Group #32985469 — subgroups

Subgroups — #32985469

Order 18688.

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

normal subgroups only

This group has 390 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	C₂×#17988985	C₂	1	1	yes	maximal, characteristic
2	#22610069	C₂	1	1	yes	maximal, characteristic
3	#22608615	C₂	1	2	yes	maximal
4	#22608611	C₂	1	2	yes	maximal
5	C₇₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556591	C₂²	1	2	yes	—
7	#17988985	—	2	2	no	—
8	#17988985	C₄	1	4	yes	—
9	C₂×#14556523	—	2	2	no	—
10	#17988611	C₂²	1	1	yes	characteristic
11	#17988611	C₂²	1	1	yes	characteristic
12	#17988352	—	2	2	no	—
13	#17988335	—	2	4	no	—
14	#17988334	—	2	4	no	—
15	#431×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#322×C₇₃	—	2	2	no	—
18	C₂²×Dic₁₄₆	—	2	2	no	—
19	C₂×#11924012	—	2	1	no	—
20	C₂×#11924012	—	2	1	no	—
21	#14556591	—	2	2	no	—
22	#14556591	—	2	2	no	—
23	#14556591	—	2	2	no	—
24	#14556591	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₇₃	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	#14556523	—	2	8	no	—
28	C₄⋊#10596831	—	2	2	no	—
29	C₄⋊#10596831	—	2	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₃	D₄	1	4	yes	—
33	C₂²⋊C₈×C₇₃	—	2	2	no	—
34	C₄×C₈×C₇₃	—	2	2	no	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₃	D₄	1	2	yes	—
37	C₂×Dic₁₄₆	—	2	4	no	—
38	C₂×Dic₁₄₆	—	4	2	no	—
39	C₂×Dic₁₄₆	—	8	4	no	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	#11924012	—	4	2	no	—
42	#11924012	—	4	2	no	—
43	C₂×#10596831	—	4	1	no	—
44	C₂×#10596831	—	4	1	no	—
45	C₄⋊Dic₇₃	—	2	4	no	—
46	C₄⋊Dic₇₃	—	2	4	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	C₄⋊Dic₇₃	—	4	2	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	#41	1	2	yes	—
51	C₂×C₈×C₇₃	—	2	2	no	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	—	2	4	no	—
54	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₃	C₂×D₄	1	2	yes	—
58	C₂³×C₇₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	4	no	—
60	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₇₃	#121	1	2	yes	—
63	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#137	1	2	yes	—
65	C₂×C₄×C₇₃	#137	1	2	yes	—
66	C₂×Dic₇₃	—	4	4	no	—
67	C₂×Dic₇₃	—	8	4	no	—
68	Dic₁₄₆	—	8	4	no	—
69	Dic₁₄₆	—	8	4	no	—
70	C₈×C₇₃	—	2	4	no	—
71	C₈×C₇₃	—	4	4	no	—
72	#10596831	—	8	1	no	—
73	#10596831	—	8	1	no	—
74	C₂²×C₇₃	—	2	2	no	—
75	C₂²×C₇₃	#390	1	1	yes	characteristic
76	C₂²×C₇₃	#491	1	2	yes	—
77	C₄×C₇₃	—	2	2	no	—
78	C₄×C₇₃	#419	1	2	yes	—
79	C₄×C₇₃	#471	1	4	yes	—
80	C₄×C₇₃	#497	1	1	yes	characteristic
81	C₄×C₇₃	#497	1	1	yes	characteristic
82	Dic₇₃	—	8	4	no	—
83	#9776	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	2	no	—
85	C₂×C₇₃	#1960	1	1	yes	characteristic
86	C₂×C₇₃	#1977	1	2	yes	—
87	C₂×#498	—	73	1	no	—
88	#1837	—	73	1	no	—
89	#1549	—	73	2	no	—
90	#1548	—	73	2	no	—
91	#1452	D₇₃	1	1	yes	characteristic
92	C₇₃	#9776	1	1	yes	characteristic, complemented, Sylow
93	C₂×C₄⋊Q₈	—	73	2	no	—
94	#498	—	73	4	no	—
95	#498	—	146	2	no	—
96	#431	C₂×D₇₃	1	2	yes	—
97	C₂×#106	—	146	2	no	—
98	#423	—	73	1	no	—
99	#423	—	73	1	no	—
100	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
101	#341	—	146	2	no	—
102	#333	—	146	4	no	—
103	#332	—	146	4	no	—
104	#322	—	2	2	no	—
105	C₂²×Q₈	—	146	2	no	—
106	C₂×M₁₆	—	146	1	no	—
107	C₂×M₁₆	—	146	1	no	—
108	C₂²×C₈	D₂₉₂	1	2	yes	—
109	C₄⋊Q₈	—	73	4	no	—
110	C₄⋊Q₈	—	146	2	no	—
111	C₄⋊Q₈	—	146	2	no	—
112	C₄⋊Q₈	—	146	2	no	—
113	C₂×C₄⋊C₄	—	146	2	no	—
114	C₂×C₄⋊C₄	—	146	2	no	—
115	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
116	#106	—	146	8	no	—
117	C₄⋊C₈	—	146	2	no	—
118	C₄⋊C₈	—	146	2	no	—
119	C₄⋊C₈	D₂₉₂	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₇₃	1	2	yes	—
123	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
124	C₂×Q₈	—	292	2	no	—
125	C₂×Q₈	—	584	4	no	—
126	C₂×Q₈	—	146	4	no	—
127	C₂²×C₄	—	292	2	no	—
128	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
129	C₂²×C₄	#11924028	1	2	yes	—
130	M₁₆	—	292	2	no	—
131	M₁₆	—	292	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	292	1	no	—
136	C₂×C₈	—	292	1	no	—
137	C₂×C₈	#11924005	1	4	yes	—
138	C₄⋊C₄	—	292	2	no	—
139	C₄⋊C₄	—	292	2	no	—
140	C₄⋊C₄	—	146	4	no	—
141	C₄⋊C₄	—	146	4	no	—
142	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
143	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
144	C₄²	C₂×D₂₉₂	1	2	yes	—
145	C₂³	#14556579	1	1	yes	characteristic
146	Q₈	—	584	4	no	—
147	Q₈	—	584	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	292	4	no	—
150	C₂×C₄	—	584	4	no	—
151	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
152	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
153	C₂×C₄	#14556565	1	2	yes	—
154	C₂×C₄	#14556588	1	1	yes	characteristic
155	C₂×C₄	#14556624	1	2	yes	—
156	C₂×C₄	#14556624	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	584	1	no	—
160	C₈	—	584	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#17988489	1	1	yes	characteristic
163	C₂²	#17988956	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	584	4	no	—
166	C₄	#17988597	1	2	yes	—
167	C₄	#17988850	1	4	yes	—
168	C₄	#17988981	1	1	yes	characteristic
169	C₄	#17988981	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#22610751	1	1	yes	characteristic
172	C₂	#22610834	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988985

