Finite Group Databasebeta

← #28393901

Subgroups — #28393901

Order 14464.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D113C211yesmaximal, characteristic
2C2×#16813605C212yesmaximal, complemented
3#20298513C214yesmaximal, complemented
4#20298477C212yesmaximal, complemented
5#20298470C212yesmaximal, complemented
6#20298454C211yesmaximal, characteristic, complemented
7#20298453C212yesmaximal, complemented
8C22×C22⋊C4×C113C211yesmaximal, characteristic, complemented
9C24×D113C2211yescharacteristic
10C22×C22⋊D113—28no—
11C22×C22⋊D113—24no—
12C22×C22⋊D113C2212yes—
13C22×C22⋊D113C2212yes—
14C23×Dic113C2211yescharacteristic
15#16813605—28no—
16C2×#14268626C2214yescomplemented
17C2×C22⋊Dic113C2212yes—
18C2×C22⋊Dic113C2212yescomplemented
19C2×C22⋊Dic113C2212yescomplemented
20C2×#14268624C2214yescomplemented
21C2×#14268624C2214yescomplemented
22#16813529C2214yescomplemented
23C25×C113C2211yescharacteristic
24C23×C4×C113C2211yescharacteristic, complemented
25C23×C4×C113C2211yescharacteristic, complemented
26C2×C22⋊C4×C113—24no—
27C2×C22⋊C4×C113C2212yes—
28C2×C22⋊C4×C113C2212yescomplemented
29C23×D113—22no—
30C23×D113—24no—
31C23×D113—24no—
32C23×D113—42no—
33C23×D113C2312yes—
34C2×C22⋊D113—232no—
35C2×C22⋊D113—24no—
36C2×C22⋊D113—24no—
37C2×C22⋊D113—24no—
38C2×C22⋊D113—24no—
39C2×C22⋊D113—24no—
40C2×C22⋊D113D418yes—
41C22×Dic113—22no—
42C22×Dic113—24no—
43C22×Dic113C2314yes—
44C22×Dic113C2312yes—
45#14268626—28no—
46C22⋊Dic113—24no—
47C22⋊Dic113—24no—
48C22⋊Dic113—42no—
49#14268624—28no—
50#14268624—44no—
51C24×C113—24no—
52C24×C113—24no—
53C24×C113—42no—
54C24×C113D414yes—
55C24×C113C2311yescharacteristic
56C24×C113C2311yescharacteristic
57C24×C113C2311yescharacteristic
58C22×C4×C113—24no—
59C22×C4×C113—22no—
60C22×C4×C113—22no—
61C22×C4×C113—42no—
62C22×C4×C113C2312yes—
63C22×C4×C113C2312yes—
64C22⋊C4×C113—24no—
65C22⋊C4×C113—24no—
66C23×C113—22no—
67C23×C113—22no—
68C23×C113—22no—
69C23×C113—22no—
70C23×C113—28no—
71C23×C113—28no—
72C23×C113—22no—
73C23×C113—22no—
74C23×C113—22no—
75C23×C113—44no—
76C23×C113—416no—
77C23×C113C2×D414yes—
78C23×C113C2×D412yes—
79C23×C113C2×D412yes—
80C23×C113C2×D412yes—
81C23×C113#4114yes—
82C23×C113C2411yescharacteristic
83C22×D113—28no—
84C22×D113—22no—
85C22×D113—22no—
86C22×D113—24no—
87C22×D113—24no—
88C22×D113—416no—
89C22×D113—44no—
90C22×D113—44no—
91C22×D113C2×D414yes—
92C2×C4×C113—22no—
93C2×C4×C113—22no—
94C2×C4×C113—24no—
95C2×C4×C113—24no—
96C2×C4×C113—44no—
97C2×C4×C113—44no—
98C22⋊D113—216no—
99C22⋊D113—216no—
100C2×Dic113—28no—
101C2×Dic113—22no—
102C2×Dic113—22no—
103C2×Dic113—24no—
104C2×Dic113—24no—
105C2×Dic113—24no—
106C2×Dic113C2×D414yes—
107C22×C113—22no—
108C22×C113—22no—
109C22×C113—22no—
110C22×C113—22no—
111C22×C113—28no—
112C22×C113—28no—
113C22×C113—22no—
114C22×C113—22no—
115C22×C113—22no—
116C22×C113—416no—
117C22×C113—44no—
118C22×C113#12014yes—
119C22×C113#12314yescomplemented
120C22×C113C22×D412yes—
121C22×C113C22×D411yescharacteristic
122C22×C113C2×#4112yes—
123C22×C113H(5,𝔽2)12yes—
124C2×D113—24no—
125C2×D113—28no—
126C2×D113—416no—
127C2×D113—42no—
128C2×D113—42no—
129C2×D113—44no—
130C4×C113—42no—
131C4×C113—42no—
132Dic113—24no—
133Dic113—44no—
134C2×C113—24no—
135C2×C113—24no—
136C2×C113—42no—
137C2×C113C2×#12011yescharacteristic
138C2×C113C2×#12311yescharacteristic
139C2×C113#53411yescharacteristic
140C2×C113#54614yes—
141D113—44no—
142#2405—1131nomaximal, Sylow
143C113#240511yescharacteristic, complemented, Sylow
