Group #32972904 — subgroups

Subgroups — #32972904

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×#334×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	—	2	1	no	—
8	C₄×#11924003	C₄	1	2	yes	—
9	#17988352	—	2	1	no	—
10	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#334×C₇₃	—	2	1	no	—
13	#334×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	2	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#104	1	2	yes	—
60	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	8	2	no	—
65	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
66	C₈×C₇₃	#97	1	1	yes	characteristic
67	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
68	#10596831	—	4	4	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	Q₈⋊C₈	1	2	yes	—
72	C₂²×C₇₃	#340	1	1	yes	characteristic
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
77	C₄×C₇₃	#336	1	1	yes	characteristic
78	C₄×C₇₃	#339	1	1	yes	characteristic
79	C₄×C₇₃	#365	1	2	yes	—
80	#7173	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	2	2	no	—
82	C₂×C₇₃	#1279	1	1	yes	characteristic
83	C₂×C₇₃	#1383	1	1	yes	characteristic
84	C₂×C₇₃	#1385	1	1	yes	characteristic
85	C₄×M₃₂	—	73	1	no	—
86	C₄×#108	—	73	1	no	—
87	C₂×#334	D₇₃	1	1	yes	characteristic
88	C₇₃	#7173	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	146	1	no	—
90	C₂×#108	—	73	2	no	—
91	C₂×C₄⋊C₈	—	2	1	no	—
92	C₄×M₁₆	—	146	1	no	—
93	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
94	#346	—	146	1	no	—
95	C₄×C₁₆	—	73	2	no	—
96	#341	—	146	1	no	—
97	#334	—	2	1	no	—
98	#334	Dic₇₃	1	2	yes	—
99	C₂×M₁₆	—	146	2	no	—
100	C₂²×C₈	—	4	1	no	—
101	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
102	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
104	M₃₂	—	146	4	no	—
105	C₂×C₁₆	—	146	2	no	—
106	#108	—	73	4	no	—
107	#108	—	146	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	4	1	no	—
110	#97	—	292	1	no	—
111	C₄×C₈	—	292	1	no	—
112	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
113	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
116	C₂²×C₄	#11924005	1	1	yes	characteristic
117	C₂²×C₄	#11924016	1	1	yes	characteristic
118	C₂²×C₄	#11924018	1	1	yes	characteristic
119	M₁₆	—	292	2	no	—
120	M₁₆	—	146	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	—	292	2	no	—
125	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
126	C₂×C₈	#11924005	1	1	yes	characteristic
127	C₂×C₈	#11924007	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
129	C₂×C₈	C₈×D₇₃	1	2	yes	—
130	C₂×C₈	#11924011	1	2	yes	—
131	C₄²	C₂²⋊Dic₇₃	1	1	yes	characteristic
132	C₄²	#11924007	1	1	yes	characteristic
133	C₄²	#11924013	1	1	yes	characteristic
134	C₄²	#11924015	1	1	yes	characteristic
135	C₁₆	—	146	4	no	—
136	C₂³	#14556514	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#14556498	1	1	yes	characteristic
140	C₂×C₄	#14556503	1	2	yes	—
141	C₂×C₄	#14556511	1	1	yes	characteristic
142	C₂×C₄	#14556512	1	1	yes	characteristic
143	C₂×C₄	#14556515	1	1	yes	characteristic
144	C₂×C₄	#14556517	1	2	yes	—
145	C₂×C₄	#14556522	1	1	yes	characteristic
146	C₂×C₄	#14556524	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	8	2	no	—
150	C₈	—	292	4	no	—
151	C₈	C₈×Dic₇₃	1	1	yes	characteristic
152	C₈	#14556501	1	1	yes	characteristic
153	C₈	#14556520	1	2	yes	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#17988317	1	2	yes	—
157	C₂²	#17988349	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#17988314	1	2	yes	—
162	C₄	#17988341	1	1	yes	characteristic
163	C₄	#17988348	1	1	yes	characteristic
164	C₄	#17988409	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#22607770	1	1	yes	characteristic
167	C₂	#22608029	1	1	yes	characteristic
168	C₂	#22608034	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×#334×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556531

—

C₂×#14556527

C₄

yes

—

C₄×#11924012

—

#17988363

—

C₄×#11924003

C₄

yes

—

#17988352

—

C₂×C₄⋊C₈×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#334×C₇₃

—

#334×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

#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

—

#7173

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×M₃₂

—

C₄×#108

—

C₂×#334

D₇₃

yes

characteristic

C₇₃

#7173

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#346

—

146

—

C₄×C₁₆

—

#341

—

146

—

#334

—

#334

Dic₇₃

yes

—

C₂×M₁₆

—

146

—

100

C₂²×C₈

—

101

C₂²×C₈

C₄×D₇₃

yes

characteristic

102

C₂²×C₈

D₂₉₂

yes

characteristic

103

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

104

M₃₂

—

146

—

105

C₂×C₁₆

—

146

—

106

#108

—

107

#108

—

146

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#97

—

292

—

111

C₄×C₈

—

292

—

112

C₄×C₈

Dic₁₄₆

yes

characteristic

113

C₄×C₈

C₄×D₇₃

yes

characteristic

114

C₄×C₈

C₂×Dic₇₃

yes

characteristic

115

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

116

C₂²×C₄

#11924005

yes

characteristic

117

C₂²×C₄

#11924016

yes

characteristic

118

C₂²×C₄

#11924018

yes

characteristic

119

M₁₆

—

292

—

120

M₁₆

—

146

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

292

—

125

C₂×C₈

C₄×Dic₇₃

yes

characteristic

126

C₂×C₈

#11924005

yes

characteristic

127

C₂×C₈

#11924007

yes

characteristic

128

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

129

C₂×C₈

C₈×D₇₃

yes

—

130

C₂×C₈

#11924011

yes

—

131

C₄²

C₂²⋊Dic₇₃

yes

characteristic

132

C₄²

#11924007

yes

characteristic

133

C₄²

#11924013

yes

characteristic

134

C₄²

#11924015

yes

characteristic

135

C₁₆

—

146

—

136

C₂³

#14556514

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#14556498

yes

characteristic

140

C₂×C₄

#14556503

yes

—

141

C₂×C₄

#14556511

yes

characteristic

142

C₂×C₄

#14556512

yes

characteristic

143

C₂×C₄

#14556515

yes

characteristic

144

C₂×C₄

#14556517

yes

—

145

C₂×C₄

#14556522

yes

characteristic

146

C₂×C₄

#14556524

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

292

—

151

C₈

C₈×Dic₇₃

yes

characteristic

152

C₈

#14556501

yes

characteristic

153

C₈

#14556520

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

#17988317

yes

—

157

C₂²

#17988349

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#17988314

yes

—

162

C₄

#17988341

yes

characteristic

163

C₄

#17988348

yes

characteristic

164

C₄

#17988409

yes

—

165

C₂

—

166

C₂

#22607770

yes

characteristic

167

C₂

#22608029

yes

characteristic

168

C₂

#22608034

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556522
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#103
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#7173
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#14556522

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₂×#334×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×#334

N	H ≅ G/N	type
18. #14556527	C₈	semidirect
88. C₇₃	#7173	semidirect

H ≅ G/N

type

18. #14556527

C₈

semidirect

88. C₇₃

#7173

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×#334×C₇₃	2
#7173	73

subgroup

index

C₄×#14556531

C₄×#14556527

C₂×#334×C₇₃

#7173

Subgroups — #32972904

Conjugacy classes of subgroups

Complements

Maximal subgroups