Group #32973285 — subgroups

Subgroups — #32973285

Order 18688.

432 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22609855	C₂	1	2	yes	maximal
2	C₇₃×#1784	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556510	—	2	2	no	—
4	C₂×#14556510	—	2	2	no	—
5	#17988551	C₄	1	4	yes	—
6	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×#101×C₇₃	—	2	2	no	—
8	#407×C₇₃	C₄	1	2	yes	—
9	C₂×#11924012	—	2	4	no	—
10	#14556510	—	4	4	no	—
11	#14556510	—	4	4	no	—
12	C₂²⋊#10596831	—	4	4	no	—
13	C₂²×Q₈×C₇₃	—	4	2	no	—
14	C₂²×Q₈×C₇₃	—	4	4	no	—
15	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
16	C₂²×Q₈×C₇₃	D₄	1	2	yes	—
17	C₂²×Q₈×C₇₃	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	—	2	2	no	—
21	#101×C₇₃	—	4	4	no	—
22	C₂²⋊C₈×C₇₃	—	4	2	no	—
23	#11924012	—	4	4	no	—
24	#11924012	—	4	4	no	—
25	C₂×#10596831	—	4	4	no	—
26	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
27	C₂×Q₈×C₇₃	—	2	4	no	—
28	C₂×Q₈×C₇₃	—	2	4	no	—
29	C₂×Q₈×C₇₃	—	2	4	no	—
30	C₂×Q₈×C₇₃	—	4	1	no	—
31	C₂×Q₈×C₇₃	—	4	2	no	—
32	C₂×Q₈×C₇₃	—	4	2	no	—
33	C₂×Q₈×C₇₃	—	4	1	no	—
34	C₂×Q₈×C₇₃	—	4	16	no	—
35	C₂²×C₄×C₇₃	—	2	2	no	—
36	C₂²×C₄×C₇₃	—	2	4	no	—
37	C₂²×C₄×C₇₃	—	4	2	no	—
38	C₂²×C₄×C₇₃	—	4	2	no	—
39	C₂²×C₄×C₇₃	—	4	2	no	—
40	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	M₁₆×C₇₃	—	4	4	no	—
45	C₂×C₈×C₇₃	—	4	2	no	—
46	C₂³×C₇₃	—	2	2	no	—
47	C₂³×C₇₃	—	2	1	no	—
48	C₂³×C₇₃	—	4	2	no	—
49	C₂³×C₇₃	#95	1	1	yes	characteristic
50	Q₈×C₇₃	—	4	4	no	—
51	Q₈×C₇₃	—	4	8	no	—
52	Q₈×C₇₃	—	4	4	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	2	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	4	8	no	—
62	C₂×C₄×C₇₃	—	4	4	no	—
63	C₂×C₄×C₇₃	#99	1	4	yes	—
64	C₂×C₄×C₇₃	#99	1	2	yes	—
65	C₈×C₇₃	—	8	2	no	—
66	#10596831	—	8	4	no	—
67	C₂²×C₇₃	—	2	4	no	—
68	C₂²×C₇₃	—	2	2	no	—
69	C₂²×C₇₃	—	4	1	no	—
70	C₂²×C₇₃	—	4	2	no	—
71	C₂²×C₇₃	—	4	1	no	—
72	C₂²×C₇₃	—	4	1	no	—
73	C₂²×C₇₃	#342	1	2	yes	—
74	C₂²×C₇₃	#342	1	1	yes	characteristic
75	C₄×C₇₃	—	2	4	no	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	—	4	1	no	—
79	#7317	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	—	4	2	no	—
83	C₂×C₇₃	#1304	1	1	yes	characteristic
84	#1784	—	73	2	no	—
85	#1784	D₇₃	1	1	yes	characteristic
86	C₇₃	#7317	1	1	yes	characteristic, complemented, Sylow
87	C₂³×Q₈	C₂×D₇₃	1	1	yes	characteristic
88	C₂×#101	—	2	2	no	—
89	C₂×#101	—	146	2	no	—
90	C₂×#101	—	146	2	no	—
91	#407	—	73	4	no	—
92	#407	Dic₇₃	1	2	yes	—
93	C₂²×Q₈	—	4	4	no	—
94	C₂²×Q₈	—	4	2	no	—
95	C₂²×Q₈	Dic₁₄₆	1	1	yes	characteristic
96	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
97	C₂²×Q₈	C₂²⋊D₇₃	1	2	yes	—
98	C₂³×C₄	C₄×D₇₃	1	2	yes	—
99	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	146	4	no	—
102	#101	—	4	4	no	—
103	#101	—	292	4	no	—
104	#101	—	292	4	no	—
105	C₂²⋊C₈	—	4	2	no	—
106	C₂²⋊C₈	—	292	4	no	—
107	C₂⁴	C₄×Dic₇₃	1	1	yes	characteristic
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	4	1	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂×Q₈	—	4	16	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#11924005	1	2	yes	—
122	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
123	C₂²×C₄	#11924007	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
125	M₁₆	—	4	4	no	—
126	M₁₆	—	292	4	no	—
127	M₁₆	—	292	4	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	292	4	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	4	2	no	—
133	C₂³	#14556498	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	Q₈	—	4	4	no	—
136	Q₈	—	4	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	#14556505	1	4	yes	—
148	C₂×C₄	#14556506	1	2	yes	—
149	C₈	—	8	2	no	—
150	C₈	—	584	4	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	#17988353	1	2	yes	—
158	C₂²	#17988354	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	4	no	—
161	C₄	—	4	2	no	—
162	C₄	—	4	1	no	—
163	C₂	—	2	2	no	—
164	C₂	—	2	1	no	—
165	C₂	—	4	2	no	—
166	C₂	#22607835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609855

