Group #30775728 — subgroups

Subgroups — #30775728

Order 17152.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22562004	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561824	C₂	1	1	yes	maximal, characteristic
3	#22560260	C₂	1	2	yes	maximal
4	#22558304	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14524049	C₂²	1	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14523979	C₂²	1	2	yes	—
9	#16863414	—	2	2	no	—
10	#16863226	—	2	2	no	—
11	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
12	#16863166	—	2	2	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
14	#427×C₆₇	C₂²	1	2	yes	—
15	C₂×#11761628	D₄	1	1	yes	characteristic
16	#14524049	—	2	2	no	—
17	#14524049	—	2	2	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	D₄	1	1	yes	characteristic
20	C₂×#11761611	—	2	1	no	—
21	C₂×C₈×D₆₇	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	1	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	#14523979	—	2	2	no	—
25	#14523979	—	2	2	no	—
26	#14523971	—	4	4	no	—
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	#117×C₆₇	—	2	2	no	—
29	M₃₂×C₆₇	—	2	2	no	—
30	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
31	#107×C₆₇	—	2	2	no	—
32	#106×C₆₇	—	2	2	no	—
33	#11761628	—	2	2	no	—
34	#11761628	—	2	2	no	—
35	C₂×Dic₁₃₄	Q₁₆	1	1	yes	characteristic
36	C₂×D₂₆₈	Q₁₆	1	1	yes	characteristic
37	C₂×C₂²⋊D₆₇	—	2	1	no	—
38	C₂×C₄×D₆₇	—	2	1	no	—
39	#11761612	—	2	2	no	—
40	#11761612	—	4	2	no	—
41	#11761612	—	4	2	no	—
42	#11761611	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	2	1	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₈×D₆₇	—	4	1	no	—
46	C₈×D₆₇	—	4	1	no	—
47	C₆₇⋊C₁₆	—	4	1	no	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	#41	1	2	yes	—
51	C₂×C₈×C₆₇	#41	1	2	yes	—
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₆₇	—	4	2	no	—
54	C₂²⋊C₄×C₆₇	—	4	2	no	—
55	C₄²×C₆₇	—	4	2	no	—
56	C₁₆×C₆₇	—	2	2	no	—
57	C₂³×C₆₇	#135	1	1	yes	characteristic
58	C₂²×D₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	8	2	no	—
60	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#123	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₂²⋊D₆₇	—	4	2	no	—
65	C₂×Dic₆₇	—	2	1	no	—
66	D₂₆₈	—	2	2	no	—
67	C₄×D₆₇	—	4	1	no	—
68	C₄×D₆₇	—	4	1	no	—
69	Dic₁₃₄	—	2	2	no	—
70	C₈×C₆₇	#123	1	1	yes	characteristic
71	C₈×C₆₇	#124	1	1	yes	characteristic
72	C₈×C₆₇	#126	1	2	yes	—
73	C₆₇⋊C₈	—	4	1	no	—
74	C₆₇⋊C₈	—	8	2	no	—
75	C₂²×C₆₇	—	4	1	no	—
76	C₂²×C₆₇	#467	1	1	yes	characteristic
77	C₂²×C₆₇	#482	1	2	yes	—
78	C₂×D₆₇	—	4	2	no	—
79	C₂×D₆₇	—	4	1	no	—
80	C₄×C₆₇	—	16	2	no	—
81	C₄×C₆₇	#397	1	1	yes	characteristic
82	C₄×C₆₇	#469	1	1	yes	characteristic
83	C₄×C₆₇	#484	1	2	yes	—
84	Dic₆₇	—	4	1	no	—
