Finite Group Databasebeta

← #27174270

Subgroups — #27174270

Order 12928.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#15692223C211yesmaximal, characteristic
2#20265080C211yesmaximal, characteristic, complemented
3#20265077C212yesmaximal, complemented
4#20265071C212yesmaximal, complemented
5#20265069C212yesmaximal, complemented
6#20265066C212yesmaximal, complemented
7#20265056C212yesmaximal, complemented
8#20265032C212yesmaximal, complemented
9#386×C101C211yesmaximal, characteristic, complemented
10C24×D101C2211yescharacteristic
11C22×C22⋊D101C2212yes—
12#15692223C418yescomplemented
13C2×C4⋊Dic101C2211yescharacteristic, complemented
14C2×#13108053C2212yescomplemented
15C2×C22⋊Dic101C2212yes—
16C2×C22⋊Dic101C2211yescharacteristic
17C2×C22⋊Dic101C2212yescomplemented
18C2×#13108051C2212yes—
19C2×#13108051C2212yes—
20C2×#13108051C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2212yes—
23C2×C4×Dic101C2211yescharacteristic, complemented
24C2×C4×Dic101C2212yescomplemented
25#15692162C2212yescomplemented
26#15692162C2212yescomplemented
27C22×D4×C101C2211yescharacteristic
28C23×C4×C101C2211yescharacteristic, complemented
29C2×C22⋊C4×C101C2212yes—
30C2×C22⋊C4×C101C2211yescharacteristic, complemented
31#95×C101C2212yescomplemented
32C23×D101—41no—
33C23×D101—41no—
34C23×D101—42no—
35C23×D101—42no—
36C23×D101C2×C414yescomplemented
37C23×D101C2312yes—
38C2×C22⋊D101—42no—
39C2×C22⋊D101—42no—
40C2×C22⋊D101C2×C418yes—
41C22×Dic101C2312yes—
42C22×Dic101C2312yes—
43C22×Dic101C2312yes—
44C4⋊Dic101—41no—
45#13108053—42no—
46C22⋊Dic101—42no—
47C22⋊Dic101—42no—
48C22⋊Dic101C2×C414yescomplemented
49#13108051—42no—
50#13108051—42no—
51#13108051—42no—
52#13108051—42no—
53#13108051C2×C418yescomplemented
54C4×Dic101—42no—
55C4×Dic101—41no—
56C24×C101C2311yescharacteristic
57C24×C101C2311yescharacteristic
58C2×D4×C101—41no—
59C2×D4×C101—41no—
60C2×D4×C101C2×C414yes—
61C22×C4×C101—42no—
62C22×C4×C101—41no—
63C22×C4×C101C2312yes—
64C22×C4×C101C2312yes—
65C22×C4×C101C2311yescharacteristic
66C22⋊C4×C101—42no—
67C22⋊C4×C101—41no—
68C23×C101—21no—
69C23×C101—22no—
70C23×C101—21no—
71C23×C101—22no—
72C23×C101—21no—
73C23×C101—21no—
74C23×C101—22no—
75C23×C101—22no—
76C23×C101C22×C412yes—
77C23×C101C22×C412yes—
78C23×C101C2411yescharacteristic
79C22×D101—22no—
80C22×D101—22no—
81C22×D101—22no—
82C22×D101—22no—
83C22×D101—22no—
84C22×D101—22no—
85C22×D101—48no—
86C22×D101—44no—
87C22×D101—48no—
88C22×D101—44no—
89C22×D101—44no—
90C22×D101C22×C414yes—
91D4×C101—44no—
92C2×C4×C101—21no—
93C2×C4×C101—21no—
94C2×C4×C101—22no—
95C2×C4×C101—22no—
96C2×C4×C101—22no—
97C2×C4×C101—22no—
98C2×C4×C101—22no—
99C2×C4×C101—22no—
100C2×C4×C101—44no—
101C2×C4×C101C22×C412yes—
102C22⋊D101—48no—
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101—22no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101—22no—
110C2×Dic101—22no—
111C2×Dic101C22×C414yes—
112C22×C101—24no—
113C22×C101—22no—
114C22×C101—22no—
115C22×C101—24no—
116C22×C101—42no—
117C22×C101—41no—
118C22×C101—42no—
119C22×C101—41no—
120C22×C101—41no—
121C22×C101—41no—
122C22×C101C23×C411yescharacteristic
123C22×C101H(5,𝔽2)11yescharacteristic
124C22×C101H(5,𝔽2)12yes—
125C22×C101H(5,𝔽2)11yescharacteristic
126C22×C101H(5,𝔽2)12yes—
127C2×D101—24no—
128C2×D101—24no—
129C2×D101—24no—
130C2×D101—416no—
131C2×D101—42no—
132C2×D101—42no—
133C2×D101—42no—
134C2×D101—42no—
135C4×C101—22no—
136C4×C101—42no—
137C4×C101—42no—
138Dic101—24no—
139Dic101—42no—
140Dic101—42no—
141C2×C101—42no—
142C2×C101—42no—
143C2×C101#51812yes—
144C2×C101#51811yescharacteristic
145C2×C101#56012yes—
146C2×C101#56011yescharacteristic
147C2×C101H(3,𝔽4)11yescharacteristic
148D101—44no—
