Group #32994670 — subgroups

Subgroups — #32994670

Order 18688.

702 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 235 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×#17988636	C₂	1	3	yes	maximal
2	#22609624	C₂	1	3	yes	maximal
3	C₂×#338×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924012	C₂²	1	3	yes	—
5	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
6	#17988636	—	2	12	no	—
7	C₂×#14556527	—	2	6	no	—
8	#17988341	—	2	6	no	—
9	C₂²×M₁₆×C₇₃	C₂²	1	3	yes	—
10	#338×C₇₃	C₄	1	4	yes	complemented
11	C₂³×Dic₇₃	Q₈	1	2	yes	—
12	C₂×#11924012	—	2	6	no	—
13	C₂×#11924012	—	2	6	no	—
14	C₂×#11924012	—	2	3	no	—
15	C₂×#11924012	D₄	1	6	yes	—
16	C₂²×#10596831	—	2	3	no	—
17	C₂×C₄×Dic₇₃	—	2	3	no	—
18	C₂×C₄×Dic₇₃	—	2	3	no	—
19	#14556527	—	2	24	no	—
20	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₇₃	—	2	6	no	—
22	C₂×M₁₆×C₇₃	—	2	6	no	—
23	C₂×M₁₆×C₇₃	C₂×C₄	1	6	yes	—
24	C₂×M₁₆×C₇₃	Q₈	1	6	yes	—
25	C₂²×C₈×C₇₃	D₄	1	6	yes	—
26	C₂²×Dic₇₃	—	2	6	no	—
27	C₂²×Dic₇₃	—	4	4	no	—
28	#11924012	—	2	12	no	—
29	#11924012	—	4	6	no	—
30	C₂×#10596831	—	2	6	no	—
31	C₂×#10596831	—	4	3	no	—
32	C₂×#10596831	—	4	3	no	—
33	C₄×Dic₇₃	—	4	3	no	—
34	C₄×Dic₇₃	—	4	1	no	—
35	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₇₃	C₂×Q₈	1	3	yes	—
40	C₂²×C₄×C₇₃	#41	1	3	yes	—
41	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
42	M₁₆×C₇₃	—	2	12	no	—
43	M₁₆×C₇₃	—	2	12	no	—
44	C₂×C₈×C₇₃	—	2	6	no	—
45	C₂×C₈×C₇₃	—	2	6	no	—
46	C₂×C₈×C₇₃	C₄⋊C₄	1	12	yes	—
47	C₂³×C₇₃	—	4	2	no	—
48	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₇₃	C₂²⋊Q₈	1	3	yes	—
50	C₂³×C₇₃	C₄⋊Q₈	1	3	yes	—
51	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
52	C₂×C₄×C₇₃	C₄×D₄	1	3	yes	—
53	C₂×C₄×C₇₃	#120	1	3	yes	—
54	C₂×C₄×C₇₃	#121	1	3	yes	—
55	C₂×C₄×C₇₃	C₂²⋊Q₈	1	6	yes	—
56	C₂×C₄×C₇₃	#123	1	3	yes	—
57	C₂×C₄×C₇₃	#125	1	3	yes	—
58	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄⋊Q₈	1	3	yes	—
60	C₂×Dic₇₃	—	2	12	no	—
61	C₂×Dic₇₃	—	4	6	no	—
62	C₂×Dic₇₃	—	4	2	no	—
63	C₈×C₇₃	—	2	12	no	—
64	#10596831	—	4	6	no	—
65	C₂²×C₇₃	—	2	6	no	—
66	C₂²×C₇₃	—	4	1	no	—
67	C₂²×C₇₃	—	4	3	no	—
68	C₂²×C₇₃	#389	1	3	yes	—
69	C₂²×C₇₃	#395	1	3	yes	—
70	C₂²×C₇₃	#399	1	1	yes	characteristic
71	C₄×C₇₃	#384	1	1	yes	characteristic
72	C₄×C₇₃	#385	1	3	yes	—
73	C₄×C₇₃	#392	1	1	yes	characteristic
74	C₄×C₇₃	#398	1	3	yes	—
75	Dic₇₃	—	4	4	no	—
76	#11338	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	3	no	—
78	C₂×C₇₃	—	2	3	no	—
