Group #30772024 — subgroups

Subgroups — #30772024

Order 17152.

354 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16863226	C₂	1	2	yes	maximal, complemented
2	#22560259	C₂	1	2	yes	maximal
3	#22558313	C₂	1	2	yes	maximal, complemented
4	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	C₂²	1	4	yes	—
6	#16863380	—	2	2	no	—
7	#16863226	—	2	4	no	—
8	C₆₇⋊#338	—	2	2	no	—
9	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
10	#504×C₆₇	—	2	4	no	—
11	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
12	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
14	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761612	—	2	4	no	—
16	#14523979	—	2	4	no	—
17	#14523979	—	2	4	no	—
18	#14523962	—	4	2	no	—
19	#14523960	—	4	2	no	—
20	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
21	#131×C₆₇	—	2	2	no	—
22	#131×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	2	no	—
28	M₃₂×C₆₇	—	2	4	no	—
29	C₂×C₁₆×C₆₇	—	2	4	no	—
30	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
34	#11761612	—	4	4	no	—
35	#11761612	—	4	4	no	—
36	C₂×C₆₇⋊C₈	—	4	4	no	—
37	#41×C₆₇	—	2	2	no	—
38	#41×C₆₇	—	2	2	no	—
39	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
40	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
41	C₂×D₄×C₆₇	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₆₇	—	2	2	no	—
45	M₁₆×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	—	2	2	no	—
47	C₂×C₈×C₆₇	—	2	2	no	—
48	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₇	#41	1	2	yes	—
53	C₁₆×C₆₇	—	2	2	no	—
54	C₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₆₇	#107	1	2	yes	—
57	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
58	Q₈×C₆₇	—	2	2	no	—
59	D₄×C₆₇	—	2	4	no	—
60	D₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	2	no	—
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	#107	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#121	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	#135	1	2	yes	—
68	C₈×C₆₇	—	2	2	no	—
69	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₆₇	#117	1	1	yes	characteristic
71	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
72	C₆₇⋊C₈	—	8	4	no	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	—	4	1	no	—
75	C₂²×C₆₇	—	4	2	no	—
76	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
77	C₂²×C₆₇	#467	1	2	yes	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	#380	1	1	yes	characteristic
80	C₄×C₆₇	#427	1	1	yes	characteristic
81	C₄×C₆₇	#466	1	2	yes	—
82	#12840	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	—	2	1	no	—
84	C₂×C₆₇	—	2	2	no	—
85	C₂×C₆₇	—	4	2	no	—
86	C₂×C₆₇	#1808	1	1	yes	characteristic
87	C₂×#504	D₆₇	1	1	yes	characteristic
88	C₂×#368	—	67	2	no	—
89	#1813	—	67	2	no	—
90	#1385	—	67	2	no	—
91	C₆₇	#12840	1	1	yes	characteristic, complemented, Sylow
92	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
93	#504	—	2	4	no	—
94	C₂×M₃₂	Dic₆₇	1	2	yes	—
95	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
96	C₂²×C₁₆	Dic₆₇	1	2	yes	—
97	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
98	C₂×#108	—	67	4	no	—
99	#413	—	134	2	no	—
100	#368	—	134	4	no	—
101	#338	—	134	2	no	—
102	C₂×#41	D₂₆₈	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	—	134	4	no	—
106	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
107	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
109	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	—	2	4	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
115	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
116	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	—
117	#108	—	134	4	no	—
118	#108	—	134	4	no	—
119	#101	—	268	2	no	—
120	#100	—	268	2	no	—
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
124	C₂×D₄	D₅₃₆	1	1	yes	characteristic
125	C₂×D₄	Dic₂₆₈	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
127	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	268	4	no	—
131	M₁₆	—	268	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	268	4	no	—
135	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
136	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
139	C₂×C₈	#11761628	1	2	yes	—
140	C₁₆	—	2	2	no	—
141	C₁₆	C₂²⋊Dic₆₇	1	2	yes	—
142	C₁₆	C₂²⋊Dic₆₇	1	2	yes	—
143	C₂³	#14523977	1	2	yes	—
144	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	#14523977	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
152	C₂×C₄	#14524017	1	2	yes	—
153	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
154	C₂×C₄	#14524065	1	2	yes	—
155	C₈	—	2	2	no	—
156	C₈	—	536	4	no	—
157	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
158	C₈	#14524002	1	1	yes	characteristic
159	C₈	#14524022	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×#14523977	1	1	yes	characteristic
164	C₂²	#16863637	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	#16863256	1	1	yes	characteristic
167	C₄	#16863434	1	1	yes	characteristic
168	C₄	#16863632	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	#22560238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863226

