Group #32979739 — subgroups

Subgroups — #32979739

Order 18688.

402 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 187 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	#22610083	C₂	1	1	yes	maximal, characteristic
2	#22609677	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988339	C₂	1	4	yes	maximal
4	C₇₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596831	—	2	2	no	—
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	C₂²	1	2	yes	—
8	#17988342	C₄	1	2	yes	—
9	#17988342	C₄	1	2	yes	—
10	#17988339	—	2	8	no	—
11	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
13	C₄³×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×#10596831	—	2	2	no	—
15	C₂²×#10596831	C₂×C₄	1	4	yes	—
16	C₄⋊#10596831	—	2	4	no	—
17	C₄⋊#10596831	—	4	4	no	—
18	C₄×#10596831	D₄	1	4	yes	—
19	C₄×#10596831	Q₈	1	4	yes	—
20	C₂²×C₈×C₇₃	—	2	2	no	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₃	D₄	1	2	yes	—
24	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
25	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₇₃	—	2	4	no	—
27	C₄⋊C₈×C₇₃	—	4	4	no	—
28	C₂×#10596831	—	2	4	no	—
29	C₂×#10596831	—	2	4	no	—
30	C₂×#10596831	—	4	2	no	—
31	C₂×#10596831	—	4	4	no	—
32	C₂×#10596831	C₄⋊C₄	1	8	yes	—
33	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	4	2	no	—
39	C₂×C₈×C₇₃	—	4	4	no	—
40	C₄²×C₇₃	—	2	2	no	—
41	C₄²×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	—	4	2	no	—
43	C₄²×C₇₃	D₈	1	4	yes	—
44	C₄²×C₇₃	Q₁₆	1	4	yes	—
45	C₄²×C₇₃	C₂×D₄	1	2	yes	—
46	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
47	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	#107	1	4	yes	—
54	C₂×C₄×C₇₃	#107	1	4	yes	—
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₇₃	#131	1	2	yes	—
60	C₂×C₄×C₇₃	#131	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
63	C₈×C₇₃	—	8	2	no	—
64	#10596831	—	2	8	no	—
65	#10596831	—	8	2	no	—
66	C₂²×C₇₃	#384	1	1	yes	characteristic
67	C₂²×C₇₃	#424	1	1	yes	characteristic
68	C₂²×C₇₃	#424	1	1	yes	characteristic
69	C₂²×C₇₃	C₂×#107	1	2	yes	—
70	C₂²×C₇₃	#430	1	2	yes	—
71	C₄×C₇₃	—	2	4	no	—
72	C₄×C₇₃	—	2	4	no	—
73	C₄×C₇₃	—	2	2	no	—
74	C₄×C₇₃	#424	1	4	yes	—
75	C₄×C₇₃	#493	1	1	yes	characteristic
76	C₄×C₇₃	#494	1	1	yes	characteristic
77	C₄×C₇₃	#500	1	2	yes	—
78	#8810	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	#1571	1	2	yes	—
80	C₂×C₇₃	#1571	1	2	yes	—
81	C₂×C₇₃	#1843	1	1	yes	characteristic
82	C₂×C₇₃	#1845	1	1	yes	characteristic
83	C₂×C₇₃	#1849	1	1	yes	characteristic
84	#1842	—	73	1	no	—
85	#1842	D₇₃	1	1	yes	characteristic
86	#1744	—	73	1	no	—
87	C₂×#335	—	73	4	no	—
88	C₇₃	#8810	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₄⋊C₈	—	2	2	no	—
90	C₂×C₄⋊C₈	—	73	2	no	—
91	C₂×C₄⋊C₈	—	146	2	no	—
92	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
93	C₂×C₄×C₈	—	73	2	no	—
94	C₄³	C₂×D₇₃	1	1	yes	characteristic
95	#336	—	73	2	no	—
96	#336	—	73	2	no	—
97	#335	—	146	8	no	—
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	73	4	no	—
100	C₂²×C₈	—	146	2	no	—
101	C₂×C₄²	Dic₁₄₆	1	2	yes	—
102	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
105	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄⋊C₈	—	292	4	no	—
109	C₄⋊C₈	—	146	4	no	—
110	C₄×C₈	—	73	4	no	—
111	C₄×C₈	—	73	4	no	—
112	C₂²×C₄	#11924007	1	2	yes	—
113	C₂²×C₄	#11924007	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	—	292	2	no	—
120	C₂×C₈	—	292	4	no	—
121	C₂×C₈	—	73	8	no	—
122	C₂×C₈	—	146	4	no	—
123	C₂×C₈	—	146	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	2	no	—
127	C₄²	#11924013	1	4	yes	—
128	C₄²	#11924018	1	4	yes	—
129	C₄²	D₄×D₇₃	1	2	yes	—
130	C₄²	Q₈×D₇₃	1	2	yes	—
131	C₂³	C₂×#11924007	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#14556524	1	4	yes	—
138	C₂×C₄	#14556524	1	4	yes	—
139	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
140	C₂×C₄	#14556587	1	1	yes	characteristic
141	C₂×C₄	#14556589	1	1	yes	characteristic
142	C₂×C₄	#14556590	1	2	yes	—
143	C₂×C₄	#14556597	1	2	yes	—
144	C₂×C₄	#14556599	1	2	yes	—
145	C₂×C₄	C₂×#11924013	1	2	yes	—
146	C₂×C₄	C₂×#11924018	1	2	yes	—
147	C₈	—	8	2	no	—
148	C₈	—	584	2	no	—
149	C₈	—	146	8	no	—
150	C₂²	#17988456	1	1	yes	characteristic
151	C₂²	#17988613	1	1	yes	characteristic
152	C₂²	#17988615	1	1	yes	characteristic
153	C₂²	C₂×#14556524	1	2	yes	—
154	C₂²	#17988635	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	#17988612	1	4	yes	—
159	C₄	#17988963	1	1	yes	characteristic
160	C₄	#17988966	1	1	yes	characteristic
161	C₄	#17988990	1	2	yes	—
162	C₂	#22608705	1	2	yes	—
163	C₂	#22608707	1	2	yes	—
164	C₂	#22610084	1	1	yes	characteristic
165	C₂	#22610090	1	1	yes	characteristic
166	C₂	#22610105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610083

