Group #33002177 — subgroups

Subgroups — #33002177

Order 18688.

354 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 143 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	2	yes	maximal, complemented
2	#22609981	C₂	1	2	yes	maximal
3	#22608035	C₂	1	2	yes	maximal, complemented
4	C₂×#504×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556527	C₂²	1	4	yes	—
6	#17988571	—	2	2	no	—
7	#17988417	—	2	4	no	—
8	#17988345	—	2	2	no	—
9	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
10	#504×C₇₃	—	2	4	no	—
11	C₂×M₃₂×C₇₃	C₄	1	2	yes	—
12	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₃	C₄	1	2	yes	—
14	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924012	—	2	4	no	—
16	#14556527	—	2	4	no	—
17	#14556527	—	2	4	no	—
18	#14556510	—	4	2	no	—
19	#14556508	—	4	2	no	—
20	C₂×#41×C₇₃	D₄	1	1	yes	characteristic
21	#131×C₇₃	—	2	2	no	—
22	#131×C₇₃	—	2	2	no	—
23	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
27	M₃₂×C₇₃	—	2	2	no	—
28	M₃₂×C₇₃	—	2	4	no	—
29	C₂×C₁₆×C₇₃	—	2	4	no	—
30	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
34	#11924012	—	4	4	no	—
35	#11924012	—	4	4	no	—
36	C₂×#10596831	—	4	4	no	—
37	#41×C₇₃	—	2	2	no	—
38	#41×C₇₃	—	2	2	no	—
39	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
40	C₂×D₄×C₇₃	D₈	1	1	yes	characteristic
41	C₂×D₄×C₇₃	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₇₃	—	2	2	no	—
45	M₁₆×C₇₃	—	2	4	no	—
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	2	yes	—
53	C₁₆×C₇₃	—	2	2	no	—
54	C₁₆×C₇₃	C₂²⋊C₄	1	2	yes	—
55	C₁₆×C₇₃	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₇₃	#107	1	2	yes	—
57	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
58	Q₈×C₇₃	—	2	2	no	—
59	D₄×C₇₃	—	2	4	no	—
60	D₄×C₇₃	—	2	2	no	—
61	C₂×C₄×C₇₃	—	2	2	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	#107	1	2	yes	—
64	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	#121	1	2	yes	—
66	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#135	1	2	yes	—
68	C₈×C₇₃	—	2	2	no	—
69	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₇₃	#117	1	1	yes	characteristic
71	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
72	#10596831	—	8	4	no	—
73	C₂²×C₇₃	—	2	2	no	—
74	C₂²×C₇₃	—	4	1	no	—
75	C₂²×C₇₃	—	4	2	no	—
76	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
77	C₂²×C₇₃	#467	1	2	yes	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	#380	1	1	yes	characteristic
80	C₄×C₇₃	#427	1	1	yes	characteristic
81	C₄×C₇₃	#466	1	2	yes	—
82	#12840	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	—	2	2	no	—
85	C₂×C₇₃	—	4	2	no	—
86	C₂×C₇₃	#1808	1	1	yes	characteristic
87	C₂×#504	D₇₃	1	1	yes	characteristic
88	C₂×#368	—	73	2	no	—
89	#1813	—	73	2	no	—
90	#1385	—	73	2	no	—
91	C₇₃	#12840	1	1	yes	characteristic, complemented, Sylow
92	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
93	#504	—	2	4	no	—
94	C₂×M₃₂	Dic₇₃	1	2	yes	—
95	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
96	C₂²×C₁₆	Dic₇₃	1	2	yes	—
97	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
98	C₂×#108	—	73	4	no	—
99	#413	—	146	2	no	—
100	#368	—	146	4	no	—
101	#338	—	146	2	no	—
102	C₂×#41	D₂₉₂	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	—	146	4	no	—
106	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
107	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
109	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	—	2	4	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
115	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
116	C₂×C₁₆	C₂²⋊D₇₃	1	4	yes	—
117	#108	—	146	4	no	—
118	#108	—	146	4	no	—
119	#101	—	292	2	no	—
120	#100	—	292	2	no	—
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
124	C₂×D₄	D₅₈₄	1	1	yes	characteristic
125	C₂×D₄	Dic₂₉₂	1	2	yes	—
126	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
127	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	292	4	no	—
131	M₁₆	—	292	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	292	4	no	—
135	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
136	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
139	C₂×C₈	#11924028	1	2	yes	—
140	C₁₆	—	2	2	no	—
141	C₁₆	C₂²⋊Dic₇₃	1	2	yes	—
142	C₁₆	C₂²⋊Dic₇₃	1	2	yes	—
143	C₂³	#14556525	1	2	yes	—
144	C₂³	C₂×Dic₂₉₂	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	#14556525	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
152	C₂×C₄	#14556565	1	2	yes	—
153	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
154	C₂×C₄	#14556613	1	2	yes	—
155	C₈	—	2	2	no	—
156	C₈	—	584	4	no	—
157	C₈	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
158	C₈	#14556550	1	1	yes	characteristic
159	C₈	#14556570	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×#14556525	1	1	yes	characteristic
164	C₂²	#17988828	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	#17988447	1	1	yes	characteristic
167	C₄	#17988625	1	1	yes	characteristic
168	C₄	#17988823	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	#22609960	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988417

