Group #34010027 — subgroups

Subgroups — #34010027

Order 18688.

342 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 118 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	#22626288	C₂	1	2	yes	maximal
2	#22624868	C₂	1	4	yes	maximal
3	C₂×#17988711	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	2	no	—
5	C₂²⋊#11924031	C₂²	1	4	yes	—
6	C₄×#11924031	—	2	2	no	—
7	C₂×C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
8	#17988711	—	2	2	no	—
9	#17988711	—	2	2	no	—
10	C₂×#14556520	C₂²	1	2	yes	—
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924031	—	2	4	no	—
15	C₂²×Dic₁₄₆	C₈	1	1	yes	characteristic
16	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
17	C₂²×#10596831	C₂³	1	1	yes	characteristic
18	C₄×Dic₁₄₆	—	2	2	no	—
19	C₄×Dic₁₄₆	—	2	2	no	—
20	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
21	C₂×#11924007	C₈	1	2	yes	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
23	#14556520	—	2	4	no	—
24	C₄×#10596831	D₄	1	2	yes	—
25	C₄×#10596831	D₄	1	2	yes	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₇₃	—	2	2	no	—
31	#11924031	—	4	4	no	—
32	C₂×Dic₁₄₆	—	2	2	no	—
33	C₂×Dic₁₄₆	—	2	2	no	—
34	C₂×Dic₁₄₆	C₁₆	1	4	yes	complemented
35	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
36	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂×D₄	1	2	yes	—
42	C₄⋊Dic₇₃	—	2	2	no	—
43	#11924007	—	2	4	no	—
44	C₄×Dic₇₃	—	2	2	no	—
45	C₄×Dic₇₃	—	2	2	no	—
46	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	4	2	no	—
51	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₇₃	M₁₆	1	2	yes	—
53	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	2	2	no	—
65	C₂×Dic₇₃	—	2	2	no	—
66	C₂×Dic₇₃	—	4	2	no	—
67	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
68	Dic₁₄₆	—	2	8	no	—
69	C₈×C₇₃	—	4	2	no	—
70	#10596831	—	2	2	no	—
71	#10596831	—	2	1	no	—
72	#10596831	#101	1	2	yes	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
76	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	C₂²⋊C₁₆	1	4	yes	—
81	C₄×C₇₃	#410	1	1	yes	characteristic
82	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
83	Dic₇₃	—	2	4	no	—
84	Dic₇₃	—	4	2	no	—
85	#11493	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	#1461	1	1	yes	characteristic
89	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
90	C₂×C₇₃	#2115	1	1	yes	characteristic
91	C₂×C₈×Q₈	—	73	1	no	—
92	#2162	—	73	2	no	—
93	#1322	—	73	4	no	—
94	C₇₃	#11493	1	1	yes	characteristic, complemented, Sylow
95	C₂×C₄×Q₈	—	73	1	no	—
96	C₈×Q₈	—	146	2	no	—
97	C₈×Q₈	—	146	2	no	—
98	C₂×C₄⋊C₈	—	73	2	no	—
99	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
100	C₂×C₄×C₈	—	73	1	no	—
101	C₂×C₄×C₈	—	73	2	no	—
102	C₄⋊C₁₆	—	146	2	no	—
103	C₂²⋊C₁₆	—	73	4	no	—
104	C₄×C₁₆	—	146	2	no	—
105	C₂²×Q₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	C₂×#64408	1	2	yes	—
109	C₄×Q₈	—	146	2	no	—
110	C₄×Q₈	—	146	2	no	—
111	C₂×C₄⋊C₄	—	73	2	no	—
112	C₂×C₄⋊C₄	—	73	1	no	—
113	C₂×C₄²	—	73	2	no	—
114	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
115	C₂×C₁₆	—	146	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	146	4	no	—
118	C₄×C₈	—	73	2	no	—
119	C₄×C₈	—	73	2	no	—
120	C₄×C₈	—	146	2	no	—
121	C₄×C₈	—	146	2	no	—
122	C₂×Q₈	—	73	4	no	—
123	C₂×Q₈	—	146	2	no	—
124	C₂×Q₈	—	146	2	no	—
125	C₂²×C₄	—	73	2	no	—
126	C₂²×C₄	—	73	2	no	—
127	C₂²×C₄	#11924036	1	1	yes	characteristic
128	C₂²×C₄	#11924036	1	1	yes	characteristic
129	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	292	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	2	no	—
136	C₂×C₈	—	146	2	no	—
137	C₄⋊C₄	—	146	4	no	—
138	C₄⋊C₄	—	146	2	no	—
139	C₄²	—	146	2	no	—
140	C₄²	—	146	2	no	—
141	C₄²	C₂²⋊#64408	1	2	yes	—
142	C₄²	#11924038	1	2	yes	—
143	C₁₆	—	292	4	no	—
144	C₂³	C₂×#11924036	1	1	yes	characteristic
145	Q₈	—	146	8	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	292	2	no	—
150	C₂×C₄	—	73	4	no	—
151	C₂×C₄	—	146	2	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	—	146	2	no	—
154	C₂×C₄	#14556648	1	2	yes	—
155	C₂×C₄	#14556648	1	2	yes	—
156	C₂×C₄	#14556664	1	2	yes	—
157	C₂×C₄	#14556666	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
159	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	73	2	no	—
162	C₈	—	146	2	no	—
163	C₈	—	146	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×#14556648	1	1	yes	characteristic
167	C₂²	C₂×#14556664	1	1	yes	characteristic
168	C₂²	C₂×#14556666	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	292	2	no	—
172	C₄	—	146	4	no	—
173	C₄	#17989482	1	4	yes	—
174	C₄	#17989567	1	1	yes	characteristic
175	C₄	D₇₃⋊#101	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#22625175	1	1	yes	characteristic
179	C₂	C₂×#17989482	1	1	yes	characteristic
180	C₂	#22626169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626288

