Group #30774439 — subgroups

Subgroups — #30774439

Order 17152.

374 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 159 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×#16863798	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal
4	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524035	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	#16863798	—	2	2	no	—
8	#16863798	C₄	1	4	yes	complemented
9	C₂×#14523977	C₂²	1	2	yes	—
10	C₂×#14523975	—	2	1	no	—
11	#16863172	—	2	2	no	—
12	C₂×#107×C₆₇	—	2	1	no	—
13	C₂×#107×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#384×C₆₇	—	2	1	no	—
16	#14524035	—	2	4	no	—
17	#14524035	—	4	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×Dic₆₇⋊C₄	—	4	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	—
23	#14523977	—	2	2	no	—
24	#14523977	C₂×C₄	1	4	yes	—
25	#14523975	—	2	4	no	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
30	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
31	#107×C₆₇	—	2	4	no	—
32	#107×C₆₇	—	4	2	no	—
33	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₇	D₄	1	2	yes	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₄⋊Dic₆₇	—	2	4	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	Dic₆₇⋊C₄	—	4	8	no	—
39	C₆₇⋊C₁₆	—	2	4	no	—
40	C₂²×C₄×C₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	D₈	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
45	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₆₇	#41	1	2	yes	—
50	C₄⋊C₄×C₆₇	—	4	2	no	—
51	C₄⋊C₄×C₆₇	—	4	2	no	—
52	C₄⋊C₄×C₆₇	—	8	4	no	—
53	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₆₇	SD₁₆	1	2	yes	—
55	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	8	2	no	—
59	C₂×C₄×C₆₇	—	8	4	no	—
60	C₂×C₄×C₆₇	#102	1	4	yes	—
61	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₆₇	—	4	4	no	—
68	C₂×Dic₆₇	—	8	2	no	—
69	C₂×Dic₆₇	—	8	2	no	—
70	C₈×C₆₇	#103	1	4	yes	—
71	C₈×C₆₇	C₄×Q₈	1	2	yes	—
72	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
74	C₂²×C₆₇	#486	1	1	yes	characteristic
75	C₂²×C₆₇	#500	1	1	yes	characteristic
76	C₂²×C₆₇	#509	1	2	yes	—
77	C₂²×C₆₇	#510	1	2	yes	—
78	C₄×C₆₇	—	2	4	no	—
79	C₄×C₆₇	—	16	2	no	—
80	C₄×C₆₇	#391	1	1	yes	characteristic
81	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
82	C₄×C₆₇	#487	1	1	yes	characteristic
83	C₄×C₆₇	#498	1	1	yes	characteristic
84	Dic₆₇	—	8	4	no	—
85	#13284	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	#1990	1	1	yes	characteristic
87	C₂×C₆₇	#2139	1	2	yes	—
88	C₂×C₆₇	#2244	1	1	yes	characteristic
89	C₂×C₆₇	#2245	1	1	yes	characteristic
90	C₂×C₆₇	#2255	1	2	yes	—
91	C₂×#499	—	67	1	no	—
92	C₂×#346	—	67	1	no	—
93	#1845	D₆₇	1	1	yes	characteristic
94	#1380	—	67	4	no	—
95	C₆₇	#13284	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	134	1	no	—
97	C₂²×C₁₆	—	67	2	no	—
98	#499	—	67	4	no	—
99	#499	—	134	2	no	—
100	C₂×#107	—	2	1	no	—
101	C₂×#107	—	67	2	no	—
102	C₂×#107	C₂×D₆₇	1	2	yes	—
103	C₂×#106	—	134	1	no	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	#384	—	2	1	no	—
106	#346	—	134	2	no	—
107	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	#125	—	268	2	no	—
110	#125	—	134	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	268	2	no	—
114	C₂×C₄⋊C₄	—	134	2	no	—
115	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂×C₁₆	—	67	4	no	—
117	C₂×C₁₆	—	134	2	no	—
118	C₂×C₁₆	—	134	2	no	—
119	#107	—	2	4	no	—
120	#107	—	4	2	no	—
121	#107	—	67	4	no	—
122	#107	—	134	2	no	—
123	#106	—	134	4	no	—
124	C₄×C₈	C₄×D₆₇	1	2	yes	—
125	C₄×C₈	D₂₆₈	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	268	2	no	—
128	C₂²×C₄	#11761613	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
130	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
131	C₂×C₈	#11761616	1	2	yes	—
132	C₂×C₈	#11761618	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₆₇	1	2	yes	—
136	C₂×C₈	#11761627	1	2	yes	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₄⋊C₄	—	268	8	no	—
141	C₄⋊C₄	—	268	2	no	—
142	C₄⋊C₄	—	134	4	no	—
143	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
144	C₄²	#11761615	1	2	yes	—
145	C₁₆	—	134	4	no	—
146	C₂³	C₂×#11761613	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	268	4	no	—
152	C₂×C₄	—	536	2	no	—
153	C₂×C₄	—	536	2	no	—
154	C₂×C₄	#14523963	1	4	yes	—
155	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#14524030	1	1	yes	characteristic
157	C₂×C₄	#14524041	1	1	yes	characteristic
158	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
159	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
160	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
161	C₈	#14523966	1	4	yes	—
162	C₈	#14524008	1	2	yes	—
163	C₈	#14524023	1	2	yes	—
164	C₂²	C₂×#14523963	1	1	yes	characteristic
165	C₂²	#16863742	1	1	yes	characteristic
166	C₂²	#16863799	1	1	yes	characteristic
167	C₂²	#16863832	1	2	yes	—
168	C₂²	#16863837	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	16	2	no	—
171	C₄	—	536	4	no	—
172	C₄	#16863301	1	1	yes	characteristic
173	C₄	C₂×#14523966	1	1	yes	characteristic
174	C₄	#16863746	1	1	yes	characteristic
175	C₄	#16863792	1	1	yes	characteristic
176	C₂	#22561169	1	1	yes	characteristic
177	C₂	#22561919	1	2	yes	—
178	C₂	#22562450	1	1	yes	characteristic
179	C₂	#22562454	1	1	yes	characteristic
180	C₂	#22562492	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863798

