Group #32972913 — subgroups

Subgroups — #32972913

Order 18688.

246 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14556531	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556527	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556531	—	2	1	no	—
5	C₂×#14556527	C₄	1	2	yes	—
6	C₄×#11924012	—	2	1	no	—
7	#17988363	C₄	1	2	yes	—
8	C₄×#11924003	—	2	1	no	—
9	#17988352	—	2	1	no	—
10	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#335×C₇₃	—	2	1	no	—
13	#335×C₇₃	C₄	1	2	yes	—
14	C₂×#11924012	—	2	2	no	—
15	#14556531	—	2	4	no	—
16	C₂×#11924003	—	2	2	no	—
17	#14556527	—	2	2	no	—
18	#14556527	C₈	1	4	yes	complemented
19	#14556502	—	4	1	no	—
20	C₄×#10596831	—	4	1	no	—
21	C₂²×C₈×C₇₃	—	4	1	no	—
22	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
25	C₄⋊C₈×C₇₃	—	2	2	no	—
26	C₄⋊C₈×C₇₃	—	4	1	no	—
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	#11924012	—	2	4	no	—
32	#11924012	—	4	2	no	—
33	C₂×#10596831	—	4	2	no	—
34	#11924003	—	2	4	no	—
35	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	—	4	2	no	—
40	C₂×C₈×C₇₃	—	8	2	no	—
41	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
47	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	D₈	1	1	yes	characteristic
50	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
51	C₂³×C₇₃	#103	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#104	1	2	yes	—
61	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	8	2	no	—
66	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
67	C₈×C₇₃	#97	1	1	yes	characteristic
68	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
69	#10596831	—	4	4	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	Q₈⋊C₈	1	2	yes	—
73	C₂²×C₇₃	#340	1	1	yes	characteristic
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
78	C₄×C₇₃	#336	1	1	yes	characteristic
79	C₄×C₇₃	#339	1	1	yes	characteristic
80	C₄×C₇₃	#361	1	1	yes	characteristic
81	C₄×C₇₃	#362	1	1	yes	characteristic
82	#7176	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	2	no	—
84	C₂×C₇₃	#1279	1	1	yes	characteristic
85	C₂×C₇₃	#1387	1	1	yes	characteristic
86	C₂×C₇₃	#1388	1	1	yes	characteristic
87	C₄×M₃₂	—	73	1	no	—
88	C₄×#108	—	73	1	no	—
89	C₂×#335	D₇₃	1	1	yes	characteristic
90	C₇₃	#7176	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	146	1	no	—
92	C₂×#108	—	73	2	no	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₄×M₁₆	—	146	1	no	—
95	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
96	#346	—	73	2	no	—
97	C₄×C₁₆	—	146	1	no	—
98	#341	—	146	1	no	—
99	#335	—	2	1	no	—
100	#335	Dic₇₃	1	2	yes	—
101	C₂×M₁₆	—	146	2	no	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
104	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
106	M₃₂	—	146	4	no	—
107	C₂×C₁₆	—	146	2	no	—
108	#108	—	73	4	no	—
109	#108	—	146	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	#97	—	292	1	no	—
113	C₄×C₈	—	292	1	no	—
114	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
115	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
118	C₂²×C₄	#11924005	1	1	yes	characteristic
119	C₂²×C₄	#11924016	1	1	yes	characteristic
120	C₂²×C₄	#11924018	1	1	yes	characteristic
121	M₁₆	—	292	2	no	—
122	M₁₆	—	146	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	8	2	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
128	C₂×C₈	#11924005	1	1	yes	characteristic
129	C₂×C₈	#11924007	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
131	C₂×C₈	C₈×D₇₃	1	2	yes	—
132	C₂×C₈	#11924011	1	2	yes	—
133	C₄²	C₂²⋊Dic₇₃	1	1	yes	characteristic
134	C₄²	#11924007	1	1	yes	characteristic
135	C₄²	#11924013	1	1	yes	characteristic
136	C₄²	#11924015	1	1	yes	characteristic
137	C₁₆	—	146	4	no	—
138	C₂³	#14556514	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	#14556498	1	1	yes	characteristic
143	C₂×C₄	#14556503	1	2	yes	—
144	C₂×C₄	#14556511	1	1	yes	characteristic
145	C₂×C₄	#14556512	1	1	yes	characteristic
146	C₂×C₄	#14556515	1	1	yes	characteristic
147	C₂×C₄	#14556517	1	2	yes	—
148	C₂×C₄	#14556522	1	1	yes	characteristic
149	C₂×C₄	#14556524	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	8	2	no	—
153	C₈	—	292	4	no	—
154	C₈	C₈×Dic₇₃	1	1	yes	characteristic
155	C₈	#14556501	1	1	yes	characteristic
156	C₈	#14556520	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#17988317	1	2	yes	—
160	C₂²	#17988349	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#17988314	1	2	yes	—
165	C₄	#17988341	1	1	yes	characteristic
166	C₄	#17988348	1	1	yes	characteristic
167	C₄	#17988400	1	1	yes	characteristic
168	C₄	#17988403	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#22607770	1	1	yes	characteristic
171	C₂	#22608039	1	1	yes	characteristic
172	C₂	#22608042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556531