85	#13499	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	2	2	no	—
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	#2076	1	1	yes	characteristic
89	D₆₇	—	8	1	no	—
90	#2161	—	67	1	no	—
91	#2113	—	67	1	no	—
92	#1814	—	67	2	no	—
93	#1384	D₆₇	1	1	yes	characteristic, complemented
94	#1382	—	67	2	no	—
95	C₆₇	#13499	1	1	yes	characteristic, complemented, Sylow
96	C₂×#131	—	67	1	no	—
97	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
98	C₂²×C₁₆	—	67	1	no	—
99	C₂×#108	—	67	2	no	—
100	#427	C₂×D₆₇	1	2	yes	—
101	#421	—	134	2	no	—
102	#368	—	134	2	no	—
103	C₂²⋊C₁₆	—	134	2	no	—
104	#344	—	134	2	no	—
105	C₂×#41	—	67	1	no	—
106	#131	—	134	2	no	—
107	#131	—	134	2	no	—
108	C₂×M₁₆	—	67	1	no	—
109	C₂×M₁₆	—	134	2	no	—
110	C₂×M₁₆	—	134	1	no	—
111	C₂²×C₈	—	134	1	no	—
112	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	#117	—	2	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	134	1	no	—
116	C₂×C₁₆	—	134	2	no	—
117	C₂×C₁₆	D₂₆₈	1	2	yes	—
118	#108	—	134	2	no	—
119	#108	—	134	2	no	—
120	#107	—	2	2	no	—
121	#106	—	2	2	no	—
122	#104	—	268	4	no	—
123	#41	—	134	2	no	—
124	#41	—	134	2	no	—
125	C₂×Q₈	—	67	1	no	—
126	C₂×D₄	—	67	1	no	—
127	C₂×D₄	—	134	1	no	—
128	C₂²×C₄	—	134	1	no	—
129	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
130	M₁₆	—	268	2	no	—
131	M₁₆	—	268	2	no	—
132	M₁₆	—	268	2	no	—
133	M₁₆	—	134	2	no	—
134	C₂×C₈	—	268	1	no	—
135	C₂×C₈	—	268	1	no	—
136	C₂×C₈	—	268	2	no	—
137	C₂×C₈	—	134	1	no	—
138	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
139	C₂×C₈	#11761627	1	2	yes	—
140	C₂×C₈	#11761628	1	2	yes	—
141	C₂×C₈	#11761629	1	1	yes	characteristic
142	C₄⋊C₄	—	4	2	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₄²	—	4	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	268	1	no	—
147	C₂³	—	134	1	no	—
148	C₂³	#14524062	1	1	yes	characteristic
149	Q₈	—	134	2	no	—
150	D₄	—	268	2	no	—
151	D₄	—	134	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	268	1	no	—
154	C₂×C₄	—	268	1	no	—
155	C₂×C₄	—	134	1	no	—
156	C₂×C₄	#14524018	1	1	yes	characteristic
157	C₂×C₄	#14524025	1	2	yes	—
158	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
159	C₂×C₄	#14524070	1	2	yes	—
160	C₈	—	268	1	no	—
161	C₈	—	536	2	no	—
162	C₈	#14524027	1	1	yes	characteristic
163	C₈	#14524031	1	1	yes	characteristic
164	C₈	#14524038	1	2	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	268	2	no	—
167	C₂²	—	268	1	no	—
168	C₂²	#16863638	1	1	yes	characteristic
169	C₂²	#16863718	1	2	yes	—
170	C₄	—	268	1	no	—
171	C₄	—	16	2	no	—
172	C₄	#16863323	1	1	yes	characteristic
173	C₄	#16863650	1	1	yes	characteristic
174	C₄	#16863730	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	536	1	no	—
178	C₂	#22561639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562004