C₂

yes

maximal

C₇₃×#1784

C₂

yes

maximal, characteristic

C₂×#14556510

—

C₂×#14556510

—

#17988551

C₄

yes

—

C₂³×Q₈×C₇₃

C₂²

yes

characteristic

C₂×#101×C₇₃

—

#407×C₇₃

C₄

yes

—

—

—

—

—

C₂²×Q₈×C₇₃

—

C₂²×Q₈×C₇₃

—

C₂²×Q₈×C₇₃

D₄

yes

characteristic

C₂²×Q₈×C₇₃

D₄

yes

—

C₂²×Q₈×C₇₃

Q₈

yes

characteristic

C₂³×C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

—

#101×C₇₃

—

C₂²⋊C₈×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₂⁴×C₇₃

C₄²

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#99

yes

—

C₂×C₄×C₇₃

#99

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#342

yes

—

C₂²×C₇₃

#342

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

#7317

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

C₇₃

#7317

yes

characteristic, complemented, Sylow

C₂³×Q₈

C₂×D₇₃

yes

characteristic

C₂×#101

—

C₂×#101

—

146

—

C₂×#101

—

146

—

#407

—

#407

Dic₇₃

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

Dic₁₄₆

yes

characteristic

C₂²×Q₈

D₂₉₂

yes

characteristic

C₂²×Q₈

C₂²⋊D₇₃

yes

—

C₂³×C₄

C₄×D₇₃

yes

—

C₂³×C₄

C₂×Dic₇₃

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

146

—

102

#101

—

103

#101

—

292

—

104

#101

—

292

—

105

C₂²⋊C₈

—

106

C₂²⋊C₈

—

292

—

107

C₂⁴

C₄×Dic₇₃

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

#11924005

yes

—

122

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

123

C₂²×C₄

#11924007

yes

—

124

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

125

M₁₆

—

126

M₁₆

—

292

—

127

M₁₆

—

292

—

128

C₂×C₈

—

129

C₂×C₈

—

292

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

#14556498

yes

characteristic

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

#14556505

yes

—

148

C₂×C₄

#14556506

yes

—

149

C₈

—

150

C₈

—

584

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#17988353

yes

—

158

C₂²

#17988354

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#22607835

yes

characteristic

	subgroup	G/H
center	C₂	#22607835
commutator, prime radical	C₂⁴×C₇₃	C₄²
Frattini	C₂³×Q₈	C₂×D₇₃
Fitting	C₇₃×#1784	C₂
socle	C₂×C₇₃	#1304
cosocle	C₂³×Q₈×C₇₃	C₂²
hypercenter	#1784	D₇₃
hypocenter	C₇₃	#7317
LCS	G ▷ C₂⁴×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ #1784

subgroup

G/H

center

C₂

#22607835

commutator, prime radical

C₂⁴×C₇₃

C₄²

Frattini

C₂³×Q₈

C₂×D₇₃

Fitting

C₇₃×#1784

C₂

socle

C₂×C₇₃

#1304

cosocle

C₂³×Q₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂⁴×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ #1784

N	H ≅ G/N	type
86. C₇₃	#7317	semidirect

H ≅ G/N

type

86. C₇₃

#7317

semidirect

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

subgroup	index
#22609855	2
C₇₃×#1784	2
#7317	73

subgroup

index

#22609855

C₇₃×#1784

#7317

Subgroups — #32973285

Conjugacy classes of subgroups

Complements

Maximal subgroups