Group #33004034 — subgroups

Subgroups — #33004034

Order 18688.

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

normal subgroups only

This group has 286 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×#17988417	C₂	1	1	yes	maximal, characteristic
2	#22611715	C₂	1	1	yes	maximal, characteristic
3	#22610679	C₂	1	2	yes	maximal
4	#22608027	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	C₂²	1	2	yes	—
7	#17988625	C₂²	1	2	yes	—
8	#17988613	—	2	2	no	—
9	#17988417	—	2	2	no	—
10	#17988415	C₄	1	2	yes	complemented
11	#17988413	C₄	1	2	yes	complemented
12	#17988347	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₇₃	—	2	2	no	—
17	C₂×#11924012	—	2	2	no	—
18	#14556550	—	2	2	no	—
19	#14556527	—	2	2	no	—
20	#14556527	—	2	2	no	—
21	#14556525	C₂×C₄	1	4	yes	—
22	#14556523	—	2	2	no	—
23	#14556520	—	4	4	no	—
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	—	2	1	no	—
28	C₂×C₁₆×C₇₃	—	2	1	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
33	C₂×C₁₆×C₇₃	Q₈	1	2	yes	—
34	C₄×C₈×C₇₃	—	2	1	no	—
35	C₄×C₈×C₇₃	—	2	1	no	—
36	#11924012	—	4	2	no	—
37	#11924012	—	4	2	no	—
38	C₂×#10596831	—	4	2	no	—
39	C₄⋊Dic₇₃	—	2	4	no	—
40	C₂²⋊Dic₇₃	—	4	2	no	—
41	C₄×Dic₇₃	—	4	2	no	—
42	C₂²×C₄×C₇₃	—	2	1	no	—
43	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	D₈	1	2	yes	—
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
53	C₄²×C₇₃	—	2	1	no	—
54	C₄²×C₇₃	Q₁₆	1	2	yes	—
55	C₁₆×C₇₃	—	2	2	no	—
56	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
58	C₂³×C₇₃	#135	1	1	yes	characteristic
59	C₂×C₄×C₇₃	—	2	4	no	—
60	C₂×C₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
67	C₂×Dic₇₃	—	4	4	no	—
68	C₈×C₇₃	—	2	2	no	—
69	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₈×C₇₃	#125	1	1	yes	characteristic
73	#10596831	—	8	2	no	—
74	C₂²×C₇₃	—	2	2	no	—
75	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
76	C₂²×C₇₃	#467	1	1	yes	characteristic
77	C₂²×C₇₃	#499	1	1	yes	characteristic
78	C₄×C₇₃	—	2	4	no	—
79	C₄×C₇₃	#389	1	1	yes	characteristic
80	C₄×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₇₃	#466	1	1	yes	characteristic
82	C₄×C₇₃	#500	1	1	yes	characteristic
83	Dic₇₃	—	8	4	no	—
84	#13169	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	#1942	1	1	yes	characteristic
88	C₂×C₇₃	#2178	1	2	yes	—
89	C₂×#368	—	73	1	no	—
90	#2158	—	73	1	no	—
91	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
92	#1945	—	73	2	no	—
93	#1382	—	73	2	no	—
94	C₇₃	#13169	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
96	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
97	C₂×#108	—	73	2	no	—
98	#427	—	73	2	no	—
99	#424	—	146	2	no	—
100	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
101	#368	—	146	2	no	—
102	#367	—	73	2	no	—
103	#366	—	73	2	no	—
104	C₄×C₁₆	—	2	2	no	—
105	#339	—	146	2	no	—
106	C₂×M₁₆	—	146	2	no	—
107	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
109	#117	—	146	2	no	—
110	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
114	C₂×C₁₆	Dic₁₄₆	1	2	yes	—
115	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
117	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
118	#108	—	146	2	no	—
119	#108	—	146	2	no	—
120	#107	—	73	4	no	—
121	#106	—	146	2	no	—
122	C₄⋊C₈	—	292	4	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
127	M₁₆	—	292	2	no	—
128	M₁₆	—	292	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	292	2	no	—
132	C₂×C₈	D₅₈₄	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
137	C₂×C₈	#11924028	1	1	yes	characteristic
138	C₂×C₈	#11924028	1	1	yes	characteristic
139	C₄⋊C₄	—	146	4	no	—
140	C₂²⋊C₄	—	292	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	292	2	no	—
143	C₄²	Dic₂₉₂	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	#11924007	1	2	yes	—
146	C₁₆	#11924007	1	2	yes	—
147	C₂³	#14556613	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	292	4	no	—
151	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
152	C₂×C₄	#14556565	1	1	yes	characteristic
153	C₂×C₄	#14556591	1	1	yes	characteristic
154	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
155	C₂×C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
156	C₂×C₄	#14556613	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	584	2	no	—
159	C₈	C₂×#11924007	1	1	yes	characteristic
160	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
161	C₈	#14556570	1	1	yes	characteristic
162	C₈	#14556583	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
165	C₂²	#17988828	1	1	yes	characteristic
166	C₂²	#17988989	1	1	yes	characteristic
167	C₄	—	2	4	no	—
168	C₄	—	584	4	no	—
169	C₄	#17988485	1	1	yes	characteristic
170	C₄	C₄×Dic₂₉₂	1	1	yes	characteristic
171	C₄	#17988823	1	1	yes	characteristic
172	C₄	#17988992	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#22610664	1	1	yes	characteristic
176	C₂	#22611813	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988417

