆₇

yes

—

129

C₄×C₈

C₂³×C₆₇

yes

—

130

C₄×C₈

C₂³×C₆₇

yes

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

C₂×D₄×C₆₇

yes

—

134

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

135

Q₁₆

—

136

C₂×C₈

—

137

C₂×C₈

C₂×D₄×C₆₇

yes

—

138

C₂×C₈

C₂×D₄×C₆₇

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

C₂×D₄×C₆₇

yes

characteristic

148

C₄²

C₂×D₄×C₆₇

yes

—

149

C₄²

C₂⁴×C₆₇

yes

characteristic

150

C₂³

#127×C₆₇

yes

characteristic

151

Q₈

—

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

#127×C₆₇

yes

—

158

C₂×C₄

#127×C₆₇

yes

characteristic

159

C₂×C₄

C₂²×D₄×C₆₇

yes

characteristic

160

C₂×C₄

C₂²×D₄×C₆₇

yes

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

C₂×#127×C₆₇

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#575×C₆₇

yes

—

171

C₄

#578×C₆₇

yes

—

172

C₂

—

173

C₂

C₆₇×#3150

yes

characteristic

174

C₂

C₆₇×#3156

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴×C₆₇

C₂³×C₆₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
3. C₆₇×#1723	C₂	semidirect
4. C₆₇×#1718	C₂	semidirect
84. #19711	C₆₇	direct (N × H)
90. #1723	C₂×C₆₇	semidirect
91. #1718	C₂×C₆₇	semidirect
93. C₆₇	#19711	direct (N × H)

H ≅ G/N

type

3. C₆₇×#1723

C₂

semidirect

4. C₆₇×#1718

C₂

semidirect

84. #19711

C₆₇

direct (N × H)

90. #1723

C₂×C₆₇

semidirect

91. #1718

C₂×C₆₇

semidirect

93. C₆₇

#19711

direct (N × H)

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

subgroup	index
C₆₇×#3231	2
C₆₇×#3170	2
C₆₇×#1723	2
C₆₇×#1718	2
C₆₇×#1452	2
#19711	67

subgroup

index

C₆₇×#3231

C₆₇×#3170

C₆₇×#1723

C₆₇×#1718

C₆₇×#1452

#19711

Subgroups — C₆₇×#19711 #30698962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#19711 #30698962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19711 #30698962