Group #32991017 — subgroups

Subgroups — #32991017

Order 18688.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22610912	C₂	1	1	yes	maximal, characteristic
2	#22610085	C₂	1	1	yes	maximal, characteristic
3	#22608702	C₂	1	2	yes	maximal
4	#22608335	C₂	1	2	yes	maximal
5	C₇₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#17989144	C₂²	1	1	yes	characteristic
7	#17988611	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10596831	—	2	2	no	—
9	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
10	#17988347	—	2	2	no	—
11	#17988338	—	2	4	no	—
12	#17988319	—	2	4	no	—
13	#431×C₇₃	C₂²	1	2	yes	—
14	#376×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924012	—	2	1	no	—
16	C₂²×#10596831	—	2	2	no	—
17	#14556591	Q₈	1	2	yes	—
18	#14556583	—	2	1	no	—
19	#14556570	—	2	2	no	—
20	C₄×Dic₁₄₆	—	2	2	no	—
21	#14556550	—	2	1	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	C₄⋊#10596831	—	4	4	no	—
26	C₂×C₄²×C₇₃	D₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	2	no	—
29	C₂²⋊C₈×C₇₃	—	2	2	no	—
30	C₄×C₈×C₇₃	D₄	1	4	yes	—
31	#95×C₇₃	—	2	2	no	—
32	C₂×Dic₁₄₆	—	2	2	no	—
33	#11924012	—	4	2	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₂×#10596831	—	4	1	no	—
36	C₂×#10596831	—	4	4	no	—
37	C₄⋊Dic₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	—	2	2	no	—
39	#11924007	—	4	2	no	—
40	C₂²⋊Dic₇₃	—	4	1	no	—
41	C₄×Dic₇₃	—	4	1	no	—
42	C₂²×C₄×C₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	#41	1	2	yes	—
45	C₂×C₈×C₇₃	—	2	4	no	—
46	C₄²×C₇₃	—	2	2	no	—
47	C₄²×C₇₃	SD₁₆	1	4	yes	—
48	C₄²×C₇₃	C₂×D₄	1	2	yes	—
49	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇₃	#41	1	1	yes	characteristic
51	C₂³×C₇₃	#123	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	2	4	no	—
53	C₂×C₄×C₇₃	—	4	2	no	—
54	C₂×C₄×C₇₃	—	4	4	no	—
55	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#123	1	2	yes	—
58	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₇₃	#137	1	2	yes	—
61	C₂×Dic₇₃	—	4	2	no	—
62	Dic₁₄₆	—	4	4	no	—
63	C₈×C₇₃	—	4	4	no	—
64	#10596831	—	8	2	no	—
65	#10596831	—	8	1	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	#399	1	1	yes	characteristic
68	C₂²×C₇₃	#483	1	2	yes	—
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	—	4	4	no	—
72	C₄×C₇₃	#475	1	2	yes	—
73	C₄×C₇₃	#492	1	1	yes	characteristic
74	C₄×C₇₃	#497	1	1	yes	characteristic
75	Dic₇₃	—	8	2	no	—
76	#10670	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	2	no	—
78	C₂×C₇₃	#2096	1	1	yes	characteristic
79	C₂×C₇₃	#2098	1	2	yes	—
80	#1995	—	73	1	no	—
81	#1843	—	73	1	no	—
82	#1570	—	73	2	no	—
83	#1481	—	73	2	no	—
84	#1274	D₇₃	1	1	yes	characteristic
85	C₇₃	#10670	1	1	yes	characteristic, complemented, Sylow
86	#533	—	73	1	no	—
87	#431	C₂×D₇₃	1	2	yes	—
88	#423	—	73	1	no	—
89	C₂×C₄⋊C₈	—	73	2	no	—
90	C₂×C₄⋊C₈	—	146	2	no	—
91	#376	C₂×D₇₃	1	1	yes	characteristic
92	#339	—	146	2	no	—
93	#334	—	146	4	no	—
94	Q₈⋊C₈	—	146	4	no	—
95	C₂×M₁₆	—	146	1	no	—
96	C₂²×C₈	—	146	2	no	—
97	C₄⋊Q₈	—	73	2	no	—
98	#125	—	146	1	no	—
99	C₂²⋊Q₈	—	146	2	no	—
100	C₄×Q₈	—	146	2	no	—
101	#117	—	146	1	no	—
102	C₂×C₄²	D₂₉₂	1	2	yes	—
103	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	292	4	no	—
106	C₄⋊C₈	—	146	2	no	—
107	C₄⋊C₈	—	146	2	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₄×C₈	D₂₉₂	1	4	yes	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	146	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
115	C₂²×C₄	#11924027	1	2	yes	—
116	M₁₆	—	292	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	292	2	no	—
119	C₂×C₈	—	292	1	no	—
120	C₂×C₈	—	292	4	no	—
121	C₄⋊C₄	—	292	2	no	—
122	C₄⋊C₄	—	146	2	no	—
123	C₄⋊C₄	—	146	2	no	—
124	C₂²⋊C₄	—	292	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	292	1	no	—
127	C₄²	#11924017	1	4	yes	—
128	C₄²	C₂×D₂₉₂	1	2	yes	—
129	C₄²	Q₈×D₇₃	1	1	yes	characteristic
130	C₄²	#11924027	1	1	yes	characteristic
131	C₂³	#14556575	1	1	yes	characteristic
132	Q₈	—	292	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	292	2	no	—
137	C₂×C₄	#14556571	1	2	yes	—
138	C₂×C₄	#14556571	1	1	yes	characteristic
139	C₂×C₄	#14556575	1	2	yes	—
140	C₂×C₄	#14556588	1	1	yes	characteristic
141	C₂×C₄	C₂×#11924017	1	2	yes	—
142	C₂×C₄	#14556624	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	584	2	no	—
145	C₈	—	584	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#17988524	1	1	yes	characteristic
148	C₂²	#17988918	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	584	2	no	—
153	C₄	#17988873	1	2	yes	—
154	C₄	#17988960	1	1	yes	characteristic
155	C₄	#17988981	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#22611473	1	1	yes	characteristic
158	C₂	#22611485	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610912

