Group #33005638 — subgroups

Subgroups — #33005638

Order 18688.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17988989	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988410	C₂	1	2	yes	maximal
3	#22610679	C₂	1	1	yes	maximal, characteristic
4	#22608037	C₂	1	2	yes	maximal
5	C₇₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556583	—	2	1	no	—
7	#17988989	—	2	4	no	—
8	C₂×#14556527	C₂²	1	1	yes	characteristic
9	#17988625	C₂²	1	1	yes	characteristic
10	C₂×#14556525	—	2	1	no	—
11	C₂×#14556523	C₂²	1	2	yes	—
12	#17988613	—	2	1	no	—
13	#17988410	—	2	4	no	—
14	#17988347	—	2	1	no	—
15	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924012	—	2	1	no	—
18	#14556583	—	2	4	no	—
19	#14556583	—	4	2	no	—
20	#14556550	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	C₂×#11924007	—	4	2	no	—
23	#14556527	—	2	2	no	—
24	#14556525	—	2	4	no	—
25	#14556525	Q₈	1	2	yes	—
26	#14556523	—	2	2	no	—
27	#14556523	—	2	1	no	—
28	#14556523	D₄	1	4	yes	—
29	#14556520	—	4	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	M₃₂×C₇₃	—	2	2	no	—
34	M₃₂×C₇₃	Q₈	1	4	yes	—
35	C₂×C₁₆×C₇₃	—	2	2	no	—
36	C₄×C₈×C₇₃	—	2	2	no	—
37	C₂²×Dic₇₃	—	4	2	no	—
38	#11924012	—	4	1	no	—
39	#11924012	—	4	1	no	—
40	C₂×#10596831	—	4	1	no	—
41	C₄⋊Dic₇₃	—	2	4	no	—
42	C₄⋊Dic₇₃	—	2	2	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	#11924007	—	4	8	no	—
45	C₂²⋊Dic₇₃	—	4	1	no	—
46	C₄×Dic₇₃	—	4	1	no	—
47	C₂²×C₄×C₇₃	—	2	1	no	—
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
52	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
56	C₄²×C₇₃	—	2	1	no	—
57	C₄²×C₇₃	SD₁₆	1	2	yes	—
58	C₁₆×C₇₃	—	2	4	no	—
59	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
68	C₂×Dic₇₃	—	4	4	no	—
69	C₂×Dic₇₃	—	4	2	no	—
70	C₂×Dic₇₃	—	8	4	no	—
71	C₈×C₇₃	—	2	2	no	—
72	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
74	#10596831	—	8	1	no	—
75	C₂²×C₇₃	—	2	2	no	—
76	C₂²×C₇₃	#450	1	1	yes	characteristic
77	C₂²×C₇₃	#476	1	1	yes	characteristic
78	C₂²×C₇₃	#501	1	1	yes	characteristic
79	C₄×C₇₃	—	4	2	no	—
80	C₄×C₇₃	#395	1	1	yes	characteristic
81	C₄×C₇₃	#448	1	1	yes	characteristic
82	C₄×C₇₃	#475	1	1	yes	characteristic
83	C₄×C₇₃	#498	1	1	yes	characteristic
84	Dic₇₃	—	8	4	no	—
85	Dic₇₃	—	8	2	no	—
86	#13458	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	#2058	1	1	yes	characteristic
90	C₂×C₇₃	#2231	1	2	yes	—
91	C₂×#499	—	73	1	no	—
92	C₂×#365	—	73	2	no	—
93	#1945	—	73	1	no	—
94	#1386	—	73	2	no	—
95	#1376	D₇₃	1	1	yes	characteristic
96	C₇₃	#13458	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	146	1	no	—
98	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
99	#499	—	146	4	no	—
100	C₂×#108	—	73	1	no	—
101	#427	—	73	1	no	—
102	C₂×#107	—	146	1	no	—
103	C₂×#106	—	73	2	no	—
104	#424	—	146	1	no	—
105	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
106	#365	—	146	4	no	—
107	#339	—	146	1	no	—
108	C₂×M₁₆	—	146	1	no	—
109	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
111	#125	—	292	2	no	—
112	#125	—	146	4	no	—
113	#117	—	146	1	no	—
114	C₂×C₄⋊C₄	—	292	2	no	—
115	C₂×C₄⋊C₄	—	146	2	no	—
116	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	Dic₁₄₆	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	#108	—	146	2	no	—
121	#107	—	73	2	no	—
122	#107	—	146	4	no	—
123	#106	—	73	4	no	—
124	#106	—	146	2	no	—
125	#106	—	146	1	no	—
126	C₄⋊C₈	—	292	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	292	2	no	—
130	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
131	M₁₆	—	292	1	no	—
132	M₁₆	—	292	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	292	1	no	—
136	C₂×C₈	#11924017	1	2	yes	—
137	C₂×C₈	D₄×D₇₃	1	2	yes	—
138	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
140	C₂×C₈	#11924027	1	1	yes	characteristic
141	C₄⋊C₄	—	292	8	no	—
142	C₄⋊C₄	—	292	2	no	—
143	C₄⋊C₄	—	146	4	no	—
144	C₄⋊C₄	—	146	2	no	—
145	C₂²⋊C₄	—	292	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	292	1	no	—
148	C₄²	#11924017	1	2	yes	—
149	C₁₆	—	2	4	no	—
150	C₂³	#14556619	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	292	2	no	—
154	C₂×C₄	—	292	4	no	—
155	C₂×C₄	—	584	4	no	—
156	C₂×C₄	#14556561	1	1	yes	characteristic
157	C₂×C₄	#14556571	1	1	yes	characteristic
158	C₂×C₄	#14556591	1	1	yes	characteristic
159	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
160	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
161	C₂×C₄	#14556624	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	584	1	no	—
164	C₈	#14556568	1	2	yes	—
165	C₈	#14556590	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#17988740	1	1	yes	characteristic
168	C₂²	#17988878	1	1	yes	characteristic
169	C₂²	#17988996	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	584	4	no	—
172	C₄	—	584	2	no	—
173	C₄	#17988508	1	1	yes	characteristic
174	C₄	#17988728	1	1	yes	characteristic
175	C₄	#17988873	1	1	yes	characteristic
176	C₄	#17988985	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#22611265	1	1	yes	characteristic
180	C₂	#22612108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988989

