Group #33002165 — subgroups

Subgroups — #33002165

Order 18688.

394 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22609969	C₂	1	2	yes	maximal
3	#22608025	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556527	C₂²	1	4	yes	—
6	#17988551	—	2	2	no	—
7	#17988417	—	2	4	no	—
8	#17988352	—	2	2	no	—
9	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₃	—	2	4	no	—
11	C₂²×C₁₆×C₇₃	C₄	1	4	yes	—
12	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#11924012	—	2	4	no	—
15	#14556527	—	2	4	no	—
16	#14556527	—	2	4	no	—
17	C₂²⋊#10596831	—	4	4	no	—
18	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	—	2	2	no	—
20	C₂²×C₈×C₇₃	—	2	2	no	—
21	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
24	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	—	2	8	no	—
27	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
30	#11924012	—	4	4	no	—
31	#11924012	—	4	4	no	—
32	C₂×#10596831	—	4	4	no	—
33	C₂⁴×C₇₃	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	—	2	2	no	—
35	C₂²×C₄×C₇₃	—	2	2	no	—
36	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₇₃	D₈	1	2	yes	—
38	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	2	no	—
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	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	#41	1	2	yes	—
49	C₁₆×C₇₃	—	2	2	no	—
50	C₁₆×C₇₃	C₂²⋊C₄	1	4	yes	—
51	C₂³×C₇₃	—	2	4	no	—
52	C₂³×C₇₃	—	2	2	no	—
53	C₂³×C₇₃	#107	1	2	yes	—
54	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	—	2	4	no	—
59	C₂×C₄×C₇₃	#107	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#121	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#135	1	2	yes	—
64	C₈×C₇₃	—	2	2	no	—
65	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₇₃	#117	1	1	yes	characteristic
67	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
68	#10596831	—	8	4	no	—
69	C₂²×C₇₃	—	2	8	no	—
70	C₂²×C₇₃	—	2	2	no	—
71	C₂²×C₇₃	—	2	2	no	—
72	C₂²×C₇₃	—	2	2	no	—
73	C₂²×C₇₃	#368	1	4	yes	complemented
74	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
75	C₂²×C₇₃	#467	1	2	yes	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	#380	1	1	yes	characteristic
78	C₄×C₇₃	#427	1	1	yes	characteristic
79	C₄×C₇₃	#466	1	2	yes	—
80	#12837	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	2	4	no	—
82	C₂×C₇₃	—	2	2	no	—
83	C₂×C₇₃	#1808	1	1	yes	characteristic
84	C₂×C₇₃	C₂×#368	1	2	yes	—
85	C₂³×C₁₆	D₇₃	1	1	yes	characteristic
86	C₂×#368	—	73	2	no	—
87	#1810	—	73	2	no	—
88	#1381	—	73	2	no	—
89	C₇₃	#12837	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
91	C₂²×C₁₆	—	2	4	no	—
92	C₂²×C₁₆	Dic₇₃	1	4	yes	—
93	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
94	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
95	C₂×#108	—	73	4	no	—
96	#407	—	146	2	no	—
97	#368	—	146	4	no	—
98	#341	—	146	2	no	—
99	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
100	C₂×M₁₆	—	146	4	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	Dic₁₄₆	1	2	yes	—
104	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
106	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	8	no	—
109	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
110	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
111	C₂×C₁₆	C₂²⋊D₇₃	1	4	yes	—
112	#108	—	146	4	no	—
113	#108	—	146	4	no	—
114	C₂²⋊C₈	—	292	4	no	—
115	C₂⁴	Dic₂₉₂	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
119	C₂²×C₄	D₅₈₄	1	2	yes	—
120	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
121	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
122	M₁₆	—	292	4	no	—
123	M₁₆	—	292	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	292	4	no	—
129	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
130	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
132	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
133	C₂×C₈	#11924028	1	2	yes	—
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊Dic₇₃	1	4	yes	—
136	C₂³	—	2	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	#14556525	1	2	yes	—
139	C₂³	C₂×Dic₂₉₂	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	#14556525	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
146	C₂×C₄	#14556565	1	2	yes	—
147	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
148	C₂×C₄	#14556613	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	584	4	no	—
151	C₈	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
152	C₈	#14556550	1	1	yes	characteristic
153	C₈	#14556570	1	2	yes	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	#17988417	1	4	yes	complemented
159	C₂²	C₂×#14556525	1	1	yes	characteristic
160	C₂²	#17988828	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	#17988447	1	1	yes	characteristic
163	C₄	#17988625	1	1	yes	characteristic
164	C₄	#17988823	1	2	yes	—
165	C₂	—	2	4	no	—
166	C₂	—	2	2	no	—
167	C₂	#22609960	1	1	yes	characteristic
168	C₂	C₂×#17988417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988417

