Group #33001382 — subgroups

Subgroups — #33001382

Order 18688.

366 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 159 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×#17988412	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₇₃	C₂	1	2	yes	maximal
4	#22608023	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	1	yes	characteristic
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×Dic₇₃	—	2	2	no	—
10	#17988412	C₄	1	8	yes	complemented
11	C₁₆×Dic₇₃	C₄	1	4	yes	complemented
12	#17988349	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₃	C₂²	1	2	yes	—
15	C₂²×#10596831	D₄	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	C₂×C₄×Dic₇₃	—	4	2	no	—
18	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
19	C₂×#11924003	C₂×C₄	1	2	yes	—
20	C₂×#11924003	C₂×C₄	1	4	yes	—
21	#14556525	—	2	2	no	—
22	#14556525	C₂×C₄	1	4	yes	—
23	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₈×Dic₇₃	Q₈	1	2	yes	—
25	#14556498	—	4	2	no	—
26	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
28	C₂×C₁₆×C₇₃	—	2	2	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
30	#107×C₇₃	—	2	2	no	—
31	#107×C₇₃	C₂×C₄	1	4	yes	—
32	C₂²×Dic₇₃	—	4	2	no	—
33	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
34	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₄⋊C₄	1	2	yes	—
37	C₄⋊Dic₇₃	—	2	4	no	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₄×Dic₇₃	—	8	4	no	—
40	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₇₃	Q₁₆	1	2	yes	—
42	#11924003	C₄²	1	4	yes	complemented
43	C₂²×C₄×C₇₃	—	4	2	no	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	#41	1	2	yes	—
49	C₄⋊C₄×C₇₃	—	2	4	no	—
50	C₄⋊C₄×C₇₃	—	4	2	no	—
51	C₁₆×C₇₃	—	2	2	no	—
52	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	4	4	no	—
54	C₂×C₄×C₇₃	—	8	2	no	—
55	C₂×C₄×C₇₃	—	8	2	no	—
56	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#135	1	2	yes	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	4	4	no	—
62	C₂×Dic₇₃	—	8	2	no	—
63	C₂×Dic₇₃	—	8	2	no	—
64	C₂×Dic₇₃	#107	1	2	yes	—
65	C₂×Dic₇₃	#107	1	2	yes	—
66	C₂×Dic₇₃	D₁₆	1	2	yes	—
67	C₂×Dic₇₃	Q₃₂	1	2	yes	—
68	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₃	#117	1	1	yes	characteristic
70	C₈×C₇₃	C₄×Q₈	1	2	yes	—
71	#10596831	—	2	2	no	—
72	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
73	C₂²×C₇₃	C₄×D₈	1	2	yes	—
74	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
75	C₂²×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
76	C₂²×C₇₃	#508	1	2	yes	—
77	C₄×C₇₃	—	8	4	no	—
78	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄×C₇₃	#427	1	1	yes	characteristic
80	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
81	Dic₇₃	—	2	2	no	—
82	Dic₇₃	—	8	4	no	—
83	Dic₇₃	#366	1	4	yes	complemented
84	C₄×#366	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
86	C₂×C₇₃	C₂×#366	1	1	yes	characteristic
87	C₂×C₇₃	#2158	1	1	yes	characteristic
88	C₂×C₇₃	C₄×D₁₆	1	2	yes	—
89	C₂×C₇₃	C₄×Q₃₂	1	2	yes	—
90	C₂×#366	—	73	2	no	—
91	C₂×C₄×C₁₆	—	73	1	no	—
92	C₄×#107	—	73	2	no	—
93	#1380	—	73	1	no	—
94	#1380	D₇₃	1	1	yes	characteristic
95	C₇₃	C₄×#366	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	73	1	no	—
97	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
98	C₂×#107	—	73	2	no	—
99	C₂×#107	C₂×D₇₃	1	2	yes	—
100	C₂×C₄×C₈	—	73	1	no	—
101	C₄×C₄⋊C₄	—	146	2	no	—
102	#366	—	73	8	no	—
103	C₄×C₁₆	—	73	4	no	—
104	#340	—	146	2	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	146	2	no	—
109	C₂×C₄²	—	292	2	no	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	73	2	no	—
113	C₂×C₁₆	—	73	4	no	—
114	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
115	#107	—	2	2	no	—
116	#107	—	73	4	no	—
117	#107	—	146	2	no	—
118	#107	C₄×D₇₃	1	4	yes	—
119	C₄×C₈	—	73	2	no	—
120	C₄×C₈	—	73	2	no	—
121	#95	—	292	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	292	2	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
131	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
132	C₂×C₈	#11924027	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	292	2	no	—
136	C₄⋊C₄	—	146	4	no	—
137	C₄²	—	584	4	no	—
138	C₄²	—	73	2	no	—
139	C₄²	—	73	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	73	4	no	—
142	C₂³	D₈×D₇₃	1	1	yes	characteristic
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	292	4	no	—
147	C₂×C₄	—	584	2	no	—
148	C₂×C₄	—	584	2	no	—
149	C₂×C₄	—	73	2	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	—	73	2	no	—
152	C₂×C₄	—	73	2	no	—
153	C₂×C₄	—	146	2	no	—
154	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
155	C₂×C₄	#14556551	1	2	yes	—
156	C₂×C₄	Q₁₆×D₇₃	1	1	yes	characteristic
157	C₂×C₄	#14556612	1	2	yes	—
158	C₈	—	146	2	no	—
159	C₈	C₄²×D₇₃	1	1	yes	characteristic
160	C₈	#14556547	1	1	yes	characteristic
161	C₈	#14556556	1	2	yes	—
162	C₂²	#107×D₇₃	1	1	yes	characteristic
163	C₂²	#17988667	1	2	yes	—
164	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
165	C₂²	Q₃₂×D₇₃	1	1	yes	characteristic
166	C₂²	#17989018	1	2	yes	—
167	C₄	—	8	4	no	—
168	C₄	—	584	4	no	—
169	C₄	—	73	4	no	—
170	C₄	—	146	2	no	—
171	C₄	#17988438	1	1	yes	characteristic
172	C₄	#17988622	1	1	yes	characteristic
173	C₄	#17988679	1	2	yes	—
174	C₂	#22609823	1	1	yes	characteristic
175	C₂	#366×D₇₃	1	1	yes	characteristic
176	C₂	#22611714	1	1	yes	characteristic
177	C₂	#22611775	1	2	yes	—
178	C₂	#22611787	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988412

