Group #33241465 — subgroups

Subgroups — C₂×#22612209 #33241465

Order 18688.

494 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 203 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×#17988988	C₂	1	2	yes	maximal
2	#22612209	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988415	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988412	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#367×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556582	—	2	2	no	—
8	C₂²×#11924003	C₂²	1	1	yes	characteristic
9	#17988988	—	2	4	no	—
10	#17988988	C₂²	1	8	yes	—
11	C₂×#14556525	C₂²	1	2	yes	—
12	C₂×#14556522	—	2	2	no	—
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	#17988415	C₂²	1	4	yes	—
15	#17988412	C₂²	1	8	yes	—
16	C₁₆×Dic₇₃	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#107×C₇₃	C₂²	1	2	yes	—
19	#367×C₇₃	C₂²	1	4	yes	—
20	C₂²×#10596831	D₄	1	1	yes	characteristic
21	#14556582	—	2	8	no	—
22	#14556582	—	4	4	no	—
23	C₂×C₄⋊Dic₇₃	—	2	2	no	—
24	C₂×C₄⋊Dic₇₃	—	4	2	no	—
25	C₂×#11924007	—	4	2	no	—
26	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	C₂×#11924003	Q₈	1	4	yes	—
28	C₂×#11924003	C₂³	1	2	yes	—
29	#14556525	—	2	2	no	—
30	#14556525	C₂³	1	4	yes	—
31	#14556522	—	2	8	no	—
32	C₈×Dic₇₃	D₄	1	2	yes	—
33	C₈×Dic₇₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
36	C₂×C₁₆×C₇₃	Q₈	1	4	yes	—
37	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
38	#107×C₇₃	—	2	2	no	—
39	#107×C₇₃	C₂³	1	4	yes	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₂²×Dic₇₃	D₈	1	2	yes	—
42	C₂×#10596831	Q₁₆	1	4	yes	—
43	C₂×#10596831	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₇₃	—	2	4	no	—
45	C₄⋊Dic₇₃	—	4	2	no	—
46	C₄⋊Dic₇₃	—	4	8	no	—
47	#11924007	—	4	8	no	—
48	C₄×Dic₇₃	D₈	1	2	yes	—
49	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
50	#11924003	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₇₃	—	4	2	no	—
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₇₃	—	2	4	no	—
58	C₄⋊C₄×C₇₃	—	4	2	no	—
59	C₁₆×C₇₃	C₂×Q₈	1	4	yes	—
60	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	4	4	no	—
62	C₂×C₄×C₇₃	—	8	4	no	—
63	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₇₃	—	2	4	no	—
68	C₂×Dic₇₃	—	4	4	no	—
69	C₂×Dic₇₃	—	8	4	no	—
70	C₂×Dic₇₃	C₂×D₈	1	4	yes	—
71	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
72	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
73	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
74	#10596831	C₂×Q₁₆	1	4	yes	—
75	C₂²×C₇₃	#500	1	2	yes	—
76	C₂²×C₇₃	#508	1	4	yes	—
77	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
78	C₄×C₇₃	—	8	4	no	—
79	C₄×C₇₃	#500	1	2	yes	—
80	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
82	Dic₇₃	—	2	4	no	—
83	Dic₇₃	—	8	4	no	—
84	C₂×#2253	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#2253	1	4	yes	complemented
86	C₂×C₇₃	C₂×#500	1	1	yes	characteristic
87	C₂×C₇₃	C₂×#508	1	2	yes	—
88	C₂×#499	—	73	2	no	—
89	#2253	—	73	8	no	—
90	C₂×#367	—	73	1	no	—
91	C₂×#367	D₇₃	1	1	yes	characteristic
92	C₂×#366	—	73	2	no	—
93	C₂×C₄×C₁₆	—	73	1	no	—
94	C₇₃	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
95	C₂×#125	—	146	2	no	—
96	C₂²×C₁₆	—	73	1	no	—
97	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
98	#499	—	73	8	no	—
99	#499	—	146	4	no	—
100	C₂×#107	—	73	2	no	—
101	C₂×#107	C₂×D₇₃	1	2	yes	—
102	C₂×#106	—	146	2	no	—
103	C₂×C₄×C₈	—	73	1	no	—
104	#367	—	73	4	no	—
105	#367	C₂×D₇₃	1	4	yes	—
106	#366	—	73	8	no	—
107	C₄×C₁₆	—	73	4	no	—
108	C₂²×C₈	—	73	1	no	—
109	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
110	#125	—	292	4	no	—
111	#125	—	146	8	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	292	2	no	—
114	C₂×C₄⋊C₄	—	292	2	no	—
115	C₂×C₄⋊C₄	—	146	2	no	—
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₁₆	—	73	4	no	—
118	C₂×C₁₆	—	73	2	no	—
119	C₂×C₁₆	Dic₁₄₆	1	4	yes	—
120	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	73	4	no	—
123	#107	—	146	2	no	—
124	#107	C₂²×D₇₃	1	4	yes	—
125	#106	—	146	8	no	—
126	C₄×C₈	—	73	2	no	—
127	C₄×C₈	—	73	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	292	2	no	—
130	C₂²×C₄	—	73	2	no	—
131	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
132	C₂×C₈	—	73	4	no	—
133	C₂×C₈	—	73	2	no	—
134	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₃	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
137	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	292	8	no	—
141	C₄⋊C₄	—	292	8	no	—
142	C₄⋊C₄	—	292	2	no	—
143	C₄⋊C₄	—	146	4	no	—
144	C₄²	—	73	2	no	—
145	C₄²	—	73	2	no	—
146	C₁₆	—	73	4	no	—
147	C₁₆	C₂×Dic₁₄₆	1	4	yes	—
148	C₂³	D₈×D₇₃	1	1	yes	characteristic
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	292	4	no	—
152	C₂×C₄	—	584	4	no	—
153	C₂×C₄	—	73	4	no	—
154	C₂×C₄	—	146	4	no	—
155	C₂×C₄	#14556590	1	2	yes	—
156	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
157	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
158	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
159	C₈	—	73	4	no	—
160	C₈	#14556590	1	2	yes	—
161	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
162	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
163	C₂²	#17988991	1	2	yes	—
164	C₂²	#17989018	1	4	yes	—
165	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
166	C₄	—	8	4	no	—
167	C₄	—	584	4	no	—
168	C₄	—	146	4	no	—
169	C₄	#17988991	1	2	yes	—
170	C₄	C₂×#14556590	1	1	yes	characteristic
171	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
172	C₂	#22612209	1	4	yes	direct product factor
173	C₂	C₂×#17988991	1	1	yes	characteristic
174	C₂	C₂×#17989018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988988

