Group #32972902 — subgroups

Subgroups — #32972902

Order 18688.

246 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14556527	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988640	—	2	1	no	—
5	C₂×#14556527	C₄	1	2	yes	—
6	C₄×#11924012	—	2	1	no	—
7	#17988352	—	2	1	no	—
8	#17988340	C₄	1	2	yes	—
9	#17988338	—	2	1	no	—
10	C₂²×C₁₆×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₇₃	—	2	1	no	—
13	C₄×C₁₆×C₇₃	C₄	1	2	yes	—
14	C₂×#11924012	—	2	2	no	—
15	#14556527	—	2	2	no	—
16	#14556527	C₈	1	4	yes	complemented
17	C₄⋊#10596831	—	2	2	no	—
18	C₂²⋊#10596831	—	4	1	no	—
19	#14556502	—	4	1	no	—
20	C₄×#10596831	—	4	1	no	—
21	C₄×#10596831	—	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	C₂×C₁₆×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	#11924012	—	2	4	no	—
32	#11924012	—	4	2	no	—
33	C₂×#10596831	—	4	2	no	—
34	C₂×#10596831	—	4	2	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	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
45	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₃	D₈	1	1	yes	characteristic
48	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
49	C₁₆×C₇₃	—	2	4	no	—
50	C₂³×C₇₃	#103	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#104	1	2	yes	—
59	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
64	C₈×C₇₃	#97	1	1	yes	characteristic
65	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
66	#10596831	—	4	4	no	—
67	#10596831	—	8	2	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	Q₈⋊C₈	1	2	yes	—
71	C₂²×C₇₃	#340	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
75	C₄×C₇₃	#336	1	1	yes	characteristic
76	C₄×C₇₃	#339	1	1	yes	characteristic
77	C₄×C₇₃	#359	1	2	yes	—
78	C₄×C₇₃	#368	1	2	yes	—
79	#7172	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	#1279	1	1	yes	characteristic
82	C₂×C₇₃	#1381	1	1	yes	characteristic
83	C₂×C₇₃	#1382	1	1	yes	characteristic
84	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
85	C₄×#108	—	73	1	no	—
86	#1564	—	73	1	no	—
87	C₇₃	#7172	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	2	1	no	—
89	#431	—	146	1	no	—
90	C₂×#108	—	73	2	no	—
91	C₄×M₁₆	—	146	1	no	—
92	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
93	C₄×C₁₆	—	2	1	no	—
94	C₄×C₁₆	Dic₇₃	1	2	yes	—
95	#341	—	146	1	no	—
96	#335	—	73	2	no	—
97	#334	—	146	1	no	—
98	C₂×M₁₆	—	146	2	no	—
99	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
101	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#108	—	73	4	no	—
105	#108	—	146	2	no	—
106	C₄⋊C₈	—	146	2	no	—
107	C₂²⋊C₈	—	292	1	no	—
108	#97	—	292	1	no	—
109	C₄×C₈	—	292	1	no	—
110	C₄×C₈	—	292	1	no	—
111	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
112	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₄	#11924005	1	1	yes	characteristic
116	C₂²×C₄	#11924017	1	1	yes	characteristic
117	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
118	M₁₆	—	292	2	no	—
119	M₁₆	—	146	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	292	2	no	—
122	C₂×C₈	—	292	2	no	—
123	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
125	C₂×C₈	#11924007	1	1	yes	characteristic
126	C₂×C₈	#11924007	1	1	yes	characteristic
127	C₂×C₈	C₈×D₇₃	1	2	yes	—
128	C₂×C₈	#11924011	1	2	yes	—
129	C₄²	#11924005	1	1	yes	characteristic
130	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
131	C₄²	D₅₈₄	1	1	yes	characteristic
132	C₄²	#11924017	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂³	#14556516	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#14556498	1	1	yes	characteristic
138	C₂×C₄	#14556503	1	2	yes	—
139	C₂×C₄	#14556513	1	1	yes	characteristic
140	C₂×C₄	#14556513	1	1	yes	characteristic
141	C₂×C₄	#14556516	1	1	yes	characteristic
142	C₂×C₄	#14556519	1	2	yes	—
143	C₂×C₄	#14556523	1	1	yes	characteristic
144	C₂×C₄	#14556525	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	292	4	no	—
148	C₈	—	584	2	no	—
149	C₈	C₈×Dic₇₃	1	1	yes	characteristic
150	C₈	#14556501	1	1	yes	characteristic
151	C₈	#14556520	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#17988319	1	2	yes	—
155	C₂²	#17988350	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#17988316	1	2	yes	—
159	C₄	#17988341	1	1	yes	characteristic
160	C₄	#17988347	1	1	yes	characteristic
161	C₄	#17988395	1	2	yes	—
162	C₄	#17988417	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#22607771	1	1	yes	characteristic
165	C₂	#22608025	1	1	yes	characteristic
166	C₂	#22608027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556527