C₂

yes

maximal, complemented

#22609981

C₂

yes

maximal

#22608035

C₂

yes

maximal, complemented

C₂×#504×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556527

C₂²

yes

—

#17988571

—

#17988417

—

#17988345

—

C₂×#131×C₇₃

C₂²

yes

characteristic

#504×C₇₃

—

C₂×M₃₂×C₇₃

C₄

yes

—

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₄

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

—

—

—

—

—

C₂×#41×C₇₃

D₄

yes

characteristic

#131×C₇₃

—

#131×C₇₃

—

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂×M₁₆×C₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

D₈

yes

characteristic

C₂×D₄×C₇₃

D₈

yes

characteristic

C₂×D₄×C₇₃

Q₁₆

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂²⋊C₄

yes

—

C₁₆×C₇₃

C₂²⋊C₄

yes

—

C₂³×C₇₃

#107

yes

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

#467

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12840

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1808

yes

characteristic

C₂×#504

D₇₃

yes

—

—

—

C₇₃

#12840

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₃

yes

characteristic

#504

—

C₂×M₃₂

Dic₇₃

yes

—

C₂×M₃₂

C₂×D₇₃

yes

characteristic

C₂²×C₁₆

Dic₇₃

yes

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×#108

—

#413

—

146

—

100

#368

—

146

—

101

#338

—

146

—

102

C₂×#41

D₂₉₂

yes

characteristic

103

#131

—

104

#131

—

105

C₂×M₁₆

—

146

—

106

C₂×M₁₆

Dic₁₄₆

yes

—

107

C₂×M₁₆

D₂₉₂

yes

characteristic

108

C₂²×C₈

C₂×Dic₇₃

yes

—

109

C₂²×C₈

C₂²×D₇₃

yes

characteristic

110

M₃₂

—

111

M₃₂

—

112

C₂×C₁₆

—

113

C₂×C₁₆

C₂×Dic₇₃

yes

—

114

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

115

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

116

C₂×C₁₆

C₂²⋊D₇₃

yes

—

117

#108

—

146

—

118

#108

—

146

—

119

#101

—

292

—

120

#100

—

292

—

121

#41

—

122

#41

—

123

C₂×Q₈

D₅₈₄

yes

characteristic

124

C₂×D₄

D₅₈₄

yes

characteristic

125

C₂×D₄

Dic₂₉₂

yes

—

126

C₂²×C₄

C₄⋊Dic₇₃

yes

—

127

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

292

—

131

M₁₆

—

292

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

292

—

135

C₂×C₈

C₄⋊Dic₇₃

yes

—

136

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

137

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

138

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

139

C₂×C₈

#11924028

yes

—

140

C₁₆

—

141

C₁₆

C₂²⋊Dic₇₃

yes

—

142

C₁₆

C₂²⋊Dic₇₃

yes

—

143

C₂³

#14556525

yes

—

144

C₂³

C₂×Dic₂₉₂

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#14556525

yes

—

151

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

152

C₂×C₄

#14556565

yes

—

153

C₂×C₄

C₂×D₅₈₄

yes

characteristic

154

C₂×C₄

#14556613

yes

—

155

C₈

—

156

C₈

—

584

—

157

C₈

C₂×C₂²⋊Dic₇₃

yes

characteristic

158

C₈

#14556550

yes

characteristic

159

C₈

#14556570

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×#14556525

yes

characteristic

164

C₂²

#17988828

yes

—

165

C₄

—

166

C₄

#17988447

yes

characteristic

167

C₄

#17988625

yes

characteristic

168

C₄

#17988823

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

#22609960

yes

characteristic

	subgroup	G/H
center	C₄	#17988625
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#504×C₇₃	C₂
socle	C₂×C₇₃	#1808
prime radical	C₂×#41×C₇₃	D₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#504	D₇₃
hypocenter	C₇₃	#12840
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#17988625

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#504×C₇₃

C₂

socle

C₂×C₇₃

#1808

prime radical

C₂×#41×C₇₃

D₄

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. C₂×#17988417	C₂	semidirect
3. #22608035	C₂	semidirect
91. C₇₃	#12840	semidirect

H ≅ G/N

type

1. C₂×#17988417

C₂

semidirect

3. #22608035

C₂

semidirect

91. C₇₃

#12840

semidirect

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

subgroup	index
C₂×#17988417	2
#22609981	2
#22608035	2
C₂×#504×C₇₃	2
#12840	73

subgroup

index

C₂×#504×C₇₃

Subgroups — #33002177

Conjugacy classes of subgroups

Complements

Maximal subgroups