C₂

yes

maximal

#22624868

C₂

yes

maximal

C₂×#17988711

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₂²⋊#11924031

C₂²

yes

—

C₄×#11924031

—

C₂×C₄×Dic₁₄₆

C₄

yes

—

—

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

C₂×C₄⋊C₈×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂²×Dic₁₄₆

C₈

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

C₈

yes

characteristic

C₂×#11924007

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

#14556520

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

#11924031

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₁₆

yes

complemented

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

Dic₁₄₆

—

C₈×C₇₃

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₁₆

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#11493

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1461

yes

characteristic

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₃

#2115

yes

characteristic

C₂×C₈×Q₈

—

#2162

—

#1322

—

C₇₃

#11493

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₈×Q₈

—

146

—

C₈×Q₈

—

146

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#64408

yes

characteristic

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

—

102

C₄⋊C₁₆

—

146

—

103

C₂²⋊C₁₆

—

104

C₄×C₁₆

—

146

—

105

C₂²×Q₈

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂×#64408

yes

—

109

C₄×Q₈

—

146

—

110

C₄×Q₈

—

146

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂×#64408

yes

characteristic

115

C₂×C₁₆

—

146

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

146

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

146

—

121

C₄×C₈

—

146

—

122

C₂×Q₈

—

123

C₂×Q₈

—

146

—

124

C₂×Q₈

—

146

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

#11924036

yes

characteristic

128

C₂²×C₄

#11924036

yes

characteristic

129

C₂²×C₄

C₂²×#64408

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

292

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

146

—

137

C₄⋊C₄

—

146

—

138

C₄⋊C₄

—

146

—

139

C₄²

—

146

—

140

C₄²

—

146

—

141

C₄²

C₂²⋊#64408

yes

—

142

C₄²

#11924038

yes

—

143

C₁₆

—

292

—

144

C₂³

C₂×#11924036

yes

characteristic

145

Q₈

—

146

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

—

146

—

154

C₂×C₄

#14556648

yes

—

155

C₂×C₄

#14556648

yes

—

156

C₂×C₄

#14556664

yes

—

157

C₂×C₄

#14556666

yes

—

158

C₂×C₄

C₂×C₂²⋊#64408

yes

characteristic

159

C₂×C₄

C₂×#11924038

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

146

—

163

C₈

—

146

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₂×#14556648

yes

characteristic

167

C₂²

C₂×#14556664

yes

characteristic

168

C₂²

C₂×#14556666

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

292

—

172

C₄

—

146

—

173

C₄

#17989482

yes

—

174

C₄

#17989567

yes

characteristic

175

C₄

D₇₃⋊#101

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#22625175

yes

characteristic

179

C₂

C₂×#17989482

yes

characteristic

180

C₂

#22626169

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#11493
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#64408

commutator

C₂²×C₇₃

C₂²×C₁₆

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×C₄⋊C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₂×C₄⋊C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
34. C₂×Dic₁₄₆	C₁₆	semidirect
94. C₇₃	#11493	semidirect

H ≅ G/N

type

34. C₂×Dic₁₄₆

C₁₆

semidirect

94. C₇₃

#11493

semidirect

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

subgroup	index
#22626288	2
#22624868	2
C₂×#17988711	2
#11493	73

subgroup

index

Subgroups — #34010027

Conjugacy classes of subgroups

Complements

Maximal subgroups