79	C₂×C₇₃	#1444	1	1	yes	characteristic
80	C₂×C₇₃	#2091	1	2	yes	—
81	C₂×#430	—	73	3	no	—
82	C₂×#338	D₇₃	1	1	yes	characteristic
83	#1728	—	73	3	no	—
84	C₇₃	#11338	1	1	yes	characteristic, complemented, Sylow
85	C₂²×M₁₆	—	73	3	no	—
86	C₂²×M₁₆	C₂×D₇₃	1	3	yes	—
87	C₂²×C₄²	—	73	1	no	—
88	#430	—	146	12	no	—
89	C₂×#108	—	146	6	no	—
90	#338	Dic₇₃	1	4	yes	complemented
91	#336	—	146	6	no	—
92	C₂³×C₄	—	73	2	no	—
93	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
94	C₂×M₁₆	—	2	6	no	—
95	C₂×M₁₆	—	2	6	no	—
96	C₂×M₁₆	—	73	6	no	—
97	C₂×M₁₆	—	146	6	no	—
98	C₂×M₁₆	—	146	6	no	—
99	C₂×M₁₆	—	146	3	no	—
100	C₂×M₁₆	Dic₁₄₆	1	6	yes	—
101	C₂×M₁₆	C₂×Dic₇₃	1	6	yes	—
102	C₂²×C₈	—	146	3	no	—
103	C₂²×C₈	D₂₉₂	1	6	yes	—
104	C₂×C₄²	—	146	3	no	—
105	C₂×C₄²	—	146	3	no	—
106	#108	—	146	24	no	—
107	C₂⁴	Q₈×D₇₃	1	1	yes	characteristic
108	C₂²×C₄	—	292	4	no	—
109	C₂²×C₄	—	146	6	no	—
110	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
111	C₂²×C₄	D₄×D₇₃	1	3	yes	—
112	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
113	C₂²×C₄	C₂×Dic₁₄₆	1	3	yes	—
114	C₂²×C₄	#11924027	1	3	yes	—
115	C₂²×C₄	#11924029	1	1	yes	characteristic
116	M₁₆	—	2	12	no	—
117	M₁₆	—	2	12	no	—
118	M₁₆	—	292	6	no	—
119	M₁₆	—	146	12	no	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	—	2	6	no	—
122	C₂×C₈	—	292	3	no	—
123	C₂×C₈	—	292	3	no	—
124	C₂×C₈	—	146	6	no	—
125	C₂×C₈	C₄⋊Dic₇₃	1	12	yes	—
126	C₄²	—	292	3	no	—
127	C₄²	—	292	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	Q₈×Dic₇₃	1	1	yes	characteristic
130	C₂³	#14556571	1	3	yes	—
131	C₂³	#14556590	1	3	yes	—
132	C₂×C₄	—	292	6	no	—
133	C₂×C₄	—	292	2	no	—
134	C₂×C₄	—	146	12	no	—
135	C₂×C₄	C₂×C₄⋊Dic₇₃	1	3	yes	—
136	C₂×C₄	D₄×Dic₇₃	1	3	yes	—
137	C₂×C₄	#14556561	1	3	yes	—
138	C₂×C₄	#14556566	1	3	yes	—
139	C₂×C₄	#14556568	1	6	yes	—
140	C₂×C₄	#14556575	1	3	yes	—
141	C₂×C₄	#14556582	1	3	yes	—
142	C₂×C₄	#14556588	1	1	yes	characteristic
143	C₂×C₄	#14556591	1	3	yes	—
144	C₈	—	2	12	no	—
145	C₈	—	292	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	3	no	—
149	C₂²	#17988486	1	3	yes	—
150	C₂²	#17988508	1	3	yes	—
151	C₂²	#17988524	1	1	yes	characteristic
152	C₄	—	292	4	no	—
153	C₄	#17988457	1	1	yes	characteristic
154	C₄	#17988462	1	3	yes	—
155	C₄	#17988497	1	1	yes	characteristic
156	C₄	#17988519	1	3	yes	—
157	C₂	—	2	3	no	—
158	C₂	—	2	3	no	—
159	C₂	#22608196	1	1	yes	characteristic
160	C₂	#22611448	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988636

