Group #33002198 — subgroups

Subgroups — #33002198

Order 18688.

454 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 171 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×C₂²⋊#11924003	C₂	1	2	yes	maximal
2	#22608009	C₂	1	4	yes	maximal, complemented
3	C₂×#405×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924003	C₂²	1	4	yes	—
5	C₂²⋊#11924003	—	2	4	no	—
6	C₂²×M₁₆×C₇₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₇₃	C₄	1	1	yes	characteristic
9	#405×C₇₃	—	2	4	no	—
10	C₂×#97×C₇₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
12	C₂×#11924003	—	2	4	no	—
13	C₂×#11924003	—	2	8	no	—
14	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₇₃	—	2	2	no	—
16	C₂×M₁₆×C₇₃	—	2	4	no	—
17	C₂²×C₈×C₇₃	—	2	4	no	—
18	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₇₃	D₄	1	4	yes	—
21	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
22	#117×C₇₃	—	2	2	no	—
23	#117×C₇₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₃	C₈	1	2	yes	—
25	C₂×C₂²⋊C₄×C₇₃	C₈	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
27	#97×C₇₃	—	2	4	no	—
28	C₄×C₈×C₇₃	—	2	4	no	—
29	#11924003	—	4	4	no	—
30	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	—	2	2	no	—
32	C₂²×C₄×C₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	M₁₆×C₇₃	—	2	8	no	—
39	C₂×C₈×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	—	2	8	no	—
43	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
45	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₇₃	#41	1	4	yes	—
47	C₄⋊C₄×C₇₃	—	2	4	no	—
48	C₂²⋊C₄×C₇₃	—	2	4	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	2	2	no	—
51	C₂³×C₇₃	—	2	2	no	—
52	C₂³×C₇₃	—	2	4	no	—
53	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	4	no	—
59	C₂×C₄×C₇₃	—	2	4	no	—
60	C₂×C₄×C₇₃	—	2	8	no	—
61	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#117	1	2	yes	—
64	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₇₃	—	2	2	no	—
67	C₈×C₇₃	—	2	4	no	—
68	C₈×C₇₃	#123	1	4	yes	—
69	C₂²×C₇₃	—	2	2	no	—
70	C₂²×C₇₃	—	2	4	no	—
71	C₂²×C₇₃	—	4	4	no	—
72	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₂²×C₇₃	#431	1	2	yes	—
74	C₂²×C₇₃	#504	1	4	yes	—
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	—	4	4	no	—
77	C₄×C₇₃	#381	1	1	yes	characteristic
78	C₄×C₇₃	#407	1	1	yes	characteristic
79	C₄×C₇₃	#432	1	2	yes	—
80	#12845	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	4	2	no	—
82	C₂×C₇₃	#1817	1	1	yes	characteristic
83	C₂×C₇₃	#2113	1	2	yes	—
84	C₂×C₇₃	#2164	1	4	yes	—
85	C₂×#405	D₇₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₁₆	—	73	2	no	—
87	#1374	—	73	4	no	—
88	C₇₃	#12845	1	1	yes	characteristic, complemented, Sylow
89	C₂²×M₁₆	Dic₇₃	1	1	yes	characteristic
90	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
91	C₂×#117	Dic₇₃	1	1	yes	characteristic
92	C₂²×C₁₆	—	73	4	no	—
93	#405	—	2	4	no	—
94	C₂×#97	Dic₇₃	1	2	yes	—
95	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
96	C₂²⋊C₁₆	—	146	4	no	—
97	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
103	C₂²×C₈	C₂²⋊D₇₃	1	4	yes	—
104	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	#10596831	1	2	yes	—
108	C₂×C₂²⋊C₄	#10596831	1	2	yes	—
109	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
110	C₂×C₁₆	—	146	4	no	—
111	C₂×C₁₆	—	146	8	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×#10596831	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
118	C₂²×C₄	C₂×#10596831	1	4	yes	—
119	C₂²×C₄	C₂×#10596831	1	1	yes	characteristic
120	C₂²×C₄	#11924012	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
128	C₂×C₈	#11924012	1	4	yes	—
129	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
130	C₂×C₈	#11924028	1	4	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	292	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	C₂²⋊#10596831	1	2	yes	—
139	C₂³	C₂²×#10596831	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₂²⋊#10596831	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
148	C₂×C₄	#14556550	1	2	yes	—
149	C₂×C₄	C₂×#11924012	1	1	yes	characteristic
150	C₂×C₄	C₂×#11924012	1	2	yes	—
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#14556574	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂²⋊#10596831	1	1	yes	characteristic
158	C₂²	#17988640	1	2	yes	—
159	C₂²	#17989004	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#17988450	1	1	yes	characteristic
163	C₄	#17988551	1	1	yes	characteristic
164	C₄	#17988644	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	#22609999	1	1	yes	characteristic
167	C₂	#22611545	1	2	yes	—
168	C₂	#22611738	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#11924003

