Group #33001402 — subgroups

Subgroups — #33001402

Order 18688.

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

normal subgroups only

This group has 306 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	C₂×#17988417	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988362	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₇₃	C₂	1	2	yes	maximal
4	#22608024	C₂	1	2	yes	maximal
5	C₂×#368×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	1	yes	characteristic
7	C₂×#14556527	C₂²	1	2	yes	—
8	M₁₆×Dic₇₃	—	2	2	no	—
9	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988417	C₄	1	4	yes	complemented
11	#17988362	C₄	1	4	yes	—
12	#17988351	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₃	C₂²	1	2	yes	—
15	#368×C₇₃	C₄	1	4	yes	complemented
16	C₂×#11924012	—	2	2	no	—
17	C₂²×#10596831	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
19	C₂×#11924003	—	2	1	no	—
20	C₂×#11924003	—	2	1	no	—
21	C₂×#11924003	C₂×C₄	1	2	yes	—
22	#14556527	—	2	2	no	—
23	#14556527	C₂×C₄	1	4	yes	—
24	#14556501	—	4	2	no	—
25	C₈×Dic₇₃	—	4	2	no	—
26	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
27	C₈×Dic₇₃	Q₈	1	2	yes	—
28	C₂×M₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
33	#108×C₇₃	—	2	2	no	—
34	#108×C₇₃	C₂×C₄	1	4	yes	—
35	C₂²×Dic₇₃	Q₁₆	1	2	yes	—
36	#11924012	—	2	4	no	—
37	#11924012	—	4	2	no	—
38	C₂×#10596831	—	2	1	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
43	C₄×Dic₇₃	D₈	1	2	yes	—
44	#11924003	—	2	2	no	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₇₃	—	2	4	no	—
47	M₁₆×C₇₃	—	4	2	no	—
48	C₂×C₈×C₇₃	—	4	2	no	—
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
55	C₁₆×C₇₃	C₄²	1	4	yes	—
56	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
63	C₂×Dic₇₃	—	4	2	no	—
64	C₂×Dic₇₃	#107	1	4	yes	—
65	C₈×C₇₃	—	4	4	no	—
66	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₇₃	#117	1	1	yes	characteristic
68	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
69	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
70	#10596831	—	2	2	no	—
71	#10596831	—	4	4	no	—
72	C₂²×C₇₃	—	2	2	no	—
73	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
74	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
75	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
76	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₇₃	#427	1	1	yes	characteristic
78	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
79	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
80	Dic₇₃	—	2	4	no	—
81	#12676	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	—	2	1	no	—
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
85	C₂×C₇₃	#2161	1	2	yes	—
86	C₂×#368	—	73	1	no	—
87	C₂×#368	D₇₃	1	1	yes	characteristic
88	C₂×#346	—	73	1	no	—
89	C₄×#108	—	73	2	no	—
90	#1381	—	73	2	no	—
91	C₇₃	#12676	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₁₆	—	73	1	no	—
93	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
94	C₂×#108	—	73	2	no	—
95	C₂×#108	C₂×D₇₃	1	2	yes	—
96	C₄×M₁₆	—	146	2	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	#368	—	73	4	no	—
99	#368	Dic₇₃	1	4	yes	complemented
100	#346	—	73	4	no	—
101	#341	—	146	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	146	2	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
106	C₂×C₄²	—	73	1	no	—
107	C₂×C₁₆	—	73	2	no	—
108	C₂×C₁₆	—	146	1	no	—
109	C₂×C₁₆	—	146	1	no	—
110	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
111	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₇₃	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	73	4	no	—
115	#108	—	146	2	no	—
116	#108	C₂×Dic₇₃	1	4	yes	—
117	#97	—	292	2	no	—
118	C₄×C₈	—	292	2	no	—
119	C₄×C₈	—	73	2	no	—
120	C₄×C₈	—	73	2	no	—
121	C₂²×C₄	—	73	2	no	—
122	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	292	2	no	—
126	M₁₆	—	146	4	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	292	2	no	—
129	C₂×C₈	—	73	2	no	—
130	C₂×C₈	—	146	1	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
134	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
136	C₂×C₈	#11924027	1	1	yes	characteristic
137	C₂×C₈	#11924029	1	1	yes	characteristic
138	C₄²	—	73	2	no	—
139	C₄²	—	73	2	no	—
140	C₁₆	—	146	2	no	—
141	C₁₆	C₄×Dic₇₃	1	4	yes	—
142	C₂³	Q₁₆×D₇₃	1	1	yes	characteristic
143	C₂×C₄	—	292	2	no	—
144	C₂×C₄	—	73	4	no	—
145	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
146	C₂×C₄	#14556553	1	1	yes	characteristic
147	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
148	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
149	C₂×C₄	#14556610	1	1	yes	characteristic
150	C₂×C₄	#14556614	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₈	—	292	4	no	—
153	C₈	—	146	2	no	—
154	C₈	C₂×C₄×Dic₇₃	1	1	yes	characteristic
155	C₈	#14556549	1	1	yes	characteristic
156	C₈	#14556556	1	1	yes	characteristic
157	C₈	Q₈×Dic₇₃	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	#107×D₇₃	1	1	yes	characteristic
160	C₂²	#17988681	1	1	yes	characteristic
161	C₂²	Q₁₆×Dic₇₃	1	1	yes	characteristic
162	C₄	—	146	4	no	—
163	C₄	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
164	C₄	#17988624	1	1	yes	characteristic
165	C₄	#17988669	1	1	yes	characteristic
166	C₄	D₈×Dic₇₃	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#107×Dic₇₃	1	1	yes	characteristic
170	C₂	#22611724	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988417

