Finite Group Databasebeta

← #28372423

Subgroups — #28372423

Order 13952.

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#20295862C211yesmaximal, characteristic, complemented
2#20295537C212yesmaximal, complemented
3#20295215C214yesmaximal, complemented
4#20295212C214yesmaximal, complemented
5#20295000C211yesmaximal, characteristic, complemented
6#20294999C212yesmaximal, complemented
7#525×C109C211yesmaximal, characteristic, complemented
8#16812747—21no—
9#16812746—22no—
10#16812745—22no—
11#16812744—21no—
12#16812744C2214yes—
13C2×#13182029—22no—
14C2×#13182029C2212yes—
15#41×D109—24no—
16#16812682C2214yescomplemented
17#16812678—22no—
18#16812673—22no—
19#16812668C2214yescomplemented
20C4×C22⋊D109—24no—
21#16812658C2212yescomplemented
22#16812614C2214yescomplemented
23#16812613C2218yescomplemented
24C2×#41×C109C2211yescharacteristic
25C23×C4×C109C2211yescharacteristic, complemented
26#123×C109C2212yescomplemented
27C22⋊Q8×C109—21no—
28#121×C109—21no—
29#120×C109C2212yescomplemented
30C4×D4×C109—22no—
31#117×C109C2211yescharacteristic, complemented
32#13182030—24no—
33#13182029—24no—
34#13182029—24no—
35#13182029—22no—
36#13182029—22no—
37#13182029—24no—
38#13182029—22no—
39#13182029—22no—
40#13182028—24no—
41#13182028—28no—
42C2×Dic218—22no—
43C2×Dic218D412yescomplemented
44Q8×D109—24no—
45C2×D436—22no—
46C2×D436D412yescomplemented
47C2×C22⋊D109—24no—
48C2×C22⋊D109—24no—
49C2×C22⋊D109C2314yes—
50D4×D109—24no—
51D4×D109—28no—
52C2×C4×D109—24no—
53C2×C4×D109D414yescomplemented
54C4⋊Dic109—22no—
55#13182008—24no—
56C22⋊Dic109—24no—
57C22⋊Dic109C2314yes—
58#13182006—24no—
59C4×Dic109—22no—
60C24×C109C2311yescharacteristic
61#41×C109—22no—
62#41×C109—22no—
63C2×Q8×C109D411yescharacteristic, complemented
64C2×D4×C109—22no—
65C2×D4×C109D411yescharacteristic, complemented
66C2×D4×C109C2312yes—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109—22no—
70C22×C4×C109—22no—
71C22×C4×C109D412yescomplemented
72C22×C4×C109C2311yescharacteristic
73C22×C4×C109C2311yescharacteristic
74C4⋊C4×C109—21no—
75C4⋊C4×C109—22no—
76C22⋊C4×C109—22no—
77C22⋊C4×C109—22no—
78C22⋊C4×C109C2312yes—
79C42×C109—21no—
80C23×C109—21no—
81C23×C109—22no—
82C23×C109—42no—
83C23×C109C2×D412yes—
84C23×C109C2411yescharacteristic
85C22×D109—24no—
86C22×D109C2×D414yes—
87Q8×C109—22no—
88D4×C109—24no—
89D4×C109—22no—
90D4×C109—44no—
91C2×C4×C109—22no—
92C2×C4×C109—22no—
93C2×C4×C109—22no—
94C2×C4×C109—21no—
95C2×C4×C109—22no—
96C2×C4×C109—41no—
97C2×C4×C109—42no—
98C2×C4×C109—41no—
99C2×C4×C109C2×D412yes—
100C2×C4×C109C2×D412yes—
101C2×C4×C109C2×D412yes—
102C2×C4×C109C2×D411yescharacteristic
103C2×C4×C109C2×D411yescharacteristic
104C22⋊D109—28no—
105C22⋊D109—24no—
106C22⋊D109—24no—
107C22⋊D109—48no—
108C2×Dic109—24no—
109C2×Dic109—24no—
110C2×Dic109C2×D414yes—
111D436—24no—
112D436—22no—
113D436—22no—
114C4×D109—28no—
115C4×D109—24no—
116C4×D109—24no—
117Dic218—24no—
118Dic218—22no—
119Dic218—22no—
120C22×C109—24no—
121C22×C109—22no—
122C22×C109—22no—
123C22×C109—41no—
124C22×C109—42no—
125C22×C109—42no—
126C22×C109—41no—
127C22×C109—41no—
128C22×C109C22×D412yes—
129C22×C109C22×D411yescharacteristic
130C2×D109—24no—
131C2×D109—44no—
132C2×D109—84no—
133C4×C109—22no—
134C4×C109—42no—
135C4×C109—41no—
136C4×C109#12011yescharacteristic
137C4×C109#12012yes—
138C4×C109#12011yescharacteristic
139Dic109—24no—
140Dic109—44no—
141C2×C109—22no—
142C2×C109—21no—
143C2×C109—42no—
144C2×C109—42no—
145C2×C109C2×#12011yescharacteristic
146D109—44no—
147#3025—1091nomaximal, Sylow
148C109#302511yescharacteristic, complemented, Sylow
149#585—1091no—
150#525—1092no—
151#525D10911yescharacteristic, complemented
152#458—1094no—
