Group #30712051 — subgroups

Subgroups — C₆₇×#32800 #30712051

Order 17152.

816 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 238 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×#2868	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2860	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2043	C₂	1	6	yes	maximal, complemented
4	C₆₇×#2013	C₂	1	6	yes	maximal
5	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
8	C₂×#126×C₆₇	—	2	1	no	—
9	C₂×#103×C₆₇	C₂²	1	12	yes	—
10	#390×C₆₇	C₂²	1	3	yes	—
11	#389×C₆₇	C₂²	1	3	yes	—
12	#388×C₆₇	—	2	3	no	—
13	#384×C₆₇	—	2	3	no	—
14	#384×C₆₇	—	2	1	no	—
15	C₄³×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	C₂²	1	6	yes	—
17	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
18	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
19	C₂×SD₁₆×C₆₇	—	4	12	no	—
20	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
21	C₄⋊Q₈×C₆₇	—	4	12	no	—
22	#126×C₆₇	—	4	4	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
27	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
28	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
29	C₂×C₄²×C₆₇	D₄	1	3	yes	—
30	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
32	#103×C₆₇	—	4	12	no	—
33	#95×C₆₇	—	2	3	no	—
34	C₂⁴×C₆₇	—	2	1	no	—
35	C₂×Q₈×C₆₇	—	2	12	no	—
36	C₂×Q₈×C₆₇	—	2	12	no	—
37	C₂×D₄×C₆₇	—	2	6	no	—
38	C₂²×C₄×C₆₇	—	2	3	no	—
39	C₂²×C₄×C₆₇	—	2	4	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
43	SD₁₆×C₆₇	—	4	16	no	—
44	C₂×C₈×C₆₇	—	2	12	no	—
45	C₄⋊C₄×C₆₇	—	2	6	no	—
46	C₄⋊C₄×C₆₇	—	4	12	no	—
47	C₄⋊C₄×C₆₇	—	4	12	no	—
48	C₄⋊C₄×C₆₇	—	4	6	no	—
49	C₂²⋊C₄×C₆₇	—	4	6	no	—
50	C₂²⋊C₄×C₆₇	—	4	6	no	—
51	C₄²×C₆₇	—	2	6	no	—
52	C₄²×C₆₇	—	2	6	no	—
53	C₄²×C₆₇	SD₁₆	1	4	yes	complemented
54	C₂³×C₆₇	—	4	3	no	—
55	C₂³×C₆₇	—	4	1	no	—
56	C₂³×C₆₇	—	4	3	no	—
57	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₆₇	—	4	4	no	—
59	Q₈×C₆₇	—	4	12	no	—
60	D₄×C₆₇	—	4	4	no	—
61	C₂×C₄×C₆₇	—	2	3	no	—
62	C₂×C₄×C₆₇	—	2	3	no	—
63	C₂×C₄×C₆₇	—	2	3	no	—
64	C₂×C₄×C₆₇	—	2	3	no	—
65	C₂×C₄×C₆₇	—	4	3	no	—
66	C₂×C₄×C₆₇	—	4	12	no	—
67	C₂×C₄×C₆₇	—	4	3	no	—
68	C₂×C₄×C₆₇	—	4	3	no	—
69	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
70	C₂×C₄×C₆₇	#137	1	6	yes	—
71	C₂×C₄×C₆₇	H(5,𝔽₂)	1	6	yes	—
72	C₈×C₆₇	—	4	4	no	—
73	C₂²×C₆₇	—	8	1	no	—
74	C₂²×C₆₇	—	8	3	no	—
75	C₂²×C₆₇	—	8	3	no	—
76	C₂²×C₆₇	#535	1	3	yes	—
77	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
78	C₂²×C₆₇	C₂×#137	1	3	yes	—
79	C₄×C₆₇	—	2	6	no	—
80	C₄×C₆₇	—	2	6	no	—
81	C₄×C₆₇	—	8	4	no	—
82	C₄×C₆₇	—	8	3	no	—
83	C₄×C₆₇	#560	1	4	yes	—
84	#32800	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₆₇	—	16	1	no	—
86	C₂×C₆₇	#2818	1	1	yes	characteristic
87	C₂×C₆₇	#3206	1	3	yes	—