C₂

yes

maximal, characteristic

#22609677

C₂

yes

maximal, characteristic

C₂×#17988339

C₂

yes

maximal

C₇₃×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

—

#17988342

C₄

yes

—

#17988342

C₄

yes

—

#17988339

—

C₂×C₄⋊C₈×C₇₃

—

C₂×C₄⋊C₈×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

—

C₂²×#10596831

C₂×C₄

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

Q₈

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

—

—

—

—

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₈

yes

—

C₄²×C₇₃

Q₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#430

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#424

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#8810

—

maximal, Sylow

C₂×C₇₃

#1571

yes

—

C₂×C₇₃

#1571

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

#1744

—

C₂×#335

—

C₇₃

#8810

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄⋊C₈

C₂×D₇₃

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₇₃

yes

—

—

—

146

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

146

—

101

C₂×C₄²

Dic₁₄₆

yes

—

102

C₂×C₄²

C₄×D₇₃

yes

characteristic

103

C₂×C₄²

C₄×D₇₃

yes

characteristic

104

C₂×C₄²

C₂²⋊D₇₃

yes

—

105

C₂×C₄²

C₂²×D₇₃

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

292

—

109

C₄⋊C₈

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₂²×C₄

#11924007

yes

—

113

C₂²×C₄

#11924007

yes

—

114

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

115

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

116

C₂²×C₄

C₂×Dic₁₄₆

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11924013

yes

—

128

C₄²

#11924018

yes

—

129

C₄²

D₄×D₇₃

yes

—

130

C₄²

Q₈×D₇₃

yes

—

131

C₂³

C₂×#11924007

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#14556524

yes

—

138

C₂×C₄

#14556524

yes

—

139

C₂×C₄

C₄⋊C₄×D₇₃

yes

—

140

C₂×C₄

#14556587

yes

characteristic

141

C₂×C₄

#14556589

yes

characteristic

142

C₂×C₄

#14556590

yes

—

143

C₂×C₄

#14556597

yes

—

144

C₂×C₄

#14556599

yes

—

145

C₂×C₄

C₂×#11924013

yes

—

146

C₂×C₄

C₂×#11924018

yes

—

147

C₈

—

148

C₈

—

584

—

149

C₈

—

146

—

150

C₂²

#17988456

yes

characteristic

151

C₂²

#17988613

yes

characteristic

152

C₂²

#17988615

yes

characteristic

153

C₂²

C₂×#14556524

yes

—

154

C₂²

#17988635

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#17988612

yes

—

159

C₄

#17988963

yes

characteristic

160

C₄

#17988966

yes

characteristic

161

C₄

#17988990

yes

—

162

C₂

#22608705

yes

—

163

C₂

#22608707

yes

—

164

C₂

#22610084

yes

characteristic

165

C₂

#22610090

yes

characteristic

166

C₂

#22610105

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1842	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1842	D₇₃
hypocenter	C₇₃	#8810
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1842

C₂

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
88. C₇₃	#8810	semidirect

H ≅ G/N

type

88. C₇₃

#8810

semidirect

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

subgroup	index
#22610083	2
#22609677	2
C₂×#17988339	2
C₇₃×#1842	2
#8810	73

subgroup

index

C₇₃×#1842

Subgroups — #32979739

Conjugacy classes of subgroups

Complements

Maximal subgroups