C₂

yes

maximal, characteristic

#22610069

C₂

yes

maximal, characteristic

#22608615

C₂

yes

maximal

#22608611

C₂

yes

maximal

C₇₃×#1452

C₂

yes

maximal, characteristic

C₂×#14556591

C₂²

yes

—

#17988985

—

#17988985

C₄

yes

—

C₂×#14556523

—

#17988611

C₂²

yes

characteristic

#17988611

C₂²

yes

—

—

—

#431×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

—

C₂²×Dic₁₄₆

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556523

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

D₄

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₂³×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

#137

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#491

yes

—

C₄×C₇₃

—

C₄×C₇₃

#419

yes

—

C₄×C₇₃

#471

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#9776

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1960

yes

characteristic

C₂×C₇₃

#1977

yes

—

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#9776

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#498

—

#498

—

146

—

#431

C₂×D₇₃

yes

—

C₂×#106

—

146

—

#423

—

#423

—

100

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

101

#341

—

146

—

102

#333

—

146

—

103

#332

—

146

—

104

#322

—

105

C₂²×Q₈

—

146

—

106

C₂×M₁₆

—

146

—

107

C₂×M₁₆

—

146

—

108

C₂²×C₈

D₂₉₂

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

146

—

111

C₄⋊Q₈

—

146

—

112

C₄⋊Q₈

—

146

—

113

C₂×C₄⋊C₄

—

146

—

114

C₂×C₄⋊C₄

—

146

—

115

C₂×C₄²

C₂²×D₇₃

yes

characteristic

116

#106

—

146

—

117

C₄⋊C₈

—

146

—

118

C₄⋊C₈

—

146

—

119

C₄⋊C₈

D₂₉₂

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₄×D₇₃

yes

—

123

C₄×C₈

C₂²⋊D₇₃

yes

—

124

C₂×Q₈

—

292

—

125

C₂×Q₈

—

584

—

126

C₂×Q₈

—

146

—

127

C₂²×C₄

—

292

—

128

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

129

C₂²×C₄

#11924028

yes

—

130

M₁₆

—

292

—

131

M₁₆

—

292

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

292

—

137

C₂×C₈

#11924005

yes

—

138

C₄⋊C₄

—

292

—

139

C₄⋊C₄

—

292

—

140

C₄⋊C₄

—

146

—

141

C₄⋊C₄

—

146

—

142

C₄²

C₂×C₄×D₇₃

yes

characteristic

143

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

144

C₄²

C₂×D₂₉₂

yes

—

145

C₂³

#14556579

yes

characteristic

146

Q₈

—

584

—

147

Q₈

—

584

—

148

C₂×C₄

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

584

—

151

C₂×C₄

C₂×#11924005

yes

characteristic

152

C₂×C₄

C₄×D₂₉₂

yes

—

153

C₂×C₄

#14556565

yes

—

154

C₂×C₄

#14556588

yes

characteristic

155

C₂×C₄

#14556624

yes

—

156

C₂×C₄

#14556624

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

584

—

160

C₈

—

584

—

161

C₂²

—

162

C₂²

#17988489

yes

characteristic

163

C₂²

#17988956

yes

—

164

C₄

—

165

C₄

—

584

—

166

C₄

#17988597

yes

—

167

C₄

#17988850

yes

—

168

C₄

#17988981

yes

characteristic

169

C₄

#17988981

yes

characteristic

170

C₂

—

171

C₂

#22610751

yes

characteristic

172

C₂

#22610834

yes

—

	subgroup	G/H
center	C₂²	#17988489
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1452	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	#124
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1452	D₇₃
hypocenter	C₇₃	#9776
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#17988489

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1452

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
92. C₇₃	#9776	semidirect

H ≅ G/N

type

92. C₇₃

#9776

semidirect

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

subgroup	index
C₂×#17988985	2
#22610069	2
#22608615	2
#22608611	2
C₇₃×#1452	2
#9776	73

subgroup

index

C₇₃×#1452

Subgroups — #32985469

Conjugacy classes of subgroups

Complements

Maximal subgroups