Group #33004035 — subgroups

Subgroups — #33004035

Order 18688.

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

normal subgroups only

This group has 286 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×#17988417	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22610680	C₂	1	2	yes	maximal
4	#22608026	C₂	1	2	yes	maximal
5	C₇₃×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	1	yes	characteristic
7	C₂×#14556527	C₂²	1	2	yes	—
8	#17988614	—	2	2	no	—
9	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988417	—	2	2	no	—
11	C₁₆×Dic₇₃	—	2	2	no	—
12	#17988346	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	#427×C₇₃	C₂²	1	2	yes	—
15	#367×C₇₃	C₄	1	2	yes	complemented
16	#366×C₇₃	C₄	1	2	yes	complemented
17	C₂×#11924012	—	2	2	no	—
18	C₂²×#10596831	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
20	C₂×#11924003	—	2	1	no	—
21	C₂×#11924003	—	2	1	no	—
22	C₂×#11924003	Q₈	1	2	yes	—
23	#14556527	—	2	2	no	—
24	#14556527	—	2	2	no	—
25	C₄⋊#10596831	—	4	4	no	—
26	C₈×Dic₇₃	—	2	1	no	—
27	C₈×Dic₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	#117×C₇₃	—	2	2	no	—
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
34	#107×C₇₃	C₂×C₄	1	4	yes	—
35	#106×C₇₃	—	2	2	no	—
36	C₂²×Dic₇₃	—	2	1	no	—
37	#11924012	—	4	2	no	—
38	#11924012	—	4	2	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	D₈	1	2	yes	—
43	C₄×Dic₇₃	—	2	1	no	—
44	C₄×Dic₇₃	Q₁₆	1	2	yes	—
45	#11924003	—	2	2	no	—
46	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₇₃	—	2	4	no	—
54	C₂²⋊C₄×C₇₃	—	4	2	no	—
55	C₄²×C₇₃	—	4	2	no	—
56	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
58	C₂³×C₇₃	#135	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	4	4	no	—
60	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
66	C₂×Dic₇₃	—	2	4	no	—
67	C₂×Dic₇₃	—	2	2	no	—
68	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
70	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₈×C₇₃	#125	1	1	yes	characteristic
72	#10596831	—	2	2	no	—
73	#10596831	—	8	2	no	—
74	C₂²×C₇₃	—	2	2	no	—
75	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
76	C₂²×C₇₃	#467	1	1	yes	characteristic
77	C₂²×C₇₃	#499	1	1	yes	characteristic
78	C₄×C₇₃	—	8	4	no	—
79	C₄×C₇₃	#389	1	1	yes	characteristic
80	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₇₃	#466	1	1	yes	characteristic
82	C₄×C₇₃	#500	1	1	yes	characteristic
83	Dic₇₃	—	2	4	no	—
84	#13169	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	#1942	1	1	yes	characteristic
88	C₂×C₇₃	#2178	1	2	yes	—
89	C₂×#368	—	73	1	no	—
90	#2158	D₇₃	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	73	1	no	—
92	#1945	—	73	2	no	—
93	#1382	—	73	2	no	—
94	C₇₃	#13169	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	73	1	no	—
96	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
97	C₂×#108	—	73	2	no	—
98	#427	C₂×D₇₃	1	2	yes	—
99	#424	—	146	2	no	—
100	C₂×C₄×C₈	—	73	1	no	—
101	#368	—	146	2	no	—
102	#367	Dic₇₃	1	2	yes	complemented
103	#366	Dic₇₃	1	2	yes	complemented
104	C₄×C₁₆	—	146	2	no	—
105	#339	—	146	2	no	—
106	C₂×M₁₆	—	146	2	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
109	#117	—	2	2	no	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₁₆	—	73	2	no	—
112	C₂×C₁₆	—	146	1	no	—
113	C₂×C₁₆	—	146	1	no	—
114	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
115	C₂×C₁₆	D₂₉₂	1	2	yes	—
116	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
117	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
118	#108	—	146	2	no	—
119	#108	—	146	2	no	—
120	#107	C₂×Dic₇₃	1	4	yes	—
121	#106	—	2	2	no	—
122	C₄⋊C₈	—	292	4	no	—
123	C₄×C₈	—	146	1	no	—
124	C₄×C₈	—	146	1	no	—
125	C₂²×C₄	—	146	1	no	—
126	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
127	M₁₆	—	292	2	no	—
128	M₁₆	—	292	2	no	—
129	C₂×C₈	—	292	2	no	—
130	C₂×C₈	—	73	2	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
134	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
137	C₂×C₈	#11924027	1	1	yes	characteristic
138	C₂×C₈	#11924029	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	4	2	no	—
142	C₄²	—	73	2	no	—
143	C₄²	—	146	1	no	—
144	C₁₆	—	146	2	no	—
145	C₁₆	C₄⋊Dic₇₃	1	2	yes	—
146	C₁₆	C₄⋊Dic₇₃	1	2	yes	—
147	C₂³	#14556610	1	1	yes	characteristic
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	146	4	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
152	C₂×C₄	#14556566	1	1	yes	characteristic
153	C₂×C₄	#14556590	1	1	yes	characteristic
154	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
155	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
156	C₂×C₄	#14556614	1	1	yes	characteristic
157	C₈	—	584	2	no	—
158	C₈	—	146	2	no	—
159	C₈	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
160	C₈	Q₈×Dic₇₃	1	1	yes	characteristic
161	C₈	#14556571	1	1	yes	characteristic
162	C₈	#14556582	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	D₈×Dic₇₃	1	1	yes	characteristic
165	C₂²	#17988829	1	1	yes	characteristic
166	C₂²	#17988988	1	1	yes	characteristic
167	C₄	—	8	4	no	—
168	C₄	—	146	4	no	—
169	C₄	#17988486	1	1	yes	characteristic
170	C₄	Q₁₆×Dic₇₃	1	1	yes	characteristic
171	C₄	#17988824	1	1	yes	characteristic
172	C₄	#17988991	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#22610665	1	1	yes	characteristic
176	C₂	#22611812	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988417

