Group #30749583 — subgroups

Subgroups — #30749583

Order 17152.

402 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 187 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	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22559955	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863146	C₂	1	4	yes	maximal
4	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#16863151	C₄	1	2	yes	—
9	#16863151	C₄	1	2	yes	—
10	#16863146	—	2	8	no	—
11	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₈	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
18	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
19	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
20	C₂²×C₈×C₆₇	—	2	2	no	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₇	D₄	1	2	yes	—
24	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₆₇	—	2	4	no	—
27	C₄⋊C₈×C₆₇	—	4	4	no	—
28	C₂×C₆₇⋊C₈	—	2	4	no	—
29	C₂×C₆₇⋊C₈	—	2	4	no	—
30	C₂×C₆₇⋊C₈	—	4	2	no	—
31	C₂×C₆₇⋊C₈	—	4	4	no	—
32	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
33	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₆₇	—	4	2	no	—
39	C₂×C₈×C₆₇	—	4	4	no	—
40	C₄²×C₆₇	—	2	2	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	—	4	2	no	—
43	C₄²×C₆₇	SD₁₆	1	4	yes	—
44	C₄²×C₆₇	SD₁₆	1	4	yes	—
45	C₄²×C₆₇	C₂×D₄	1	2	yes	—
46	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	#106	1	4	yes	—
54	C₂×C₄×C₆₇	#106	1	4	yes	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₆₇	#131	1	2	yes	—
60	C₂×C₄×C₆₇	#131	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
63	C₈×C₆₇	—	8	2	no	—
64	C₆₇⋊C₈	—	2	8	no	—
65	C₆₇⋊C₈	—	8	2	no	—
66	C₂²×C₆₇	#384	1	1	yes	characteristic
67	C₂²×C₆₇	#424	1	1	yes	characteristic
68	C₂²×C₆₇	#424	1	1	yes	characteristic
69	C₂²×C₆₇	C₂×#106	1	2	yes	—
70	C₂²×C₆₇	#430	1	2	yes	—
71	C₄×C₆₇	—	2	4	no	—
72	C₄×C₆₇	—	2	4	no	—
73	C₄×C₆₇	—	2	2	no	—
74	C₄×C₆₇	#424	1	4	yes	—
75	C₄×C₆₇	#492	1	1	yes	characteristic
76	C₄×C₆₇	#492	1	1	yes	characteristic
77	C₄×C₆₇	#498	1	2	yes	—
78	#8809	—	67	1	no	maximal, Sylow
79	C₂×C₆₇	#1570	1	2	yes	—
80	C₂×C₆₇	#1570	1	2	yes	—
81	C₂×C₆₇	#1843	1	1	yes	characteristic
82	C₂×C₆₇	#1844	1	1	yes	characteristic
83	C₂×C₆₇	#1849	1	1	yes	characteristic
84	#1842	—	67	1	no	—
85	#1842	D₆₇	1	1	yes	characteristic
86	#1744	—	67	1	no	—
87	C₂×#334	—	67	4	no	—
88	C₆₇	#8809	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₄⋊C₈	—	2	2	no	—
90	C₂×C₄⋊C₈	—	67	2	no	—
91	C₂×C₄⋊C₈	—	134	2	no	—
92	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
93	C₂×C₄×C₈	—	67	2	no	—
94	C₄³	C₂×D₆₇	1	1	yes	characteristic
95	#336	—	67	2	no	—
96	#336	—	67	2	no	—
97	#334	—	134	8	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	67	4	no	—
100	C₂²×C₈	—	134	2	no	—
101	C₂×C₄²	Dic₁₃₄	1	2	yes	—
102	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
105	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄⋊C₈	—	268	4	no	—
109	C₄⋊C₈	—	134	4	no	—
110	C₄×C₈	—	67	4	no	—
111	C₄×C₈	—	67	4	no	—
112	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
113	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	—	268	2	no	—
120	C₂×C₈	—	268	4	no	—
121	C₂×C₈	—	67	8	no	—
122	C₂×C₈	—	134	4	no	—
123	C₂×C₈	—	134	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	2	no	—
127	C₄²	#11761615	1	4	yes	—
128	C₄²	#11761616	1	4	yes	—
129	C₄²	D₄×D₆₇	1	2	yes	—
130	C₄²	Q₈×D₆₇	1	2	yes	—
131	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#14523974	1	4	yes	—
138	C₂×C₄	#14523974	1	4	yes	—
139	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
140	C₂×C₄	#14524039	1	1	yes	characteristic
141	C₂×C₄	#14524041	1	1	yes	characteristic
142	C₂×C₄	#14524042	1	2	yes	—
143	C₂×C₄	#14524049	1	2	yes	—
144	C₂×C₄	#14524051	1	2	yes	—
145	C₂×C₄	C₂×#11761615	1	2	yes	—
146	C₂×C₄	C₂×#11761616	1	2	yes	—
147	C₈	—	8	2	no	—
148	C₈	—	536	2	no	—
149	C₈	—	134	8	no	—
150	C₂²	#16863265	1	1	yes	characteristic
151	C₂²	#16863422	1	1	yes	characteristic
152	C₂²	#16863424	1	1	yes	characteristic
153	C₂²	C₂×#14523974	1	2	yes	—
154	C₂²	#16863444	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	#16863421	1	4	yes	—
159	C₄	#16863768	1	1	yes	characteristic
160	C₄	#16863770	1	1	yes	characteristic
161	C₄	#16863792	1	2	yes	—
162	C₂	#22558979	1	2	yes	—
163	C₂	#22558981	1	2	yes	—
164	C₂	#22560362	1	1	yes	characteristic
165	C₂	#22560365	1	1	yes	characteristic
166	C₂	#22560383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560361

