Group #33006579 — subgroups

Subgroups — #33006579

Order 18688.

188 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 65 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×#17988427	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#22611789	C₂	1	1	yes	maximal, characteristic
4	#22608132	C₂	1	1	yes	maximal, characteristic
5	#22608113	C₂	1	1	yes	maximal, characteristic
6	#22608064	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₇₃	—	2	1	no	—
10	#17988681	—	2	1	no	—
11	#17988427	C₄	1	4	yes	complemented
12	C₂×#14556497	C₂²	1	1	yes	characteristic
13	#17988413	C₂²	1	1	yes	characteristic
14	#17988391	—	2	1	no	—
15	#17988390	—	2	1	no	—
16	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
19	#366×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₉₂	—	2	1	no	—
21	Q₈×Dic₇₃	—	4	1	no	—
22	#14556556	—	4	1	no	—
23	Dic₅₈₄	—	2	1	no	—
24	Dic₅₈₄	C₂×C₄	1	2	yes	—
25	C₂×#11924003	D₄	1	1	yes	characteristic
26	#14556525	—	2	1	no	—
27	#14556515	—	4	1	no	—
28	#14556514	—	4	1	no	—
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	#14556497	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₇₃	—	2	1	no	—
32	Q₃₂×C₇₃	—	2	1	no	—
33	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
35	#107×C₇₃	—	2	1	no	—
36	C₃₂×C₇₃	—	2	1	no	—
37	C₂×Dic₁₄₆	—	4	1	no	—
38	Dic₂₉₂	—	2	2	no	—
39	Dic₂₉₂	—	4	1	no	—
40	C₂×#10596831	D₈	1	1	yes	characteristic
41	C₄⋊Dic₇₃	—	4	1	no	—
42	C₄⋊Dic₇₃	—	8	1	no	—
43	#11924007	—	8	1	no	—
44	C₄×Dic₇₃	—	8	1	no	—
45	C₄×Dic₇₃	—	8	1	no	—
46	C₄×Dic₇₃	D₈	1	1	yes	characteristic
47	#11924003	—	2	1	no	—
48	C₂×Q₈×C₇₃	—	4	1	no	—
49	Q₁₆×C₇₃	—	2	2	no	—
50	Q₁₆×C₇₃	—	4	1	no	—
51	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₇₃	—	4	1	no	—
53	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₁₆×C₇₃	#41	1	1	yes	characteristic
55	Q₈×C₇₃	—	4	2	no	—
56	Q₈×C₇₃	—	8	1	no	—
57	C₂×C₄×C₇₃	—	8	1	no	—
58	C₂×C₄×C₇₃	—	8	1	no	—
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₇₃	—	8	1	no	—
61	C₂×Dic₇₃	—	8	1	no	—
62	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
63	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
64	Dic₁₄₆	—	4	2	no	—
65	Dic₁₄₆	—	8	1	no	—
66	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
67	C₈×C₇₃	#135	1	1	yes	characteristic
68	#10596831	—	2	1	no	—
69	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₇₃	—	8	2	no	—
71	C₄×C₇₃	—	16	1	no	—
72	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
73	C₄×C₇₃	#508	1	1	yes	characteristic
74	Dic₇₃	—	2	1	no	—
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	—	8	2	no	—
77	Dic₇₃	—	16	1	no	—
78	#13653	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
80	C₂×C₇₃	#2264	1	1	yes	characteristic
81	C₂×C₇₃	#2264	1	1	yes	characteristic
82	C₂×Q₆₄	—	73	1	no	—
83	C₄×Q₃₂	—	73	1	no	—
84	C₄×Q₃₂	—	73	1	no	—
85	#1424	—	73	1	no	—
86	#1416	—	73	1	no	—
87	#1416	D₇₃	1	1	yes	characteristic
88	#1397	—	73	1	no	—
89	C₇₃	#13653	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₃₂	—	73	1	no	—
91	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
92	C₄×Q₁₆	—	146	1	no	—
93	C₄×Q₁₆	—	146	1	no	—
94	Q₆₄	—	73	4	no	—
95	C₂×C₃₂	—	73	1	no	—
96	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
97	#366	—	73	1	no	—
98	#366	C₂×D₇₃	1	1	yes	characteristic
99	#358	—	146	1	no	—
100	#358	—	146	1	no	—
101	C₄×C₁₆	—	73	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	146	1	no	—
104	C₄×Q₈	—	292	1	no	—
105	C₄×Q₈	—	292	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	73	2	no	—
108	Q₃₂	—	146	1	no	—
109	Q₃₂	C₄×D₇₃	1	2	yes	—
110	C₂×C₁₆	—	73	1	no	—
111	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	146	1	no	—
114	#103	—	292	1	no	—
115	#103	—	292	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	73	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	292	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	292	1	no	—
124	Q₁₆	—	146	2	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	292	1	no	—
129	C₄⋊C₄	—	584	1	no	—
130	C₄⋊C₄	—	584	1	no	—
131	C₄²	—	584	1	no	—
132	C₄²	—	584	1	no	—
133	C₄²	—	73	1	no	—
134	C₁₆	—	146	1	no	—
135	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
136	C₁₆	#11924027	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	292	2	no	—
140	Q₈	—	584	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	584	1	no	—
144	C₂×C₄	—	584	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
148	C₈	—	146	1	no	—
149	C₈	#14556551	1	1	yes	characteristic
150	C₈	#14556612	1	1	yes	characteristic
151	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	584	2	no	—
155	C₄	—	1168	1	no	—
156	C₄	—	146	1	no	—
157	C₄	—	146	1	no	—
158	C₄	#17988667	1	1	yes	characteristic
159	C₄	#17989018	1	1	yes	characteristic
160	C₂	#22611775	1	1	yes	characteristic
161	C₂	#22612247	1	1	yes	characteristic
162	C₂	#22612250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988427

