Group #32976435 — subgroups

Subgroups — #32976435

Order 18688.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22609617	C₂	1	1	yes	maximal, characteristic
2	#22608223	C₂	1	1	yes	maximal, characteristic
3	#22607761	C₂	1	2	yes	maximal
4	#22607759	C₂	1	2	yes	maximal
5	C₇₃×#1452	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556502	C₂²	1	1	yes	characteristic
8	C₂×#14556501	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
10	#17988433	C₂²	1	1	yes	characteristic
11	#17988309	C₄	1	4	yes	complemented
12	#431×C₇₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#322×C₇₃	C₄	1	4	yes	complemented
15	C₂²×#10596831	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
17	C₄⋊#10596831	—	2	2	no	—
18	C₂²⋊#10596831	—	2	2	no	—
19	#14556502	—	2	2	no	—
20	#14556501	—	2	4	no	—
21	C₄×#10596831	C₂×C₄	1	4	yes	—
22	C₈×Dic₇₃	—	2	4	no	—
23	#14556498	—	2	2	no	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₇₃	—	2	2	no	—
28	C₂²⋊C₈×C₇₃	—	2	2	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×Dic₇₃	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	—	2	2	no	—
35	C₂×#10596831	—	2	4	no	—
36	C₂×#10596831	M₁₆	1	4	yes	—
37	C₄×Dic₇₃	—	2	2	no	—
38	C₄×Dic₇₃	M₁₆	1	4	yes	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	#41	1	2	yes	—
41	C₂×C₈×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	—	2	4	no	—
44	C₂×C₈×C₇₃	C₄²	1	2	yes	—
45	C₂×C₈×C₇₃	C₄²	1	2	yes	—
46	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₃	#41	1	2	yes	—
49	C₂³×C₇₃	#117	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#117	1	2	yes	—
53	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#117	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₇₃	—	2	4	no	—
58	C₂×Dic₇₃	—	4	4	no	—
59	C₈×C₇₃	—	2	2	no	—
60	C₈×C₇₃	—	2	2	no	—
61	C₈×C₇₃	—	4	4	no	—
62	#10596831	—	2	4	no	—
63	#10596831	—	4	4	no	—
64	C₂²×C₇₃	—	2	2	no	—
65	C₂²×C₇₃	#376	1	1	yes	characteristic
66	C₂²×C₇₃	#432	1	2	yes	—
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
71	C₄×C₇₃	#432	1	2	yes	—
72	Dic₇₃	—	4	4	no	—
73	#8140	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	—	2	2	no	—
75	C₂×C₇₃	#1745	1	1	yes	characteristic
76	C₂×C₇₃	#1747	1	2	yes	—
77	#1726	—	73	1	no	—
78	#1452	—	73	1	no	—
79	#1452	D₇₃	1	1	yes	characteristic
80	#1275	—	73	2	no	—
81	#1274	—	73	2	no	—
82	C₇₃	#8140	1	1	yes	characteristic, complemented, Sylow
83	#431	—	73	2	no	—
84	#431	C₂×D₇₃	1	2	yes	—
85	C₂×#97	—	73	2	no	—
86	C₂×#97	—	73	1	no	—
87	C₂×C₄×C₈	—	73	2	no	—
88	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
89	#376	—	73	1	no	—
90	#322	—	73	4	no	—
91	#322	Dic₇₃	1	4	yes	complemented
92	C₂²×C₈	—	73	2	no	—
93	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
95	C₂×C₄²	—	73	2	no	—
96	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	146	2	no	—
99	C₂²⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	—	146	2	no	—
101	#97	—	146	4	no	—
102	#97	—	146	2	no	—
103	C₄×C₈	—	73	4	no	—
104	C₄×C₈	—	146	4	no	—
105	C₄×C₈	C₄×D₇₃	1	2	yes	—
106	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
107	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
108	#95	—	146	2	no	—
109	C₂²×C₄	—	146	2	no	—
110	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
111	C₂²×C₄	#11924028	1	2	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	146	4	no	—
117	C₂×C₈	—	146	2	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
120	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
121	C₄²	—	73	4	no	—
122	C₄²	—	146	2	no	—
123	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
124	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
125	C₄²	#11924028	1	2	yes	—
126	C₂³	#14556548	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	292	4	no	—
129	C₂×C₄	—	146	4	no	—
130	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
131	C₂×C₄	#14556548	1	2	yes	—
132	C₂×C₄	#14556548	1	1	yes	characteristic
133	C₂×C₄	#14556550	1	2	yes	—
134	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
135	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	4	4	no	—
139	C₈	—	292	4	no	—
140	C₈	—	146	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#17988433	1	1	yes	characteristic
143	C₂²	#17988642	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	292	4	no	—
147	C₄	M₁₆×Dic₇₃	1	1	yes	characteristic
148	C₄	M₁₆×Dic₇₃	1	1	yes	characteristic
149	C₄	#17988642	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#22609679	1	1	yes	characteristic
152	C₂	#22609686	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22609617