C₂

yes

maximal, complemented

#22609969

C₂

yes

maximal

#22608025

C₂

yes

maximal, complemented

C₂³×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556527

C₂²

yes

—

#17988551

—

#17988417

—

#17988352

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

—

C₂²×C₁₆×C₇₃

C₄

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₁₆×C₇₃

C₂²

yes

—

—

—

—

C₂³×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₂⁴×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×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

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#107

yes

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×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

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12837

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1808

yes

characteristic

C₂×C₇₃

C₂×#368

yes

—

C₂³×C₁₆

D₇₃

yes

—

—

—

C₇₃

#12837

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₃

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

Dic₇₃

yes

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×#108

—

#407

—

146

—

#368

—

146

—

#341

—

146

—

C₂³×C₄

D₂₉₂

yes

characteristic

100

C₂×M₁₆

—

146

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₁₄₆

yes

—

104

C₂²×C₈

D₂₉₂

yes

characteristic

105

C₂²×C₈

C₂×Dic₇₃

yes

—

106

C₂²×C₈

C₂²×D₇₃

yes

characteristic

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₂×Dic₇₃

yes

—

110

C₂×C₁₆

C₂×Dic₇₃

yes

—

111

C₂×C₁₆

C₂²⋊D₇₃

yes

—

112

#108

—

146

—

113

#108

—

146

—

114

C₂²⋊C₈

—

292

—

115

C₂⁴

Dic₂₉₂

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₄⋊Dic₇₃

yes

—

119

C₂²×C₄

D₅₈₄

yes

—

120

C₂²×C₄

Dic₂₉₂

yes

characteristic

121

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

122

M₁₆

—

292

—

123

M₁₆

—

292

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

292

—

129

C₂×C₈

C₄⋊Dic₇₃

yes

—

130

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

132

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

133

C₂×C₈

#11924028

yes

—

134

C₁₆

—

135

C₁₆

C₂²⋊Dic₇₃

yes

—

136

C₂³

—

137

C₂³

—

138

C₂³

#14556525

yes

—

139

C₂³

C₂×Dic₂₉₂

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#14556525

yes

—

145

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

146

C₂×C₄

#14556565

yes

—

147

C₂×C₄

C₂×D₅₈₄

yes

characteristic

148

C₂×C₄

#14556613

yes

—

149

C₈

—

150

C₈

—

584

—

151

C₈

C₂×C₂²⋊Dic₇₃

yes

characteristic

152

C₈

#14556550

yes

characteristic

153

C₈

#14556570

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#17988417

yes

complemented

159

C₂²

C₂×#14556525

yes

characteristic

160

C₂²

#17988828

yes

—

161

C₄

—

162

C₄

#17988447

yes

characteristic

163

C₄

#17988625

yes

characteristic

164

C₄

#17988823

yes

—

165

C₂

—

166

C₂

—

167

C₂

#22609960

yes

characteristic

168

C₂

C₂×#17988417

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂³×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂⁴×C₇₃	Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂³×C₁₆	D₇₃
hypocenter	C₇₃	#12837
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂³×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂⁴×C₇₃

Q₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

hypercenter

C₂³×C₁₆

D₇₃

hypocenter

C₇₃

#12837

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

N	H ≅ G/N	type
1. C₂×#17988417	C₂	semidirect
3. #22608025	C₂	semidirect
73. C₂²×C₇₃	#368	semidirect
89. C₇₃	#12837	semidirect
158. C₂²	#17988417	semidirect

H ≅ G/N

type

1. C₂×#17988417

C₂

semidirect

3. #22608025

C₂

semidirect

73. C₂²×C₇₃

#368

semidirect

89. C₇₃

#12837

semidirect

158. C₂²

#17988417

semidirect

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

subgroup	index
C₂×#17988417	2
#22609969	2
#22608025	2
C₂³×C₁₆×C₇₃	2
#12837	73

subgroup

index

C₂³×C₁₆×C₇₃

Subgroups — #33002165

Conjugacy classes of subgroups

Complements

Maximal subgroups