C₂

yes

maximal, characteristic, complemented

#22561824

C₂

yes

maximal, characteristic

#22560260

C₂

yes

maximal

#22558304

C₂

yes

maximal, complemented

C₆₇×#1384

C₂

yes

maximal, characteristic, complemented

C₂×#14524049

C₂²

yes

characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

#16863414

—

#16863226

—

(C₂×D₆₇)⋊C₁₆

—

#16863166

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

#427×C₆₇

C₂²

yes

—

C₂×#11761628

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11761611

—

C₂×C₈×D₆₇

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

#14523979

—

#14523979

—

#14523971

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#106×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

Q₁₆

yes

characteristic

C₂×D₂₆₈

Q₁₆

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₈×D₆₇

—

C₈×D₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

#123

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#467

yes

characteristic

C₂²×C₆₇

#482

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#13499

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#2076

yes

characteristic

D₆₇

—

—

—

—

D₆₇

yes

characteristic, complemented

#1382

—

C₆₇

#13499

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

100

#427

C₂×D₆₇

yes

—

101

#421

—

134

—

102

#368

—

134

—

103

C₂²⋊C₁₆

—

134

—

104

#344

—

134

—

105

C₂×#41

—

106

#131

—

134

—

107

#131

—

134

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

134

—

110

C₂×M₁₆

—

134

—

111

C₂²×C₈

—

134

—

112

C₂²×C₈

C₂²×D₆₇

yes

characteristic

113

#117

—

114

M₃₂

—

115

C₂×C₁₆

—

134

—

116

C₂×C₁₆

—

134

—

117

C₂×C₁₆

D₂₆₈

yes

—

118

#108

—

134

—

119

#108

—

134

—

120

#107

—

121

#106

—

122

#104

—

268

—

123

#41

—

134

—

124

#41

—

134

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

134

—

128

C₂²×C₄

—

134

—

129

C₂²×C₄

D₄×D₆₇

yes

characteristic

130

M₁₆

—

268

—

131

M₁₆

—

268

—

132

M₁₆

—

268

—

133

M₁₆

—

134

—

134

C₂×C₈

—

268

—

135

C₂×C₈

—

268

—

136

C₂×C₈

—

268

—

137

C₂×C₈

—

134

—

138

C₂×C₈

C₂×D₂₆₈

yes

characteristic

139

C₂×C₈

#11761627

yes

—

140

C₂×C₈

#11761628

yes

—

141

C₂×C₈

#11761629

yes

characteristic

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₁₆

—

146

C₁₆

—

268

—

147

C₂³

—

134

—

148

C₂³

#14524062

yes

characteristic

149

Q₈

—

134

—

150

D₄

—

268

—

151

D₄

—

134

—

152

C₂×C₄

—

153

C₂×C₄

—

268

—

154

C₂×C₄

—

268

—

155

C₂×C₄

—

134

—

156

C₂×C₄

#14524018

yes

characteristic

157

C₂×C₄

#14524025

yes

—

158

C₂×C₄

Q₁₆×D₆₇

yes

characteristic

159

C₂×C₄

#14524070

yes

—

160

C₈

—

268

—

161

C₈

—

536

—

162

C₈

#14524027

yes

characteristic

163

C₈

#14524031

yes

characteristic

164

C₈

#14524038

yes

—

165

C₂²

—

166

C₂²

—

268

—

167

C₂²

—

268

—

168

C₂²

#16863638

yes

characteristic

169

C₂²

#16863718

yes

—

170

C₄

—

268

—

171

C₄

—

172

C₄

#16863323

yes

characteristic

173

C₄

#16863650

yes

characteristic

174

C₄

#16863730

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

536

—

178

C₂

#22561639

yes

characteristic

	subgroup	G/H
center	C₄	#16863650
commutator, cosocle	C₂²×C₈×C₆₇	C₂³
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1384	C₂
socle	C₂×C₆₇	#2076
prime radical	C₂×#11761628	D₄
hypercenter	#1384	D₆₇
hypocenter	C₆₇	#13499
LCS	G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#16863650

commutator, cosocle

C₂²×C₈×C₆₇

C₂³

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1384

C₂

C₂×C₆₇

D₄

D₆₇

C₆₇

G ▷ C₂²×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
1. #22562004	C₂	semidirect
4. #22558304	C₂	semidirect
5. C₆₇×#1384	C₂	semidirect
93. #1384	D₆₇	semidirect
95. C₆₇	#13499	semidirect

H ≅ G/N

type

1. #22562004

C₂

semidirect

4. #22558304

C₂

semidirect

5. C₆₇×#1384

C₂

semidirect

93. #1384

D₆₇

semidirect

95. C₆₇

#13499

semidirect

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

subgroup	index
#22562004	2
#22561824	2
#22560260	2
#22558304	2
C₆₇×#1384	2
#13499	67

subgroup

index

C₆₇×#1384

Subgroups — #30775728

Conjugacy classes of subgroups

Complements

Maximal subgroups