Group #32972911 — subgroups

Subgroups — #32972911

Order 18688.

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

normal subgroups only

This group has 262 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×#14556523	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988640	—	2	1	no	—
5	C₂×#14556523	C₄	1	2	yes	—
6	C₄×C₄⋊Dic₇₃	—	2	1	no	—
7	#17988350	—	2	1	no	—
8	#17988340	C₄	1	2	yes	complemented
9	#17988338	—	2	1	no	—
10	C₂×M₃₂×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#346×C₇₃	C₄	1	2	yes	complemented
13	C₄×C₁₆×C₇₃	—	2	1	no	—
14	C₂×C₄⋊Dic₇₃	—	2	2	no	—
15	C₂×C₄×Dic₇₃	—	4	1	no	—
16	#14556523	—	2	2	no	—
17	#14556523	C₈	1	4	yes	complemented
18	C₄⋊#10596831	—	2	2	no	—
19	C₂²⋊#10596831	—	4	1	no	—
20	C₄×#10596831	—	4	1	no	—
21	#14556498	—	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	M₃₂×C₇₃	—	2	4	no	—
26	C₂×C₁₆×C₇₃	—	2	2	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	C₂²×Dic₇₃	—	4	2	no	—
32	C₂×#10596831	—	4	2	no	—
33	C₄⋊Dic₇₃	—	2	4	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄×Dic₇₃	—	8	2	no	—
36	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
46	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
50	C₁₆×C₇₃	—	2	4	no	—
51	C₂³×C₇₃	#102	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	#103	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	C₂×Dic₇₃	—	4	4	no	—
63	C₂×Dic₇₃	—	8	2	no	—
64	C₂×Dic₇₃	—	8	2	no	—
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
68	C₈×C₇₃	#97	1	1	yes	characteristic
69	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
70	#10596831	—	8	2	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	D₄⋊C₈	1	2	yes	—
74	C₂²×C₇₃	#340	1	1	yes	characteristic
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
79	C₄×C₇₃	#336	1	1	yes	characteristic
80	C₄×C₇₃	#339	1	1	yes	characteristic
81	C₄×C₇₃	#360	1	2	yes	—
82	Dic₇₃	—	8	4	no	—
83	#7175	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	2	no	—
85	C₂×C₇₃	#1279	1	1	yes	characteristic
86	C₂×C₇₃	#1386	1	1	yes	characteristic
87	C₂×C₇₃	#1389	1	1	yes	characteristic
88	C₄×M₃₂	D₇₃	1	1	yes	characteristic
89	C₄×#106	—	73	1	no	—
90	#1564	—	73	1	no	—
91	C₇₃	#7175	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	2	1	no	—
93	#431	—	146	1	no	—
94	C₂×#106	—	73	2	no	—
95	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
96	C₄×C₄⋊C₄	—	146	1	no	—
97	#346	Dic₇₃	1	2	yes	complemented
98	C₄×C₁₆	—	2	1	no	—
99	#340	—	146	1	no	—
100	#335	—	73	2	no	—
101	#334	—	146	1	no	—
102	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	146	2	no	—
105	C₂×C₄²	—	292	1	no	—
106	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
107	M₃₂	—	2	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#106	—	73	4	no	—
110	#106	—	146	2	no	—
111	C₄⋊C₈	—	146	2	no	—
112	C₂²⋊C₈	—	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	#95	—	292	1	no	—
119	C₂²×C₄	—	292	2	no	—
120	C₂²×C₄	#11924005	1	1	yes	characteristic
121	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
122	C₂²×C₄	#11924017	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	292	2	no	—
125	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
127	C₂×C₈	#11924007	1	1	yes	characteristic
128	C₂×C₈	#11924007	1	1	yes	characteristic
129	C₂×C₈	C₈×D₇₃	1	2	yes	—
130	C₂×C₈	#11924011	1	2	yes	—
131	C₄⋊C₄	—	292	2	no	—
132	C₄⋊C₄	—	146	4	no	—
133	C₄²	—	584	2	no	—
134	C₄²	#11924005	1	1	yes	characteristic
135	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
136	C₄²	#11924017	1	1	yes	characteristic
137	C₄²	Dic₂₉₂	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₂³	#14556513	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	292	4	no	—
143	C₂×C₄	—	584	2	no	—
144	C₂×C₄	—	584	2	no	—
145	C₂×C₄	#14556498	1	1	yes	characteristic
146	C₂×C₄	#14556503	1	2	yes	—
147	C₂×C₄	#14556513	1	1	yes	characteristic
148	C₂×C₄	#14556516	1	1	yes	characteristic
149	C₂×C₄	#14556516	1	1	yes	characteristic
150	C₂×C₄	#14556519	1	2	yes	—
151	C₂×C₄	#14556523	1	1	yes	characteristic
152	C₂×C₄	#14556525	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	584	2	no	—
156	C₈	C₈×Dic₇₃	1	1	yes	characteristic
157	C₈	#14556501	1	1	yes	characteristic
158	C₈	#14556520	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	#17988316	1	2	yes	—
162	C₂²	#17988350	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	584	4	no	—
167	C₄	#17988319	1	2	yes	—
168	C₄	#17988341	1	1	yes	characteristic
169	C₄	#17988347	1	1	yes	characteristic
170	C₄	#17988398	1	2	yes	—
171	C₂	—	2	2	no	—
172	C₂	#22607771	1	1	yes	characteristic
173	C₂	#22608037	1	1	yes	characteristic
174	C₂	#22608046	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556523

