Group #30755316 — subgroups

Subgroups — #30755316

Order 17152.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16863794	C₂	1	1	yes	maximal, characteristic
2	#22560347	C₂	1	1	yes	maximal, characteristic
3	#22558893	C₂	1	2	yes	maximal
4	#22558889	C₂	1	2	yes	maximal
5	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	2	yes	—
7	#16863794	—	2	2	no	—
8	#16863794	C₄	1	4	yes	—
9	C₂×#14523975	—	2	2	no	—
10	#16863420	C₂²	1	1	yes	characteristic
11	#16863420	C₂²	1	1	yes	characteristic
12	#16863161	—	2	2	no	—
13	#16863144	—	2	4	no	—
14	#16863143	—	2	4	no	—
15	#431×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#322×C₆₇	—	2	2	no	—
18	C₂²×Dic₁₃₄	—	2	2	no	—
19	C₂×#11761612	—	2	1	no	—
20	C₂×#11761612	—	2	1	no	—
21	#14524043	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	#14524043	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	#14523975	—	2	8	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₂²×C₈×C₆₇	D₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
33	C₂²⋊C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₂×Dic₁₃₄	—	2	4	no	—
38	C₂×Dic₁₃₄	—	4	2	no	—
39	C₂×Dic₁₃₄	—	8	4	no	—
40	C₂²×Dic₆₇	—	4	2	no	—
41	#11761612	—	4	2	no	—
42	#11761612	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	1	no	—
44	C₂×C₆₇⋊C₈	—	4	1	no	—
45	C₄⋊Dic₆₇	—	2	4	no	—
46	C₄⋊Dic₆₇	—	2	4	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₄⋊Dic₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	#41	1	2	yes	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₆₇	C₂×D₄	1	2	yes	—
58	C₂³×C₆₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₇	—	2	4	no	—
60	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₆₇	#121	1	2	yes	—
63	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#137	1	2	yes	—
65	C₂×C₄×C₆₇	#137	1	2	yes	—
66	C₂×Dic₆₇	—	4	4	no	—
67	C₂×Dic₆₇	—	8	4	no	—
68	Dic₁₃₄	—	8	4	no	—
69	Dic₁₃₄	—	8	4	no	—
70	C₈×C₆₇	—	2	4	no	—
71	C₈×C₆₇	—	4	4	no	—
72	C₆₇⋊C₈	—	8	1	no	—
73	C₆₇⋊C₈	—	8	1	no	—
74	C₂²×C₆₇	—	2	2	no	—
75	C₂²×C₆₇	#390	1	1	yes	characteristic
76	C₂²×C₆₇	#491	1	2	yes	—
77	C₄×C₆₇	—	2	2	no	—
78	C₄×C₆₇	#419	1	2	yes	—
79	C₄×C₆₇	#471	1	4	yes	—
80	C₄×C₆₇	#497	1	1	yes	characteristic
81	C₄×C₆₇	#497	1	1	yes	characteristic
82	Dic₆₇	—	8	4	no	—
83	#9776	—	67	1	no	maximal, Sylow
84	C₂×C₆₇	—	2	2	no	—
85	C₂×C₆₇	#1960	1	1	yes	characteristic
86	C₂×C₆₇	#1977	1	2	yes	—
87	C₂×#498	—	67	1	no	—
88	#1837	—	67	1	no	—
89	#1549	—	67	2	no	—
90	#1548	—	67	2	no	—
91	#1452	D₆₇	1	1	yes	characteristic
92	C₆₇	#9776	1	1	yes	characteristic, complemented, Sylow
93	C₂×C₄⋊Q₈	—	67	2	no	—
94	#498	—	67	4	no	—
95	#498	—	134	2	no	—
96	#431	C₂×D₆₇	1	2	yes	—
97	C₂×#106	—	134	2	no	—
98	#423	—	67	1	no	—
99	#423	—	67	1	no	—
100	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	#341	—	134	2	no	—
102	#333	—	134	4	no	—
103	#332	—	134	4	no	—
104	#322	—	2	2	no	—
105	C₂²×Q₈	—	134	2	no	—
106	C₂×M₁₆	—	134	1	no	—
107	C₂×M₁₆	—	134	1	no	—
108	C₂²×C₈	D₂₆₈	1	2	yes	—
109	C₄⋊Q₈	—	67	4	no	—
110	C₄⋊Q₈	—	134	2	no	—
111	C₄⋊Q₈	—	134	2	no	—
112	C₄⋊Q₈	—	134	2	no	—
113	C₂×C₄⋊C₄	—	134	2	no	—
114	C₂×C₄⋊C₄	—	134	2	no	—
115	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	#106	—	134	8	no	—
117	C₄⋊C₈	—	134	2	no	—
118	C₄⋊C₈	—	134	2	no	—
119	C₄⋊C₈	D₂₆₈	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₆₇	1	2	yes	—
123	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
124	C₂×Q₈	—	268	2	no	—
125	C₂×Q₈	—	536	4	no	—
126	C₂×Q₈	—	134	4	no	—
127	C₂²×C₄	—	268	2	no	—
128	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
129	C₂²×C₄	#11761628	1	2	yes	—
130	M₁₆	—	268	2	no	—
131	M₁₆	—	268	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	268	1	no	—
136	C₂×C₈	—	268	1	no	—
137	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	268	2	no	—
139	C₄⋊C₄	—	268	2	no	—
140	C₄⋊C₄	—	134	4	no	—
141	C₄⋊C₄	—	134	4	no	—
142	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
143	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₄²	C₂×D₂₆₈	1	2	yes	—
145	C₂³	#14524031	1	1	yes	characteristic
146	Q₈	—	536	4	no	—
147	Q₈	—	536	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	268	4	no	—
150	C₂×C₄	—	536	4	no	—
151	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
153	C₂×C₄	#14524017	1	2	yes	—
154	C₂×C₄	#14524040	1	1	yes	characteristic
155	C₂×C₄	#14524076	1	2	yes	—
156	C₂×C₄	#14524076	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	536	1	no	—
160	C₈	—	536	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#16863298	1	1	yes	characteristic
163	C₂²	#16863765	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	536	4	no	—
166	C₄	#16863406	1	2	yes	—
167	C₄	#16863659	1	4	yes	—
168	C₄	#16863790	1	1	yes	characteristic
169	C₄	#16863790	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#22561029	1	1	yes	characteristic
172	C₂	#22561112	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863794

