Group #33955204 — subgroups

Subgroups — C₂²×#17988956 #33955204

Order 18688.

2142 conjugacy classes of subgroups; only the 747 normal subgroups are shown.

normal subgroups only

This group has 2142 conjugacy classes of subgroups — too many to draw. Only its 747 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	C₂²×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556583	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988956	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556516	C₂	1	4	yes	maximal
5	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₄₆	C₂²	1	2	yes	—
7	C₂×#14556591	—	2	6	no	—
8	C₂×#14556591	C₂²	1	12	yes	—
9	C₂×#14556583	—	2	6	no	—
10	C₂×#14556583	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₇₃	—	2	2	no	—
12	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	C₂²×#11924007	—	2	2	no	—
14	#17988956	C₂²	1	64	yes	direct product factor
15	C₂×#14556516	C₂²	1	48	yes	—
16	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₃	C₂²	1	12	yes	—
19	C₂²×Dic₁₄₆	—	2	6	no	—
20	C₂²×Dic₁₄₆	—	2	16	no	—
21	C₂²×Dic₁₄₆	C₂³	1	12	yes	—
22	C₂³×Dic₇₃	—	2	2	no	—
23	C₂³×Dic₇₃	—	2	2	no	—
24	#14556591	—	2	24	no	—
25	#14556591	C₂³	1	16	yes	—
26	#14556583	—	2	24	no	—
27	#14556583	C₂³	1	16	yes	—
28	C₂×C₄⋊Dic₇₃	—	2	24	no	—
29	C₂×C₄⋊Dic₇₃	—	2	6	no	—
30	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
31	C₂×#11924007	—	2	24	no	—
32	#14556516	C₂³	1	64	yes	—
33	C₂³×C₄×C₇₃	D₄	1	2	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	—	2	8	no	—
36	C₂²×C₈×C₇₃	C₂³	1	12	yes	—
37	C₂×C₄²×C₇₃	D₄	1	6	yes	—
38	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
39	#97×C₇₃	C₂³	1	16	yes	—
40	C₂×Dic₁₄₆	—	2	48	no	—
41	C₂×Dic₁₄₆	—	2	12	no	—
42	C₂×Dic₁₄₆	—	4	24	no	—
43	C₂×Dic₁₄₆	C₂⁴	1	8	yes	—
44	C₂²×Dic₇₃	—	2	12	no	—
45	C₂²×Dic₇₃	—	2	12	no	—
46	C₂²×Dic₇₃	—	4	8	no	—
47	C₂²×Dic₇₃	—	4	8	no	—
48	C₄⋊Dic₇₃	—	2	32	no	—
49	C₄⋊Dic₇₃	—	2	12	no	—
50	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
51	#11924007	—	2	32	no	—
52	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	—	2	8	no	—
54	C₂²×C₄×C₇₃	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₇₃	#41	1	8	yes	—
57	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
58	C₂×C₈×C₇₃	—	2	24	no	—
59	C₂×C₈×C₇₃	C₂⁴	1	8	yes	—
60	C₄²×C₇₃	C₂×D₄	1	12	yes	—
61	C₄²×C₇₃	C₂⁴	1	4	yes	—
62	C₂³×C₇₃	#124	1	4	yes	—
63	C₂³×C₇₃	#137	1	8	yes	—
64	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
65	C₂×C₄×C₇₃	—	2	24	no	—
66	C₂×C₄×C₇₃	#124	1	12	yes	—
67	C₂×C₄×C₇₃	C₂²×D₄	1	8	yes	—
68	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
69	C₂×C₄×C₇₃	C₂×#41	1	12	yes	—
70	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
71	C₂×Dic₇₃	—	2	8	no	—
72	C₂×Dic₇₃	—	2	8	no	—
73	C₂×Dic₇₃	—	4	24	no	—
74	C₂×Dic₇₃	—	4	24	no	—
75	Dic₁₄₆	—	2	16	no	—
76	Dic₁₄₆	—	4	24	no	—
77	C₈×C₇₃	—	2	8	no	—
78	C₂²×C₇₃	#491	1	16	yes	complemented
79	C₂²×C₇₃	C₂×#124	1	6	yes	—
80	C₂²×C₇₃	C₂×#137	1	12	yes	—
81	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
82	C₄×C₇₃	—	2	8	no	—
