Group #33000734 — subgroups

Subgroups — #33000734

Order 18688.

340 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612180	C₂	1	2	yes	maximal, complemented
2	#22612156	C₂	1	2	yes	maximal, complemented
3	#22609546	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607849	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556532	—	2	4	no	—
7	#17988964	—	2	1	no	—
8	#17988964	—	2	1	no	—
9	#17988964	C₂²	1	2	yes	—
10	#17988387	—	2	4	no	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	#500×C₇₃	C₂²	1	1	yes	characteristic
13	#493×C₇₃	C₂²	1	1	yes	characteristic
14	#492×C₇₃	—	2	1	no	—
15	#486×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₅₈₄	—	2	2	no	—
18	C₂×D₅₈₄	—	2	2	no	—
19	C₂×D₅₈₄	—	4	2	no	—
20	C₂×D₅₈₄	—	4	2	no	—
21	#14556588	—	2	2	no	—
22	#14556532	—	4	8	no	—
23	C₂×#11924003	D₄	1	4	yes	—
24	C₂×SD₁₆×C₇₃	—	4	2	no	—
25	C₂×D₈×C₇₃	—	2	2	no	—
26	C₄⋊Q₈×C₇₃	—	2	1	no	—
27	#127×C₇₃	—	2	1	no	—
28	#107×C₇₃	—	2	2	no	—
29	#102×C₇₃	—	4	2	no	—
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×D₂₉₂	—	4	2	no	—
34	C₂×D₂₉₂	—	4	2	no	—
35	C₂×D₂₉₂	—	4	2	no	—
36	D₅₈₄	—	4	4	no	—
37	D₅₈₄	—	4	4	no	—
38	D₅₈₄	—	8	4	no	—
39	D₅₈₄	—	8	4	no	—
40	#11924003	—	2	4	no	—
41	C₂×Q₈×C₇₃	—	4	1	no	—
42	C₂×D₄×C₇₃	—	4	1	no	—
43	C₂×D₄×C₇₃	—	4	1	no	—
44	C₂×D₄×C₇₃	—	4	1	no	—
45	SD₁₆×C₇₃	—	8	4	no	—
46	D₈×C₇₃	—	4	4	no	—
47	C₂×C₈×C₇₃	D₈	1	2	yes	—
48	C₂×C₈×C₇₃	D₈	1	2	yes	—
49	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
50	C₄⋊C₄×C₇₃	—	4	1	no	—
51	C₄⋊C₄×C₇₃	—	4	1	no	—
52	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₇₃	—	8	1	no	—
54	C₂²×D₇₃	—	8	2	no	—
55	Q₈×C₇₃	—	8	2	no	—
56	D₄×C₇₃	—	8	2	no	—
57	D₄×C₇₃	—	8	2	no	—
58	D₄×C₇₃	—	8	2	no	—
59	C₂×C₄×C₇₃	—	8	1	no	—
60	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
63	D₂₉₂	—	8	2	no	—
64	D₂₉₂	—	8	2	no	—
65	D₂₉₂	—	8	4	no	—
66	D₂₉₂	—	8	4	no	—
67	C₈×C₇₃	—	2	2	no	—
68	C₈×C₇₃	D₁₆	1	4	yes	complemented
69	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
70	C₂²×C₇₃	—	16	1	no	—
71	C₂²×C₇₃	—	16	1	no	—
72	C₂²×C₇₃	—	16	1	no	—
73	C₂²×C₇₃	#493	1	1	yes	characteristic
74	C₂×D₇₃	—	16	2	no	—
75	C₂×D₇₃	—	16	2	no	—
76	C₂×D₇₃	—	16	2	no	—
77	C₄×C₇₃	—	16	1	no	—
78	C₄×C₇₃	#496	1	2	yes	—
79	C₄×C₇₃	C₂×D₁₆	1	2	yes	—
80	C₄×C₇₃	#509	1	2	yes	—
81	#12525	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	—	32	1	no	—
83	C₂×C₇₃	#1721	1	1	yes	characteristic
