Group #30747000 — subgroups

Subgroups — #30747000

Order 17152.

366 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄×#14523954	C₂	1	1	yes	maximal, characteristic
2	#22558955	C₂	1	2	yes	maximal
3	#22558502	C₂	1	2	yes	maximal
4	#22558042	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₄	1	2	yes	—
8	C₂×#14523954	C₄	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16863148	C₄	1	4	yes	complemented
12	#16863146	C₄	1	4	yes	complemented
13	#16863118	—	2	2	no	—
14	#16863117	—	2	2	no	—
15	#431×C₆₇	C₂²	1	2	yes	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
22	#14523954	—	2	4	no	—
23	#14523954	—	2	4	no	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₇	—	2	2	no	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	C₂×C₆₇⋊C₈	—	2	2	no	—
37	C₂×C₆₇⋊C₈	—	2	2	no	—
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₆₇⋊C₈	C₄²	1	1	yes	characteristic
42	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
43	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
46	C₂²×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	#41	1	2	yes	—
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₄²×C₆₇	—	2	2	no	—
51	C₄²×C₆₇	—	4	4	no	—
52	C₄²×C₆₇	M₁₆	1	4	yes	—
53	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
54	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₆₇	#117	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	4	no	—
59	C₂×C₄×C₆₇	—	4	4	no	—
60	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
66	C₈×C₆₇	—	4	4	no	—
67	C₆₇⋊C₈	—	2	1	no	—
68	C₆₇⋊C₈	—	2	1	no	—
69	C₆₇⋊C₈	—	2	2	no	—
70	C₆₇⋊C₈	—	2	2	no	—
71	C₆₇⋊C₈	—	4	4	no	—
72	C₆₇⋊C₈	#97	1	4	yes	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₆₇	#432	1	2	yes	—
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	—	4	4	no	—
78	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
79	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₆₇	#423	1	2	yes	—
81	C₄×C₆₇	#444	1	4	yes	—
82	#8269	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	—	2	2	no	—
84	C₂×C₆₇	#1767	1	1	yes	characteristic
85	C₂×C₆₇	#1773	1	2	yes	—
86	C₄×#97	—	67	1	no	—
87	#1564	—	67	2	no	—
88	#1452	—	67	2	no	—
89	#1276	—	67	1	no	—
90	#1276	D₆₇	1	1	yes	characteristic
91	C₆₇	#8269	1	1	yes	characteristic, complemented, Sylow
92	#431	—	67	2	no	—
93	#431	C₂×D₆₇	1	2	yes	—
94	C₂×#97	—	67	2	no	—
95	C₂×#97	—	67	2	no	—
96	C₂×C₄×C₈	—	67	1	no	—
97	C₂×C₄×C₈	—	67	1	no	—
98	C₄³	C₂×D₆₇	1	1	yes	characteristic
99	#335	—	67	4	no	—
100	#334	—	67	4	no	—
101	#322	—	134	2	no	—
102	#322	—	134	2	no	—
103	C₂²×C₈	—	67	1	no	—
104	C₂²×C₈	—	67	1	no	—
105	C₂²×C₈	—	67	2	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₄×D₆₇	1	2	yes	—
108	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
109	C₄⋊C₈	—	67	4	no	—
110	C₄⋊C₈	C₄×D₆₇	1	4	yes	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	134	2	no	—
113	#97	—	134	4	no	—
114	#97	—	134	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	67	2	no	—
117	C₄×C₈	—	67	2	no	—
118	C₄×C₈	—	67	1	no	—
119	C₄×C₈	—	67	1	no	—
120	C₄×C₈	—	67	2	no	—
121	C₄×C₈	—	134	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
124	C₂²×C₄	#11761627	1	2	yes	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	67	2	no	—
127	C₂×C₈	—	67	4	no	—
128	C₂×C₈	—	67	1	no	—
129	C₂×C₈	—	67	1	no	—
130	C₂×C₈	—	67	2	no	—
131	C₂×C₈	—	67	2	no	—
132	C₂×C₈	—	134	4	no	—
133	C₂×C₈	—	134	2	no	—
134	C₂×C₈	—	134	2	no	—
135	C₂×C₈	—	134	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	#11761611	1	4	yes	—
139	C₄²	C₂×C₄×D₆₇	1	2	yes	—
140	C₄²	D₄×D₆₇	1	1	yes	characteristic
141	C₄²	Q₈×D₆₇	1	1	yes	characteristic
142	C₂³	#14523999	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
147	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
148	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
149	C₂×C₄	#14524008	1	2	yes	—
150	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
151	C₂×C₄	C₂×#11761611	1	2	yes	—
152	C₈	—	4	4	no	—
153	C₈	—	268	4	no	—
154	C₈	—	67	4	no	—
155	C₈	—	134	1	no	—
156	C₈	—	134	1	no	—
157	C₈	—	134	2	no	—
158	C₈	—	134	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#16863247	1	1	yes	characteristic
161	C₂²	#16863450	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	4	4	no	—
164	C₄	#97×D₆₇	1	1	yes	characteristic
165	C₄	C₄×#11761611	1	1	yes	characteristic
166	C₄	#16863418	1	2	yes	—
167	C₄	#16863515	1	4	yes	—
168	C₂	—	2	2	no	—
169	C₂	#22560061	1	1	yes	characteristic
170	C₂	#22560092	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523954

