Group #32997618 — subgroups

Subgroups — #32997618

Order 18688.

232 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22611524	C₂	1	1	yes	maximal, characteristic
2	#22608685	C₂	1	2	yes	maximal
3	#22607997	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607996	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556530	C₄	1	1	yes	characteristic
7	C₂×#14556530	C₄	1	1	yes	characteristic
8	#17988609	—	2	2	no	—
9	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#17988363	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
13	#17988338	C₂²	1	2	yes	—
14	#17988337	—	2	2	no	—
15	#346×C₇₃	C₂²	1	1	yes	characteristic
16	#334×C₇₃	C₂²	1	2	yes	—
17	C₂×#11924011	—	4	2	no	—
18	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
20	C₄²×D₇₃	D₄	1	1	yes	characteristic
21	#14556530	—	2	2	no	—
22	#14556530	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
25	C₄⋊#10596831	—	2	2	no	—
26	#14556520	—	4	2	no	—
27	C₄×#10596831	D₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₇₃	—	2	2	no	—
33	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
36	#11924011	—	8	4	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₈×D₇₃	—	2	1	no	—
41	C₈×D₇₃	—	2	1	no	—
42	C₈×D₇₃	—	2	2	no	—
43	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	SD₁₆	1	2	yes	—
45	#11924003	—	2	1	no	—
46	#11924003	M₁₆	1	2	yes	complemented
47	C₂×C₈×C₇₃	—	4	2	no	—
48	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₁₆×C₇₃	—	2	1	no	—
52	C₁₆×C₇₃	M₁₆	1	2	yes	complemented
53	C₂²×D₇₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
56	C₂×Dic₇₃	#106	1	1	yes	characteristic
57	C₂×Dic₇₃	#106	1	2	yes	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	4	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	8	2	no	—
63	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
65	#10596831	—	2	1	no	—
66	#10596831	—	8	2	no	—
67	#10596831	#108	1	2	yes	—
68	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	2	no	—
71	C₄×C₇₃	—	2	2	no	—
72	C₄×C₇₃	#423	1	1	yes	characteristic
73	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
74	Dic₇₃	—	2	1	no	—
75	Dic₇₃	#365	1	4	yes	—
76	#12020	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	#1566	1	1	yes	characteristic
78	C₂×C₇₃	C₂×#365	1	2	yes	—
79	D₇₃	—	4	1	no	—
80	C₄×M₃₂	—	73	1	no	—
81	#1566	—	73	2	no	—
82	#1368	—	73	1	no	—
83	#1368	—	73	2	no	—
84	#1368	D₇₃	1	1	yes	characteristic, complemented
85	C₇₃	#12020	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	73	1	no	—
87	C₂×M₃₂	—	73	1	no	—
88	#423	—	146	2	no	—
89	C₂×C₄×C₈	—	73	1	no	—
90	#346	—	73	1	no	—
91	#346	C₂×D₇₃	1	1	yes	characteristic
92	C₄×C₁₆	—	73	1	no	—
93	C₄×C₁₆	—	73	1	no	—
94	#334	—	73	2	no	—
95	#334	—	146	2	no	—
96	#334	C₂×D₇₃	1	2	yes	—
97	C₂×M₁₆	—	292	2	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂×C₄²	—	73	1	no	—
101	M₃₂	—	146	2	no	—
102	M₃₂	—	146	2	no	—
103	C₂×C₁₆	—	73	1	no	—
104	C₂×C₁₆	—	73	1	no	—
105	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
106	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	292	2	no	—
109	C₄⋊C₈	—	146	2	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	2	no	—
116	M₁₆	—	584	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	292	2	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
125	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
126	C₄²	—	73	1	no	—
127	C₄²	—	73	2	no	—
128	C₄²	D₄×D₇₃	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	73	2	no	—
131	C₁₆	—	146	1	no	—
132	C₁₆	#11924011	1	2	yes	complemented
133	C₂³	—	73	1	no	—
134	C₂×C₄	—	73	2	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	146	4	no	—
137	C₂×C₄	—	146	1	no	—
138	C₂×C₄	—	146	1	no	—
139	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
140	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	584	2	no	—
144	C₈	—	73	2	no	—
145	C₈	—	146	1	no	—
146	C₈	M₁₆×D₇₃	1	1	yes	characteristic
147	C₈	C₂×#11924011	1	1	yes	characteristic
148	C₂²	—	146	1	no	—
149	C₂²	—	146	2	no	—
150	C₂²	#106×D₇₃	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	73	4	no	—
153	C₄	—	146	1	no	—
154	C₄	#17988609	1	1	yes	characteristic
155	C₄	#108×D₇₃	1	1	yes	characteristic
156	C₂	—	292	1	no	—
157	C₂	#22608685	1	1	yes	characteristic
158	C₂	#365×D₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22611524