88	C₂×C₆₇	#3218	1	3	yes	—
89	#2868	C₂×C₆₇	1	1	yes	characteristic, complemented
90	#2860	C₂×C₆₇	1	1	yes	characteristic
91	#2043	C₂×C₆₇	1	6	yes	complemented
92	#2013	C₂×C₆₇	1	6	yes	—
93	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
94	C₆₇	#32800	1	1	yes	characteristic, direct product factor, Sylow
95	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
96	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
97	C₂×#126	—	2	1	no	—
98	C₂×#103	C₂²×C₆₇	1	12	yes	—
99	#390	C₂²×C₆₇	1	3	yes	—
100	#389	C₂²×C₆₇	1	3	yes	—
101	#388	—	2	3	no	—
102	#384	—	2	3	no	—
103	#384	—	2	1	no	—
104	C₄³	C₂²×C₆₇	1	1	yes	characteristic
105	#336	C₂²×C₆₇	1	6	yes	—
106	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
107	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
108	C₂×SD₁₆	—	4	12	no	—
109	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
110	C₄⋊Q₈	—	4	12	no	—
111	#126	—	4	4	no	—
112	C₂×C₄⋊C₄	—	2	3	no	—
113	C₂×C₄⋊C₄	—	2	6	no	—
114	C₂×C₄⋊C₄	—	2	6	no	—
115	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
116	C₂×C₂²⋊C₄	—	2	3	no	—
117	C₂×C₂²⋊C₄	—	2	3	no	—
118	C₂×C₄²	D₄×C₆₇	1	3	yes	—
119	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
121	#103	—	4	12	no	—
122	#95	—	2	3	no	—
123	C₂⁴	—	2	1	no	—
124	C₂×Q₈	—	2	12	no	—
125	C₂×Q₈	—	2	12	no	—
126	C₂×D₄	—	2	6	no	—
127	C₂²×C₄	—	2	3	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
130	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
131	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
132	SD₁₆	—	4	16	no	—
133	C₂×C₈	—	2	12	no	—
134	C₄⋊C₄	—	2	6	no	—
135	C₄⋊C₄	—	4	12	no	—
136	C₄⋊C₄	—	4	12	no	—
137	C₄⋊C₄	—	4	6	no	—
138	C₂²⋊C₄	—	4	6	no	—
139	C₂²⋊C₄	—	4	6	no	—
140	C₄²	—	2	6	no	—
141	C₄²	—	2	6	no	—
142	C₄²	SD₁₆×C₆₇	1	4	yes	complemented
143	C₂³	—	4	3	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	3	no	—
146	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
147	Q₈	—	4	4	no	—
148	Q₈	—	4	12	no	—
149	D₄	—	4	4	no	—
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	2	3	no	—
154	C₂×C₄	—	4	3	no	—
155	C₂×C₄	—	4	12	no	—
156	C₂×C₄	—	4	3	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂×SD₁₆×C₆₇	1	6	yes	—
159	C₂×C₄	#137×C₆₇	1	6	yes	—
160	C₂×C₄	H(5,𝔽₂)×C₆₇	1	6	yes	—
161	C₈	—	4	4	no	—
162	C₂²	—	8	1	no	—
163	C₂²	—	8	3	no	—
164	C₂²	—	8	3	no	—
165	C₂²	#535×C₆₇	1	3	yes	—
166	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
167	C₂²	C₂×#137×C₆₇	1	3	yes	—
168	C₄	—	2	6	no	—
169	C₄	—	2	6	no	—
170	C₄	—	8	4	no	—
171	C₄	—	8	3	no	—
172	C₄	#560×C₆₇	1	4	yes	—
173	C₂	—	16	1	no	—
174	C₂	C₆₇×#2818	1	1	yes	characteristic
175	C₂	C₆₇×#3206	1	3	yes	—
176	C₂	C₆₇×#3218	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2868

