Finite Group Databasebeta

← #37535366

Subgroups — #37535366

Order 23168.

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

normal subgroups only

This group has 280 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#26041840C211yesmaximal, characteristic
2#100×D181C211yesmaximal, characteristic, complemented
3#99×D181C211yesmaximal, characteristic, complemented
4#26041069C211yesmaximal, characteristic, complemented
5#26041067C211yesmaximal, characteristic, complemented
6#26041065C211yesmaximal, characteristic, complemented
7#352×C181C211yesmaximal, characteristic, complemented
8C22×D724C411yescharacteristic, complemented
9C2×D4×D181C2211yescharacteristic
10M16×D181—21no—
11#19153801—21no—
12#19153799C411yescharacteristic, complemented
13#19153791—21no—
14#19153789—21no—
15#19153786—21no—
16#19153786C411yescharacteristic, complemented
17#19153784C2211yescharacteristic
18C22⋊C4×D181—21no—
19C22⋊C4×D181C411yescharacteristic, complemented
20#19153733C2211yescharacteristic
21#19153732—21no—
22#19153731C2211yescharacteristic
23#19153730—21no—
24#123×C181C2211yescharacteristic
25#100×C181C2211yescharacteristic
26#99×C181C2211yescharacteristic
27C23×D181C2×C411yescharacteristic
28C23×D181C2×C411yescharacteristic, complemented
29C23×D181D411yescharacteristic
30C2×D724—21no—
31C2×D724—21no—
32C2×D724—21no—
33C2×D724—41no—
34C2×D724—41no—
35C2×D724D411yescharacteristic
36C2×C22⋊D181—21no—
37C2×C22⋊D181C2×C411yescharacteristic
38C2×C22⋊D181D411yescharacteristic
39D4×D181—41no—
40D4×D181—41no—
41C2×C4×D181—21no—
42C2×C4×D181—41no—
43C2×C4×D181D411yescharacteristic
44#15668433—21no—
45#15668432—41no—
46C8×D181—41no—
47#15668428—21no—
48C22⋊Dic181—21no—
49C22⋊Dic181C2×C411yescharacteristic, complemented
50#15668426—21no—
51#15668426—21no—
52#15668426—21no—
53#15668426—21no—
54#15668426—41no—
55C2×D4×C181C2311yescharacteristic
56C22×C4×C181C2×C411yescharacteristic, complemented
57M16×C181—21no—
58C4⋊C4×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic, complemented
62C22×D181—21no—
63C22×D181—21no—
64C22×D181—21no—
65C22×D181—21no—
66C22×D181—21no—
67C22×D181—41no—
68C22×D181—81no—
69C22×D181—81no—
70C22×D181—81no—
71C22×D181C22⋊C411yescharacteristic
72C22×D181C22⋊C411yescharacteristic
73C22×D181C22⋊C411yescharacteristic
74D724—41no—
75D724—42no—
76D724—41no—
77D724—41no—
78D724—81no—
79C22⋊D181—41no—
80C22⋊D181—41no—
81C22⋊D181—81no—
82C2×Dic181—21no—
83C2×Dic181—41no—
84C2×Dic181C22⋊C411yescharacteristic
85C4×D181—21no—
86C4×D181—21no—
87C4×D181—81no—
88C4×D181—82no—
89#13089678—41no—
90C23×C181C22×C411yescharacteristic
91C23×C181C2×D411yescharacteristic
92D4×C181—41no—
93C2×C4×C181—21no—
94C2×C4×C181—21no—
95C2×C4×C181—41no—
96C2×C4×C181—41no—
97C2×C4×C181C2×D411yescharacteristic
98C8×C181—41no—
99C22×C181—21no—
100C22×C181—21no—
101C22×C181—41no—
102C22×C181—41no—
103C22×C181C2×C22⋊C411yescharacteristic
104C2×D181—21no—
105C2×D181—21no—
106C2×D181—41no—
107C2×D181—42no—
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—41no—
111C2×D181—41no—
112C2×D181—41no—
113C2×D181—81no—
114C2×D181—81no—
115C2×D181—81no—
116C2×D181—82no—
117C2×D181#9911yescharacteristic
118C2×D181#9911yescharacteristic
119C4×C181—21no—
120C4×C181—41no—
121C4×C181—41no—
122C4×C181—81no—
123Dic181—21no—
124Dic181—41no—
125Dic181—81no—
126C2×C181—21no—
127C2×C181—41no—
128C2×C181—41no—
129C2×C181C2×#9911yescharacteristic