83	C₄×C₇₃	C₂×#124	1	6	yes	—
84	C₄×C₇₃	C₂²×#41	1	2	yes	—
85	Dic₇₃	—	4	8	no	—
86	Dic₇₃	—	4	8	no	—
87	C₂²×#491	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	C₂×#491	1	12	yes	complemented
89	C₂×C₇₃	C₂²×#124	1	1	yes	characteristic
90	C₂×C₇₃	C₂²×#137	1	2	yes	—
91	C₂²×C₄⋊Q₈	—	73	1	no	—
92	C₂²×#125	—	73	1	no	—
93	C₂×#491	—	73	24	no	—
94	C₂²×#103	—	73	4	no	—
95	C₂²×#97	D₇₃	1	1	yes	characteristic
96	C₇₃	C₂²×#491	1	1	yes	characteristic, complemented, Sylow
97	C₂³×Q₈	—	73	2	no	—
98	C₂³×C₈	C₂×D₇₃	1	2	yes	—
99	C₂×C₄⋊Q₈	—	73	12	no	—
100	C₂×C₄⋊Q₈	—	146	6	no	—
101	C₂×#125	—	73	12	no	—
102	C₂×#125	—	146	6	no	—
103	C₂²×C₄⋊C₄	—	73	2	no	—
104	C₂²×C₄⋊C₄	—	146	2	no	—
105	C₂²×C₄⋊C₄	—	146	2	no	—
106	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
107	#491	—	73	64	no	—
108	C₂×#103	—	73	48	no	—
109	C₂×#97	C₂×D₇₃	1	12	yes	—
110	C₂²×Q₈	—	73	12	no	—
111	C₂²×Q₈	—	146	16	no	—
112	C₂²×Q₈	—	146	6	no	—
113	C₂³×C₄	—	146	2	no	—
114	C₂³×C₄	—	146	2	no	—
115	C₂³×C₄	D₂₉₂	1	2	yes	—
116	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	C₂²×D₇₃	1	12	yes	—
119	C₄⋊Q₈	—	73	16	no	—
120	C₄⋊Q₈	—	146	24	no	—
121	#125	—	73	16	no	—
122	#125	—	146	24	no	—
123	C₂×C₄⋊C₄	—	73	12	no	—
124	C₂×C₄⋊C₄	—	146	6	no	—
125	C₂×C₄⋊C₄	—	146	24	no	—
126	C₂×C₄⋊C₄	—	146	24	no	—
127	C₂×C₄²	D₂₉₂	1	6	yes	—
128	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
129	#103	—	73	64	no	—
130	#97	C₂²×D₇₃	1	16	yes	—
131	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
132	C₂×Q₈	—	292	24	no	—
133	C₂×Q₈	—	73	8	no	—
134	C₂×Q₈	—	146	48	no	—
135	C₂×Q₈	—	146	12	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	292	8	no	—
138	C₂²×C₄	—	292	8	no	—
139	C₂²×C₄	—	146	12	no	—
140	C₂²×C₄	—	146	12	no	—
141	C₂²×C₄	C₂×D₂₉₂	1	12	yes	—
142	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
143	C₂²×C₄	#11924028	1	8	yes	—
144	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
145	C₂×C₈	—	2	24	no	—
146	C₂×C₈	C₂³×D₇₃	1	8	yes	—
147	C₄⋊C₄	—	73	8	no	—
148	C₄⋊C₄	—	146	32	no	—
149	C₄⋊C₄	—	146	12	no	—
150	C₄⋊C₄	—	146	32	no	—
151	C₄²	C₂×D₂₉₂	1	12	yes	—
152	C₄²	C₂³×D₇₃	1	4	yes	—
153	C₂³	#14556579	1	4	yes	—
154	C₂³	#14556624	1	8	yes	—
155	C₂³	C₂²×D₂₉₂	1	3	yes	—
156	Q₈	—	292	24	no	—
157	Q₈	—	146	16	no	—
158	C₂×C₄	—	2	24	no	—
159	C₂×C₄	—	292	24	no	—
160	C₂×C₄	—	292	24	no	—
161	C₂×C₄	—	146	8	no	—
162	C₂×C₄	—	146	8	no	—
163	C₂×C₄	#14556579	1	12	yes	—
164	C₂×C₄	C₂²×D₂₉₂	1	8	yes	—
165	C₂×C₄	C₂²×D₂₉₂	1	3	yes	—
166	C₂×C₄	C₂×#11924028	1	12	yes	—
167	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
168	C₈	—	2	8	no	—
169	C₂²	#17988956	1	16	yes	direct product factor
170	C₂²	C₂×#14556579	1	6	yes	—
171	C₂²	C₂×#14556624	1	12	yes	—
172	C₂²	C₂³×D₂₉₂	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	292	8	no	—
175	C₄	—	292	8	no	—
176	C₄	C₂×#14556579	1	6	yes	—
177	C₄	C₂²×#11924028	1	2	yes	—
178	C₂	C₂×#17988956	1	12	yes	direct product factor
179	C₂	C₂²×#14556579	1	1	yes	characteristic
180	C₂	C₂²×#14556624	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14556591