C₂

yes

maximal, characteristic

#22608679

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆×C₇₃

C₂

yes

maximal, characteristic

#17988640

—

C₂×#14556527

C₄

yes

—

C₄×#11924012

—

#17988352

—

#17988340

C₄

yes

—

#17988338

—

C₂²×C₁₆×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

—

C₄×C₁₆×C₇₃

C₄

yes

—

C₂×#11924012

—

#14556527

—

#14556527

C₈

yes

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×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

—

—

—

—

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

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

C₁₆×C₇₃

—

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

C₄×C₈

yes

characteristic

C₈×C₇₃

#97

yes

characteristic

C₈×C₇₃

C₄⋊C₈

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

Q₈⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₄×C₇₃

#368

yes

—

#7172

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₄×C₁₆

D₇₃

yes

characteristic

C₄×#108

—

#1564

—

C₇₃

#7172

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

146

—

C₂×#108

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

Dic₇₃

yes

—

#341

—

146

—

#335

—

#334

—

146

—

C₂×M₁₆

—

146

—

C₂²×C₈

C₄×D₇₃

yes

characteristic

100

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

101

C₂×C₄²

D₂₉₂

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

#108

—

105

#108

—

146

—

106

C₄⋊C₈

—

146

—

107

C₂²⋊C₈

—

292

—

108

#97

—

292

—

109

C₄×C₈

—

292

—

110

C₄×C₈

—

292

—

111

C₄×C₈

Dic₁₄₆

yes

characteristic

112

C₄×C₈

C₄×D₇₃

yes

characteristic

113

C₄×C₈

C₂×Dic₇₃

yes

characteristic

114

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₄

#11924005

yes

characteristic

116

C₂²×C₄

#11924017

yes

characteristic

117

C₂²×C₄

Dic₂₉₂

yes

characteristic

118

M₁₆

—

292

—

119

M₁₆

—

146

—

120

C₂×C₈

—

121

C₂×C₈

—

292

—

122

C₂×C₈

—

292

—

123

C₂×C₈

C₄×Dic₇₃

yes

characteristic

124

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

125

C₂×C₈

#11924007

yes

characteristic

126

C₂×C₈

#11924007

yes

characteristic

127

C₂×C₈

C₈×D₇₃

yes

—

128

C₂×C₈

#11924011

yes

—

129

C₄²

#11924005

yes

characteristic

130

C₄²

C₄⋊Dic₇₃

yes

characteristic

131

C₄²

D₅₈₄

yes

characteristic

132

C₄²

#11924017

yes

characteristic

133

C₁₆

—

134

C₂³

#14556516

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14556498

yes

characteristic

138

C₂×C₄

#14556503

yes

—

139

C₂×C₄

#14556513

yes

characteristic

140

C₂×C₄

#14556513

yes

characteristic

141

C₂×C₄

#14556516

yes

characteristic

142

C₂×C₄

#14556519

yes

—

143

C₂×C₄

#14556523

yes

characteristic

144

C₂×C₄

#14556525

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

292

—

148

C₈

—

584

—

149

C₈

C₈×Dic₇₃

yes

characteristic

150

C₈

#14556501

yes

characteristic

151

C₈

#14556520

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

#17988319

yes

—

155

C₂²

#17988350

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

#17988316

yes

—

159

C₄

#17988341

yes

characteristic

160

C₄

#17988347

yes

characteristic

161

C₄

#17988395

yes

—

162

C₄

#17988417

yes

—

163

C₂

—

164

C₂

#22607771

yes

characteristic

165

C₂

#22608025

yes

characteristic

166

C₂

#22608027

yes

characteristic

	subgroup	G/H
center	C₄²	D₅₈₄
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#103
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#7172
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₄²

D₅₈₄

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₂×C₄×C₁₆×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

hypercenter

C₂×C₄×C₁₆

D₇₃

hypocenter

C₇₃

#7172

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
16. #14556527	C₈	semidirect
87. C₇₃	#7172	semidirect

H ≅ G/N

type

16. #14556527

C₈

semidirect

87. C₇₃

#7172

semidirect

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

subgroup	index
C₄×#14556527	2
#22608679	2
C₂×C₄×C₁₆×C₇₃	2
#7172	73

subgroup

index

C₄×#14556527

#22608679

C₂×C₄×C₁₆×C₇₃

#7172

Subgroups — #32972902

Conjugacy classes of subgroups

Complements

Maximal subgroups