Group #34034206 — subgroups

Subgroups — #34034206

Order 18688.

322 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 113 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×#17989690	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic
3	#22627001	C₂	1	4	yes	maximal
4	#22625588	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924040	C₂²	1	2	yes	—
6	C₂²×#11924040	C₂²	1	2	yes	—
7	#17989690	—	2	2	no	—
8	C₄⋊#11924040	—	2	2	no	—
9	C₂×#14556646	C₄	1	2	yes	—
10	C₂×#14556646	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	2	yes	—
12	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
13	#17988432	C₄	1	1	yes	characteristic
14	C₂×#11924040	—	2	2	no	—
15	C₂×#11924040	—	2	2	no	—
16	C₂×#11924040	—	2	4	no	—
17	C₂×#11924040	—	2	4	no	—
18	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	2	yes	—
20	C₂²×#10596833	C₂³	1	1	yes	characteristic
21	#14556646	—	2	4	no	—
22	#14556646	—	2	2	no	—
23	C₄×#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	D₄	1	2	yes	—
26	C₄×#10596833	Q₈	1	2	yes	—
27	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	#14556498	C₈	1	2	yes	—
30	#14556498	C₈	1	1	yes	characteristic
31	#95×C₇₃	C₈	1	1	yes	characteristic
32	#11924040	—	4	2	no	—
33	#11924040	—	4	2	no	—
34	C₂×#10596833	—	2	2	no	—
35	C₂×#10596833	—	2	2	no	—
36	C₂×#10596833	—	2	2	no	—
37	C₂×#10596833	C₄⋊C₄	1	2	yes	—
38	C₂×#10596833	M₁₆	1	4	yes	—
39	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
40	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
41	C₂×#10596833	#41	1	2	yes	—
42	C₂×#10596833	#41	1	2	yes	—
43	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
44	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₄×Dic₇₃	—	2	2	no	—
47	C₄×Dic₇₃	—	2	2	no	—
48	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
49	C₄×Dic₇₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	—	2	2	no	—
63	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
64	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×Dic₇₃	#117	1	2	yes	—
66	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
67	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
68	#10596833	—	2	2	no	—
69	#10596833	—	2	4	no	—
70	#10596833	#124	1	1	yes	characteristic
71	#10596833	#124	1	1	yes	characteristic
72	#10596833	#125	1	2	yes	—
73	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₃	#431	1	2	yes	—
75	C₂²×C₇₃	#504	1	2	yes	—
76	C₂²×C₇₃	#504	1	1	yes	characteristic
77	C₂²×C₇₃	#504	1	1	yes	characteristic
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	4	2	no	—
80	Dic₇₃	—	2	2	no	—
81	Dic₇₃	—	4	2	no	—
82	Dic₇₃	#382	1	1	yes	characteristic
83	Dic₇₃	#423	1	1	yes	characteristic
84	Dic₇₃	#432	1	2	yes	—
85	#12880	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	#1831	1	1	yes	characteristic
87	C₂×C₇₃	#2151	1	1	yes	characteristic
88	C₂×C₇₃	#2151	1	1	yes	characteristic
89	C₂×C₇₃	#2164	1	2	yes	—
90	C₂×C₇₃	#2164	1	2	yes	—
91	C₂×C₄⋊C₁₆	—	73	1	no	—
92	C₂×C₄⋊C₁₆	—	73	1	no	—
93	#1726	—	73	1	no	—
94	#1374	—	73	4	no	—
95	C₇₃	#12880	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	73	2	no	—
97	C₂²×C₁₆	—	73	2	no	—
98	C₂×#97	—	73	1	no	—
99	C₂×#97	—	73	2	no	—
100	C₂×C₄×C₈	—	73	1	no	—
101	C₂×C₄×C₈	—	73	2	no	—
102	#376	—	73	1	no	—
103	C₄⋊C₁₆	—	146	2	no	—
104	C₄⋊C₁₆	—	146	2	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	2	no	—
108	C₂×C₄²	—	73	1	no	—
109	C₂×C₄²	—	73	2	no	—
110	C₂×C₁₆	—	146	2	no	—
111	C₂×C₁₆	—	146	2	no	—
112	C₂×C₁₆	—	146	4	no	—
113	C₂×C₁₆	—	146	4	no	—
114	#97	—	146	2	no	—
115	#97	—	146	4	no	—
116	C₄×C₈	—	73	2	no	—
117	C₄×C₈	—	73	2	no	—
118	C₄×C₈	—	146	2	no	—
119	C₄×C₈	—	146	2	no	—
120	#95	—	73	2	no	—
121	#95	—	73	1	no	—
122	#95	#10596842	1	1	yes	characteristic
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	2	no	—
126	C₂²×C₄	C₂×#10596842	1	2	yes	—
127	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	2	no	—
131	C₂×C₈	—	73	2	no	—
132	C₂×C₈	—	73	2	no	—
133	C₂×C₈	—	73	4	no	—
134	C₂×C₈	—	146	2	no	—
135	C₂×C₈	—	146	2	no	—
136	C₂×C₈	—	146	2	no	—
137	C₄²	—	73	2	no	—
138	C₄²	—	73	2	no	—
139	C₄²	—	146	2	no	—
140	C₄²	—	146	2	no	—
141	C₁₆	—	292	2	no	—
142	C₁₆	—	292	2	no	—
143	C₂³	C₂²×#10596842	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	—	73	2	no	—
152	C₂×C₄	—	73	2	no	—
153	C₂×C₄	—	146	2	no	—
154	C₂×C₄	—	146	2	no	—
155	C₂×C₄	—	146	2	no	—
156	C₂×C₄	—	146	1	no	—
157	C₂×C₄	—	146	1	no	—
158	C₂×C₄	C₄⋊#10596842	1	2	yes	—
159	C₈	—	73	1	no	—
160	C₈	—	73	1	no	—
161	C₈	—	73	2	no	—
162	C₈	—	146	2	no	—
163	C₈	—	146	4	no	—
164	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
165	C₂²	#17989700	1	2	yes	—
166	C₂²	#17989708	1	2	yes	—
167	C₂²	#17989708	1	1	yes	characteristic
168	C₂²	#17989709	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	292	2	no	—
172	C₄	—	73	1	no	—
173	C₄	—	73	1	no	—
174	C₄	—	73	2	no	—
175	C₄	—	146	2	no	—
176	C₂	#22627073	1	1	yes	characteristic
177	C₂	#22627122	1	1	yes	characteristic
178	C₂	#22627123	1	1	yes	characteristic
179	C₂	#22627136	1	2	yes	—
180	C₂	#22627137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989690