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₇₃

C₂

yes

maximal, characteristic

#22610680

C₂

yes

maximal

#22608026

C₂

yes

maximal

C₇₃×#2158

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556527

C₂²

yes

—

#17988614

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988417

—

C₁₆×Dic₇₃

—

#17988346

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#427×C₇₃

C₂²

yes

—

#367×C₇₃

C₄

yes

complemented

#366×C₇₃

C₄

yes

complemented

C₂×#11924012

—

C₂²×#10596831

D₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

—

Q₈

yes

—

#14556527

—

#14556527

—

C₄⋊#10596831

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#117×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

Q₈

yes

—

#107×C₇₃

C₂×C₄

yes

—

#106×C₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

D₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

Q₁₆

yes

—

#11924003

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

C₄⋊C₄

yes

—

C₁₆×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

#135

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

#389

yes

characteristic

C₄×C₇₃

C₄×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#13169

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1942

yes

characteristic

C₂×C₇₃

#2178

yes

—

C₂×#368

—

#2158

D₇₃

yes

characteristic

C₂×C₄×C₁₆

—

#1945

—

#1382

—

C₇₃

#13169

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×#108

—

#427

C₂×D₇₃

yes

—

#424

—

146

—

100

C₂×C₄×C₈

—

101

#368

—

146

—

102

#367

Dic₇₃

yes

complemented

103

#366

Dic₇₃

yes

complemented

104

C₄×C₁₆

—

146

—

105

#339

—

146

—

106

C₂×M₁₆

—

146

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²×D₇₃

yes

characteristic

109

#117

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

146

—

113

C₂×C₁₆

—

146

—

114

C₂×C₁₆

Dic₁₄₆

yes

—

115

C₂×C₁₆

D₂₉₂

yes

—

116

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

117

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

118

#108

—

146

—

119

#108

—

146

—

120

#107

C₂×Dic₇₃

yes

—

121

#106

—

122

C₄⋊C₈

—

292

—

123

C₄×C₈

—

146

—

124

C₄×C₈

—

146

—

125

C₂²×C₄

—

146

—

126

C₂²×C₄

D₄×D₇₃

yes

characteristic

127

M₁₆

—

292

—

128

M₁₆

—

292

—

129

C₂×C₈

—

292

—

130

C₂×C₈

—

131

C₂×C₈

—

146

—

132

C₂×C₈

—

146

—

133

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

134

C₂×C₈

C₂×D₂₉₂

yes

characteristic

135

C₂×C₈

Q₈×D₇₃

yes

characteristic

136

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

137

C₂×C₈

#11924027

yes

characteristic

138

C₂×C₈

#11924029

yes

characteristic

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

146

—

144

C₁₆

—

146

—

145

C₁₆

C₄⋊Dic₇₃

yes

—

146

C₁₆

C₄⋊Dic₇₃

yes

—

147

C₂³

#14556610

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

D₄×Dic₇₃

yes

characteristic

152

C₂×C₄

#14556566

yes

characteristic

153

C₂×C₄

#14556590

yes

characteristic

154

C₂×C₄

D₈×D₇₃

yes

characteristic

155

C₂×C₄

Q₁₆×D₇₃

yes

characteristic

156

C₂×C₄

#14556614

yes

characteristic

157

C₈

—

584

—

158

C₈

—

146

—

159

C₈

C₂×C₄⋊Dic₇₃

yes

characteristic

160

C₈

Q₈×Dic₇₃

yes

characteristic

161

C₈

#14556571

yes

characteristic

162

C₈

#14556582

yes

characteristic

163

C₂²

—

164

C₂²

D₈×Dic₇₃

yes

characteristic

165

C₂²

#17988829

yes

characteristic

166

C₂²

#17988988

yes

characteristic

167

C₄

—

168

C₄

—

146

—

169

C₄

#17988486

yes

characteristic

170

C₄

Q₁₆×Dic₇₃

yes

characteristic

171

C₄

#17988824

yes

characteristic

172

C₄

#17988991

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#22610665

yes

characteristic

176

C₂

#22611812

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2158	C₂
socle	C₂²×C₇₃	C₄×D₈
prime radical	C₂³×C₇₃	#135
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2158	D₇₃
hypocenter	C₇₃	#13169
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#2158

C₂

socle

C₂²×C₇₃

C₄×D₈

prime radical

C₂³×C₇₃

#135

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
15. #367×C₇₃	C₄	semidirect
16. #366×C₇₃	C₄	semidirect
94. C₇₃	#13169	semidirect
102. #367	Dic₇₃	semidirect
103. #366	Dic₇₃	semidirect

H ≅ G/N

type

15. #367×C₇₃

C₄

semidirect

16. #366×C₇₃

C₄

semidirect

94. C₇₃

#13169

semidirect

102. #367

Dic₇₃

semidirect

103. #366

Dic₇₃

semidirect

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

subgroup	index
C₂×#17988417	2
C₂×C₁₆×Dic₇₃	2
#22610680	2
#22608026	2
C₇₃×#2158	2
#13169	73

subgroup

index

C₂×C₁₆×Dic₇₃

C₇₃×#2158

Subgroups — #33004035

Conjugacy classes of subgroups

Complements

Maximal subgroups