Finite Group Databasebeta

← #27180523

Subgroups — #27180523

Order 12928.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#20266013C211yesmaximal, characteristic, complemented
2#20265688C212yesmaximal, complemented
3#20265366C214yesmaximal, complemented
4#20265363C214yesmaximal, complemented
5#20265151C211yesmaximal, characteristic, complemented
6#20265150C212yesmaximal, complemented
7#525×C101C211yesmaximal, characteristic, complemented
8#15692303—21no—
9#15692302—22no—
10#15692301—22no—
11#15692300—21no—
12#15692300C2214yes—
13C2×#13108074—22no—
14C2×#13108074C2212yes—
15#41×D101—24no—
16#15692238C2214yescomplemented
17#15692234—22no—
18#15692229—22no—
19#15692224C2214yescomplemented
20C4×C22⋊D101—24no—
21#15692214C2212yescomplemented
22#15692170C2214yescomplemented
23#15692169C2218yescomplemented
24C2×#41×C101C2211yescharacteristic
25C23×C4×C101C2211yescharacteristic, complemented
26#123×C101C2212yescomplemented
27C22⋊Q8×C101—21no—
28#121×C101—21no—
29#120×C101C2212yescomplemented
30C4×D4×C101—22no—
31#117×C101C2211yescharacteristic, complemented
32#13108075—24no—
33#13108074—24no—
34#13108074—24no—
35#13108074—22no—
36#13108074—22no—
37#13108074—24no—
38#13108074—22no—
39#13108074—22no—
40#13108073—24no—
41#13108073—28no—
42C2×Dic202—22no—
43C2×Dic202D412yescomplemented
44Q8×D101—24no—
45C2×D404—22no—
46C2×D404D412yescomplemented
47C2×C22⋊D101—24no—
48C2×C22⋊D101—24no—
49C2×C22⋊D101C2314yes—
50D4×D101—24no—
51D4×D101—28no—
52C2×C4×D101—24no—
53C2×C4×D101D414yescomplemented
54C4⋊Dic101—22no—
55#13108053—24no—
56C22⋊Dic101—24no—
57C22⋊Dic101C2314yes—
58#13108051—24no—
59C4×Dic101—22no—
60C24×C101C2311yescharacteristic
61#41×C101—22no—
62#41×C101—22no—
63C2×Q8×C101D411yescharacteristic, complemented
64C2×D4×C101—22no—
65C2×D4×C101D411yescharacteristic, complemented
66C2×D4×C101C2312yes—
67C22×C4×C101—21no—
68C22×C4×C101—21no—
69C22×C4×C101—22no—
70C22×C4×C101—22no—
71C22×C4×C101D412yescomplemented
72C22×C4×C101C2311yescharacteristic
73C22×C4×C101C2311yescharacteristic
74C4⋊C4×C101—21no—
75C4⋊C4×C101—22no—
76C22⋊C4×C101—22no—
77C22⋊C4×C101—22no—
78C22⋊C4×C101C2312yes—
79C42×C101—21no—
80C23×C101—21no—
81C23×C101—22no—
82C23×C101—42no—
83C23×C101C2×D412yes—
84C23×C101C2411yescharacteristic
85C22×D101—24no—
86C22×D101C2×D414yes—
87Q8×C101—22no—
88D4×C101—24no—
89D4×C101—22no—
90D4×C101—44no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—22no—
94C2×C4×C101—21no—
95C2×C4×C101—22no—
96C2×C4×C101—41no—
97C2×C4×C101—42no—
98C2×C4×C101—41no—
99C2×C4×C101C2×D412yes—
100C2×C4×C101C2×D412yes—
101C2×C4×C101C2×D412yes—
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2×D411yescharacteristic
104C22⋊D101—28no—
105C22⋊D101—24no—
106C22⋊D101—24no—
107C22⋊D101—48no—
108C2×Dic101—24no—
109C2×Dic101—24no—
110C2×Dic101C2×D414yes—
111D404—24no—
112D404—22no—
113D404—22no—
114C4×D101—28no—
115C4×D101—24no—
116C4×D101—24no—
117Dic202—24no—
118Dic202—22no—
119Dic202—22no—
120C22×C101—24no—
121C22×C101—22no—
122C22×C101—22no—
123C22×C101—41no—
124C22×C101—42no—
125C22×C101—42no—
126C22×C101—41no—
127C22×C101—41no—
128C22×C101C22×D412yes—
129C22×C101C22×D411yescharacteristic
130C2×D101—24no—
131C2×D101—44no—
132C2×D101—84no—
133C4×C101—22no—
134C4×C101—42no—
135C4×C101—41no—
136C4×C101#12011yescharacteristic
137C4×C101#12012yes—
138C4×C101#12011yescharacteristic
139Dic101—24no—
140Dic101—44no—
141C2×C101—22no—
142C2×C101—21no—
143C2×C101—42no—
144C2×C101—42no—
145C2×C101C2×#12011yescharacteristic
146D101—44no—
147#3025—1011nomaximal, Sylow
148C101#302511yescharacteristic, complemented, Sylow
149#585—1011no—
150#525—1012no—
151#525D10111yescharacteristic, complemented
152#458—1014no—