C₂

yes

maximal, characteristic

Q₃₂×Dic₇₃

C₂

yes

maximal, characteristic

#22611789

C₂

yes

maximal, characteristic

#22608132

C₂

yes

maximal, characteristic

#22608113

C₂

yes

maximal, characteristic

#22608064

C₂

yes

maximal, characteristic

C₇₃×#1416

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

C₂²

yes

characteristic

Q₁₆×Dic₇₃

—

#17988681

—

#17988427

C₄

yes

complemented

C₂×#14556497

C₂²

yes

characteristic

#17988413

C₂²

yes

characteristic

#17988391

—

#17988390

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

#366×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

Q₈×Dic₇₃

—

#14556556

—

Dic₅₈₄

—

Dic₅₈₄

C₂×C₄

yes

—

C₂×#11924003

D₄

yes

—

—

—

C₈×Dic₇₃

D₄

yes

characteristic

#14556497

C₂×C₄

yes

—

C₂×Q₁₆×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

C₃₂×C₇₃

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₂×#10596831

D₈

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₁₆

yes

characteristic

C₂×Dic₇₃

D₁₆

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₄×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#13653

—

maximal, Sylow

C₂×C₇₃

C₄×D₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1416

D₇₃

yes

characteristic

#1397

—

C₇₃

#13653

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₇₃

yes

characteristic

C₄×Q₁₆

—

146

—

C₄×Q₁₆

—

146

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

#366

—

#366

C₂×D₇₃

yes

characteristic

#358

—

146

—

100

#358

—

146

—

101

C₄×C₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

146

—

104

C₄×Q₈

—

292

—

105

C₄×Q₈

—

292

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

146

—

109

Q₃₂

C₄×D₇₃

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

112

#107

—

113

#107

—

146

—

114

#103

—

292

—

115

#103

—

292

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

292

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

292

—

124

Q₁₆

—

146

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₇₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

292

—

129

C₄⋊C₄

—

584

—

130

C₄⋊C₄

—

584

—

131

C₄²

—

584

—

132

C₄²

—

584

—

133

C₄²

—

134

C₁₆

—

146

—

135

C₁₆

C₂×C₄×D₇₃

yes

characteristic

136

C₁₆

#11924027

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

292

—

140

Q₈

—

584

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

584

—

144

C₂×C₄

—

584

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

D₈×D₇₃

yes

characteristic

148

C₈

—

146

—

149

C₈

#14556551

yes

characteristic

150

C₈

#14556612

yes

characteristic

151

C₂²

D₁₆×D₇₃

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

584

—

155

C₄

—

1168

—

156

C₄

—

146

—

157

C₄

—

146

—

158

C₄

#17988667

yes

characteristic

159

C₄

#17989018

yes

characteristic

160

C₂

#22611775

yes

characteristic

161

C₂

#22612247

yes

characteristic

162

C₂

#22612250

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1416	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#1416	D₇₃
hypocenter	C₇₃	#13653
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₇₃

commutator

C₁₆×C₇₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#1416

C₂

socle, prime radical

C₂²×C₇₃

C₂×D₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
11. #17988427	C₄	semidirect
89. C₇₃	#13653	semidirect

H ≅ G/N

type

11. #17988427

C₄

semidirect

89. C₇₃

#13653

semidirect

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

subgroup	index
C₂×#17988427	2
Q₃₂×Dic₇₃	2
#22611789	2
#22608132	2
#22608113	2
#22608064	2
C₇₃×#1416	2
#13653	73

subgroup

index

Q₃₂×Dic₇₃

C₇₃×#1416

Subgroups — #33006579

Conjugacy classes of subgroups

Complements

Maximal subgroups