Finite Group Databasebeta

← #37535713

Subgroups — #37535713

Order 23168.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041960C211yesmaximal, characteristic, complemented
2#26041470C211yesmaximal, characteristic, complemented
3#26041465C211yesmaximal, characteristic
4#26041459C211yesmaximal, characteristic, complemented
5#26041073C211yesmaximal, characteristic, complemented
6#26041070C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C181C211yesmaximal, characteristic, complemented
8#19153861C2211yescharacteristic
9#19153849—21no—
10#19153809—21no—
11#19153803C2211yescharacteristic
12#19153800—21no—
13#19153797—21no—
14#19153797C2211yescharacteristic
15#19153793—21no—
16#19153785—21no—
17D4×Dic181—21no—
18#19153772—21no—
19C2×C22⋊Dic181—21no—
20#19153742—21no—
21#19153731C2211yescharacteristic
22#19153730—21no—
23#19153730—21no—
24#120×C181C2211yescharacteristic
25#100×C181C2211yescharacteristic
26#99×C181C2211yescharacteristic
27#15668449—21no—
28#15668448—41no—
29C2×Dic362D411yescharacteristic, complemented
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C22⋊D181—41no—
33D4×D181—41no—
34C22×Dic181—21no—
35C22×Dic181—21no—
36C22×Dic181D411yescharacteristic, complemented
37Dic724—41no—
38#15668438—41no—
39C4⋊Dic181—21no—
40C4⋊Dic181—41no—
41#15668428—21no—
42#15668428—21no—
43#15668428—21no—
44C22⋊Dic181—21no—
45C22⋊Dic181—21no—
46C22⋊Dic181—21no—
47C22⋊Dic181—21no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181—41no—
51C22⋊Dic181D411yescharacteristic, complemented
52#15668426—41no—
53C4×Dic181—21no—
54C4×Dic181D411yescharacteristic, complemented
55C24×C181D411yescharacteristic
56C2×D4×C181—21no—
57C2×D4×C181C2311yescharacteristic
58M16×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181D411yescharacteristic
62C22×D181—21no—
63C22×D181—21no—
64Dic362—21no—
65Dic362—41no—
66D724—21no—
67C22⋊D181—41no—
68C22⋊D181—41no—
69C22⋊D181—41no—
70C22⋊D181—82no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—21no—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—42no—
78C2×Dic181—41no—
79C2×Dic181—41no—
80C2×Dic181—81no—
81C4×D181—41no—
82C23×C181—21no—
83C23×C181—42no—
84C23×C181—41no—
85C23×C181C2×D411yescharacteristic
86C23×C181C2×D411yescharacteristic
87D4×C181—41no—
88D4×C181—42no—
89C2×C4×C181—21no—
90C2×C4×C181—41no—
91C2×C4×C181C2×D411yescharacteristic
92C8×C181—41no—
93C22×C181—22no—
94C22×C181—21no—
95C22×C181—21no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181—42no—
100C22×C181—41no—
101C22×C181—81no—
102C22×C181#12011yescharacteristic
103C2×D181—41no—
104C2×D181—81no—
105C2×D181—81no—
106C4×C181—21no—
107C4×C181—41no—
108C4×C181—81no—
109Dic181—41no—
110Dic181—41no—
111Dic181—41no—
112Dic181—41no—
113Dic181—41no—
114Dic181—81no—
115C2×C181—21no—
116C2×C181—41no—
117C2×C181—41no—
118C2×C181—42no—
119C2×C181B(4,𝔽2)11yescharacteristic
120D181—81no—
121C181#219711yescharacteristic, complemented, Sylow
122#2197—1811nomaximal, Sylow
123#551—1811no—
124#458—1811no—
125B(4,𝔽2)—1811no—
126#455—1811no—
127#354—1811no—
128#352—1811no—
129AΣL(1,𝔽16)D18111yescharacteristic, complemented
130H(5,𝔽2)—1811no—
131#137—3621no—
132#126—3621no—
133#124—1811no—
134#123—3621no—
135#123—3621no—
136#123—1811no—
137C22⋊Q8—3621no—
138#120—3621no—
139#120C2×D18111yescharacteristic
140C4×D4—3621no—
141#117—3621no—
142C2×C22⋊C4—3621no—
143#104—3621no—
144#100C2×D18111yescharacteristic
145#99—3621no—
146#99—3621no—
147#99—1811no—
148#99C2×D18111yescharacteristic
149C24D72411yescharacteristic
150#41—3621no—
151#41—7241no—
152C2×Q8—1811no—
153C2×D4—21no—
154C2×D4—3621no—
155C2×D4—3621no—
156C2×D4—7241no—
157C2×D4—7241no—
158C2×D4C22×D18111yescharacteristic
159C22×C4—3621no—
160C22×C4—3621no—
161C22×C4—1811no—
162Q16—7241no—
163SD16—7241no—
164M16—21no—
165C4⋊C4—3621no—
166C4⋊C4—3621no—
167C4⋊C4—3621no—
168C4⋊C4—3621no—
169C4⋊C4—7241no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3621no—
173C22⋊C4—3621no—
174C22⋊C4—3621no—
175C22⋊C4—3621no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—1811no—
179C22⋊C4—7241no—
180C22⋊C4—7241no—
181C22⋊C4D72411yescharacteristic
182C42—3621no—
183C42—1811no—
184C23—21no—
185C23—41no—
186C23—42no—
187C23—3621no—
188C23—3621no—
189C23D4×D18111yescharacteristic
190C23C2×D72411yescharacteristic
191Q8—3621no—
192Q8—7241no—
193D4—41no—
194D4—42no—
195D4—3621no—
196D4—14482no—
197D4—7241no—
198D4—7241no—
199D4—7241no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—3621no—
203C2×C4—3621no—
204C2×C4—3621no—
205C2×C4—3621no—
206C2×C4—3621no—
207C2×C4—3621no—
208C2×C4—14481no—
209C2×C4—7241no—
210C2×C4—7241no—
211C2×C4—7242no—
212C2×C4—7241no—
213C2×C4D4×D18111yescharacteristic
214C8—41no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—14481no—
225C22—14481no—
226C22—7241no—
227C22#1915378611yescharacteristic
228C4—21no—
229C4—41no—
230C4—81no—
231C4—14481no—
232C4—7241no—
233C4—7241no—
234C4—7241no—
235C4—7241no—
236C4—7241no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—14481no—
242C2#2604146711yescharacteristic
subgroupG/H
centerC2#26041467
commutator, cosocleC2×D4×C181C23
FrattiniC2×D4C22×D181
FittingAΣL(1,𝔽16)×C181C2
socleC2×C181B(4,𝔽2)
prime radical#26041465C2
hypercenterAΣL(1,𝔽16)D181
hypocenterC181#2197
LCSG ▷ C2×D4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26041960C2semidirect
2. #26041470C2semidirect
4. #26041459C2semidirect
5. #26041073C2semidirect
6. #26041070C2semidirect
7. AΣL(1,𝔽16)×C181C2semidirect
29. C2×Dic362D4semidirect
36. C22×Dic181D4semidirect
51. C22⋊Dic181D4semidirect
54. C4×Dic181D4semidirect
121. C181#2197semidirect
129. AΣL(1,𝔽16)D181semidirect

Maximal subgroups

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

subgroupindex
#260419602
#260414702
#260414652
#260414592
#260410732
#260410702
AΣL(1,𝔽16)×C1812
#2197181