C₂

yes

maximal, characteristic

C₄×#14556527

C₂

yes

maximal, characteristic

C₂×#335×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556531

—

C₂×#14556527

C₄

yes

—

C₄×#11924012

—

#17988363

C₄

yes

—

C₄×#11924003

—

#17988352

—

C₂×C₄⋊C₈×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#335×C₇₃

—

#335×C₇₃

C₄

yes

—

—

—

—

—

C₈

yes

complemented

#14556502

—

C₄×#10596831

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₂³×C₇₃

#103

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

characteristic

C₈×C₇₃

#97

yes

characteristic

C₈×C₇₃

C₄⋊C₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

Q₈⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×M₃₂

—

C₄×#108

—

C₂×#335

D₇₃

yes

characteristic

C₇₃

#7176

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#346

—

C₄×C₁₆

—

146

—

#341

—

146

—

#335

—

100

#335

Dic₇₃

yes

—

101

C₂×M₁₆

—

146

—

102

C₂²×C₈

—

103

C₂²×C₈

C₄×D₇₃

yes

characteristic

104

C₂²×C₈

D₂₉₂

yes

characteristic

105

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

106

M₃₂

—

146

—

107

C₂×C₁₆

—

146

—

108

#108

—

109

#108

—

146

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#97

—

292

—

113

C₄×C₈

—

292

—

114

C₄×C₈

Dic₁₄₆

yes

characteristic

115

C₄×C₈

C₄×D₇₃

yes

characteristic

116

C₄×C₈

C₂×Dic₇₃

yes

characteristic

117

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

118

C₂²×C₄

#11924005

yes

characteristic

119

C₂²×C₄

#11924016

yes

characteristic

120

C₂²×C₄

#11924018

yes

characteristic

121

M₁₆

—

292

—

122

M₁₆

—

146

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

292

—

127

C₂×C₈

C₄×Dic₇₃

yes

characteristic

128

C₂×C₈

#11924005

yes

characteristic

129

C₂×C₈

#11924007

yes

characteristic

130

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

131

C₂×C₈

C₈×D₇₃

yes

—

132

C₂×C₈

#11924011

yes

—

133

C₄²

C₂²⋊Dic₇₃

yes

characteristic

134

C₄²

#11924007

yes

characteristic

135

C₄²

#11924013

yes

characteristic

136

C₄²

#11924015

yes

characteristic

137

C₁₆

—

146

—

138

C₂³

#14556514

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#14556498

yes

characteristic

143

C₂×C₄

#14556503

yes

—

144

C₂×C₄

#14556511

yes

characteristic

145

C₂×C₄

#14556512

yes

characteristic

146

C₂×C₄

#14556515

yes

characteristic

147

C₂×C₄

#14556517

yes

—

148

C₂×C₄

#14556522

yes

characteristic

149

C₂×C₄

#14556524

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

292

—

154

C₈

C₈×Dic₇₃

yes

characteristic

155

C₈

#14556501

yes

characteristic

156

C₈

#14556520

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

#17988317

yes

—

160

C₂²

#17988349

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

#17988314

yes

—

165

C₄

#17988341

yes

characteristic

166

C₄

#17988348

yes

characteristic

167

C₄

#17988400

yes

characteristic

168

C₄

#17988403

yes

characteristic

169

C₂

—

170

C₂

#22607770

yes

characteristic

171

C₂

#22608039

yes

characteristic

172

C₂

#22608042

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556512
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#103
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	#7176
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

subgroup

G/H

center

C₂×C₄

#14556512

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₂×#335×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

N	H ≅ G/N	type
18. #14556527	C₈	semidirect
90. C₇₃	#7176	semidirect

H ≅ G/N

type

18. #14556527

C₈

semidirect

90. C₇₃

#7176

semidirect

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

subgroup	index
C₄×#14556531	2
C₄×#14556527	2
C₂×#335×C₇₃	2
#7176	73

subgroup

index

C₄×#14556531

C₄×#14556527

C₂×#335×C₇₃

#7176

Subgroups — #32972913

Conjugacy classes of subgroups

Complements

Maximal subgroups