C₂

yes

maximal, characteristic

#22558955

C₂

yes

maximal

#22558502

C₂

yes

maximal

#22558042

C₂

yes

maximal, characteristic

C₆₇×#1276

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523954

C₄

yes

—

C₂×#14523954

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#16863148

C₄

yes

complemented

#16863146

C₄

yes

complemented

#16863118

—

#16863117

—

#431×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

C₂×C₄

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄²

yes

characteristic

C₂×C₆₇⋊C₈

C₄²

yes

characteristic

C₂×C₆₇⋊C₈

C₄²

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

M₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

C₄×Q₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#97

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#97

yes

characteristic

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

#423

yes

—

C₄×C₆₇

#444

yes

—

#8269

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1767

yes

characteristic

C₂×C₆₇

#1773

yes

—

C₄×#97

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#8269

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂×D₆₇

yes

characteristic

#335

—

100

#334

—

101

#322

—

134

—

102

#322

—

134

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₄²

C₄×D₆₇

yes

—

108

C₂×C₄²

C₂²×D₆₇

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

C₄×D₆₇

yes

—

111

C₂²⋊C₈

—

112

C₂²⋊C₈

—

134

—

113

#97

—

134

—

114

#97

—

134

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

134

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

124

C₂²×C₄

#11761627

yes

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

134

—

133

C₂×C₈

—

134

—

134

C₂×C₈

—

134

—

135

C₂×C₈

—

134

—

136

C₄²

—

137

C₄²

—

138

C₄²

#11761611

yes

—

139

C₄²

C₂×C₄×D₆₇

yes

—

140

C₄²

D₄×D₆₇

yes

characteristic

141

C₄²

Q₈×D₆₇

yes

characteristic

142

C₂³

#14523999

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₄²×D₆₇

yes

characteristic

147

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

148

C₂×C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

149

C₂×C₄

#14524008

yes

—

150

C₂×C₄

M₁₆×D₆₇

yes

—

151

C₂×C₄

C₂×#11761611

yes

—

152

C₈

—

153

C₈

—

268

—

154

C₈

—

155

C₈

—

134

—

156

C₈

—

134

—

157

C₈

—

134

—

158

C₈

—

134

—

159

C₂²

—

160

C₂²

#16863247

yes

characteristic

161

C₂²

#16863450

yes

—

162

C₄

—

163

C₄

—

164

C₄

#97×D₆₇

yes

characteristic

165

C₄

C₄×#11761611

yes

characteristic

166

C₄

#16863418

yes

—

167

C₄

#16863515

yes

—

168

C₂

—

169

C₂

#22560061

yes

characteristic

170

C₂

#22560092

yes

—

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1276	C₂
socle	C₂²×C₆₇	C₄×C₄⋊C₄
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1276	D₆₇
hypocenter	C₆₇	#8269
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1276

C₂

socle

C₂²×C₆₇

C₄×C₄⋊C₄

prime radical

C₂³×C₆₇

#117

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #16863148	C₄	semidirect
12. #16863146	C₄	semidirect
91. C₆₇	#8269	semidirect

H ≅ G/N

type

11. #16863148

C₄

semidirect

12. #16863146

C₄

semidirect

91. C₆₇

#8269

semidirect

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

subgroup	index
C₄×#14523954	2
#22558955	2
#22558502	2
#22558042	2
C₆₇×#1276	2
#8269	67

subgroup

index

C₆₇×#1276

Subgroups — #30747000

Conjugacy classes of subgroups

Complements

Maximal subgroups