Group #33005879 — subgroups

Subgroups — #33005879

Order 18688.

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

normal subgroups only

This group has 276 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	#22611726	C₂	1	1	yes	maximal, characteristic, complemented
2	#22609985	C₂	1	2	yes	maximal
3	#22608033	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608027	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2113	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	C₂²	1	2	yes	—
7	#17988625	C₂²	1	2	yes	—
8	#17988604	—	2	2	no	—
9	#17988417	—	2	2	no	—
10	#17988359	—	2	2	no	—
11	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
15	C₂×#11924012	—	2	2	no	—
16	#14556550	—	2	2	no	—
17	#14556527	—	2	2	no	—
18	#14556527	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556523	—	2	2	no	—
21	#14556518	—	4	4	no	—
22	C₂×#41×C₇₃	D₄	1	1	yes	characteristic
23	#131×C₇₃	—	2	2	no	—
24	#131×C₇₃	—	2	2	no	—
25	C₂×M₁₆×C₇₃	—	2	1	no	—
26	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	M₃₂×C₇₃	—	2	2	no	—
30	C₂×C₁₆×C₇₃	—	2	1	no	—
31	C₂×C₁₆×C₇₃	—	2	2	no	—
32	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
33	#11924012	—	4	2	no	—
34	#11924012	—	4	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₂²⋊Dic₇₃	—	4	2	no	—
38	C₄×Dic₇₃	—	4	2	no	—
39	#41×C₇₃	—	2	2	no	—
40	#41×C₇₃	—	2	2	no	—
41	C₂×Q₈×C₇₃	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₇₃	—	2	1	no	—
43	C₂×D₄×C₇₃	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	—	2	1	no	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₇₃	—	2	2	no	—
47	M₁₆×C₇₃	—	4	2	no	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	—	4	1	no	—
50	C₂×C₈×C₇₃	—	4	1	no	—
51	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	2	yes	—
53	C₂×C₈×C₇₃	#41	1	2	yes	—
54	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
55	C₁₆×C₇₃	—	2	2	no	—
56	C₁₆×C₇₃	—	4	1	no	—
57	C₂³×C₇₃	—	2	1	no	—
58	C₂³×C₇₃	#135	1	1	yes	characteristic
59	Q₈×C₇₃	—	2	2	no	—
60	D₄×C₇₃	—	2	2	no	—
61	D₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	—	2	1	no	—
63	C₂×C₄×C₇₃	—	4	1	no	—
64	C₂×C₄×C₇₃	—	4	1	no	—
65	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#123	1	2	yes	—
67	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×Dic₇₃	—	8	2	no	—
70	C₈×C₇₃	—	4	1	no	—
71	C₈×C₇₃	#123	1	1	yes	characteristic
72	C₈×C₇₃	#124	1	1	yes	characteristic
73	C₈×C₇₃	#126	1	2	yes	—
74	#10596831	—	8	2	no	—
75	C₂²×C₇₃	—	4	1	no	—
76	C₂²×C₇₃	—	4	2	no	—
77	C₂²×C₇₃	—	4	1	no	—
78	C₂²×C₇₃	#467	1	1	yes	characteristic
79	C₂²×C₇₃	#482	1	2	yes	—
80	C₄×C₇₃	—	4	1	no	—
81	C₄×C₇₃	#397	1	1	yes	characteristic
82	C₄×C₇₃	#469	1	1	yes	characteristic
83	C₄×C₇₃	#484	1	2	yes	—
84	Dic₇₃	—	16	2	no	—
85	#13499	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	—	2	2	no	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	—	8	1	no	—
89	C₂×C₇₃	#2076	1	1	yes	characteristic
90	#2161	—	73	1	no	—
91	#2113	D₇₃	1	1	yes	characteristic
92	#1814	—	73	2	no	—
93	#1384	—	73	1	no	—
94	#1382	—	73	2	no	—
95	C₇₃	#13499	1	1	yes	characteristic, complemented, Sylow
96	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
97	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
98	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
99	C₂×#108	—	73	2	no	—
100	#427	—	73	2	no	—
101	#421	—	146	2	no	—
102	#368	—	146	2	no	—
103	C₂²⋊C₁₆	—	2	2	no	—
104	#344	—	146	2	no	—
105	C₂×#41	D₂₉₂	1	1	yes	characteristic
106	#131	—	2	2	no	—
107	#131	—	2	2	no	—
108	C₂×M₁₆	—	2	1	no	—
109	C₂×M₁₆	—	146	2	no	—
110	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
113	#117	—	146	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
118	#108	—	146	2	no	—
119	#108	—	146	2	no	—
120	#107	—	146	2	no	—
121	#106	—	146	2	no	—
122	#104	—	292	4	no	—
123	#41	—	2	2	no	—
124	#41	—	2	2	no	—
125	C₂×Q₈	Dic₂₉₂	1	1	yes	characteristic
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	Dic₂₉₂	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
130	M₁₆	—	2	2	no	—
131	M₁₆	—	4	2	no	—
132	M₁₆	—	292	2	no	—
133	M₁₆	—	292	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	292	2	no	—
138	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂×C₈	#11924027	1	2	yes	—
140	C₂×C₈	#11924027	1	2	yes	—
141	C₂×C₈	#11924028	1	1	yes	characteristic
142	C₄⋊C₄	—	292	2	no	—
143	C₂²⋊C₄	—	292	2	no	—
144	C₄²	—	292	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	4	1	no	—
147	C₂³	—	2	1	no	—
148	C₂³	#14556613	1	1	yes	characteristic
149	Q₈	—	2	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	584	2	no	—
156	C₂×C₄	#14556565	1	1	yes	characteristic
157	C₂×C₄	#14556575	1	2	yes	—
158	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
159	C₂×C₄	#14556619	1	2	yes	—
160	C₈	—	4	1	no	—
161	C₈	—	584	2	no	—
162	C₈	#14556572	1	1	yes	characteristic
163	C₈	#14556578	1	1	yes	characteristic
164	C₈	#14556584	1	2	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	#17988828	1	1	yes	characteristic
169	C₂²	#17988912	1	2	yes	—
170	C₄	—	4	1	no	—
171	C₄	—	1168	2	no	—
172	C₄	#17988512	1	1	yes	characteristic
173	C₄	#17988840	1	1	yes	characteristic
174	C₄	#17988923	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	#22611359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611726

