Finite Group Databasebeta

← #37535359

Subgroups — #37535359

Order 23168.

264 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 55 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#26041785C211yesmaximal, characteristic
2#26041260C211yesmaximal, characteristic, complemented
3#26041251C211yesmaximal, characteristic, complemented
4#26041070C211yesmaximal, characteristic, complemented
5#26041068C211yesmaximal, characteristic, complemented
6#26041066C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C181C211yesmaximal, characteristic, complemented
8C2×#15668448C2211yescharacteristic
9C23×Dic181C411yescharacteristic, complemented
10M16×D181—21no—
11#19153800C411yescharacteristic, complemented
12#19153797C2211yescharacteristic
13#19153793—21no—
14#19153788—21no—
15#19153785C411yescharacteristic, complemented
16D4×Dic181—21no—
17#19153776—21no—
18#19153772—21no—
19#19153772C411yescharacteristic, complemented
20#19153734—21no—
21#19153733C2211yescharacteristic
22#19153731C2211yescharacteristic
23#19153730—21no—
24#120×C181C2211yescharacteristic
25#100×C181C2211yescharacteristic
26#99×C181C2211yescharacteristic
27#15668448—41no—
28#15668448—41no—
29C2×Dic362D411yescharacteristic
30C2×C22⋊D181—21no—
31C2×C22⋊D181C2×C411yescharacteristic
32C2×C22⋊D181D411yescharacteristic
33C22×Dic181—21no—
34C22×Dic181—21no—
35C22×Dic181—41no—
36C22×Dic181—41no—
37C22×Dic181C2×C411yescharacteristic
38C22×Dic181C2×C411yescharacteristic
39C22×Dic181D411yescharacteristic
40C2×C4×D181D411yescharacteristic
41#15668433—21no—
42#15668432—41no—
43C8×D181—41no—
44C4⋊Dic181—21no—
45#15668428—21no—
46#15668428—21no—
47#15668428—41no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181—21no—
51C22⋊Dic181C2×C411yescharacteristic
52#15668426—21no—
53C4×Dic181—21no—
54C4×Dic181—41no—
55C24×C181C2×C411yescharacteristic
56C2×D4×C181—21no—
57C2×D4×C181C2311yescharacteristic
58M16×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic
62C22×D181C22⋊C411yescharacteristic
63Dic362—41no—
64C22⋊D181—41no—
65C22⋊D181—41no—
66C22⋊D181—42no—
67C2×Dic181—21no—
68C2×Dic181—21no—
69C2×Dic181—21no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—42no—
74C2×Dic181—41no—
75C2×Dic181—41no—
76C2×Dic181—41no—
77C2×Dic181—41no—
78C2×Dic181C22⋊C411yescharacteristic
79C2×Dic181C22⋊C411yescharacteristic
80C2×Dic181C22⋊C411yescharacteristic
81C4×D181—21no—
82C4×D181—21no—
83#13089678—41no—
84C23×C181—21no—
85C23×C181—42no—
86C23×C181—41no—
87C23×C181C22×C411yescharacteristic
88C23×C181C2×D411yescharacteristic
89D4×C181—41no—
90D4×C181—42no—
91C2×C4×C181—21no—
92C2×C4×C181—41no—
93C2×C4×C181C2×D411yescharacteristic
94C8×C181—41no—
95C22×C181—22no—
96C22×C181—21no—
97C22×C181—21no—
98C22×C181—41no—
99C22×C181—41no—
100C22×C181—41no—
101C22×C181—42no—
102C22×C181—42no—
103C22×C181—41no—
104C22×C181C2×C22⋊C411yescharacteristic
105C2×D181—21no—
106C2×D181—41no—
107C4×C181—21no—
108C4×C181—41no—
109C4×C181—81no—
110Dic181—21no—
111Dic181—21no—
112Dic181—41no—
113Dic181—41no—
114Dic181—81no—
115Dic181#9911yescharacteristic
116Dic181#9911yescharacteristic