C₂

yes

maximal, characteristic

#22608679

C₂

yes

maximal, characteristic

C₄×M₃₂×C₇₃

C₂

yes

maximal, characteristic

#17988640

—

C₂×#14556523

C₄

yes

—

C₄×C₄⋊Dic₇₃

—

#17988350

—

#17988340

C₄

yes

complemented

#17988338

—

C₂×M₃₂×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₄

yes

complemented

C₄×C₁₆×C₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

#14556523

—

#14556523

C₈

yes

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×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

characteristic

C₂²×Dic₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

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

SD₁₆

yes

characteristic

C₄²×C₇₃

Q₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#102

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

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

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

D₄⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

Q₈⋊C₈

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#7175

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×M₃₂

D₇₃

yes

characteristic

C₄×#106

—

#1564

—

C₇₃

#7175

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#431

—

146

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

146

—

#346

Dic₇₃

yes

complemented

C₄×C₁₆

—

#340

—

146

—

100

#335

—

101

#334

—

146

—

102

C₂²×C₈

C₄×D₇₃

yes

characteristic

103

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

104

C₂×C₄⋊C₄

—

146

—

105

C₂×C₄²

—

292

—

106

C₂×C₄²

D₂₉₂

yes

characteristic

107

M₃₂

—

108

C₂×C₁₆

—

109

#106

—

110

#106

—

146

—

111

C₄⋊C₈

—

146

—

112

C₂²⋊C₈

—

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

#95

—

292

—

119

C₂²×C₄

—

292

—

120

C₂²×C₄

#11924005

yes

characteristic

121

C₂²×C₄

D₅₈₄

yes

characteristic

122

C₂²×C₄

#11924017

yes

characteristic

123

C₂×C₈

—

124

C₂×C₈

—

292

—

125

C₂×C₈

C₄×Dic₇₃

yes

characteristic

126

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

127

C₂×C₈

#11924007

yes

characteristic

128

C₂×C₈

#11924007

yes

characteristic

129

C₂×C₈

C₈×D₇₃

yes

—

130

C₂×C₈

#11924011

yes

—

131

C₄⋊C₄

—

292

—

132

C₄⋊C₄

—

146

—

133

C₄²

—

584

—

134

C₄²

#11924005

yes

characteristic

135

C₄²

C₄⋊Dic₇₃

yes

characteristic

136

C₄²

#11924017

yes

characteristic

137

C₄²

Dic₂₉₂

yes

characteristic

138

C₁₆

—

139

C₂³

#14556513

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

584

—

144

C₂×C₄

—

584

—

145

C₂×C₄

#14556498

yes

characteristic

146

C₂×C₄

#14556503

yes

—

147

C₂×C₄

#14556513

yes

characteristic

148

C₂×C₄

#14556516

yes

characteristic

149

C₂×C₄

#14556516

yes

characteristic

150

C₂×C₄

#14556519

yes

—

151

C₂×C₄

#14556523

yes

characteristic

152

C₂×C₄

#14556525

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

584

—

156

C₈

C₈×Dic₇₃

yes

characteristic

157

C₈

#14556501

yes

characteristic

158

C₈

#14556520

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

#17988316

yes

—

162

C₂²

#17988350

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

584

—

167

C₄

#17988319

yes

—

168

C₄

#17988341

yes

characteristic

169

C₄

#17988347

yes

characteristic

170

C₄

#17988398

yes

—

171

C₂

—

172

C₂

#22607771

yes

characteristic

173

C₂

#22608037

yes

characteristic

174

C₂

#22608046

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556513
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#7175
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂×C₄

#14556513

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₄×M₃₂×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

hypercenter

C₄×M₃₂

D₇₃

hypocenter

C₇₃

#7175

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
8. #17988340	C₄	semidirect
12. #346×C₇₃	C₄	semidirect
17. #14556523	C₈	semidirect
91. C₇₃	#7175	semidirect
97. #346	Dic₇₃	semidirect

H ≅ G/N

type

8. #17988340

C₄

semidirect

12. #346×C₇₃

C₄

semidirect

17. #14556523

C₈

semidirect

91. C₇₃

#7175

semidirect

97. #346

Dic₇₃

semidirect

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

subgroup	index
C₄×#14556523	2
#22608679	2
C₄×M₃₂×C₇₃	2
#7175	73

subgroup

index

C₄×#14556523

#22608679

C₄×M₃₂×C₇₃

#7175

Subgroups — #32972911

Conjugacy classes of subgroups

Complements

Maximal subgroups