C₂

yes

maximal, characteristic

C₂²×#14556583

C₂

yes

maximal, characteristic

C₂×#17988956

C₂

yes

maximal, direct product factor

C₂²×#14556516

C₂

yes

maximal

C₂²×#97×C₇₃

C₂

yes

maximal, characteristic

C₂³×Dic₁₄₆

C₂²

yes

—

C₂×#14556591

—

C₂×#14556591

C₂²

yes

—

C₂×#14556583

—

C₂×#14556583

C₂²

yes

—

C₂²×C₄⋊Dic₇₃

—

C₂²×C₄⋊Dic₇₃

C₂²

yes

—

C₂²×#11924007

—

#17988956

C₂²

yes

direct product factor

C₂×#14556516

C₂²

yes

—

C₂³×C₈×C₇₃

C₂²

yes

—

C₂²×C₄²×C₇₃

C₂²

yes

characteristic

C₂×#97×C₇₃

C₂²

yes

—

C₂²×Dic₁₄₆

—

C₂²×Dic₁₄₆

—

C₂²×Dic₁₄₆

C₂³

yes

—

C₂³×Dic₇₃

—

C₂³×Dic₇₃

—

#14556591

—

#14556591

C₂³

yes

—

#14556583

—

#14556583

C₂³

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂³

yes

—

C₂×#11924007

—

#14556516

C₂³

yes

—

C₂³×C₄×C₇₃

D₄

yes

—

C₂³×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

—

#97×C₇₃

C₂³

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂⁴

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂⁴

yes

—

#11924007

—

C₂⁴×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

#41

yes

—

C₂²×C₄×C₇₃

C₂⁴

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂⁴

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂⁴

yes

—

C₂³×C₇₃

#124

yes

—

C₂³×C₇₃

#137

yes

—

C₂³×C₇₃

C₂²×D₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

—

C₂×C₄×C₇₃

C₂²×D₄

yes

—

C₂×C₄×C₇₃

C₂×#41

yes

—

C₂×C₄×C₇₃

C₂⁵

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₂²×C₇₃

#491

yes

complemented

C₂²×C₇₃

C₂×#124

yes

—

C₂²×C₇₃

C₂×#137

yes

—

C₂²×C₇₃

C₂³×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×#124

yes

—

C₄×C₇₃

C₂²×#41

yes

—

Dic₇₃

—

Dic₇₃

—

C₂²×#491

—

maximal, Sylow

C₂×C₇₃

C₂×#491

yes

complemented

C₂×C₇₃

C₂²×#124

yes

characteristic

C₂×C₇₃

C₂²×#137

yes

—

C₂²×C₄⋊Q₈

—

C₂²×#125

—

C₂×#491

—

C₂²×#103

—

C₂²×#97

D₇₃

yes

characteristic

C₇₃

C₂²×#491

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

C₂³×C₈

C₂×D₇₃

yes

—

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

—

146

—

101

C₂×#125

—

102

C₂×#125

—

146

—

103

C₂²×C₄⋊C₄

—

104

C₂²×C₄⋊C₄

—

146

—

105

C₂²×C₄⋊C₄

—

146

—

106

C₂²×C₄²

C₂×D₇₃

yes

characteristic

107

#491

—

108

C₂×#103

—

109

C₂×#97

C₂×D₇₃

yes

—

110

C₂²×Q₈

—

111

C₂²×Q₈

—

146

—

112

C₂²×Q₈

—

146

—

113

C₂³×C₄

—

146

—

114

C₂³×C₄

—

146

—

115

C₂³×C₄

D₂₉₂

yes

—

116

C₂³×C₄

C₂²×D₇₃

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

C₂²×D₇₃

yes

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

146

—

121

#125

—

122

#125

—

146

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

146

—

125

C₂×C₄⋊C₄

—

146

—

126

C₂×C₄⋊C₄

—

146

—

127

C₂×C₄²

