Group #32979766 — subgroups

Subgroups — #32979766

Order 18688.

438 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 203 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	#22610091	C₂	1	1	yes	maximal, characteristic
2	#22610087	C₂	1	2	yes	maximal
3	#22610058	C₂	1	2	yes	maximal
4	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556525	—	2	2	no	—
7	C₂×#14556522	—	2	4	no	—
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	#17988457	C₂²	1	1	yes	characteristic
10	#17988457	C₂²	1	1	yes	characteristic
11	#17988349	—	2	4	no	—
12	#17988308	C₄	1	4	yes	complemented
13	C₂×#106×C₇₃	—	2	2	no	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#384×C₇₃	C₂²	1	2	yes	—
16	C₂²×#10596831	D₄	1	4	yes	—
17	C₂×C₄⋊Dic₇₃	—	2	2	no	—
18	C₂×C₄⋊Dic₇₃	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	—	2	1	no	—
20	C₂×C₄⋊Dic₇₃	—	2	2	no	—
21	#14556525	—	4	4	no	—
22	#14556522	—	4	8	no	—
23	C₄×#10596831	C₂×C₄	1	4	yes	—
24	C₄×#10596831	Q₈	1	4	yes	—
25	C₂²×C₈×C₇₃	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
29	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
30	#106×C₇₃	—	4	4	no	—
31	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₇₃	Q₈	1	2	yes	—
33	C₂²×Dic₇₃	—	4	1	no	—
34	C₂²×Dic₇₃	—	4	1	no	—
35	C₂×#10596831	—	2	4	no	—
36	C₂×#10596831	—	2	4	no	—
37	C₂×#10596831	C₄⋊C₄	1	8	yes	—
38	C₄⋊Dic₇₃	—	4	2	no	—
39	C₄⋊Dic₇₃	—	4	2	no	—
40	C₄⋊Dic₇₃	—	4	4	no	—
41	C₄⋊Dic₇₃	—	4	4	no	—
42	C₂²×C₄×C₇₃	—	4	2	no	—
43	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
46	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
47	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
48	C₄⋊C₄×C₇₃	—	4	4	no	—
49	C₄⋊C₄×C₇₃	—	4	4	no	—
50	C₄⋊C₄×C₇₃	—	4	4	no	—
51	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
53	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₇₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	8	2	no	—
56	C₂×C₄×C₇₃	—	8	2	no	—
57	C₂×C₄×C₇₃	—	8	2	no	—
58	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
65	C₂×C₄×C₇₃	#137	1	4	yes	—
66	C₂×Dic₇₃	—	8	1	no	—
67	C₂×Dic₇₃	—	8	1	no	—
68	C₂×Dic₇₃	—	8	2	no	—
69	C₂×Dic₇₃	—	8	2	no	—
70	C₈×C₇₃	#106	1	8	yes	complemented
71	#10596831	—	2	8	no	—
72	C₂²×C₇₃	#384	1	1	yes	characteristic
73	C₂²×C₇₃	#492	1	1	yes	characteristic
74	C₂²×C₇₃	#492	1	1	yes	characteristic
75	C₂²×C₇₃	#496	1	2	yes	—
76	C₂²×C₇₃	#497	1	2	yes	—
77	C₄×C₇₃	—	16	2	no	—
78	C₄×C₇₃	C₂×#106	1	2	yes	—
79	C₄×C₇₃	#428	1	4	yes	—
80	C₄×C₇₃	#498	1	2	yes	—
81	C₄×C₇₃	#501	1	4	yes	—
82	Dic₇₃	—	16	1	no	—
83	Dic₇₃	—	16	1	no	—
84	#8819	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#1714	1	2	yes	—
86	C₂×C₇₃	#1715	1	1	yes	characteristic
87	C₂×C₇₃	#1716	1	1	yes	characteristic