C₂

yes

maximal, characteristic

C₂×C₄⋊#11924040

C₂

yes

maximal, characteristic

#22627001

C₂

yes

maximal

#22625588

C₂

yes

maximal, characteristic

C₂²×#11924040

C₂²

yes

—

C₂²×#11924040

C₂²

yes

—

#17989690

—

C₄⋊#11924040

—

C₂×#14556646

C₄

yes

—

C₂×#14556646

C₄

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

—

C₂×C₄×#10596833

C₂²

yes

characteristic

#17988432

C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂²×#10596833

C₂×C₄

yes

—

C₂²×#10596833

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#10596833

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

#14556498

C₈

yes

—

#14556498

C₈

yes

characteristic

#95×C₇₃

C₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596833

M₁₆

yes

—

C₂×#10596833

C₂×D₄

yes

characteristic

C₂×#10596833

C₂×Q₈

yes

characteristic

C₂×#10596833

#41

yes

—

C₂×#10596833

#41

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

characteristic

C₂²×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#117

yes

—

C₂×Dic₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

C₂×M₁₆

yes

—

—

yes

yes

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

#504

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2164

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

#1726

—

#1374

—

C₇₃

#12880

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

—

102

#376

—

103

C₄⋊C₁₆

—

146

—

104

C₄⋊C₁₆

—

146

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₁₆

—

146

—

111

C₂×C₁₆

—

146

—

112

C₂×C₁₆

—

146

—

113

C₂×C₁₆

—

146

—

114

#97

—

146

—

115

#97

—

146

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

146

—

119

C₄×C₈

—

146

—

120

#95

—

121

#95

—

122

#95

#10596842

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×#10596842

yes

—

127

C₂²×C₄

C₂×#10596842

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

146

—

135

C₂×C₈

—

146

—

136

C₂×C₈

—

146

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

146

—

140

C₄²

—

146

—

141

C₁₆

—

292

—

142

C₁₆

—

292

—

143

C₂³

C₂²×#10596842

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

146

—

154

C₂×C₄

—

146

—

155

C₂×C₄

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

—

146

—

158

C₂×C₄

C₄⋊#10596842

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

146

—

163

C₈

—

146

—

164

C₂²

C₂×C₄⋊#10596842

yes

characteristic

165

C₂²

#17989700

yes

—

166

C₂²

#17989708

yes

—

167

C₂²

#17989708

yes

characteristic

168

C₂²

#17989709

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

292

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

146

—

176

C₂

#22627073

yes

characteristic

177

C₂

#22627122

yes

characteristic

178

C₂

#22627123

yes

characteristic

179

C₂

#22627136

yes

—

180

C₂

#22627137

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator, socle, prime radical	C₂³×C₇₃	C₂²×C₈
Fitting	#95×C₇₃	C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	#95	#10596842
hypocenter	C₇₃	#12880
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#10596842

commutator, socle, prime radical

C₂³×C₇₃

C₂²×C₈

Fitting

#95×C₇₃

C₈

C₂³

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
95. C₇₃	#12880	semidirect

H ≅ G/N

type

95. C₇₃

#12880

semidirect

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

subgroup	index
C₂×#17989690	2
C₂×C₄⋊#11924040	2
#22627001	2
#22625588	2
#12880	73

subgroup

index

Subgroups — #34034206

Conjugacy classes of subgroups

Complements

Maximal subgroups