C₂

yes

maximal

C₂×C₁₆×Dic₇₃

C₂

yes

maximal, characteristic

#107×Dic₇₃

C₂

yes

maximal

#22608023

C₂

yes

maximal, characteristic

C₇₃×#1380

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

C₄⋊C₄×Dic₇₃

—

#17988412

C₄

yes

complemented

C₁₆×Dic₇₃

C₄

yes

complemented

#17988349

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#107×C₇₃

C₂²

yes

—

C₂²×#10596831

D₄

yes

characteristic

C₂×C₄⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

—

#14556525

—

#14556525

C₂×C₄

yes

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

#14556498

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

#107×C₇₃

—

#107×C₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₈

yes

characteristic

C₂²×Dic₇₃

D₈

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

Q₁₆

yes

—

#11924003

C₄²

yes

complemented

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#107

yes

—

C₂×Dic₇₃

#107

yes

—

C₂×Dic₇₃

D₁₆

yes

—

C₂×Dic₇₃

Q₃₂

yes

—

C₈×C₇₃

C₂×C₄²

yes

characteristic

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

—

#10596831

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

C₄×D₈

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₂²×C₇₃

C₂×Q₃₂

yes

characteristic

C₂²×C₇₃

#508

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₃

#427

yes

characteristic

C₄×C₇₃

C₄×Q₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

C₄×D₁₆

yes

—

C₂×C₇₃

C₄×Q₃₂

yes

—

C₂×#366

—

C₂×C₄×C₁₆

—

C₄×#107

—

#1380

—

#1380

D₇₃

yes

characteristic

C₇₃

C₄×#366

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₇₃

yes

—

100

C₂×C₄×C₈

—

101

C₄×C₄⋊C₄

—

146

—

102

#366

—

103

C₄×C₁₆

—

104

#340

—

146

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₇₃

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

146

—

109

C₂×C₄²

—

292

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

C₄×D₇₃

yes

—

115

#107

—

116

#107

—

117

#107

—

146

—

118

#107

C₄×D₇₃

yes

—

119

C₄×C₈

—

120

C₄×C₈

—

121

#95

—

292

—

122

C₂²×C₄

—

123

C₂²×C₄

—

292

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₄×D₇₃

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

146

—

129

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

130

C₂×C₈

C₂×C₄×D₇₃

yes

—

131

C₂×C₈

Q₈×D₇₃

yes

characteristic

132

C₂×C₈

#11924027

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

292

—

136

C₄⋊C₄

—

146

—

137

C₄²

—

584

—

138

C₄²

—

139

C₄²

—

140

C₁₆

—

141

C₁₆

—

142

C₂³

D₈×D₇₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

584

—

148

C₂×C₄

—

584

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

146

—

154

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

155

C₂×C₄

#14556551

yes

—

156

C₂×C₄

Q₁₆×D₇₃

yes

characteristic

157

C₂×C₄

#14556612

yes

—

158

C₈

—

146

—

159

C₈

C₄²×D₇₃

yes

characteristic

160

C₈

#14556547

yes

characteristic

161

C₈

#14556556

yes

—

162

C₂²

#107×D₇₃

yes

characteristic

163

C₂²

#17988667

yes

—

164

C₂²

D₁₆×D₇₃

yes

characteristic

165

C₂²

Q₃₂×D₇₃

yes

characteristic

166

C₂²

#17989018

yes

—

167

C₄

—

168

C₄

—

584

—

169

C₄

—

170

C₄

—

146

—

171

C₄

#17988438

yes

characteristic

172

C₄

#17988622

yes

characteristic

173

C₄

#17988679

yes

—

174

C₂

#22609823

yes

characteristic

175

C₂

#366×D₇₃

yes

characteristic

176

C₂

#22611714

yes

characteristic

177

C₂

#22611775

yes

—

178

C₂

#22611787

yes

—

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1380	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1380	D₇₃
hypocenter	C₇₃	C₄×#366
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₇₃

commutator

C₈×C₇₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1380

C₂

socle, prime radical

C₂³×C₇₃

C₂×D₈

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
10. #17988412	C₄	semidirect
11. C₁₆×Dic₇₃	C₄	semidirect
42. #11924003	C₄²	semidirect
83. Dic₇₃	#366	semidirect
95. C₇₃	C₄×#366	semidirect

H ≅ G/N

type

10. #17988412

C₄

semidirect

11. C₁₆×Dic₇₃

C₄

semidirect

42. #11924003

C₄²

semidirect

83. Dic₇₃

#366

semidirect

95. C₇₃

C₄×#366

semidirect

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

subgroup	index
C₂×#17988412	2
C₂×C₁₆×Dic₇₃	2
#107×Dic₇₃	2
#22608023	2
C₇₃×#1380	2
C₄×#366	73

subgroup

index

C₂×#17988412

C₂×C₁₆×Dic₇₃

#107×Dic₇₃

#22608023

C₇₃×#1380

C₄×#366

Subgroups — #33001382

Conjugacy classes of subgroups

Complements

Maximal subgroups