C₂

yes

maximal, characteristic

#22611715

C₂

yes

maximal, characteristic

#22610679

C₂

yes

maximal

#22608027

C₂

yes

maximal

C₂×C₄×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556527

C₂²

yes

—

#17988625

C₂²

yes

—

#17988613

—

#17988417

—

#17988415

C₄

yes

complemented

#17988413

C₄

yes

complemented

#17988347

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

—

—

—

—

—

C₂×C₄

yes

—

#14556523

—

#14556520

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

Q₈

yes

—

C₂×C₁₆×C₇₃

Q₈

yes

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

C₄²×C₇₃

—

C₄²×C₇₃

Q₁₆

yes

—

C₁₆×C₇₃

—

C₁₆×C₇₃

C₄⋊C₄

yes

—

C₁₆×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

#135

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

#389

yes

characteristic

C₄×C₇₃

C₄×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#13169

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1942

yes

characteristic

C₂×C₇₃

#2178

yes

—

C₂×#368

—

#2158

—

C₂×C₄×C₁₆

D₇₃

yes

characteristic

#1945

—

#1382

—

C₇₃

#13169

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₁₆

C₂×D₇₃

yes

—

—

—

146

—

100

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

101

#368

—

146

—

102

#367

—

103

#366

—

104

C₄×C₁₆

—

105

#339

—

146

—

106

C₂×M₁₆

—

146

—

107

C₂²×C₈

D₂₉₂

yes

characteristic

108

C₂²×C₈

C₂²×D₇₃

yes

characteristic

109

#117

—

146

—

110

C₂×C₄²

D₂₉₂

yes

characteristic

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

Dic₁₄₆

yes

—

114

C₂×C₁₆

Dic₁₄₆

yes

—

115

C₂×C₁₆

C₄×D₇₃

yes

characteristic

116

C₂×C₁₆

C₄×D₇₃

yes

characteristic

117

C₂×C₁₆

C₂²⋊D₇₃

yes

—

118

#108

—

146

—

119

#108

—

146

—

120

#107

—

121

#106

—

146

—

122

C₄⋊C₈

—

292

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

127

M₁₆

—

292

—

128

M₁₆

—

292

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

292

—

132

C₂×C₈

D₅₈₄

yes

—

133

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

134

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

135

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

136

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

137

C₂×C₈

#11924028

yes

characteristic

138

C₂×C₈

#11924028

yes

characteristic

139

C₄⋊C₄

—

146

—

140

C₂²⋊C₄

—

292

—

141

C₄²

—

142

C₄²

—

292

—

143

C₄²

Dic₂₉₂

yes

—

144

C₁₆

—

145

C₁₆

#11924007

yes

—

146

C₁₆

#11924007

yes

—

147

C₂³

#14556613

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

292

—

151

C₂×C₄

C₄×D₂₉₂

yes

characteristic

152

C₂×C₄

#14556565

yes

characteristic

153

C₂×C₄

#14556591

yes

characteristic

154

C₂×C₄

C₂×D₅₈₄

yes

characteristic

155

C₂×C₄

C₂×Dic₂₉₂

yes

characteristic

156

C₂×C₄

#14556613

yes

characteristic

157

C₈

—

158

C₈

—

584

—

159

C₈

C₂×#11924007

yes

characteristic

160

C₈

C₄×Dic₁₄₆

yes

characteristic

161

C₈

#14556570

yes

characteristic

162

C₈

#14556583

yes

characteristic

163

C₂²

—

164

C₂²

C₄×D₅₈₄

yes

characteristic

165

C₂²

#17988828

yes

characteristic

166

C₂²

#17988989

yes

characteristic

167

C₄

—

168

C₄

—

584

—

169

C₄

#17988485

yes

characteristic

170

C₄

C₄×Dic₂₉₂

yes

characteristic

171

C₄

#17988823

yes

characteristic

172

C₄

#17988992

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#22610664

yes

characteristic

176

C₂

#22611813

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₈
prime radical	C₂³×C₇₃	#135
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	#13169
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×C₄×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₄×D₈

prime radical

C₂³×C₇₃

#135

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₇₃

hypocenter

C₇₃

#13169

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

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

N	H ≅ G/N	type
10. #17988415	C₄	semidirect
11. #17988413	C₄	semidirect
94. C₇₃	#13169	semidirect

H ≅ G/N

type

10. #17988415

C₄

semidirect

11. #17988413

C₄

semidirect

94. C₇₃

#13169

semidirect

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

subgroup	index
C₂×#17988417	2
#22611715	2
#22610679	2
#22608027	2
C₂×C₄×C₁₆×C₇₃	2
#13169	73

subgroup

index

C₂×C₄×C₁₆×C₇₃

Subgroups — #33004034

Conjugacy classes of subgroups

Complements

Maximal subgroups