C₂

yes

maximal, characteristic

#22608685

C₂

yes

maximal

#22607997

C₂

yes

maximal, characteristic, complemented

#22607996

C₂

yes

maximal, complemented

C₇₃×#1368

C₂

yes

maximal, characteristic, complemented

C₂×#14556530

C₄

yes

characteristic

C₂×#14556530

C₄

yes

characteristic

#17988609

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₂²

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

characteristic

#17988338

C₂²

yes

—

#17988337

—

#346×C₇₃

C₂²

yes

characteristic

#334×C₇₃

C₂²

yes

—

C₂×#11924011

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

—

#11924011

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

—

#11924003

—

#11924003

M₁₆

yes

complemented

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

complemented

C₂²×D₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×Dic₇₃

#106

yes

characteristic

C₂×Dic₇₃

#106

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

—

—

yes

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

#365

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₄×M₃₂

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#12020

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

#423

—

146

—

C₂×C₄×C₈

—

#346

—

#346

C₂×D₇₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#334

—

#334

—

146

—

#334

C₂×D₇₃

yes

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

M₃₂

—

146

—

102

M₃₂

—

146

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

C₄×D₇₃

yes

characteristic

106

C₂×C₁₆

C₄×D₇₃

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

292

—

109

C₄⋊C₈

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂²×D₇₃

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

M₁₆

—

584

—

117

C₂×C₈

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

125

C₂×C₈

Q₈×D₇₃

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

D₄×D₇₃

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

146

—

132

C₁₆

#11924011

yes

complemented

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

140

C₂×C₄

SD₁₆×D₇₃

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

584

—

144

C₈

—

145

C₈

—

146

—

146

C₈

M₁₆×D₇₃

yes

characteristic

147

C₈

C₂×#11924011

yes

characteristic

148

C₂²

—

146

—

149

C₂²

—

146

—

150

C₂²

#106×D₇₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

146

—

154

C₄

#17988609

yes

characteristic

155

C₄

#108×D₇₃

yes

characteristic

156

C₂

—

292

—

157

C₂

#22608685

yes

characteristic

158

C₂

#365×D₇₃

yes

—

	subgroup	G/H
center	C₂²	#106×D₇₃
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1368	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂²×D₇₃	#106
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1368	D₇₃
hypocenter	C₇₃	#12020
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#106×D₇₃

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1368

C₂

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
3. #22607997	C₂	semidirect
4. #22607996	C₂	semidirect
5. C₇₃×#1368	C₂	semidirect
46. #11924003	M₁₆	semidirect
52. C₁₆×C₇₃	M₁₆	semidirect
84. #1368	D₇₃	semidirect
85. C₇₃	#12020	semidirect
132. C₁₆	#11924011	semidirect

H ≅ G/N

type

3. #22607997

C₂

semidirect

4. #22607996

C₂

semidirect

5. C₇₃×#1368

C₂

semidirect

46. #11924003

M₁₆

semidirect

52. C₁₆×C₇₃

M₁₆

semidirect

84. #1368

D₇₃

semidirect

85. C₇₃

#12020

semidirect

132. C₁₆

#11924011

semidirect

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

subgroup	index
#22611524	2
#22608685	2
#22607997	2
#22607996	2
C₇₃×#1368	2
#12020	73

subgroup

index

C₇₃×#1368

Subgroups — #32997618

Conjugacy classes of subgroups

Complements

Maximal subgroups