C₂

yes

maximal

#22608009

C₂

yes

maximal, complemented

C₂×#405×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

—

C₂²⋊#11924003

—

C₂²×M₁₆×C₇₃

C₄

yes

characteristic

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂×#117×C₇₃

C₄

yes

characteristic

#405×C₇₃

—

C₂×#97×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂×#11924003

—

C₂×#11924003

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#117×C₇₃

—

#117×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₈

yes

—

C₂×C₂²⋊C₄×C₇₃

C₈

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

#97×C₇₃

—

C₄×C₈×C₇₃

—

#11924003

—

C₂⁴×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#123

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

#504

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12845

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1817

yes

characteristic

C₂×C₇₃

#2113

yes

—

C₂×C₇₃

#2164

yes

—

C₂×#405

D₇₃

yes

characteristic

C₂×C₂²⋊C₁₆

—

#1374

—

C₇₃

#12845

yes

characteristic, complemented, Sylow

C₂²×M₁₆

Dic₇₃

yes

characteristic

C₂³×C₈

C₂×D₇₃

yes

characteristic

C₂×#117

Dic₇₃

yes

characteristic

C₂²×C₁₆

—

#405

—

C₂×#97

Dic₇₃

yes

—

C₂×C₄×C₈

C₂×D₇₃

yes

—

C₂²⋊C₁₆

—

146

—

C₂³×C₄

C₂×Dic₇₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

102

C₂²×C₈

C₂×Dic₇₃

yes

—

103

C₂²×C₈

C₂²⋊D₇₃

yes

—

104

C₂²×C₈

C₂²×D₇₃

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

#10596831

yes

—

108

C₂×C₂²⋊C₄

#10596831

yes

—

109

C₂×C₄²

C₂×Dic₇₃

yes

—

110

C₂×C₁₆

—

146

—

111

C₂×C₁₆

—

146

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×#10596831

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊Dic₇₃

yes

—

118

C₂²×C₄

C₂×#10596831

yes

—

119

C₂²×C₄

C₂×#10596831

yes

characteristic

120

C₂²×C₄

#11924012

yes

—

121

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊Dic₇₃

yes

—

128

C₂×C₈

#11924012

yes

—

129

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

130

C₂×C₈

#11924028

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

292

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊#10596831

yes

—

139

C₂³

C₂²×#10596831

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#10596831

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

148

C₂×C₄

#14556550

yes

—

149

C₂×C₄

C₂×#11924012

yes

characteristic

150

C₂×C₄

C₂×#11924012

yes

—

151

C₈

—

152

C₈

—

153

C₈

#14556574

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#10596831

yes

characteristic

158

C₂²

#17988640

yes

—

159

C₂²

#17989004

yes

—

160

C₄

—

161

C₄

—

162

C₄

#17988450

yes

characteristic

163

C₄

#17988551

yes

characteristic

164

C₄

#17988644

yes

—

165

C₂

—

166

C₂

#22609999

yes

characteristic

167

C₂

#22611545

yes

—

168

C₂

#22611738

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator, socle	C₂³×C₇₃	C₂²×C₈
Fitting	C₂×#405×C₇₃	C₂
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#405	D₇₃
hypocenter	C₇₃	#12845
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₃

commutator, socle

C₂³×C₇₃

C₂²×C₈

Fitting

C₂×#405×C₇₃

C₂

prime radical

C₂⁴×C₇₃

C₂×C₈

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #22608009	C₂	semidirect
88. C₇₃	#12845	semidirect

H ≅ G/N

type

2. #22608009

C₂

semidirect

88. C₇₃

#12845

semidirect

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

subgroup	index
C₂×C₂²⋊#11924003	2
#22608009	2
C₂×#405×C₇₃	2
#12845	73

subgroup

index

C₂×C₂²⋊#11924003

#22608009

C₂×#405×C₇₃

#12845

Subgroups — #33002198

Conjugacy classes of subgroups

Complements

Maximal subgroups