C₂

yes

maximal, characteristic

#22559955

C₂

yes

maximal, characteristic

C₂×#16863146

C₂

yes

maximal

C₆₇×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₆₇⋊C₈)

—

C₂×C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863151

C₄

yes

—

#16863151

C₄

yes

—

#16863146

—

C₂×C₄⋊C₈×C₆₇

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

SD₁₆

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#106

yes

—

C₂×C₄×C₆₇

#106

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#430

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#424

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#8809

—

maximal, Sylow

C₂×C₆₇

#1570

yes

—

C₂×C₆₇

#1570

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

#1744

—

C₂×#334

—

C₆₇

#8809

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

134

—

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₆₇

yes

—

—

—

134

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

134

—

101

C₂×C₄²

Dic₁₃₄

yes

—

102

C₂×C₄²

C₄×D₆₇

yes

characteristic

103

C₂×C₄²

C₄×D₆₇

yes

characteristic

104

C₂×C₄²

C₂²⋊D₆₇

yes

—

105

C₂×C₄²

C₂²×D₆₇

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

268

—

109

C₄⋊C₈

—

134

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₂²×C₄

Dic₆₇⋊C₄

yes

—

113

C₂²×C₄

Dic₆₇⋊C₄

yes

—

114

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

115

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

116

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

268

—

120

C₂×C₈

—

268

—

121

C₂×C₈

—

122

C₂×C₈

—

134

—

123

C₂×C₈

—

134

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11761615

yes

—

128

C₄²

#11761616

yes

—

129

C₄²

D₄×D₆₇

yes

—

130

C₄²

Q₈×D₆₇

yes

—

131

C₂³

C₂×Dic₆₇⋊C₄

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14523974

yes

—

138

C₂×C₄

#14523974

yes

—

139

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

140

C₂×C₄

#14524039

yes

characteristic

141

C₂×C₄

#14524041

yes

characteristic

142

C₂×C₄

#14524042

yes

—

143

C₂×C₄

#14524049

yes

—

144

C₂×C₄

#14524051

yes

—

145

C₂×C₄

C₂×#11761615

yes

—

146

C₂×C₄

C₂×#11761616

yes

—

147

C₈

—

148

C₈

—

536

—

149

C₈

—

134

—

150

C₂²

#16863265

yes

characteristic

151

C₂²

#16863422

yes

characteristic

152

C₂²

#16863424

yes

characteristic

153

C₂²

C₂×#14523974

yes

—

154

C₂²

#16863444

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#16863421

yes

—

159

C₄

#16863768

yes

characteristic

160

C₄

#16863770

yes

characteristic

161

C₄

#16863792

yes

—

162

C₂

#22558979

yes

—

163

C₂

#22558981

yes

—

164

C₂

#22560362

yes

characteristic

165

C₂

#22560365

yes

characteristic

166

C₂

#22560383

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1842	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1842	D₆₇
hypocenter	C₆₇	#8809
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₆₇

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1842

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
88. C₆₇	#8809	semidirect

H ≅ G/N

type

88. C₆₇

#8809

semidirect

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

subgroup	index
#22560361	2
#22559955	2
C₂×#16863146	2
C₆₇×#1842	2
#8809	67

subgroup

index

C₆₇×#1842

Subgroups — #30749583

Conjugacy classes of subgroups

Complements

Maximal subgroups