84	C₂×C₇₃	#2246	1	1	yes	characteristic
85	C₂×C₇₃	#2250	1	1	yes	characteristic
86	D₇₃	—	32	2	no	—
87	#2246	—	73	2	no	—
88	#2240	—	73	2	no	—
89	#1712	—	73	1	no	—
90	#1706	D₇₃	1	1	yes	characteristic, complemented
91	#1312	—	73	1	no	—
92	C₇₃	#12525	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₁₆	—	146	4	no	—
94	#500	C₂×D₇₃	1	1	yes	characteristic
95	#493	—	73	2	no	—
96	#493	—	146	1	no	—
97	#493	—	146	1	no	—
98	#493	C₂×D₇₃	1	1	yes	characteristic
99	#492	—	2	1	no	—
100	#486	—	2	1	no	—
101	#357	—	146	4	no	—
102	C₄×C₁₆	—	73	2	no	—
103	C₈²	C₂×D₇₃	1	1	yes	characteristic
104	C₂×SD₁₆	—	4	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	292	2	no	—
107	C₂×D₈	—	292	2	no	—
108	C₂×D₈	—	146	2	no	—
109	C₂×D₈	—	146	2	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	#127	—	2	1	no	—
112	#127	—	146	2	no	—
113	D₁₆	—	292	8	no	—
114	C₂×C₁₆	—	73	4	no	—
115	#107	—	2	2	no	—
116	#102	—	4	2	no	—
117	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
120	C₂×Q₈	—	4	1	no	—
121	C₂×D₄	—	4	1	no	—
122	C₂×D₄	—	4	1	no	—
123	C₂×D₄	—	4	1	no	—
124	C₂×D₄	—	292	2	no	—
125	C₂×D₄	—	292	2	no	—
126	C₂×D₄	—	292	2	no	—
127	SD₁₆	—	8	4	no	—
128	D₈	—	4	4	no	—
129	D₈	—	292	4	no	—
130	D₈	—	292	4	no	—
131	D₈	—	584	4	no	—
132	D₈	—	584	4	no	—
133	C₂×C₈	#11924013	1	2	yes	—
134	C₂×C₈	#11924013	1	2	yes	—
135	C₂×C₈	D₄×D₇₃	1	2	yes	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	C₁₆	—	146	4	no	—
140	C₂³	—	8	1	no	—
141	C₂³	—	584	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	8	2	no	—
146	D₄	—	584	2	no	—
147	D₄	—	584	2	no	—
148	D₄	—	584	4	no	—
149	D₄	—	584	4	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	#14556587	1	1	yes	characteristic
152	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
153	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	#14556532	1	4	yes	complemented
156	C₈	#14556618	1	4	yes	—
157	C₂²	—	16	1	no	—
158	C₂²	—	16	1	no	—
159	C₂²	—	16	1	no	—
160	C₂²	—	1168	2	no	—
161	C₂²	—	1168	2	no	—
162	C₂²	—	1168	2	no	—
163	C₂²	#17988963	1	1	yes	characteristic
164	C₄	—	16	1	no	—
165	C₄	#17988973	1	2	yes	—
166	C₄	C₂×#14556532	1	2	yes	—
167	C₄	#17989022	1	2	yes	—
168	C₂	—	32	1	no	—
169	C₂	—	2336	2	no	—
170	C₂	#22609595	1	1	yes	characteristic
171	C₂	#22612180	1	1	yes	characteristic
172	C₂	#22612198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612180