88	C₂×C₇₃	#1844	1	1	yes	characteristic
89	C₂×C₇₃	#1847	1	2	yes	—
90	#1845	—	73	1	no	—
91	#1844	—	73	2	no	—
92	#1844	D₇₃	1	1	yes	characteristic
93	#1834	—	73	2	no	—
94	C₂×#322	—	73	1	no	—
95	C₇₃	#8819	1	1	yes	characteristic, complemented, Sylow
96	C₂×#107	—	146	2	no	—
97	C₂×#106	—	2	2	no	—
98	C₂×#106	—	146	4	no	—
99	C₂×C₄×C₈	—	73	2	no	—
100	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
101	#384	—	73	1	no	—
102	#384	—	73	1	no	—
103	#384	C₂×D₇₃	1	2	yes	—
104	#340	—	146	4	no	—
105	#322	—	73	4	no	—
106	C₂²×C₈	—	73	4	no	—
107	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	146	2	no	—
112	C₂×C₄⋊C₄	—	146	2	no	—
113	C₂×C₄⋊C₄	—	146	1	no	—
114	C₂×C₄⋊C₄	—	146	1	no	—
115	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
116	#107	—	292	4	no	—
117	#106	—	4	4	no	—
118	#106	—	292	8	no	—
119	C₄×C₈	—	73	4	no	—
120	C₄×C₈	—	73	4	no	—
121	C₄×C₈	Dic₁₄₆	1	2	yes	—
122	C₄×C₈	C₄×D₇₃	1	2	yes	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	292	1	no	—
125	C₂²×C₄	—	292	1	no	—
126	C₂²×C₄	D₄×D₇₃	1	2	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
128	C₂×C₈	—	73	8	no	—
129	C₂×C₈	—	146	4	no	—
130	C₂×C₈	—	146	4	no	—
131	C₂×C₈	#11924007	1	4	yes	—
132	C₂×C₈	#11924015	1	4	yes	—
133	C₂×C₈	#11924016	1	4	yes	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	292	2	no	—
138	C₄⋊C₄	—	292	2	no	—
139	C₄⋊C₄	—	292	4	no	—
140	C₄⋊C₄	—	292	4	no	—
141	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
142	C₄²	Q₈×D₇₃	1	2	yes	—
143	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
144	C₂³	#14556587	1	1	yes	characteristic
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	584	1	no	—
149	C₂×C₄	—	584	1	no	—
150	C₂×C₄	—	584	2	no	—
151	C₂×C₄	—	584	2	no	—
152	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
153	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
154	C₂×C₄	#14556589	1	1	yes	characteristic
155	C₂×C₄	#14556590	1	2	yes	—
156	C₂×C₄	C₂×#11924015	1	2	yes	—
157	C₂×C₄	C₂×#11924016	1	2	yes	—
158	C₂×C₄	#14556618	1	4	yes	—
159	C₂×C₄	#14556620	1	4	yes	—
160	C₈	—	146	8	no	—
161	C₈	#14556522	1	8	yes	complemented
162	C₂²	#17988456	1	1	yes	characteristic
163	C₂²	#17988959	1	1	yes	characteristic
164	C₂²	#17988961	1	1	yes	characteristic
165	C₂²	#17988973	1	2	yes	—
166	C₂²	#17988978	1	2	yes	—
167	C₄	—	16	2	no	—
168	C₄	—	1168	1	no	—
169	C₄	—	1168	1	no	—
170	C₄	C₂×#14556522	1	2	yes	—
171	C₄	#17988626	1	4	yes	—
172	C₄	#17988983	1	2	yes	—
173	C₄	#17988994	1	4	yes	—
174	C₂	#22609550	1	2	yes	—
175	C₂	#22609560	1	1	yes	characteristic
176	C₂	#22609565	1	1	yes	characteristic
177	C₂	#22610087	1	1	yes	characteristic
178	C₂	#22610096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610091