117C2×C181—21no—
118C2×C181—41no—
119C2×C181—42no—
120C2×C181—41no—
121C2×C181C2×#9911yescharacteristic
122D181—41no—
123C181#212011yescharacteristic, complemented, Sylow
124#2120—1811nomaximal, Sylow
125#525—1811no—
126#413—1811no—
127#410—1811no—
128#352—1811no—
129#352—1811no—
130AΣL(1,𝔽16)—1811no—
131AΣL(1,𝔽16)D18111yescharacteristic, complemented
132C2×#41—1811no—
133C23×C4—1811no—
134C2×M16—3621no—
135#123—1811no—
136#123—1811no—
137C22⋊Q8—3621no—
138#121—3621no—
139#120—1811no—
140#120C2×D18111yescharacteristic
141C4×D4—3621no—
142C4×D4—3621no—
143#117—3621no—
144#117—1811no—
145#101—3621no—
146#100—1811no—
147#100C2×D18111yescharacteristic
148#99—3621no—
149#99—1811no—
150#99C2×D18111yescharacteristic
151C24C4×D18111yescharacteristic
152#41—7241no—
153#41—7241no—
154C2×Q8—1811no—
155C2×D4—21no—
156C2×D4—3621no—
157C2×D4—1811no—
158C2×D4—1811no—
159C2×D4C22×D18111yescharacteristic
160C22×C4—3621no—
161C22×C4—3621no—
162C22×C4—1811no—
163C22×C4—1811no—
164C22×C4—1811no—
165C22×C4—1811no—
166C22×C4—7241no—
167C22×C4—7241no—
168M16—21no—
169M16—3621no—
170M16—7241no—
171C2×C8—7241no—
172C4⋊C4—3621no—
173C4⋊C4—3621no—
174C4⋊C4—3621no—
175C4⋊C4—7241no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—3621no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—1811no—
183C22⋊C4C4×D18111yescharacteristic
184C42—3621no—
185C42—7241no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23—1811no—
190C23C2×C4×D18111yescharacteristic
191C23D4×D18111yescharacteristic
192Q8—7241no—
193D4—41no—
194D4—42no—
195D4—7241no—
196D4—7241no—
197D4—7242no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—3621no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—3621no—
204C2×C4—3621no—
205C2×C4—3621no—
206C2×C4—3621no—
207C2×C4—3621no—
208C2×C4—1811no—
209C2×C4—1811no—
210C2×C4—1811no—
211C2×C4—7242no—
212C2×C4—7241no—
213C2×C4—7241no—
214C2×C4—7241no—
215C2×C4—7241no—
216C2×C4D4×D18111yescharacteristic
217C8—41no—
218C8—7241no—
219C22—22no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—3621no—
229C22—7241no—
230C22C22⋊C4×D18111yescharacteristic
231C4—21no—
232C4—41no—
233C4—81no—
234C4—3621no—
235C4—3621no—
236C4—14481no—
237C4—1811no—
238C4—1811no—
239C4—7241no—
240C4—7241no—
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—7241no—
246C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC23×C181C22×C4
FrattiniC2×D4C22×D181
FittingAΣL(1,𝔽16)×C181C2
socleC2×C181C2×#99
prime radical#26041785C2
cosocleC2×D4×C181C23
hypercenterAΣL(1,𝔽16)D181
hypocenterC181#2120
LCSG ▷ C23×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
2. #26041260C2semidirect
3. #26041251C2semidirect
4. #26041070C2semidirect
5. #26041068C2semidirect
6. #26041066C2semidirect
7. AΣL(1,𝔽16)×C181C2semidirect
9. C23×Dic181C4semidirect
11. #19153800C4semidirect
15. #19153785C4semidirect
19. #19153772C4semidirect
123. C181#2120semidirect
131. AΣL(1,𝔽16)D181semidirect

Maximal subgroups

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

subgroupindex
#260417852
#260412602
#260412512
#260410702
#260410682
#260410662
AΣL(1,𝔽16)×C1812
#2120181