Group #30772180 — subgroups

Subgroups — #30772180

Order 17152.

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

normal subgroups only

This group has 322 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	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22559895	C₂	1	1	yes	maximal, characteristic
3	#22558286	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523954	C₄	1	1	yes	characteristic
7	C₂×#14523953	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	#16863242	C₄	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
11	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	2	no	—
14	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
16	C₂×C₆₇⋊C₁₆	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	—	2	4	no	—
18	#14523954	—	2	2	no	—
19	#14523953	—	2	4	no	—
20	C₈×Dic₆₇	—	2	2	no	—
21	C₈×Dic₆₇	—	2	2	no	—
22	C₆₇⋊#95	C₈	1	2	yes	—
23	C₆₇⋊#95	C₈	1	2	yes	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	—	2	4	no	—
29	C₄×C₈×C₆₇	D₄	1	2	yes	—
30	C₄×C₈×C₆₇	Q₈	1	2	yes	—
31	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
35	C₄×Dic₆₇	—	2	2	no	—
36	C₄×Dic₆₇	—	2	2	no	—
37	C₆₇⋊C₁₆	—	4	2	no	—
38	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	—	2	2	no	—
42	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	#41	1	2	yes	—
46	C₂×C₈×C₆₇	#41	1	2	yes	—
47	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₄²×C₆₇	M₁₆	1	2	yes	—
49	C₁₆×C₆₇	—	4	2	no	—
50	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#117	1	2	yes	—
57	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂×Dic₆₇	—	2	4	no	—
60	C₂×Dic₆₇	—	2	4	no	—
61	C₂×Dic₆₇	—	2	4	no	—
62	C₈×C₆₇	—	2	2	no	—
63	C₈×C₆₇	#124	1	1	yes	characteristic
64	C₈×C₆₇	#124	1	1	yes	characteristic
65	C₈×C₆₇	#125	1	2	yes	—
66	C₆₇⋊C₈	—	2	4	no	—
67	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₆₇	#431	1	2	yes	—
69	C₂²×C₆₇	#504	1	2	yes	—
70	C₂²×C₆₇	#504	1	2	yes	—
71	C₄×C₆₇	—	2	2	no	—
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	#382	1	1	yes	characteristic
74	C₄×C₆₇	#423	1	1	yes	characteristic
75	C₄×C₆₇	#432	1	2	yes	—
76	Dic₆₇	—	4	4	no	—
77	#12880	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	#1831	1	1	yes	characteristic
79	C₂×C₆₇	#2151	1	1	yes	characteristic
80	C₂×C₆₇	#2151	1	1	yes	characteristic
81	C₂×C₆₇	#2164	1	4	yes	—
82	C₂×C₄⋊C₁₆	—	67	1	no	—
83	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
84	#1726	—	67	1	no	—
85	#1374	—	67	4	no	—
86	C₆₇	#12880	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₁₆	—	67	2	no	—
88	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
89	C₂×#97	—	67	2	no	—
90	C₂×#97	—	67	1	no	—
91	C₂×C₄×C₈	—	67	2	no	—
92	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	#376	—	67	1	no	—
94	C₄⋊C₁₆	—	2	2	no	—
95	C₄⋊C₁₆	—	134	2	no	—
96	C₂²×C₈	—	67	2	no	—
97	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	C₂×C₄²	—	67	2	no	—
100	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	134	2	no	—
104	C₂×C₁₆	—	134	4	no	—
105	#97	—	134	4	no	—
106	#97	—	134	2	no	—
107	C₄×C₈	—	134	2	no	—
108	C₄×C₈	—	134	2	no	—
109	C₄×C₈	Dic₁₃₄	1	2	yes	—
110	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
111	#95	—	67	2	no	—
112	#95	—	67	2	no	—
113	C₂²×C₄	—	67	4	no	—
114	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
115	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	67	4	no	—
119	C₂×C₈	—	134	2	no	—
120	C₂×C₈	—	134	2	no	—
121	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
122	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
124	C₂×C₈	#11761627	1	2	yes	—
125	C₂×C₈	#11761629	1	2	yes	—
126	C₄²	—	134	2	no	—
127	C₄²	—	134	2	no	—
128	C₄²	Dic₆₇⋊C₄	1	2	yes	—
129	C₄²	#11761611	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	268	2	no	—
132	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	134	4	no	—
136	C₂×C₄	—	134	4	no	—
137	C₂×C₄	—	134	4	no	—
138	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
139	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
141	C₂×C₄	#14524001	1	2	yes	—
142	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
143	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	134	4	no	—
146	C₈	#14524030	1	1	yes	characteristic
147	C₈	#14524032	1	1	yes	characteristic
148	C₈	#14524034	1	2	yes	—
149	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
150	C₂²	#16863448	1	2	yes	—
151	C₂²	#16863812	1	2	yes	—
152	C₂²	#16863814	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	268	4	no	—
156	C₄	#16863261	1	1	yes	characteristic
157	C₄	#16863419	1	1	yes	characteristic
158	C₄	#16863452	1	2	yes	—
159	C₂	#22560325	1	1	yes	characteristic
160	C₂	#22561967	1	1	yes	characteristic
161	C₂	#22561969	1	1	yes	characteristic
162	C₂	#22562015	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic

#22559895

C₂

yes

maximal, characteristic

#22558286

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

C₂×#14523954

C₄

yes

characteristic

C₂×#14523953

C₄

yes

—

C₂×C₈×Dic₆₇

C₂²

yes

—

#16863242

C₄

yes

characteristic

C₄⋊(C₆₇⋊C₁₆)

—

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

#14523954

—

#14523953

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₆₇⋊#95

C₈

yes

—

C₆₇⋊#95

C₈

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

M₁₆

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

M₁₆

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₇

#431

yes

—

C₂²×C₆₇

#504

yes

—

C₂²×C₆₇

#504

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₆₇

yes

characteristic

#1726

—

#1374

—

C₆₇

#12880

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₄×D₆₇

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

134

—

104

C₂×C₁₆

—

134

—

105

#97

—

134

—

106

#97

—

134

—

107

C₄×C₈

—

134

—

108

C₄×C₈

—

134

—

109

C₄×C₈

Dic₁₃₄

yes

—

110

C₄×C₈

C₂²⋊D₆₇

yes

—

111

#95

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₈×D₆₇

yes

characteristic

115

C₂²×C₄

C₈×D₆₇

yes

characteristic

116

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

134

—

120

C₂×C₈

—

134

—

121

C₂×C₈

Dic₆₇⋊C₄

yes

—

122

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

123

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

124

C₂×C₈

#11761627

yes

—

125

C₂×C₈

#11761629

yes

—

126

C₄²

—

134

—

127

C₄²

—

134

—

128

C₄²

Dic₆₇⋊C₄

yes

—

129

C₄²

#11761611

yes

—

130

C₁₆

—

131

C₁₆

—

268

—

132

C₂³

C₂×C₈×D₆₇

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

134

—

136

C₂×C₄

—

134

—

137

C₂×C₄

—

134

—

138

C₂×C₄

Dic₆₇⋊C₈

yes

—

139

C₂×C₄

Dic₆₇⋊C₈

yes

—

140

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

141

C₂×C₄

#14524001

yes

—

142

C₂×C₄

M₁₆×D₆₇

yes

—

143

C₂×C₄

C₂×#11761611

yes

characteristic

144

C₈

—

145

C₈

—

134

—

146

C₈

#14524030

yes

characteristic

147

C₈

#14524032

yes

characteristic

148

C₈

#14524034

yes

—

149

C₂²

C₂×Dic₆₇⋊C₈

yes

characteristic

150

C₂²

#16863448

yes

—

151

C₂²

#16863812

yes

—

152

C₂²

#16863814

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

268

—

156

C₄

#16863261

yes

characteristic

157

C₄

#16863419

yes

characteristic

158

C₄

#16863452

yes

—

159

C₂

#22560325

yes

characteristic

160

C₂

#22561967

yes

characteristic

161

C₂

#22561969

yes

characteristic

162

C₂

#22562015

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#12880
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₇

commutator, socle, prime radical

C₂³×C₆₇

C₂²×C₈

Fitting

C₂×C₄⋊C₁₆×C₆₇

C₂

cosocle

C₂²×C₈×C₆₇

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
86. C₆₇	#12880	semidirect

H ≅ G/N

type

86. C₆₇

#12880

semidirect

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22559895	2
#22558286	2
C₂×C₄⋊C₁₆×C₆₇	2
#12880	67

subgroup

index

C₂×C₄⋊(C₆₇⋊C₁₆)

#22559895

#22558286

C₂×C₄⋊C₁₆×C₆₇

#12880

Subgroups — #30772180

Conjugacy classes of subgroups

Complements

Maximal subgroups