C₂

yes

maximal, characteristic

#22560347

C₂

yes

maximal, characteristic

#22558893

C₂

yes

maximal

#22558889

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

—

#16863794

—

#16863794

C₄

yes

—

C₂×#14523975

—

#16863420

C₂²

yes

characteristic

#16863420

C₂²

yes

—

—

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

—

C₂²×Dic₁₃₄

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523975

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

D₄

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#491

yes

—

C₄×C₆₇

—

C₄×C₆₇

#419

yes

—

C₄×C₆₇

#471

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

#9776

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1960

yes

characteristic

C₂×C₆₇

#1977

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#9776

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#498

—

#498

—

134

—

#431

C₂×D₆₇

yes

—

C₂×#106

—

134

—

#423

—

#423

—

100

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

101

#341

—

134

—

102

#333

—

134

—

103

#332

—

134

—

104

#322

—

105

C₂²×Q₈

—

134

—

106

C₂×M₁₆

—

134

—

107

C₂×M₁₆

—

134

—

108

C₂²×C₈

D₂₆₈

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

134

—

111

C₄⋊Q₈

—

134

—

112

C₄⋊Q₈

—

134

—

113

C₂×C₄⋊C₄

—

134

—

114

C₂×C₄⋊C₄

—

134

—

115

C₂×C₄²

C₂²×D₆₇

yes

characteristic

116

#106

—

134

—

117

C₄⋊C₈

—

134

—

118

C₄⋊C₈

—

134

—

119

C₄⋊C₈

D₂₆₈

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₄×D₆₇

yes

—

123

C₄×C₈

C₂²⋊D₆₇

yes

—

124

C₂×Q₈

—

268

—

125

C₂×Q₈

—

536

—

126

C₂×Q₈

—

134

—

127

C₂²×C₄

—

268

—

128

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

129

C₂²×C₄

#11761628

yes

—

130

M₁₆

—

268

—

131

M₁₆

—

268

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

268

—

136

C₂×C₈

—

268

—

137

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

138

C₄⋊C₄

—

268

—

139

C₄⋊C₄

—

268

—

140

C₄⋊C₄

—

134

—

141

C₄⋊C₄

—

134

—

142

C₄²

C₂×C₄×D₆₇

yes

characteristic

143

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

144

C₄²

C₂×D₂₆₈

yes

—

145

C₂³

#14524031

yes

characteristic

146

Q₈

—

536

—

147

Q₈

—

536

—

148

C₂×C₄

—

149

C₂×C₄

—

268

—

150

C₂×C₄

—

536

—

151

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

152

C₂×C₄

C₄×D₂₆₈

yes

—

153

C₂×C₄

#14524017

yes

—

154

C₂×C₄

#14524040

yes

characteristic

155

C₂×C₄

#14524076

yes

—

156

C₂×C₄

#14524076

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

536

—

160

C₈

—

536

—

161

C₂²

—

162

C₂²

#16863298

yes

characteristic

163

C₂²

#16863765

yes

—

164

C₄

—

165

C₄

—

536

—

166

C₄

#16863406

yes

—

167

C₄

#16863659

yes

—

168

C₄

#16863790

yes

characteristic

169

C₄

#16863790

yes

characteristic

170

C₂

—

171

C₂

#22561029

yes

characteristic

172

C₂

#22561112

yes

—

	subgroup	G/H
center	C₂²	#16863298
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#9776
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16863298

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
92. C₆₇	#9776	semidirect

H ≅ G/N

type

92. C₆₇

#9776

semidirect

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

subgroup	index
C₂×#16863794	2
#22560347	2
#22558893	2
#22558889	2
C₆₇×#1452	2
#9776	67

subgroup

index

C₆₇×#1452

Subgroups — #30755316

Conjugacy classes of subgroups

Complements

Maximal subgroups