C₂

yes

maximal, characteristic

#22608223

C₂

yes

maximal, characteristic

#22607761

C₂

yes

maximal

#22607759

C₂

yes

maximal

C₇₃×#1452

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556502

C₂²

yes

characteristic

C₂×#14556501

C₄

yes

—

C₂×C₈×Dic₇₃

C₄

yes

—

#17988433

C₂²

yes

characteristic

#17988309

C₄

yes

complemented

#431×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

C₄

yes

complemented

C₂²×#10596831

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₈×Dic₇₃

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

—

—

—

—

M₁₆

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

—

C₂×C₈×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#117

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#117

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#376

yes

characteristic

C₂²×C₇₃

#432

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×M₁₆

yes

characteristic

C₄×C₇₃

C₄×M₁₆

yes

characteristic

C₄×C₇₃

#432

yes

—

Dic₇₃

—

#8140

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1745

yes

characteristic

C₂×C₇₃

#1747

yes

—

#1726

—

#1452

—

#1452

D₇₃

yes

characteristic

#1275

—

#1274

—

C₇₃

#8140

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₃

yes

—

—

Dic₇₃

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₇₃

yes

characteristic

C₂²×C₈

C₄×D₇₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

146

—

101

#97

—

146

—

102

#97

—

146

—

103

C₄×C₈

—

104

C₄×C₈

—

146

—

105

C₄×C₈

C₄×D₇₃

yes

—

106

C₄×C₈

C₂×Dic₇₃

yes

—

107

C₄×C₈

C₂×Dic₇₃

yes

—

108

#95

—

146

—

109

C₂²×C₄

—

146

—

110

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

111

C₂²×C₄

#11924028

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

146

—

117

C₂×C₈

—

146

—

118

C₂×C₈

—

146

—

119

C₂×C₈

C₄×Dic₇₃

yes

—

120

C₂×C₈

C₄×Dic₇₃

yes

—

121

C₄²

—

122

C₄²

—

146

—

123

C₄²

C₂×C₄×D₇₃

yes

characteristic

124

C₄²

C₂²×Dic₇₃

yes

characteristic

125

C₄²

#11924028

yes

—

126

C₂³

#14556548

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

146

—

130

C₂×C₄

C₂×C₄×Dic₇₃

yes

characteristic

131

C₂×C₄

#14556548

yes

—

132

C₂×C₄

#14556548

yes

characteristic

133

C₂×C₄

#14556550

yes

—

134

C₂×C₄

M₁₆×D₇₃

yes

—

135

C₂×C₄

M₁₆×D₇₃

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

292

—

140

C₈

—

146

—

141

C₂²

—

142

C₂²

#17988433

yes

characteristic

143

C₂²

#17988642

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

292

—

147

C₄

M₁₆×Dic₇₃

yes

characteristic

148

C₄

M₁₆×Dic₇₃

yes

characteristic

149

C₄

#17988642

yes

—

150

C₂

—

151

C₂

#22609679

yes

characteristic

152

C₂

#22609686

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1452	C₂
socle	C₂²×C₇₃	#376
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1452	D₇₃
hypocenter	C₇₃	#8140
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₇₃

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1452

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
11. #17988309	C₄	semidirect
14. #322×C₇₃	C₄	semidirect
82. C₇₃	#8140	semidirect
91. #322	Dic₇₃	semidirect

H ≅ G/N

type

11. #17988309

C₄

semidirect

14. #322×C₇₃

C₄

semidirect

82. C₇₃

#8140

semidirect

91. #322

Dic₇₃

semidirect

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

subgroup	index
#22609617	2
#22608223	2
#22607761	2
#22607759	2
C₇₃×#1452	2
#8140	73

subgroup

index

C₇₃×#1452

Subgroups — #32976435

Conjugacy classes of subgroups

Complements

Maximal subgroups