C₂

yes

maximal, characteristic

C₂×#17988362

C₂

yes

maximal, characteristic

#108×Dic₇₃

C₂

yes

maximal

#22608024

C₂

yes

maximal

C₂×#368×C₇₃

C₂

yes

maximal, characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×#14556527

C₂²

yes

—

M₁₆×Dic₇₃

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988417

C₄

yes

complemented

#17988362

C₄

yes

—

#17988351

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

#368×C₇₃

C₄

yes

complemented

C₂×#11924012

—

C₂²×#10596831

D₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

—

C₂×C₄

yes

—

#14556527

—

#14556527

C₂×C₄

yes

—

#14556501

—

C₈×Dic₇₃

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

#108×C₇₃

—

#108×C₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

Q₁₆

yes

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

D₈

yes

—

#11924003

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

C₂×C₈×C₇₃

yes

C₂×C₈×C₇₃

yes

C₁₆×C₇₃

C₄²

yes

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

#107

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₄²

yes

characteristic

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

C₈×C₇₃

C₄×Q₈

yes

characteristic

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

C₄×Q₁₆

yes

characteristic

C₂²×C₇₃

C₄×Q₁₆

yes

characteristic

C₄×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₃

#427

yes

characteristic

C₄×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

C₄×D₈

yes

characteristic

Dic₇₃

—

#12676

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₄×#107

yes

characteristic

C₂×C₇₃

#2161

yes

—

C₂×#368

—

C₂×#368

D₇₃

yes

—

—

—

C₇₃

#12676

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₇₃

yes

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

—

#368

—

#368

Dic₇₃

yes

complemented

100

#346

—

101

#341

—

146

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

146

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²×D₇₃

yes

characteristic

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

146

—

109

C₂×C₁₆

—

146

—

110

C₂×C₁₆

C₄×D₇₃

yes

—

111

C₂×C₁₆

C₄×D₇₃

yes

—

112

C₂×C₁₆

C₂×Dic₇₃

yes

—

113

#108

—

114

#108

—

115

#108

—

146

—

116

#108

C₂×Dic₇₃

yes

—

117

#97

—

292

—

118

C₄×C₈

—

292

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₂²×C₄

—

122

C₂²×C₄

D₄×D₇₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

292

—

126

M₁₆

—

146

—

127

C₂×C₈

—

128

C₂×C₈

—

292

—

129

C₂×C₈

—

130

C₂×C₈

—

146

—

131

C₂×C₈

—

146

—

132

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

133

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

134

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

135

C₂×C₈

Q₈×D₇₃

yes

characteristic

136

C₂×C₈

#11924027

yes

characteristic

137

C₂×C₈

#11924029

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₁₆

—

146

—

141

C₁₆

C₄×Dic₇₃

yes

—

142

C₂³

Q₁₆×D₇₃

yes

characteristic

143

C₂×C₄

—

292

—

144

C₂×C₄

—

145

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

146

C₂×C₄

#14556553

yes

characteristic

147

C₂×C₄

D₄×Dic₇₃

yes

characteristic

148

C₂×C₄

D₈×D₇₃

yes

characteristic

149

C₂×C₄

#14556610

yes

characteristic

150

C₂×C₄

#14556614

yes

characteristic

151

C₈

—

152

C₈

—

292

—

153

C₈

—

146

—

154

C₈

C₂×C₄×Dic₇₃

yes

characteristic

155

C₈

#14556549

yes

characteristic

156

C₈

#14556556

yes

characteristic

157

C₈

Q₈×Dic₇₃

yes

characteristic

158

C₂²

—

159

C₂²

#107×D₇₃

yes

characteristic

160

C₂²

#17988681

yes

characteristic

161

C₂²

Q₁₆×Dic₇₃

yes

characteristic

162

C₄

—

146

—

163

C₄

C₄⋊C₄×Dic₇₃

yes

characteristic

164

C₄

#17988624

yes

characteristic

165

C₄

#17988669

yes

characteristic

166

C₄

D₈×Dic₇₃

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#107×Dic₇₃

yes

characteristic

170

C₂

#22611724

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#368×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×C₇₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#368	D₇₃
hypocenter	C₇₃	#12676
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₇₃

commutator

C₈×C₇₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#368×C₇₃

C₂

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂³×C₇₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #17988417	C₄	semidirect
15. #368×C₇₃	C₄	semidirect
91. C₇₃	#12676	semidirect
99. #368	Dic₇₃	semidirect

H ≅ G/N

type

10. #17988417

C₄

semidirect

15. #368×C₇₃

C₄

semidirect

91. C₇₃

#12676

semidirect

99. #368

Dic₇₃

semidirect

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

subgroup	index
C₂×#17988417	2
C₂×#17988362	2
#108×Dic₇₃	2
#22608024	2
C₂×#368×C₇₃	2
#12676	73

subgroup

index

#108×Dic₇₃

C₂×#368×C₇₃

Subgroups — #33001402

Conjugacy classes of subgroups

Complements

Maximal subgroups