D₂₉₂

yes

—

128

C₂×C₄²

C₂²×D₇₃

yes

—

129

#103

—

130

#97

C₂²×D₇₃

yes

—

131

C₂⁴

C₂×D₂₉₂

yes

characteristic

132

C₂×Q₈

—

292

—

133

C₂×Q₈

—

134

C₂×Q₈

—

146

—

135

C₂×Q₈

—

146

—

136

C₂²×C₄

—

137

C₂²×C₄

—

292

—

138

C₂²×C₄

—

292

—

139

C₂²×C₄

—

146

—

140

C₂²×C₄

—

146

—

141

C₂²×C₄

C₂×D₂₉₂

yes

—

142

C₂²×C₄

C₂×D₂₉₂

yes

—

143

C₂²×C₄

#11924028

yes

—

144

C₂²×C₄

C₂³×D₇₃

yes

—

145

C₂×C₈

—

146

C₂×C₈

C₂³×D₇₃

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

146

—

149

C₄⋊C₄

—

146

—

150

C₄⋊C₄

—

146

—

151

C₄²

C₂×D₂₉₂

yes

—

152

C₄²

C₂³×D₇₃

yes

—

153

C₂³

#14556579

yes

—

154

C₂³

#14556624

yes

—

155

C₂³

C₂²×D₂₉₂

yes

—

156

Q₈

—

292

—

157

Q₈

—

146

—

158

C₂×C₄

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

292

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

#14556579

yes

—

164

C₂×C₄

C₂²×D₂₉₂

yes

—

165

C₂×C₄

C₂²×D₂₉₂

yes

—

166

C₂×C₄

C₂×#11924028

yes

—

167

C₂×C₄

C₂⁴×D₇₃

yes

characteristic

168

C₈

—

169

C₂²

#17988956

yes

direct product factor

170

C₂²

C₂×#14556579

yes

—

171

C₂²

C₂×#14556624

yes

—

172

C₂²

C₂³×D₂₉₂

yes

characteristic

173

C₄

—

174

C₄

—

292

—

175

C₄

—

292

—

176

C₄

C₂×#14556579

yes

—

177

C₄

C₂²×#11924028

yes

—

178

C₂

C₂×#17988956

yes

direct product factor

179

C₂

C₂²×#14556579

yes

characteristic

180

C₂

C₂²×#14556624

yes

—

	subgroup	G/H
center	C₂⁴	C₂×D₂₉₂
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	C₂²×#491
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

subgroup

G/H

center

C₂⁴

C₂×D₂₉₂

commutator, cosocle

C₂×C₄×C₇₃

C₂⁵

Frattini

C₂×C₄

C₂⁴×D₇₃

Fitting

C₂²×#97×C₇₃

C₂

socle, prime radical

C₂⁴×C₇₃

C₂×D₄

hypercenter

C₂²×#97

D₇₃

hypocenter

C₇₃

C₂²×#491

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

N	H ≅ G/N	type
3. C₂×#17988956	C₂	direct (N × H)
14. #17988956	C₂²	direct (N × H)
78. C₂²×C₇₃	#491	semidirect
88. C₂×C₇₃	C₂×#491	semidirect
96. C₇₃	C₂²×#491	semidirect
169. C₂²	#17988956	direct (N × H)
178. C₂	C₂×#17988956	direct (N × H)

H ≅ G/N

type

3. C₂×#17988956

C₂

direct (N × H)

14. #17988956

C₂²

direct (N × H)

78. C₂²×C₇₃

#491

semidirect

88. C₂×C₇₃

C₂×#491

semidirect

96. C₇₃

C₂²×#491

semidirect

169. C₂²

#17988956

direct (N × H)

178. C₂

C₂×#17988956

direct (N × H)

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

subgroup	index
C₂²×#14556591	2
C₂²×#14556583	2
C₂×#17988956	2
C₂²×#14556516	2
C₂²×#97×C₇₃	2
C₂²×#491	73

subgroup

index

C₂²×#97×C₇₃

C₂²×#491

Subgroups — C₂²×#17988956 #33955204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×#17988956 #33955204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988956 #33955204