Group #33004515 — subgroups

Subgroups — #33004515

Order 18688.

342 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22618865	C₂	1	1	yes	maximal, characteristic
2	#22611687	C₂	1	1	yes	maximal, characteristic
3	#22608019	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989144	C₄	1	1	yes	characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	#17988717	—	2	2	no	—
8	C₈×Dic₁₄₆	—	2	2	no	—
9	#17988613	C₄	1	2	yes	—
10	#17988611	C₄	1	1	yes	characteristic
11	#17988542	C₂²	1	2	yes	—
12	C₄⋊#11924003	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₃	—	2	2	no	—
16	C₂×#11924012	C₂×C₄	1	2	yes	—
17	#14556591	C₈	1	2	yes	—
18	#14556583	C₈	1	2	yes	—
19	#14556570	—	2	2	no	—
20	C₄×Dic₁₄₆	—	2	2	no	—
21	#14556550	C₂×C₄	1	2	yes	—
22	#14556531	—	2	4	no	—
23	C₂×#11924003	—	2	2	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	#14556520	—	2	4	no	—
26	#14556501	—	2	2	no	—
27	C₈×Dic₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	—	2	2	no	—
32	C₂×C₁₆×C₇₃	—	2	2	no	—
33	C₂×C₁₆×C₇₃	Q₈	1	4	yes	complemented
34	C₄×C₈×C₇₃	D₄	1	2	yes	—
35	C₄×C₈×C₇₃	D₄	1	2	yes	—
36	C₂×Dic₁₄₆	—	2	2	no	—
37	#11924012	—	2	4	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₄⋊Dic₇₃	—	2	2	no	—
40	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
41	#11924007	—	2	4	no	—
42	C₂²⋊Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	—	2	2	no	—
44	#11924003	—	4	2	no	—
45	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₇₃	#41	1	2	yes	—
53	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₁₆×C₇₃	—	2	4	no	—
57	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₃	—	2	4	no	—
59	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#131	1	2	yes	—
65	C₂×Dic₇₃	—	2	4	no	—
66	Dic₁₄₆	—	4	4	no	—
67	C₈×C₇₃	—	2	2	no	—
68	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₇₃	#124	1	1	yes	characteristic
70	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
71	#10596831	—	4	2	no	—
72	C₂²×C₇₃	—	2	2	no	—
73	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₃	#446	1	2	yes	—
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	#350	1	4	yes	—
77	C₄×C₇₃	#391	1	1	yes	characteristic
78	C₄×C₇₃	#408	1	1	yes	characteristic
79	C₄×C₇₃	C₈×Q₈	1	2	yes	—
80	Dic₇₃	—	4	4	no	—
81	#13266	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	—	2	2	no	—
83	C₂×C₇₃	#1984	1	1	yes	characteristic
84	C₂×C₇₃	C₂×#350	1	2	yes	—
85	#2960	—	73	1	no	—
86	#2150	—	73	1	no	—
87	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
88	#1379	—	73	4	no	—
89	C₇₃	#13266	1	1	yes	characteristic, complemented, Sylow
90	#533	—	73	1	no	—
91	C₂×M₃₂	—	73	2	no	—
92	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
93	#446	—	146	2	no	—
94	C₈×Q₈	—	146	2	no	—
95	#424	—	73	2	no	—
96	#423	—	73	1	no	—
97	#405	—	73	2	no	—
98	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	C₄⋊C₁₆	—	146	2	no	—
100	C₄×C₁₆	—	2	2	no	—
101	C₂×M₁₆	—	73	2	no	—
102	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₄⋊Q₈	—	73	2	no	—
105	#125	—	73	2	no	—
106	C₂²⋊Q₈	—	146	2	no	—
107	C₄×Q₈	—	146	2	no	—
108	#117	—	73	2	no	—
109	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
110	M₃₂	—	146	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	146	2	no	—
114	C₂×C₁₆	Dic₁₄₆	1	4	yes	complemented
115	C₄⋊C₈	—	146	4	no	—
116	C₄⋊C₈	—	146	2	no	—
117	#97	—	146	2	no	—
118	C₄×C₈	—	146	2	no	—
119	C₄×C₈	D₂₉₂	1	2	yes	—
120	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
121	C₂×Q₈	—	146	2	no	—
122	C₂²×C₄	#11924011	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
124	M₁₆	—	146	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	#11924005	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
131	C₂×C₈	#11924028	1	2	yes	—
132	C₄⋊C₄	—	73	4	no	—
133	C₄⋊C₄	—	146	4	no	—
134	C₄⋊C₄	—	146	2	no	—
135	C₂²⋊C₄	—	146	2	no	—
136	C₄²	—	146	2	no	—
137	C₄²	#11924005	1	2	yes	—
138	C₄²	C₈×D₇₃	1	1	yes	characteristic
139	C₄²	C₈×D₇₃	1	1	yes	characteristic
140	C₁₆	—	2	4	no	—
141	C₁₆	—	292	2	no	—
142	C₂³	C₂×#11924011	1	1	yes	characteristic
143	Q₈	—	292	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	146	4	no	—
146	C₂×C₄	#14556503	1	2	yes	—
147	C₂×C₄	#14556503	1	2	yes	—
148	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
149	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
151	C₂×C₄	#14556597	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	292	2	no	—
154	C₈	#14556570	1	2	yes	—
155	C₈	#14556579	1	1	yes	characteristic
156	C₈	#14556591	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×#14556503	1	1	yes	characteristic
159	C₂²	#17988717	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	292	4	no	—
162	C₄	#17988370	1	4	yes	—
163	C₄	#17988493	1	1	yes	characteristic
164	C₄	#17988553	1	1	yes	characteristic
165	C₄	C₈×Dic₁₄₆	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#22610867	1	1	yes	characteristic
168	C₂	C₂×#17988370	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618865

