Group #32985488 — subgroups

Subgroups — #32985488

Order 18688.

558 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 123 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×#17988964	C₂	1	1	yes	maximal, characteristic
2	#22610069	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608598	C₂	1	4	yes	maximal, complemented
4	C₇₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₈₄	—	2	2	no	—
6	C₂×#14556588	C₂²	1	2	yes	—
7	#17988964	—	2	2	no	—
8	#17988964	C₄	1	4	yes	—
9	#17988611	C₂²	1	2	yes	—
10	#17988352	—	2	2	no	—
11	#17988330	—	2	8	no	—
12	#431×C₇₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#322×C₇₃	—	2	2	no	—
15	C₂²×D₂₉₂	—	2	2	no	—
16	C₂²×D₂₉₂	—	2	4	no	—
17	C₂×D₅₈₄	—	2	8	no	—
18	C₂×D₅₈₄	—	4	8	no	—
19	C₂×#11924012	—	2	2	no	—
20	#14556588	—	2	2	no	—
21	#14556588	—	2	4	no	—
22	#14556588	C₂×C₄	1	4	yes	—
23	C₄⋊#10596831	—	2	4	no	—
24	C₂²×C₈×C₇₃	D₄	1	2	yes	—
25	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₇₃	D₄	1	4	yes	—
27	C₂²⋊C₈×C₇₃	—	2	2	no	—
28	C₄×C₈×C₇₃	—	2	2	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₃	D₄	1	2	yes	—
31	C₂³×D₇₃	—	4	2	no	—
32	C₂×D₂₉₂	—	2	4	no	—
33	C₂×D₂₉₂	—	2	8	no	—
34	C₂×D₂₉₂	—	4	8	no	—
35	C₂×D₂₉₂	—	4	4	no	—
36	C₂×D₂₉₂	—	4	2	no	—
37	C₂×D₂₉₂	—	8	8	no	—
38	D₅₈₄	—	4	16	no	—
39	#11924012	—	4	4	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	#41	1	2	yes	—
43	C₂×C₈×C₇₃	—	2	4	no	—
44	C₂×C₈×C₇₃	—	2	4	no	—
45	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
46	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₂×D₄	1	2	yes	—
49	C₂³×C₇₃	#124	1	1	yes	characteristic
50	C₂²×D₇₃	—	4	4	no	—
51	C₂²×D₇₃	—	8	8	no	—
52	C₂²×D₇₃	—	8	4	no	—
53	C₂×C₄×C₇₃	—	2	4	no	—
54	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₇₃	#121	1	2	yes	—
57	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
59	D₂₉₂	—	4	8	no	—
60	D₂₉₂	—	8	8	no	—
61	D₂₉₂	—	8	8	no	—
62	D₂₉₂	—	8	4	no	—
63	C₈×C₇₃	—	2	4	no	—
64	C₈×C₇₃	—	4	4	no	—
65	#10596831	—	8	2	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	#390	1	1	yes	characteristic
68	C₂²×C₇₃	#490	1	2	yes	—
69	C₂×D₇₃	—	8	4	no	—
70	C₂×D₇₃	—	8	8	no	—
71	C₂×D₇₃	—	16	8	no	—
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	#418	1	2	yes	—
74	C₄×C₇₃	#472	1	4	yes	—
75	C₄×C₇₃	#496	1	2	yes	—
76	#9780	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	2	no	—
78	C₂×C₇₃	#1961	1	1	yes	characteristic
79	C₂×C₇₃	#1978	1	2	yes	—
80	D₇₃	—	16	4	no	—
81	C₂×#493	—	73	1	no	—
82	#1837	—	73	1	no	—
83	#1545	—	73	4	no	—
84	#1452	D₇₃	1	1	yes	characteristic, complemented
85	C₇₃	#9780	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₈	—	146	2	no	—
87	C₂×#127	—	73	2	no	—
88	#493	—	73	4	no	—
89	#493	—	146	2	no	—
90	#431	C₂×D₇₃	1	2	yes	—
91	#423	—	73	2	no	—
92	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
93	#341	—	146	2	no	—
94	#331	—	146	8	no	—
95	#322	—	2	2	no	—
96	C₂²×D₄	—	146	2	no	—
97	C₂²×D₄	—	146	4	no	—
98	C₂×D₈	—	292	8	no	—
99	C₂×D₈	—	146	8	no	—
100	C₂×M₁₆	—	146	2	no	—
101	C₂²×C₈	D₂₉₂	1	2	yes	—
102	#127	—	73	4	no	—
103	#127	—	146	2	no	—
104	#127	—	146	4	no	—
105	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
106	C₄⋊C₈	—	146	4	no	—
107	C₄⋊C₈	D₂₉₂	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	C₄×D₇₃	1	2	yes	—
111	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
112	C₂⁴	—	292	2	no	—
113	C₂×D₄	—	292	4	no	—
114	C₂×D₄	—	292	2	no	—
115	C₂×D₄	—	292	8	no	—
116	C₂×D₄	—	584	8	no	—
117	C₂×D₄	—	146	4	no	—
118	C₂×D₄	—	146	8	no	—
119	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
120	C₂²×C₄	#11924028	1	2	yes	—
121	D₈	—	292	16	no	—
122	M₁₆	—	292	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	292	2	no	—
126	C₂×C₈	#11924005	1	4	yes	—
127	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
128	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
129	C₄²	C₂×D₂₉₂	1	2	yes	—
130	C₂³	—	292	4	no	—
131	C₂³	—	584	8	no	—
132	C₂³	—	584	4	no	—
133	C₂³	#14556579	1	1	yes	characteristic
134	D₄	—	292	8	no	—
135	D₄	—	584	8	no	—
136	D₄	—	584	4	no	—
137	D₄	—	584	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
140	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
141	C₂×C₄	#14556565	1	2	yes	—
142	C₂×C₄	#14556588	1	1	yes	characteristic
143	C₂×C₄	#14556619	1	4	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	4	4	no	—
146	C₈	—	584	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	584	4	no	—
149	C₂²	—	584	8	no	—
150	C₂²	—	1168	8	no	—
151	C₂²	#17988489	1	1	yes	characteristic
152	C₂²	#17988952	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#17988593	1	2	yes	—
155	C₄	#17988857	1	4	yes	—
156	C₄	#17988976	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	1168	4	no	—
159	C₂	#22610755	1	1	yes	characteristic
160	C₂	#22610841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988964