144C23×D4—1131no—
145C2×#123—1132no—
146C22×C22⋊C4D11311yescharacteristic, complemented
147#394—1134no—
148#386—1132no—
149#385—1132no—
150#379—1131no—
151#379—1132no—
152C25—1131no—
153C25C2×D11311yescharacteristic
154C22×D4—1132no—
155C22×D4—1132no—
156C22×D4—2268no—
157C22×D4—2264no—
158C23×C4—1131no—
159C23×C4C2×D11311yescharacteristic, complemented
160C23×C4C2×D11311yescharacteristic, complemented
161#123—2268no—
162C2×C4⋊C4—1134no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—1132no—
165C2×C22⋊C4—1134no—
166C2×C22⋊C4—1132no—
167C2×C22⋊C4—1132no—
168C2×C22⋊C4—1134no—
169C2×C22⋊C4C2×D11312yes—
170C2×C22⋊C4C2×D11312yescomplemented
171#95—1134no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—1132no—
176C24—4522no—
177C24—2264no—
178C24—2264no—
179C24—2262no—
180C24C22⋊D11314yes—
181C24C22×D11311yescharacteristic
182C24C22×D11311yescharacteristic
183C24C22×D11311yescharacteristic
184C2×D4—1138no—
185C2×D4—2264no—
186C2×D4—2264no—
187C2×D4—22632no—
188C2×D4—2264no—
189C2×D4—2264no—
190C2×D4—2264no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—1132no—
196C22×C4—1134no—
197C22×C4—2264no—
198C22×C4—2262no—
199C22×C4C22×D11312yes—
200C22×C4C22×D11312yes—
201C4⋊C4—2268no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—4524no—
205C22⋊C4—4522no—
206C22⋊C4—2268no—
207C22⋊C4—2264no—
208C22⋊C4—2264no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—1134no—
221C23—4524no—
222C23—45216no—
223C23—4524no—
224C23—2268no—
225C23—2262no—
226C23—2262no—
227C23—2264no—
228C23—2264no—
229C23D4×D11314yes—
230C23C2×C22⋊D11312yes—
231C23C2×C22⋊D11312yes—
232C23C2×C22⋊D11312yes—
233C23#1426864714yes—
234C23C23×D11311yescharacteristic
235D4—22616no—
236D4—22616no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—1134no—
244C2×C4—2264no—
245C2×C4—2264no—
246C2×C4—2264no—
247C2×C4—2268no—
248C2×C4—2262no—
249C2×C4—2262no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—45216no—
262C22—4522no—
263C22—4522no—
264C22—4524no—
265C22—2264no—
266C22—2268no—
267C22#1681359014yes—
268C22#1681360514yescomplemented
269C22C2×D4×D11312yes—
270C22C22×C22⋊D11311yescharacteristic
271C22C2×#1426864712yes—
272C22#1681366712yes—
273C4—42no—
274C4—42no—
275C4—4524no—
276C4—2264no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—4524no—
281C2C2×#1681359011yescharacteristic
282C2C2×#1681360511yescharacteristic
283C2#2029915711yescharacteristic
284C2#2029923814yes—
subgroupG/H
center, FrattiniC23C23×D113
commutator, socle, cosocleC23×C113C24
FittingC22×C22⋊C4×C113C2
prime radicalC23×C22⋊D113C2
hypercenterC22×C22⋊C4D113
hypocenterC113#2405
LCSG ▷ C23×C113 ▷ C113
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16813605C2semidirect
3. #20298513C2semidirect
4. #20298477C2semidirect
5. #20298470C2semidirect
6. #20298454C2semidirect
7. #20298453C2semidirect
8. C22×C22⋊C4×C113C2semidirect
16. C2×#14268626C22semidirect
18. C2×C22⋊Dic113C22semidirect
19. C2×C22⋊Dic113C22semidirect
20. C2×#14268624C22semidirect
21. C2×#14268624C22semidirect
22. #16813529C22semidirect
24. C23×C4×C113C22semidirect
25. C23×C4×C113C22semidirect
28. C2×C22⋊C4×C113C22semidirect
119. C22×C113#123semidirect
143. C113#2405semidirect
146. C22×C22⋊C4D113semidirect
159. C23×C4C2×D113semidirect
160. C23×C4C2×D113semidirect
170. C2×C22⋊C4C2×D113semidirect
268. C22#16813605semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1132
C2×#168136052
#202985132
#202984772
#202984702
#202984542
#202984532
C22×C22⋊C4×C1132
#2405113