C₂

yes

maximal, characteristic

C₂×#17988410

C₂

yes

maximal

#22610679

C₂

yes

maximal, characteristic

#22608037

C₂

yes

maximal

C₇₃×#1376

C₂

yes

maximal, characteristic

C₂×#14556583

—

#17988989

—

C₂×#14556527

C₂²

yes

characteristic

#17988625

C₂²

yes

characteristic

C₂×#14556525

—

C₂×#14556523

C₂²

yes

—

#17988613

—

#17988410

—

#17988347

—

C₂×M₃₂×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

—

—

C₂×C₄⋊Dic₇₃

—

—

—

—

Q₈

yes

—

#14556523

—

#14556523

—

#14556523

D₄

yes

—

#14556520

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

Q₈

yes

—

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

#137

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#13458

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2058

yes

characteristic

C₂×C₇₃

#2231

yes

—

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#13458

yes

characteristic, complemented, Sylow

C₂×#125

—

146

—

C₂×M₃₂

C₂×D₇₃

yes

—

#499

—

146

—

100

C₂×#108

—

101

#427

—

102

C₂×#107

—

146

—

103

C₂×#106

—

104

#424

—

146

—

105

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

106

#365

—

146

—

107

#339

—

146

—

108

C₂×M₁₆

—

146

—

109

C₂²×C₈

D₂₉₂

yes

characteristic

110

C₂²×C₈

C₂²×D₇₃

yes

characteristic

111

#125

—

292

—

112

#125

—

146

—

113

#117

—

146

—

114

C₂×C₄⋊C₄

—

292

—

115

C₂×C₄⋊C₄

—

146

—

116

C₂×C₄²

D₂₉₂

yes

characteristic

117

M₃₂

—

118

M₃₂

Dic₁₄₆

yes

—

119

C₂×C₁₆

—

120

#108

—

146

—

121

#107

—

122

#107

—

146

—

123

#106

—

124

#106

—

146

—

125

#106

—

146

—

126

C₄⋊C₈

—

292

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

292

—

130

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

131

M₁₆

—

292

—

132

M₁₆

—

292

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

292

—

136

C₂×C₈

#11924017

yes

—

137

C₂×C₈

D₄×D₇₃

yes

—

138

C₂×C₈

Q₈×D₇₃

yes

characteristic

139

C₂×C₈

C₂×Dic₁₄₆

yes

—

140

C₂×C₈

#11924027

yes

characteristic

141

C₄⋊C₄

—

292

—

142

C₄⋊C₄

—

292

—

143

C₄⋊C₄

—

146

—

144

C₄⋊C₄

—

146

—

145

C₂²⋊C₄

—

292

—

146

C₄²

—

147

C₄²

—

292

—

148

C₄²

#11924017

yes

—

149

C₁₆

—

150

C₂³

#14556619

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

292

—

155

C₂×C₄

—

584

—

156

C₂×C₄

#14556561

yes

characteristic

157

C₂×C₄

#14556571

yes

characteristic

158

C₂×C₄

#14556591

yes

characteristic

159

C₂×C₄

C₂×#11924017

yes

characteristic

160

C₂×C₄

C₂×#11924017

yes

characteristic

161

C₂×C₄

#14556624

yes

characteristic

162

C₈

—

163

C₈

—

584

—

164

C₈

#14556568

yes

—

165

C₈

#14556590

yes

—

166

C₂²

—

167

C₂²

#17988740

yes

characteristic

168

C₂²

#17988878

yes

characteristic

169

C₂²

#17988996

yes

characteristic

170

C₄

—

171

C₄

—

584

—

172

C₄

—

584

—

173

C₄

#17988508

yes

characteristic

174

C₄

#17988728

yes

characteristic

175

C₄

#17988873

yes

characteristic

176

C₄

#17988985

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#22611265

yes

characteristic

180

C₂

#22612108

yes

—

	subgroup	G/H
center	C₂²	#17988878
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1376	C₂
socle	C₂²×C₇₃	#476
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
hypercenter	#1376	D₇₃
hypocenter	C₇₃	#13458
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#17988878

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1376

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
96. C₇₃	#13458	semidirect

H ≅ G/N

type

96. C₇₃

#13458

semidirect

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

subgroup	index
C₂×#17988989	2
C₂×#17988410	2
#22610679	2
#22608037	2
C₇₃×#1376	2
#13458	73

subgroup

index

C₇₃×#1376

Subgroups — #33005638

Conjugacy classes of subgroups

Complements

Maximal subgroups