C₂

yes

maximal

#22612209

C₂

yes

maximal, direct product factor

C₂×#17988415

C₂

yes

maximal, characteristic

C₂×#17988412

C₂

yes

maximal

C₂×C₁₆×Dic₇₃

C₂

yes

maximal, characteristic

C₂×#367×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556582

—

C₂²×#11924003

C₂²

yes

characteristic

#17988988

—

#17988988

C₂²

yes

—

C₂×#14556525

C₂²

yes

—

C₂×#14556522

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988415

C₂²

yes

—

#17988412

C₂²

yes

—

C₁₆×Dic₇₃

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

—

#367×C₇₃

C₂²

yes

—

C₂²×#10596831

D₄

yes

characteristic

#14556582

—

#14556582

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×#11924007

—

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×#11924003

Q₈

yes

—

C₂×#11924003

C₂³

yes

—

#14556525

—

#14556525

C₂³

yes

—

#14556522

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

C₂³

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₁₆×C₇₃

Q₈

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

—

#107×C₇₃

—

#107×C₇₃

C₂³

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

—

C₂×#10596831

Q₁₆

yes

—

C₂×#10596831

C₂×D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

D₈

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

#11924003

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₈

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

—

C₈×C₇₃

C₂²×Q₈

yes

characteristic

C₈×C₇₃

C₂²×Q₈

yes

characteristic

#10596831

C₂×Q₁₆

yes

—

C₂²×C₇₃

#500

yes

—

C₂²×C₇₃

#508

yes

—

C₂²×C₇₃

C₂²×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#500

yes

—

C₄×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²×Q₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