153B(4,𝔽2)—1094no—
154#410—1091no—
155#410—1092no—
1562−1+4—2182no—
1572−1+4—2181no—
158H(5,𝔽2)—1094no—
159H(5,𝔽2)—2182no—
160H(5,𝔽2)—2181no—
161C2×#41—1092no—
162C2×#41—2184no—
163C2×#41—2182no—
164C2×#41C2×D10911yescharacteristic
165C23×C4C2×D10911yescharacteristic, complemented
166#123—1094no—
167#123C2×D10912yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—2182no—
170#121—21no—
171#121—2182no—
172#120—1094no—
173#120C2×D10912yescomplemented
174C4×D4—22no—
175C4×D4—2184no—
176#117—1092no—
177#117C2×D10911yescharacteristic, complemented
178#99—1094no—
179#99—1098no—
180C24C22×D10911yescharacteristic
181#41—22no—
182#41—22no—
183#41—2184no—
184#41—2184no—
185#41—2188no—
186#41—2184no—
187#41—2184no—
188#41—2182no—
189#41—2182no—
190#41—2182no—
191#41—2182no—
192#41—2184no—
193C2×Q8—1092no—
194C2×Q8—2182no—
195C2×Q8—2184no—
196C2×Q8C22⋊D10911yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—1094no—
199C2×D4—1092no—
200C2×D4—2182no—
201C2×D4—2188no—
202C2×D4—2184no—
203C2×D4—2184no—
204C2×D4—2184no—
205C2×D4C22⋊D10911yescharacteristic, complemented
206C2×D4C22×D10912yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—1094no—
212C22×C4—2184no—
213C22×C4C22⋊D10912yescomplemented
214C22×C4C22×D10911yescharacteristic
215C22×C4C22×D10911yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—2184no—
219C4⋊C4—2182no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—1094no—
223C22⋊C4—2184no—
224C22⋊C4—2184no—
225C22⋊C4C22×D10912yes—
226C42—21no—
227C42—2182no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—1094no—
232C23—2184no—
233C23C2×C22⋊D10912yes—
234C23C23×D10911yescharacteristic
235Q8—22no—
236Q8—2184no—
237Q8—2182no—
238Q8—2182no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—4368no—
243D4—2184no—
244D4—2188no—
245D4—2184no—
246D4—2184no—
247D4—2182no—
248D4—2182no—
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—1094no—
258C2×C4—2188no—
259C2×C4—2184no—
260C2×C4—2184no—
261C2×C4—2184no—
262C2×C4—2184no—
263C2×C4D4×D10912yes—
264C2×C4D4×D10912yes—
265C2×C4C2×C22⋊D10912yes—
266C2×C4C2×C22⋊D10911yescharacteristic
267C2×C4C2×C22⋊D10911yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—8724no—
277C22—4364no—
278C22—2184no—
279C22C2×D4×D10912yes—
280C22C22×C22⋊D10911yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—4364no—
285C4—2184no—
286C4#1681266711yescharacteristic
287C4#1681266712yes—
288C4#1681266711yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—4364no—
294C2C2×#1681266711yescharacteristic
subgroupG/H
centerC4#16812667
commutator, cosocleC23×C109C24
FrattiniC23C23×D109
Fitting#525×C109C2
socleC2×C109C2×#120
hypercenter#525D109
hypocenterC109#3025
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20295862C2semidirect
2. #20295537C2semidirect
3. #20295215C2semidirect
4. #20295212C2semidirect
5. #20295000C2semidirect
6. #20294999C2semidirect
7. #525×C109C2semidirect
16. #16812682C22semidirect
19. #16812668C22semidirect
21. #16812658C22semidirect
22. #16812614C22semidirect
23. #16812613C22semidirect
25. C23×C4×C109C22semidirect
26. #123×C109C22semidirect
29. #120×C109C22semidirect
31. #117×C109C22semidirect
43. C2×Dic218D4semidirect
46. C2×D436D4semidirect
53. C2×C4×D109D4semidirect
63. C2×Q8×C109D4semidirect
65. C2×D4×C109D4semidirect
71. C22×C4×C109D4semidirect
148. C109#3025semidirect
151. #525D109semidirect
165. C23×C4C2×D109semidirect
167. #123C2×D109semidirect
173. #120C2×D109semidirect
177. #117C2×D109semidirect
196. C2×Q8C22⋊D109semidirect
205. C2×D4C22⋊D109semidirect
213. C22×C4C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202958622
#202955372
#202952152
#202952122
#202950002
#202949992
#525×C1092
#3025109