C₂

yes

maximal, characteristic

C₂×#16863172

C₂

yes

maximal, characteristic

#22558300

C₂

yes

maximal

C₆₇×#1845

C₂

yes

maximal, characteristic

C₂×#14524035

—

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

#16863798

—

#16863798

C₄

yes

complemented

C₂×#14523977

C₂²

yes

—

C₂×#14523975

—

#16863172

—

C₂×#107×C₆₇

—

C₂×#107×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

—

#14524035

—

#14524035

—

C₂×C₄⋊Dic₆₇

—

C₂×Dic₆₇⋊C₄

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

Q₈

yes

—

#14523977

—

#14523977

C₂×C₄

yes

—

#14523975

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₈

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

Q₁₆

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

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

SD₁₆

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#124

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

C₂×Q₁₆

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

C₄×Q₈

yes

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#510

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×#499

—

C₂×#346

—

#1845

D₆₇

yes

characteristic

#1380

—

C₆₇

#13284

yes

characteristic, complemented, Sylow

C₂×#125

—

134

—

C₂²×C₁₆

—

#499

—

#499

—

134

—

100

C₂×#107

—

101

C₂×#107

—

102

C₂×#107

C₂×D₆₇

yes

—

103

C₂×#106

—

134

—

104

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

105

#384

—

106

#346

—

134

—

107

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

108

C₂²×C₈

C₂²×D₆₇

yes

characteristic

109

#125

—

268

—

110

#125

—

134

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

268

—

114

C₂×C₄⋊C₄

—

134

—

115

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

116

C₂×C₁₆

—

117

C₂×C₁₆

—

134

—

118

C₂×C₁₆

—

134

—

119

#107

—

120

#107

—

121

#107

—

122

#107

—

134

—

123

#106

—

134

—

124

C₄×C₈

C₄×D₆₇

yes

—

125

C₄×C₈

D₂₆₈

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

268

—

128

C₂²×C₄

#11761613

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

130

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

131

C₂×C₈

#11761616

yes

—

132

C₂×C₈

#11761618

yes

—

133

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

134

C₂×C₈

C₂×D₂₆₈

yes

characteristic

135

C₂×C₈

Q₈×D₆₇

yes

—

136

C₂×C₈

#11761627

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

268

—

141

C₄⋊C₄

—

268

—

142

C₄⋊C₄

—

134

—

143

C₄²

(C₂×D₆₇)⋊C₄

yes

—

144

C₄²

#11761615

yes

—

145

C₁₆

—

134

—

146

C₂³

C₂×#11761613

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

268

—

152

C₂×C₄

—

536

—

153

C₂×C₄

—

536

—

154

C₂×C₄

#14523963

yes

—

155

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

156

C₂×C₄

#14524030

yes

characteristic

157

C₂×C₄

#14524041

yes

characteristic

158

C₂×C₄

C₂×#11761615

yes

characteristic

159

C₂×C₄

C₂×#11761616

yes

characteristic

160

C₂×C₄

C₂×#11761618

yes

characteristic

161

C₈

#14523966

yes

—

162

C₈

#14524008

yes

—

163

C₈

#14524023

yes

—

164

C₂²

C₂×#14523963

yes

characteristic

165

C₂²

#16863742

yes

characteristic

166

C₂²

#16863799

yes

characteristic

167

C₂²

#16863832

yes

—

168

C₂²

#16863837

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

536

—

172

C₄

#16863301

yes

characteristic

173

C₄

C₂×#14523966

yes

characteristic

174

C₄

#16863746

yes

characteristic

175

C₄

#16863792

yes

characteristic

176

C₂

#22561169

yes

characteristic

177

C₂

#22561919

yes

—

178

C₂

#22562450

yes

characteristic

179

C₂

#22562454

yes

characteristic

180

C₂

#22562492

yes

—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#13284
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1845

C₂

socle, prime radical

C₂³×C₆₇

C₂×D₈

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
8. #16863798	C₄	semidirect
95. C₆₇	#13284	semidirect

H ≅ G/N

type

8. #16863798

C₄

semidirect

95. C₆₇

#13284

semidirect

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

subgroup	index
C₂×#16863798	2
C₂×#16863172	2
#22558300	2
C₆₇×#1845	2
#13284	67

subgroup

index

C₆₇×#1845

Subgroups — #30774439

Conjugacy classes of subgroups

Complements

Maximal subgroups