C₂

yes

maximal, characteristic

#22611687

C₂

yes

maximal, characteristic

#22608019

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₃

C₂

yes

maximal, characteristic

#17989144

C₄

yes

characteristic

C₂×#14556531

C₂²

yes

—

#17988717

—

C₈×Dic₁₄₆

—

#17988613

C₄

yes

—

#17988611

C₄

yes

characteristic

#17988542

C₂²

yes

—

C₄⋊#11924003

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

—

C₂×#11924012

C₂×C₄

yes

—

#14556591

C₈

yes

—

#14556583

C₈

yes

—

#14556570

—

C₄×Dic₁₄₆

—

#14556550

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

Q₈

yes

complemented

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₈

yes

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

#124

yes

characteristic

C₈×C₇₃

C₄⋊Q₈

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#446

yes

—

C₄×C₇₃

—

C₄×C₇₃

#350

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₈×Q₈

yes

—

Dic₇₃

—

#13266

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1984

yes

characteristic

C₂×C₇₃

C₂×#350

yes

—

#2960

—

#2150

—

C₂×C₄×C₁₆

D₇₃

yes

characteristic

#1379

—

C₇₃

#13266

yes

characteristic, complemented, Sylow

#533

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇₃

yes

—

#446

—

146

—

C₈×Q₈

—

146

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₁₆

—

146

—

100

C₄×C₁₆

—

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₇₃

yes

characteristic

103

C₂²×C₈

C₂²×D₇₃

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

146

—

107

C₄×Q₈

—

146

—

108

#117

—

109

C₂×C₄²

C₄×D₇₃

yes

characteristic

110

M₃₂

—

146

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

146

—

114

C₂×C₁₆

Dic₁₄₆

yes

complemented

115

C₄⋊C₈

—

146

—

116

C₄⋊C₈

—

146

—

117

#97

—

146

—

118

C₄×C₈

—

146

—

119

C₄×C₈

D₂₉₂

yes

—

120

C₄×C₈

C₂²⋊D₇₃

yes

—

121

C₂×Q₈

—

146

—

122

C₂²×C₄

#11924011

yes

—

123

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

124

M₁₆

—

146

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₂×C₈

#11924005

yes

—

128

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

129

C₂×C₈

C₂×D₂₉₂

yes

characteristic

130

C₂×C₈

C₂×Dic₁₄₆

yes

—

131

C₂×C₈

#11924028

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

—

146

—

135

C₂²⋊C₄

—

146

—

136

C₄²

—

146

—

137

C₄²

#11924005

yes

—

138

C₄²

C₈×D₇₃

yes

characteristic

139

C₄²

C₈×D₇₃

yes

characteristic

140

C₁₆

—

141

C₁₆

—

292

—

142

C₂³

C₂×#11924011

yes

characteristic

143

Q₈

—

292

—

144

C₂×C₄

—

145

C₂×C₄

—

146

—

146

C₂×C₄

#14556503

yes

—

147

C₂×C₄

#14556503

yes

—

148

C₂×C₄

C₂×#11924005

yes

characteristic

149

C₂×C₄

C₄×Dic₁₄₆

yes

—

150

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

151

C₂×C₄

#14556597

yes

—

152

C₈

—

153

C₈

—

292

—

154

C₈

#14556570

yes

—

155

C₈

#14556579

yes

characteristic

156

C₈

#14556591

yes

characteristic

157

C₂²

—

158

C₂²

C₂×#14556503

yes

characteristic

159

C₂²

#17988717

yes

—

160

C₄

—

161

C₄

—

292

—

162

C₄

#17988370

yes

—

163

C₄

#17988493

yes

characteristic

164

C₄

#17988553

yes

characteristic

165

C₄

C₈×Dic₁₄₆

yes

—

166

C₂

—

167

C₂

#22610867

yes

characteristic

168

C₂

C₂×#17988370

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#13266
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×C₄×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₇₃

C₂×M₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₃

hypocenter

C₇₃

#13266

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
33. C₂×C₁₆×C₇₃	Q₈	semidirect
89. C₇₃	#13266	semidirect
114. C₂×C₁₆	Dic₁₄₆	semidirect

H ≅ G/N

type

33. C₂×C₁₆×C₇₃

Q₈

semidirect

89. C₇₃

#13266

semidirect

114. C₂×C₁₆

Dic₁₄₆

semidirect

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

subgroup	index
#22618865	2
#22611687	2
#22608019	2
C₂×C₄×C₁₆×C₇₃	2
#13266	73

subgroup

index

C₂×C₄×C₁₆×C₇₃

Subgroups — #33004515

Conjugacy classes of subgroups

Complements

Maximal subgroups