C₂

yes

maximal, characteristic, complemented

#22609985

C₂

yes

maximal

#22608033

C₂

yes

maximal, characteristic, complemented

#22608027

C₂

yes

maximal, complemented

C₇₃×#2113

C₂

yes

maximal, characteristic

C₂×#14556527

C₂²

yes

—

#17988625

C₂²

yes

—

#17988604

—

#17988417

—

#17988359

—

C₂×#131×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₃

—

—

—

—

—

—

—

—

C₂×#41×C₇₃

D₄

yes

characteristic

#131×C₇₃

—

#131×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

Q₁₆

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#135

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#467

yes

characteristic

C₂²×C₇₃

#482

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#13499

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

C₇₃

#13499

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₃

yes

characteristic

C₂×M₃₂

C₂×D₇₃

yes

characteristic

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×#108

—

100

#427

—

101

#421

—

146

—

102

#368

—

146

—

103

C₂²⋊C₁₆

—

104

#344

—

146

—

105

C₂×#41

D₂₉₂

yes

characteristic

106

#131

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

146

—

110

C₂×M₁₆

D₂₉₂

yes

characteristic

111

C₂²×C₈

—

112

C₂²×C₈

C₂²×D₇₃

yes

characteristic

113

#117

—

146

—

114

M₃₂

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

C₂²⋊D₇₃

yes

—

118

#108

—

146

—

119

#108

—

146

—

120

#107

—

146

—

121

#106

—

146

—

122

#104

—

292

—

123

#41

—

124

#41

—

125

C₂×Q₈

Dic₂₉₂

yes

characteristic

126

C₂×D₄

—

127

C₂×D₄

Dic₂₉₂

yes

characteristic

128

C₂²×C₄

—

129

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

292

—

133

M₁₆

—

292

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

292

—

138

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

139

C₂×C₈

#11924027

yes

—

140

C₂×C₈

#11924027

yes

—

141

C₂×C₈

#11924028

yes

characteristic

142

C₄⋊C₄

—

292

—

143

C₂²⋊C₄

—

292

—

144

C₄²

—

292

—

145

C₁₆

—

146

C₁₆

—

147

C₂³

—

148

C₂³

#14556613

yes

characteristic

149

Q₈

—

150

D₄

—

151

D₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

584

—

156

C₂×C₄

#14556565

yes

characteristic

157

C₂×C₄

#14556575

yes

—

158

C₂×C₄

C₂×Dic₂₉₂

yes

characteristic

159

C₂×C₄

#14556619

yes

—

160

C₈

—

161

C₈

—

584

—

162

C₈

#14556572

yes

characteristic

163

C₈

#14556578

yes

characteristic

164

C₈

#14556584

yes

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#17988828

yes

characteristic

169

C₂²

#17988912

yes

—

170

C₄

—

171

C₄

—

1168

—

172

C₄

#17988512

yes

characteristic

173

C₄

#17988840

yes

characteristic

174

C₄

#17988923

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

178

C₂

#22611359

yes

characteristic

	subgroup	G/H
center	C₄	#17988840
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2113	C₂
socle	C₂×C₇₃	#2076
prime radical	C₂×#41×C₇₃	D₄
hypercenter	#2113	D₇₃
hypocenter	C₇₃	#13499
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#17988840

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#2113

C₂

socle

C₂×C₇₃

#2076

prime radical

C₂×#41×C₇₃

D₄

D₇₃

C₇₃

G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
1. #22611726	C₂	semidirect
3. #22608033	C₂	semidirect
4. #22608027	C₂	semidirect
95. C₇₃	#13499	semidirect

H ≅ G/N

type

1. #22611726

C₂

semidirect

3. #22608033

C₂

semidirect

4. #22608027

C₂

semidirect

95. C₇₃

#13499

semidirect

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

subgroup	index
#22611726	2
#22609985	2
#22608033	2
#22608027	2
C₇₃×#2113	2
#13499	73

subgroup

index

C₇₃×#2113

Subgroups — #33005879

Conjugacy classes of subgroups

Complements

Maximal subgroups