D₇₃

yes

characteristic

C₂×#366

—

C₂×C₄×C₁₆

—

C₇₃

C₂×#2253

yes

characteristic, complemented, Sylow

C₂×#125

—

146

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

#499

—

#499

—

146

—

100

C₂×#107

—

101

C₂×#107

C₂×D₇₃

yes

—

102

C₂×#106

—

146

—

103

C₂×C₄×C₈

—

104

#367

—

105

#367

C₂×D₇₃

yes

—

106

#366

—

107

C₄×C₁₆

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂²×D₇₃

yes

characteristic

110

#125

—

292

—

111

#125

—

146

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

292

—

114

C₂×C₄⋊C₄

—

292

—

115

C₂×C₄⋊C₄

—

146

—

116

C₂×C₄²

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

Dic₁₄₆

yes

—

120

C₂×C₁₆

C₂²×D₇₃

yes

—

121

#107

—

122

#107

—

123

#107

—

146

—

124

#107

C₂²×D₇₃

yes

—

125

#106

—

146

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

292

—

130

C₂²×C₄

—

131

C₂²×C₄

D₄×D₇₃

yes

characteristic

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

D₄×D₇₃

yes

characteristic

135

C₂×C₈

Q₈×D₇₃

yes

—

136

C₂×C₈

C₂×Dic₁₄₆

yes

—

137

C₂×C₈

C₂³×D₇₃

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

292

—

141

C₄⋊C₄

—

292

—

142

C₄⋊C₄

—

292

—

143

C₄⋊C₄

—

146

—

144

C₄²

—

145

C₄²

—

146

C₁₆

—

147

C₁₆

C₂×Dic₁₄₆

yes

—

148

C₂³

D₈×D₇₃

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

584

—

153

C₂×C₄

—

154

C₂×C₄

—

146

—

155

C₂×C₄

#14556590

yes

—

156

C₂×C₄

D₈×D₇₃

yes

characteristic

157

C₂×C₄

Q₁₆×D₇₃

yes

—

158

C₂×C₄

C₂×D₄×D₇₃

yes

characteristic

159

C₈

—

160

C₈

#14556590

yes

—

161

C₈

C₂×Q₈×D₇₃

yes

characteristic

162

C₈

C₂²×Dic₁₄₆

yes

characteristic

163

C₂²

#17988991

yes

—

164

C₂²

#17989018

yes

—

165

C₂²

C₂×D₈×D₇₃

yes

characteristic

166

C₄

—

167

C₄

—

584

—

168

C₄

—

146

—

169

C₄

#17988991

yes

—

170

C₄

C₂×#14556590

yes

characteristic

171

C₄

C₂×Q₁₆×D₇₃

yes

characteristic

172

C₂

#22612209

yes

direct product factor

173

C₂

C₂×#17988991

yes

characteristic

174

C₂

C₂×#17989018

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#367×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₂×#367	D₇₃
hypocenter	C₇₃	C₂×#2253
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₇₃

commutator, cosocle

C₂×C₈×C₇₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₃

Fitting

C₂×#367×C₇₃

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
2. #22612209	C₂	direct (N × H)
85. C₂×C₇₃	#2253	semidirect
94. C₇₃	C₂×#2253	semidirect
172. C₂	#22612209	direct (N × H)

H ≅ G/N

type

2. #22612209

C₂

direct (N × H)

85. C₂×C₇₃

#2253

semidirect

94. C₇₃

C₂×#2253

semidirect

172. C₂

#22612209

direct (N × H)

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

subgroup	index
C₂×#17988988	2
#22612209	2
C₂×#17988415	2
C₂×#17988412	2
C₂×C₁₆×Dic₇₃	2
C₂×#367×C₇₃	2
C₂×#2253	73

subgroup

index

C₂×C₁₆×Dic₇₃

C₂×#367×C₇₃

C₂×#2253

Subgroups — C₂×#22612209 #33241465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22612209 #33241465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612209 #33241465