C₂

yes

maximal, characteristic

#22610069

C₂

yes

maximal, characteristic, complemented

#22608598

C₂

yes

maximal, complemented

C₇₃×#1452

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₈₄

—

C₂×#14556588

C₂²

yes

—

#17988964

—

#17988964

C₄

yes

—

#17988611

C₂²

yes

—

#17988352

—

#17988330

—

#431×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

—

C₂²×D₂₉₂

—

C₂²×D₂₉₂

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

—

—

—

C₂×C₄

yes

—

C₄⋊#10596831

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

D₄

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

#11924012

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₂³×C₇₃

#124

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#490

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#418

yes

—

C₄×C₇₃

#472

yes

—

C₄×C₇₃

#496

yes

—

#9780

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1961

yes

characteristic

C₂×C₇₃

#1978

yes

—

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#9780

yes

characteristic, complemented, Sylow

C₂²×D₈

—

146

—

C₂×#127

—

#493

—

#493

—

146

—

#431

C₂×D₇₃

yes

—

#423

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#341

—

146

—

#331

—

146

—

#322

—

C₂²×D₄

—

146

—

C₂²×D₄

—

146

—

C₂×D₈

—

292

—

C₂×D₈

—

146

—

100

C₂×M₁₆

—

146

—

101

C₂²×C₈

D₂₉₂

yes

—

102

#127

—

103

#127

—

146

—

104

#127

—

146

—

105

C₂×C₄²

C₂²×D₇₃

yes

characteristic

106

C₄⋊C₈

—

146

—

107

C₄⋊C₈

D₂₉₂

yes

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₇₃

yes

—

111

C₄×C₈

C₂²⋊D₇₃

yes

—

112

C₂⁴

—

292

—

113

C₂×D₄

—

292

—

114

C₂×D₄

—

292

—

115

C₂×D₄

—

292

—

116

C₂×D₄

—

584

—

117

C₂×D₄

—

146

—

118

C₂×D₄

—

146

—

119

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

120

C₂²×C₄

#11924028

yes

—

121

D₈

—

292

—

122

M₁₆

—

292

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

292

—

126

C₂×C₈

#11924005

yes

—

127

C₄²

C₂×C₄×D₇₃

yes

characteristic

128

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

129

C₄²

C₂×D₂₉₂

yes

—

130

C₂³

—

292

—

131

C₂³

—

584

—

132

C₂³

—

584

—

133

C₂³

#14556579

yes

characteristic

134

D₄

—

292

—

135

D₄

—

584

—

136

D₄

—

584

—

137

D₄

—

584

—

138

C₂×C₄

—

139

C₂×C₄

C₂×#11924005

yes

characteristic

140

C₂×C₄

C₄×D₂₉₂

yes

—

141

C₂×C₄

#14556565

yes

—

142

C₂×C₄

#14556588

yes

characteristic

143

C₂×C₄

#14556619

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

584

—

147

C₂²

—

148

C₂²

—

584

—

149

C₂²

—

584

—

150

C₂²

—

1168

—

151

C₂²

#17988489

yes

characteristic

152

C₂²

#17988952

yes

—

153

C₄

—

154

C₄

#17988593

yes

—

155

C₄

#17988857

yes

—

156

C₄

#17988976

yes

—

157

C₂

—

158

C₂

—

1168

—

159

C₂

#22610755

yes

characteristic

160

C₂

#22610841

yes

—

	subgroup	G/H
center	C₂²	#17988489
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1452	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂×#17988964	C₂
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1452	D₇₃
hypocenter	C₇₃	#9780
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#17988489

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1452

C₂

C₂²×C₇₃

C₂

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
2. #22610069	C₂	semidirect
3. #22608598	C₂	semidirect
4. C₇₃×#1452	C₂	semidirect
84. #1452	D₇₃	semidirect
85. C₇₃	#9780	semidirect

H ≅ G/N

type

2. #22610069

C₂

semidirect

3. #22608598

C₂

semidirect

4. C₇₃×#1452

C₂

semidirect

84. #1452

D₇₃

semidirect

85. C₇₃

#9780

semidirect

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

subgroup	index
C₂×#17988964	2
#22610069	2
#22608598	2
C₇₃×#1452	2
#9780	73

subgroup

index

C₇₃×#1452

Subgroups — #32985488

Conjugacy classes of subgroups

Complements

Maximal subgroups