C₂

yes

maximal, complemented

#22612156

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic, complemented

#22607849

C₂

yes

maximal, characteristic, complemented

C₇₃×#1706

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

#17988387

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

C₂²

yes

characteristic

#493×C₇₃

C₂²

yes

characteristic

#492×C₇₃

—

#486×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

#14556532

—

C₂×#11924003

D₄

yes

—

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#127×C₇₃

—

#107×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

D₅₈₄

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

D₁₆

yes

complemented

C₈×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#496

yes

—

C₄×C₇₃

C₂×D₁₆

yes

—

C₄×C₇₃

#509

yes

—

#12525

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

#1312

—

C₇₃

#12525

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

146

—

#500

C₂×D₇₃

yes

characteristic

#493

—

#493

—

146

—

#493

—

146

—

#493

C₂×D₇₃

yes

characteristic

#492

—

100

#486

—

101

#357

—

146

—

102

C₄×C₁₆

—

103

C₈²

C₂×D₇₃

yes

characteristic

104

C₂×SD₁₆

—

105

C₂×D₈

—

106

C₂×D₈

—

292

—

107

C₂×D₈

—

292

—

108

C₂×D₈

—

146

—

109

C₂×D₈

—

146

—

110

C₄⋊Q₈

—

111

#127

—

112

#127

—

146

—

113

D₁₆

—

292

—

114

C₂×C₁₆

—

115

#107

—

116

#102

—

117

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

118

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

119

C₄×C₈

C₂²×D₇₃

yes

characteristic

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

292

—

125

C₂×D₄

—

292

—

126

C₂×D₄

—

292

—

127

SD₁₆

—

128

D₈

—

129

D₈

—

292

—

130

D₈

—

292

—

131

D₈

—

584

—

132

D₈

—

584

—

133

C₂×C₈

#11924013

yes

—

134

C₂×C₈

#11924013

yes

—

135

C₂×C₈

D₄×D₇₃

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

139

C₁₆

—

146

—

140

C₂³

—

141

C₂³

—

584

—

142

Q₈

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

584

—

147

D₄

—

584

—

148

D₄

—

584

—

149

D₄

—

584

—

150

C₂×C₄

—

151

C₂×C₄

#14556587

yes

characteristic

152

C₂×C₄

C₂×#11924013

yes

characteristic

153

C₂×C₄

C₂×#11924013

yes

characteristic

154

C₈

—

155

C₈

#14556532

yes

complemented

156

C₈

#14556618

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

1168

—

161

C₂²

—

1168

—

162

C₂²

—

1168

—

163

C₂²

#17988963

yes

characteristic

164

C₄

—

165

C₄

#17988973

yes

—

166

C₄

C₂×#14556532

yes

—

167

C₄

#17989022

yes

—

168

C₂

—

169

C₂

—

2336

—

170

C₂

#22609595

yes

characteristic

171

C₂

#22612180

yes

characteristic

172

C₂

#22612198

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1706	C₂
socle	C₂²×C₇₃	#493
hypercenter	#1706	D₇₃
hypocenter	C₇₃	#12525
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1706

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

N	H ≅ G/N	type
1. #22612180	C₂	semidirect
2. #22612156	C₂	semidirect
3. #22609546	C₂	semidirect
4. #22607849	C₂	semidirect
5. C₇₃×#1706	C₂	semidirect
68. C₈×C₇₃	D₁₆	semidirect
90. #1706	D₇₃	semidirect
92. C₇₃	#12525	semidirect
155. C₈	#14556532	semidirect

H ≅ G/N

type

1. #22612180

C₂

semidirect

2. #22612156

C₂

semidirect

3. #22609546

C₂

semidirect

4. #22607849

C₂

semidirect

5. C₇₃×#1706

C₂

semidirect

68. C₈×C₇₃

D₁₆

semidirect

90. #1706

D₇₃

semidirect

92. C₇₃

#12525

semidirect

155. C₈

#14556532

semidirect

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

subgroup	index
#22612180	2
#22612156	2
#22609546	2
#22607849	2
C₇₃×#1706	2
#12525	73

subgroup

index

C₇₃×#1706

Subgroups — #33000734

Conjugacy classes of subgroups

Complements

Maximal subgroups