130D181—21no—
131D181—21no—
132D181—41no—
133D181—41no—
134D181—81no—
135C181#212111yescharacteristic, complemented, Sylow
136#2121—1811nomaximal, Sylow
137#534—1811no—
138C2×#100—1811no—
139C2×#99—1811no—
140#352—1811no—
141#352D18111yescharacteristic, complemented
142AΣL(1,𝔽16)—1811no—
143AΣL(1,𝔽16)—1811no—
144C22×D4—1811no—
145C22×D4—1811no—
146C2×M16—3621no—
147#123—3621no—
148#123—1811no—
149#123C2×D18111yescharacteristic
150#121—3621no—
151#121—3621no—
152#120—3621no—
153#120—1811no—
154#120—1811no—
155C2×C22⋊C4—3621no—
156C2×C22⋊C4—1811no—
157#100—3621no—
158#100—1811no—
159#100C2×D18111yescharacteristic
160#99—3621no—
161#99—1811no—
162#99C2×D18111yescharacteristic
163C24—1811no—
164C24—1811no—
165C24—1811no—
166C2×D4—3621no—
167C2×D4—3621no—
168C2×D4—3621no—
169C2×D4—3621no—
170C2×D4—1811no—
171C2×D4—1811no—
172C2×D4—1811no—
173C2×D4—7241no—
174C2×D4—7241no—
175C2×D4—7241no—
176C2×D4—7241no—
177C2×D4C22×D18111yescharacteristic
178C22×C4—3621no—
179C22×C4—1811no—
180C22×C4—7241no—
181C22×C4C4×D18111yescharacteristic, complemented
182M16—21no—
183M16—3621no—
184M16—7241no—
185C2×C8—7241no—
186C4⋊C4—21no—
187C4⋊C4—3621no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—3621no—
191C22⋊C4—3621no—
192C22⋊C4—3621no—
193C22⋊C4—3621no—
194C22⋊C4—3621no—
195C22⋊C4—1811no—
196C22⋊C4—7241no—
197C22⋊C4C4×D18111yescharacteristic, complemented
198C23—3621no—
199C23—3621no—
200C23—3621no—
201C23—3621no—
202C23—3621no—
203C23—14481no—
204C23—14481no—
205C23—14481no—
206C23—1811no—
207C23—1811no—
208C23—1811no—
209C23—7241no—
210C23C2×C4×D18111yescharacteristic
211C23D4×D18111yescharacteristic
212D4—41no—
213D4—14481no—
214D4—14481no—
215D4—7241no—
216D4—7242no—
217D4—7241no—
218D4—7241no—
219D4—7241no—
220D4—7241no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—3621no—
226C2×C4—3621no—
227C2×C4—3621no—
228C2×C4—14481no—
229C2×C4—14482no—
230C2×C4—1811no—
231C2×C4—7241no—
232C2×C4D4×D18111yescharacteristic
233C8—41no—
234C8—7241no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—3621no—
240C22—3621no—
241C22—14481no—
242C22—14481no—
243C22—14481no—
244C22—14482no—
245C22—1811no—
246C22—1811no—
247C22—7241no—
248C22—7241no—
249C22—7241no—
250C22—7241no—
251C22—7241no—
252C22—7242no—
253C22—7241no—
254C22C22⋊C4×D18111yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—3621no—
260C4—14481no—
261C4—7241no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—3621no—
266C2—3621no—
267C2—14481no—
268C2—7241no—
269C2—7241no—
270C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC23×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#352×C181C2
socleC2×C181C2×#99
prime radical#26041840C2
cosocleC2×D4×C181C23
hypercenter#352D181
hypocenterC181#2121
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D181C2semidirect
3. #99×D181C2semidirect
4. #26041069C2semidirect
5. #26041067C2semidirect
6. #26041065C2semidirect
7. #352×C181C2semidirect
8. C22×D724C4semidirect
12. #19153799C4semidirect
16. #19153786C4semidirect
19. C22⋊C4×D181C4semidirect
28. C23×D181C2×C4semidirect
49. C22⋊Dic181C2×C4semidirect
56. C22×C4×C181C2×C4semidirect
61. C22⋊C4×C181C2×C4semidirect
135. C181#2121semidirect
141. #352D181semidirect
181. C22×C4C4×D181semidirect
197. C22⋊C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
#260418402
#100×D1812
#99×D1812
#260410692
#260410672
#260410652
#352×C1812
#2121181