C₂

yes

maximal, characteristic

#22610087

C₂

yes

maximal

#22610058

C₂

yes

maximal

C₂×#17988308

C₂

yes

maximal, characteristic

C₇₃×#1844

C₂

yes

maximal, characteristic

C₂×#14556525

—

C₂×#14556522

—

C₂×C₄×#10596831

C₂²

yes

—

#17988457

C₂²

yes

characteristic

#17988457

C₂²

yes

characteristic

#17988349

—

#17988308

C₄

yes

complemented

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

—

C₂²×#10596831

D₄

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

#14556525

—

#14556522

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

Q₈

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

#106

yes

complemented

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#497

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

#428

yes

—

C₄×C₇₃

#498

yes

—

C₄×C₇₃

#501

yes

—

Dic₇₃

—

Dic₇₃

—

#8819

—

maximal, Sylow

C₂×C₇₃

#1714

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

#1845

—

#1844

—

#1844

D₇₃

yes

characteristic

#1834

—

C₂×#322

—

C₇₃

#8819

yes

characteristic, complemented, Sylow

C₂×#107

—

146

—

C₂×#106

—

C₂×#106

—

146

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

101

#384

—

102

#384

—

103

#384

C₂×D₇₃

yes

—

104

#340

—

146

—

105

#322

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂²⋊D₇₃

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

146

—

112

C₂×C₄⋊C₄

—

146

—

113

C₂×C₄⋊C₄

—

146

—

114

C₂×C₄⋊C₄

—

146

—

115

C₂×C₄²

C₂²×D₇₃

yes

characteristic

116

#107

—

292

—

117

#106

—

118

#106

—

292

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₁₄₆

yes

—

122

C₄×C₈

C₄×D₇₃

yes

—

123

C₂²×C₄

—

124

C₂²×C₄

—

292

—

125

C₂²×C₄

—

292

—

126

C₂²×C₄

D₄×D₇₃

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₂×C₈

#11924007

yes

—

132

C₂×C₈

#11924015

yes

—

133

C₂×C₈

#11924016

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

292

—

138

C₄⋊C₄

—

292

—

139

C₄⋊C₄

—

292

—

140

C₄⋊C₄

—

292

—

141

C₄²

C₂×C₄×D₇₃

yes

characteristic

142

C₄²

Q₈×D₇₃

yes

—

143

C₄²

C₂×Dic₁₄₆

yes

characteristic

144

C₂³

#14556587

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

584

—

149

C₂×C₄

—

584

—

150

C₂×C₄

—

584

—

151

C₂×C₄

—

584

—

152

C₂×C₄

C₄⋊C₄×D₇₃

yes

—

153

C₂×C₄

C₂×#11924007

yes

characteristic

154

C₂×C₄

#14556589

yes

characteristic

155

C₂×C₄

#14556590

yes

—

156

C₂×C₄

C₂×#11924015

yes

—

157

C₂×C₄

C₂×#11924016

yes

—

158

C₂×C₄

#14556618

yes

—

159

C₂×C₄

#14556620

yes

—

160

C₈

—

146

—

161

C₈

#14556522

yes

complemented

162

C₂²

#17988456

yes

characteristic

163

C₂²

#17988959

yes

characteristic

164

C₂²

#17988961

yes

characteristic

165

C₂²

#17988973

yes

—

166

C₂²

#17988978

yes

—

167

C₄

—

168

C₄

—

1168

—

169

C₄

—

1168

—

170

C₄

C₂×#14556522

yes

—

171

C₄

#17988626

yes

—

172

C₄

#17988983

yes

—

173

C₄

#17988994

yes

—

174

C₂

#22609550

yes

—

175

C₂

#22609560

yes

characteristic

176

C₂

#22609565

yes

characteristic

177

C₂

#22610087

yes

characteristic

178

C₂

#22610096

yes

—

	subgroup	G/H
center	C₂³	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1844	C₂
socle, prime radical	C₂³×C₇₃	#127
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1844	D₇₃
hypocenter	C₇₃	#8819
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14556587

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1844

C₂

socle, prime radical

C₂³×C₇₃

#127

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
12. #17988308	C₄	semidirect
70. C₈×C₇₃	#106	semidirect
95. C₇₃	#8819	semidirect
161. C₈	#14556522	semidirect

H ≅ G/N

type

12. #17988308

C₄

semidirect

70. C₈×C₇₃

#106

semidirect

95. C₇₃

#8819

semidirect

161. C₈

#14556522

semidirect

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

subgroup	index
#22610091	2
#22610087	2
#22610058	2
C₂×#17988308	2
C₇₃×#1844	2
#8819	73

subgroup

index

C₇₃×#1844

Subgroups — #32979766

Conjugacy classes of subgroups

Complements

Maximal subgroups