153B(4,𝔽2)—1014no—
154#410—1011no—
155#410—1012no—
1562−1+4—2022no—
1572−1+4—2021no—
158H(5,𝔽2)—1014no—
159H(5,𝔽2)—2022no—
160H(5,𝔽2)—2021no—
161C2×#41—1012no—
162C2×#41—2024no—
163C2×#41—2022no—
164C2×#41C2×D10111yescharacteristic
165C23×C4C2×D10111yescharacteristic, complemented
166#123—1014no—
167#123C2×D10112yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—2022no—
170#121—21no—
171#121—2022no—
172#120—1014no—
173#120C2×D10112yescomplemented
174C4×D4—22no—
175C4×D4—2024no—
176#117—1012no—
177#117C2×D10111yescharacteristic, complemented
178#99—1014no—
179#99—1018no—
180C24C22×D10111yescharacteristic
181#41—22no—
182#41—22no—
183#41—2024no—
184#41—2024no—
185#41—2028no—
186#41—2024no—
187#41—2024no—
188#41—2022no—
189#41—2022no—
190#41—2022no—
191#41—2022no—
192#41—2024no—
193C2×Q8—1012no—
194C2×Q8—2022no—
195C2×Q8—2024no—
196C2×Q8C22⋊D10111yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—1014no—
199C2×D4—1012no—
200C2×D4—2022no—
201C2×D4—2028no—
202C2×D4—2024no—
203C2×D4—2024no—
204C2×D4—2024no—
205C2×D4C22⋊D10111yescharacteristic, complemented
206C2×D4C22×D10112yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—1014no—
212C22×C4—2024no—
213C22×C4C22⋊D10112yescomplemented
214C22×C4C22×D10111yescharacteristic
215C22×C4C22×D10111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—2024no—
219C4⋊C4—2022no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—1014no—
223C22⋊C4—2024no—
224C22⋊C4—2024no—
225C22⋊C4C22×D10112yes—
226C42—21no—
227C42—2022no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—1014no—
232C23—2024no—
233C23C2×C22⋊D10112yes—
234C23C23×D10111yescharacteristic
235Q8—22no—
236Q8—2024no—
237Q8—2022no—
238Q8—2022no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—4048no—
243D4—2024no—
244D4—2028no—
245D4—2024no—
246D4—2024no—
247D4—2022no—
248D4—2022no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—1014no—
258C2×C4—2028no—
259C2×C4—2024no—
260C2×C4—2024no—
261C2×C4—2024no—
262C2×C4—2024no—
263C2×C4D4×D10112yes—
264C2×C4D4×D10112yes—
265C2×C4C2×C22⋊D10112yes—
266C2×C4C2×C22⋊D10111yescharacteristic
267C2×C4C2×C22⋊D10111yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—8084no—
277C22—4044no—
278C22—2024no—
279C22C2×D4×D10112yes—
280C22C22×C22⋊D10111yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—4044no—
285C4—2024no—
286C4#1569222311yescharacteristic
287C4#1569222312yes—
288C4#1569222311yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—4044no—
294C2C2×#1569222311yescharacteristic
subgroupG/H
centerC4#15692223
commutator, cosocleC23×C101C24
FrattiniC23C23×D101
Fitting#525×C101C2
socleC2×C101C2×#120
hypercenter#525D101
hypocenterC101#3025
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20266013C2semidirect
2. #20265688C2semidirect
3. #20265366C2semidirect
4. #20265363C2semidirect
5. #20265151C2semidirect
6. #20265150C2semidirect
7. #525×C101C2semidirect
16. #15692238C22semidirect
19. #15692224C22semidirect
21. #15692214C22semidirect
22. #15692170C22semidirect
23. #15692169C22semidirect
25. C23×C4×C101C22semidirect
26. #123×C101C22semidirect
29. #120×C101C22semidirect
31. #117×C101C22semidirect
43. C2×Dic202D4semidirect
46. C2×D404D4semidirect
53. C2×C4×D101D4semidirect
63. C2×Q8×C101D4semidirect
65. C2×D4×C101D4semidirect
71. C22×C4×C101D4semidirect
148. C101#3025semidirect
151. #525D101semidirect
165. C23×C4C2×D101semidirect
167. #123C2×D101semidirect
173. #120C2×D101semidirect
177. #117C2×D101semidirect
196. C2×Q8C22⋊D101semidirect
205. C2×D4C22⋊D101semidirect
213. C22×C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
#202656882
#202653662
#202653632
#202651512
#202651502
#525×C1012
#3025101