C₂

yes

maximal, complemented

#22560259

C₂

yes

maximal

#22558313

C₂

yes

maximal, complemented

C₂×#504×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523979

C₂²

yes

—

#16863380

—

#16863226

—

C₆₇⋊#338

—

C₂×#131×C₆₇

C₂²

yes

characteristic

#504×C₆₇

—

C₂×M₃₂×C₆₇

C₄

yes

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₄

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

—

—

—

—

—

C₂×#41×C₆₇

D₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

D₈

yes

characteristic

C₂×D₄×C₆₇

D₈

yes

characteristic

C₂×D₄×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂²⋊C₄

yes

—

C₁₆×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

#107

yes

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

#117

yes

characteristic

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#467

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#12840

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1808

yes

characteristic

C₂×#504

D₆₇

yes

—

—

—

C₆₇

#12840

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₆₇

yes

characteristic

#504

—

C₂×M₃₂

Dic₆₇

yes

—

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

Dic₆₇

yes

—

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

C₂×#108

—

#413

—

134

—

100

#368

—

134

—

101

#338

—

134

—

102

C₂×#41

D₂₆₈

yes

characteristic

103

#131

—

104

#131

—

105

C₂×M₁₆

—

134

—

106

C₂×M₁₆

Dic₁₃₄

yes

—

107

C₂×M₁₆

D₂₆₈

yes

characteristic

108

C₂²×C₈

C₂×Dic₆₇

yes

—

109

C₂²×C₈

C₂²×D₆₇

yes

characteristic

110

M₃₂

—

111

M₃₂

—

112

C₂×C₁₆

—

113

C₂×C₁₆

C₂×Dic₆₇

yes

—

114

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

115

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

116

C₂×C₁₆

C₂²⋊D₆₇

yes

—

117

#108

—

134

—

118

#108

—

134

—

119

#101

—

268

—

120

#100

—

268

—

121

#41

—

122

#41

—

123

C₂×Q₈

D₅₃₆

yes

characteristic

124

C₂×D₄

D₅₃₆

yes

characteristic

125

C₂×D₄

Dic₂₆₈

yes

—

126

C₂²×C₄

C₄⋊Dic₆₇

yes

—

127

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

268

—

131

M₁₆

—

268

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

268

—

135

C₂×C₈

C₄⋊Dic₆₇

yes

—

136

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

137

C₂×C₈

C₂×C₂²⋊D₆₇

yes

—

138

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

139

C₂×C₈

#11761628

yes

—

140

C₁₆

—

141

C₁₆

C₂²⋊Dic₆₇

yes

—

142

C₁₆

C₂²⋊Dic₆₇

yes

—

143

C₂³

#14523977

yes

—

144

C₂³

C₂×Dic₂₆₈

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#14523977

yes

—

151

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

152

C₂×C₄

#14524017

yes

—

153

C₂×C₄

C₂×D₅₃₆

yes

characteristic

154

C₂×C₄

#14524065

yes

—

155

C₈

—

156

C₈

—

536

—

157

C₈

C₂×C₂²⋊Dic₆₇

yes

characteristic

158

C₈

#14524002

yes

characteristic

159

C₈

#14524022

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×#14523977

yes

characteristic

164

C₂²

#16863637

yes

—

165

C₄

—

166

C₄

#16863256

yes

characteristic

167

C₄

#16863434

yes

characteristic

168

C₄

#16863632

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

#22560238

yes

characteristic

	subgroup	G/H
center	C₄	#16863434
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂×C₆₇	#1808
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#12840
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#16863434

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#504×C₆₇

C₂

socle

C₂×C₆₇

#1808

prime radical

C₂×#41×C₆₇

D₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. C₂×#16863226	C₂	semidirect
3. #22558313	C₂	semidirect
91. C₆₇	#12840	semidirect

H ≅ G/N

type

1. C₂×#16863226

C₂

semidirect

3. #22558313

C₂

semidirect

91. C₆₇

#12840

semidirect

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

subgroup	index
C₂×#16863226	2
#22560259	2
#22558313	2
C₂×#504×C₆₇	2
#12840	67

subgroup

index

C₂×#504×C₆₇

Subgroups — #30772024

Conjugacy classes of subgroups

Complements

Maximal subgroups