C₂

yes

maximal, characteristic

#22610085

C₂

yes

maximal, characteristic

#22608702

C₂

yes

maximal

#22608335

C₂

yes

maximal

C₇₃×#1274

C₂

yes

maximal, characteristic

#17989144

C₂²

yes

characteristic

#17988611

C₂²

yes

characteristic

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₂²

yes

—

#17988347

—

#17988338

—

#17988319

—

#431×C₇₃

C₂²

yes

—

#376×C₇₃

C₂²

yes

—

—

Q₈

yes

—

#14556583

—

#14556570

—

C₄×Dic₁₄₆

—

—

—

—

—

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

#95×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×Q₈

yes

C₄²×C₇₃

yes

C₂³×C₇₃

yes

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#399

yes

characteristic

C₂²×C₇₃

#483

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#475

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#10670

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2096

yes

characteristic

C₂×C₇₃

#2098

yes

—

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#10670

yes

characteristic, complemented, Sylow

#533

—

#431

C₂×D₇₃

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

#376

C₂×D₇₃

yes

characteristic

#339

—

146

—

#334

—

146

—

Q₈⋊C₈

—

146

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

#125

—

146

—

C₂²⋊Q₈

—

146

—

100

C₄×Q₈

—

146

—

101

#117

—

146

—

102

C₂×C₄²

D₂₉₂

yes

—

103

C₂×C₄²

C₂²×D₇₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

292

—

106

C₄⋊C₈

—

146

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

—

146

—

109

C₂²⋊C₈

—

110

C₄×C₈

D₂₉₂

yes

—

111

#95

—

112

C₂×Q₈

—

146

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

115

C₂²×C₄

#11924027

yes

—

116

M₁₆

—

292

—

117

C₂×C₈

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₄⋊C₄

—

292

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₂²⋊C₄

—

292

—

125

C₄²

—

126

C₄²

—

292

—

127

C₄²

#11924017

yes

—

128

C₄²

C₂×D₂₉₂

yes

—

129

C₄²

Q₈×D₇₃

yes

characteristic

130

C₄²

#11924027

yes

characteristic

131

C₂³

#14556575

yes

characteristic

132

Q₈

—

292

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

292

—

137

C₂×C₄

#14556571

yes

—

138

C₂×C₄

#14556571

yes

characteristic

139

C₂×C₄

#14556575

yes

—

140

C₂×C₄

#14556588

yes

characteristic

141

C₂×C₄

C₂×#11924017

yes

—

142

C₂×C₄

#14556624

yes

—

143

C₈

—

144

C₈

—

584

—

145

C₈

—

584

—

146

C₂²

—

147

C₂²

#17988524

yes

characteristic

148

C₂²

#17988918

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

584

—

153

C₄

#17988873

yes

—

154

C₄

#17988960

yes

characteristic

155

C₄

#17988981

yes

characteristic

156

C₂

—

157

C₂

#22611473

yes

characteristic

158

C₂

#22611485

yes

—

	subgroup	G/H
center	C₂×C₄	#14556588
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1274	C₂
socle	C₂²×C₇₃	#399
prime radical	C₂³×C₇₃	#123
hypercenter	#1274	D₇₃
hypocenter	C₇₃	#10670
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14556588

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1274

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
85. C₇₃	#10670	semidirect

H ≅ G/N

type

85. C₇₃

#10670

semidirect

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

subgroup	index
#22610912	2
#22610085	2
#22608702	2
#22608335	2
C₇₃×#1274	2
#10670	73

subgroup

index

C₇₃×#1274

Subgroups — #32991017

Conjugacy classes of subgroups

Complements

Maximal subgroups