149#2375—1011nomaximal, Sylow
150C101#237511yescharacteristic, complemented, Sylow
151C2×#120—1011no—
152#390—1011no—
153#390—1012no—
154#388—1012no—
155#388—1012no—
156#388—1012no—
157#386—1012no—
158#386D10111yescharacteristic, complemented
159#379—1012no—
160C25—1011no—
161C22×D4—1012no—
162C22×D4C2×D10111yescharacteristic
163C23×C4C2×D10111yescharacteristic, complemented
164#120—1018no—
165C2×C4⋊C4—1012no—
166C2×C4⋊C4—1011no—
167C2×C22⋊C4—1011no—
168C2×C22⋊C4—1012no—
169C2×C22⋊C4—1012no—
170C2×C22⋊C4—1012no—
171C2×C22⋊C4—1012no—
172C2×C22⋊C4—1012no—
173C2×C22⋊C4—1012no—
174C2×C22⋊C4—1012no—
175C2×C22⋊C4C2×D10112yes—
176C2×C22⋊C4C2×D10111yescharacteristic, complemented
177C2×C42—1012no—
178C2×C42—1011no—
179#95—1012no—
180#95—1012no—
181#95C2×D10112yescomplemented
182C24—1014no—
183C24—1012no—
184C24—4042no—
185C24—4041no—
186C24—4041no—
187C24—4042no—
188C24C22×D10111yescharacteristic
189C24C22×D10111yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—1018no—
193C2×D4—4042no—
194C2×D4—4042no—
195C2×D4C4×D10114yes—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—1012no—
199C22×C4—1012no—
200C22×C4—1012no—
201C22×C4C22×D10112yes—
202C22×C4C22×D10112yes—
203C22×C4C22×D10111yescharacteristic
204C4⋊C4—4042no—
205C4⋊C4—4041no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—1014no—
209C22⋊C4—1018no—
210C22⋊C4—4042no—
211C22⋊C4—4042no—
212C22⋊C4—4042no—
213C22⋊C4—4042no—
214C22⋊C4—4042no—
215C22⋊C4—4042no—
216C42—4041no—
217C42—4042no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—1014no—
227C23—4048no—
228C23—4044no—
229C23—4048no—
230C23—4044no—
231C23—4044no—
232C23—2022no—
233C23—2022no—
234C23—2022no—
235C23—2022no—
236C23—2022no—
237C23—2022no—
238C23C2×C4×D10112yes—
239C23C2×C4×D10112yes—
240C23C23×D10111yescharacteristic
241D4—44no—
242D4—4048no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—1014no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—2022no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2022no—
260C2×C4—2022no—
261C2×C4C2×C4×D10112yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—40416no—
273C22—4042no—
274C22—4042no—
275C22—4042no—
276C22—4042no—
277C22—2024no—
278C22—2024no—
279C22—2024no—
280C22C22×C4×D10111yescharacteristic
281C22#1569230012yes—
282C22#1569230011yescharacteristic
283C22#1569230011yescharacteristic
284C22#1569230112yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—4042no—
289C4—4042no—
290C4—2024no—
291C2—42no—
292C2—42no—
293C2—4044no—
294C2#2026564912yes—
295C2#2026565011yescharacteristic
296C2#2026591512yes—
297C2#2026591711yescharacteristic
298C2#2026592011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#386×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692223C2
hypercenter#386D101
hypocenterC101#2375
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20265080C2semidirect
3. #20265077C2semidirect
4. #20265071C2semidirect
5. #20265069C2semidirect
6. #20265066C2semidirect
7. #20265056C2semidirect
8. #20265032C2semidirect
9. #386×C101C2semidirect
12. #15692223C4semidirect
13. C2×C4⋊Dic101C22semidirect
14. C2×#13108053C22semidirect
17. C2×C22⋊Dic101C22semidirect
23. C2×C4×Dic101C22semidirect
24. C2×C4×Dic101C22semidirect
25. #15692162C22semidirect
26. #15692162C22semidirect
28. C23×C4×C101C22semidirect
30. C2×C22⋊C4×C101C22semidirect
31. #95×C101C22semidirect
36. C23×D101C2×C4semidirect
48. C22⋊Dic101C2×C4semidirect
53. #13108051C2×C4semidirect
150. C101#2375semidirect
158. #386D101semidirect
163. C23×C4C2×D101semidirect
176. C2×C22⋊C4C2×D101semidirect
181. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922232
#202650802
#202650772
#202650712
#202650692
#202650662
#202650562
#202650322
#386×C1012
#2375101