C₂

yes

maximal

#22609624

C₂

yes

maximal

C₂×#338×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924012

C₂²

yes

—

C₂²×C₄×Dic₇₃

C₂²

yes

—

—

—

C₂²×M₁₆×C₇₃

C₂²

yes

—

#338×C₇₃

C₄

yes

complemented

C₂³×Dic₇₃

Q₈

yes

—

—

—

—

D₄

yes

—

C₂²×#10596831

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

#14556527

—

C₂³×C₄×C₇₃

C₂³

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

C₂×M₁₆×C₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂⁴×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

#41

yes

—

C₂²×C₄×C₇₃

#41

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

C₄×Q₈

yes

characteristic

C₂³×C₇₃

C₂²⋊Q₈

yes

—

C₂³×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#120

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#125

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#389

yes

—

C₂²×C₇₃

#395

yes

—

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₄×C₇₃

#392

yes

characteristic

C₄×C₇₃

#398

yes

—

Dic₇₃

—

#11338

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1444

yes

characteristic

C₂×C₇₃

#2091

yes

—

C₂×#430

—

C₂×#338

D₇₃

yes

characteristic

#1728

—

C₇₃

#11338

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×M₁₆

C₂×D₇₃

yes

—

C₂²×C₄²

—

#430

—

146

—

C₂×#108

—

146

—

#338

Dic₇₃

yes

complemented

#336

—

146

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₇₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₂×M₁₆

—

146

—

C₂×M₁₆

—

146

—

100

C₂×M₁₆

Dic₁₄₆

yes

—

101

C₂×M₁₆

C₂×Dic₇₃

yes

—

102

C₂²×C₈

—

146

—

103

C₂²×C₈

D₂₉₂

yes

—

104

C₂×C₄²

—

146

—

105

C₂×C₄²

—

146

—

106

#108

—

146

—

107

C₂⁴

Q₈×D₇₃

yes

characteristic

108

C₂²×C₄

—

292

—

109

C₂²×C₄

—

146

—

110

C₂²×C₄

C₂²×Dic₇₃

yes

characteristic

111

C₂²×C₄

D₄×D₇₃

yes

—

112

C₂²×C₄

C₂×D₂₉₂

yes

—

113

C₂²×C₄

C₂×Dic₁₄₆

yes

—

114

C₂²×C₄

#11924027

yes

—

115

C₂²×C₄

#11924029

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

292

—

119

M₁₆

—

146

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

146

—

125

C₂×C₈

C₄⋊Dic₇₃

yes

—

126

C₄²

—

292

—

127

C₄²

—

292

—

128

C₂³

—

129

C₂³

Q₈×Dic₇₃

yes

characteristic

130

C₂³

#14556571

yes

—

131

C₂³

#14556590

yes

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

146

—

135

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

—

136

C₂×C₄

D₄×Dic₇₃

yes

—

137

C₂×C₄

#14556561

yes

—

138

C₂×C₄

#14556566

yes

—

139

C₂×C₄

#14556568

yes

—

140

C₂×C₄

#14556575

yes

—

141

C₂×C₄

#14556582

yes

—

142

C₂×C₄

#14556588

yes

characteristic

143

C₂×C₄

#14556591

yes

—

144

C₈

—

145

C₈

—

292

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#17988486

yes

—

150

C₂²

#17988508

yes

—

151

C₂²

#17988524

yes

characteristic

152

C₄

—

292

—

153

C₄

#17988457

yes

characteristic

154

C₄

#17988462

yes

—

155

C₄

#17988497

yes

characteristic

156

C₄

#17988519

yes

—

157

C₂

—

158

C₂

—

159

C₂

#22608196

yes

characteristic

160

C₂

#22611448

yes

—

	subgroup	G/H
center	C₂×C₄	#14556588
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×#338×C₇₃	C₂
socle	C₂²×C₇₃	#399
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×#338	D₇₃
hypocenter	C₇₃	#11338
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#14556588

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₇₃

Fitting

C₂×#338×C₇₃

C₂

socle

C₂²×C₇₃

#399

prime radical

C₂⁴×C₇₃

C₂×Q₈

cosocle

C₂³×C₄×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. #338×C₇₃	C₄	semidirect
84. C₇₃	#11338	semidirect
90. #338	Dic₇₃	semidirect

H ≅ G/N

type

10. #338×C₇₃

C₄

semidirect

84. C₇₃

#11338

semidirect

90. #338

Dic₇₃

semidirect

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

subgroup	index
C₂×#17988636	2
#22609624	2
C₂×#338×C₇₃	2
#11338	73

subgroup

index

C₂×#17988636

#22609624

C₂×#338×C₇₃

#11338

Subgroups — #32994670

Conjugacy classes of subgroups

Complements

Maximal subgroups