C₂

yes

maximal, characteristic, complemented

C₆₇×#2860

C₂

yes

maximal, characteristic

C₆₇×#2043

C₂

yes

maximal, complemented

C₆₇×#2013

C₂

yes

maximal

C₆₇×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×SD₁₆×C₆₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×#126×C₆₇

—

C₂×#103×C₆₇

C₂²

yes

—

#390×C₆₇

C₂²

yes

—

#389×C₆₇

C₂²

yes

—

#388×C₆₇

—

#384×C₆₇

—

#384×C₆₇

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂³

yes

—

C₂²×D₄×C₆₇

C₂³

yes

characteristic

C₂×SD₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

#126×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

—

#103×C₆₇

—

#95×C₆₇

—

C₂⁴×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

SD₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

complemented

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

H(5,𝔽₂)

yes

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#535

yes

—

C₂²×C₆₇

C₂²×SD₁₆

yes

characteristic

C₂²×C₆₇

C₂×#137

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#560

yes

—

#32800

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2818

yes

characteristic

C₂×C₆₇

#3206

yes

—

C₂×C₆₇

#3218

yes

—

#2868

C₂×C₆₇

yes

characteristic, complemented

#2860

C₂×C₆₇

yes

characteristic

#2043

C₂×C₆₇

yes

complemented

#2013

C₂×C₆₇

yes

—

#1839

C₂×C₆₇

yes

characteristic, complemented

C₆₇

#32800

yes

characteristic, direct product factor, Sylow

C₂²×SD₁₆

C₂²×C₆₇

yes

—

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

C₂×#126

—

C₂×#103

C₂²×C₆₇

yes

—

#390

C₂²×C₆₇

yes

—

100

#389

C₂²×C₆₇

yes

—

101

#388

—

102

#384

—

103

#384

—

104

C₄³

C₂²×C₆₇

yes

characteristic

105

#336

C₂²×C₆₇

yes

—

106

C₂²×Q₈

C₂³×C₆₇

yes

—

107

C₂²×D₄

C₂³×C₆₇

yes

characteristic

108

C₂×SD₁₆

—

109

C₂²×C₈

C₂³×C₆₇

yes

—

110

C₄⋊Q₈

—

111

#126

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₄²

D₄×C₆₇

yes

—

119

C₂×C₄²

D₄×C₆₇

yes

characteristic

120

C₂×C₄²

C₂³×C₆₇

yes

—

121

#103

—

122

#95

—

123

C₂⁴

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂×D₄×C₆₇

yes

—

130

C₂²×C₄

C₂×D₄×C₆₇

yes

—

131

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

132

SD₁₆

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

SD₁₆×C₆₇

yes

complemented

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

C₂²×D₄×C₆₇

yes

characteristic

147

Q₈

—

148

Q₈

—

149

D₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₂×SD₁₆×C₆₇

yes

—

159

C₂×C₄

#137×C₆₇

yes

—

160

C₂×C₄

H(5,𝔽₂)×C₆₇

yes

—

161

C₈

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#535×C₆₇

yes

—

166

C₂²

C₂²×SD₁₆×C₆₇

yes

characteristic

167

C₂²

C₂×#137×C₆₇

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#560×C₆₇

yes

—

173

C₂

—

174

C₂

C₆₇×#2818

yes

characteristic

175

C₂

C₆₇×#3206

yes

—

176

C₂

C₆₇×#3218

yes

—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

prime radical

C₂²×D₄×C₆₇

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

N	H ≅ G/N	type
1. C₆₇×#2868	C₂	semidirect
3. C₆₇×#2043	C₂	semidirect
5. C₆₇×#1839	C₂	semidirect
53. C₄²×C₆₇	SD₁₆	semidirect
84. #32800	C₆₇	direct (N × H)
89. #2868	C₂×C₆₇	semidirect
91. #2043	C₂×C₆₇	semidirect
93. #1839	C₂×C₆₇	semidirect
94. C₆₇	#32800	direct (N × H)
142. C₄²	SD₁₆×C₆₇	semidirect

H ≅ G/N

type

1. C₆₇×#2868

C₂

semidirect

3. C₆₇×#2043

C₂

semidirect

5. C₆₇×#1839

C₂

semidirect

53. C₄²×C₆₇

SD₁₆

semidirect

84. #32800

C₆₇

direct (N × H)

89. #2868

C₂×C₆₇

semidirect

91. #2043

C₂×C₆₇

semidirect

93. #1839

C₂×C₆₇

semidirect

94. C₆₇

#32800

direct (N × H)

142. C₄²

SD₁₆×C₆₇

semidirect

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

subgroup	index
C₆₇×#2868	2
C₆₇×#2860	2
C₆₇×#2043	2
C₆₇×#2013	2
C₆₇×#1839	2
#32800	67

subgroup

index

C₆₇×#2868

C₆₇×#2860

C₆₇×#2043

C₆₇×#2013

C₆₇×#1839

#32800

Subgroups — C₆₇×#32800 #30712051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#32800 #30712051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32800 #30712051