Finite Group Databasebeta

← #49130652

Subgroups — #49130652

Order 36224.

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#31827301C211yesmaximal, characteristic, complemented
2#31826976C212yesmaximal, complemented
3#31826654C214yesmaximal, complemented
4#31826651C214yesmaximal, complemented
5#31826439C211yesmaximal, characteristic, complemented
6#31826438C212yesmaximal, complemented
7#525×C283C211yesmaximal, characteristic, complemented
8#22583658—21no—
9#22583657—22no—
10#22583656—22no—
11#22583655—21no—
12#22583655C2214yes—
13C2×#17985392—22no—
14C2×#17985392C2212yes—
15#41×D283—24no—
16#22583593C2214yescomplemented
17#22583589—22no—
18#22583584—22no—
19#22583579C2214yescomplemented
20C4×C22⋊D283—24no—
21#22583569C2212yescomplemented
22#22583525C2214yescomplemented
23#22583524C2218yescomplemented
24C2×#41×C283C2211yescharacteristic
25C23×C4×C283C2211yescharacteristic, complemented
26#123×C283C2212yescomplemented
27C22⋊Q8×C283—21no—
28#121×C283—21no—
29#120×C283C2212yescomplemented
30C4×D4×C283—22no—
31#117×C283C2211yescharacteristic, complemented
32#17985393—24no—
33#17985392—24no—
34#17985392—24no—
35#17985392—22no—
36#17985392—22no—
37#17985392—24no—
38#17985392—22no—
39#17985392—22no—
40#17985391—24no—
41#17985391—28no—
42C2×Dic566—22no—
43C2×Dic566D412yescomplemented
44Q8×D283—24no—
45C2×D1132—22no—
46C2×D1132D412yescomplemented
47C2×C22⋊D283—24no—
48C2×C22⋊D283—24no—
49C2×C22⋊D283C2314yes—
50D4×D283—24no—
51D4×D283—28no—
52C2×C4×D283—24no—
53C2×C4×D283D414yescomplemented
54C4⋊Dic283—22no—
55Dic283⋊C4—24no—
56C22⋊Dic283—24no—
57C22⋊Dic283C2314yes—
58(C2×D283)⋊C4—24no—
59C4×Dic283—22no—
60C24×C283C2311yescharacteristic
61#41×C283—22no—
62#41×C283—22no—
63C2×Q8×C283D411yescharacteristic, complemented
64C2×D4×C283—22no—
65C2×D4×C283D411yescharacteristic, complemented
66C2×D4×C283C2312yes—
67C22×C4×C283—22no—
68C22×C4×C283—22no—
69C22×C4×C283—21no—
70C22×C4×C283—21no—
71C22×C4×C283D412yescomplemented
72C22×C4×C283C2311yescharacteristic
73C22×C4×C283C2311yescharacteristic
74C4⋊C4×C283—21no—
75C4⋊C4×C283—22no—
76C22⋊C4×C283—22no—
77C22⋊C4×C283—22no—
78C22⋊C4×C283C2312yes—
79C42×C283—21no—
80C22×D283—24no—
81C22×D283C2×D414yes—
82Dic566—24no—
83Dic566—22no—
84Dic566—22no—
85D1132—24no—
86D1132—22no—
87D1132—22no—
88C22⋊D283—28no—
89C22⋊D283—24no—
90C22⋊D283—24no—
91C22⋊D283—48no—
92C2×Dic283—24no—
93C2×Dic283—24no—
94C2×Dic283C2×D414yes—
95C4×D283—28no—
96C4×D283—24no—
97C4×D283—24no—
98C23×C283—21no—
99C23×C283—22no—
100C23×C283—42no—
101C23×C283C2×D412yes—
102C23×C283C2411yescharacteristic
103Q8×C283—22no—
104D4×C283—24no—
105D4×C283—22no—
106D4×C283—44no—
107C2×C4×C283—22no—
108C2×C4×C283—22no—
109C2×C4×C283—22no—
110C2×C4×C283—21no—
111C2×C4×C283—22no—
112C2×C4×C283—41no—
113C2×C4×C283—42no—
114C2×C4×C283—41no—
115C2×C4×C283C2×D412yes—
116C2×C4×C283C2×D412yes—
117C2×C4×C283C2×D412yes—
118C2×C4×C283C2×D411yescharacteristic
119C2×C4×C283C2×D411yescharacteristic
120C22×C283—22no—
121C22×C283—24no—
122C22×C283—22no—
123C22×C283—41no—
124C22×C283—42no—
125C22×C283—42no—
126C22×C283—41no—
127C22×C283—41no—
128C22×C283C22×D412yes—
129C22×C283C22×D411yescharacteristic
130C2×D283—24no—
131C2×D283—44no—
132C2×D283—84no—
133C4×C283—22no—
134C4×C283—42no—
135C4×C283—41no—
136C4×C283#12011yescharacteristic
137C4×C283#12012yes—
138C4×C283#12011yescharacteristic
139Dic283—24no—
140Dic283—44no—
141C2×C283—22no—
142C2×C283—21no—
143C2×C283—42no—
144C2×C283—42no—
145C2×C283C2×#12011yescharacteristic
146D283—44no—
147C283#302511yescharacteristic, complemented, Sylow
148#3025—2831nomaximal, Sylow
149#585—2831no—
150#525—2832no—
151#525D28311yescharacteristic, complemented
152#458—2834no—
153B(4,𝔽2)—2834no—
154#410—2831no—
155#410—2832no—
1562−1+4—5662no—
1572−1+4—5661no—
158H(5,𝔽2)—2834no—
159H(5,𝔽2)—5662no—
160H(5,𝔽2)—5661no—
161C2×#41—2832no—
162C2×#41—5664no—
163C2×#41—5662no—
164C2×#41C2×D28311yescharacteristic
165C23×C4C2×D28311yescharacteristic, complemented
166#123—2834no—
167#123C2×D28312yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—5662no—
170#121—21no—
171#121—5662no—
172#120—2834no—
173#120C2×D28312yescomplemented
174C4×D4—22no—
175C4×D4—5664no—
176#117—2832no—
177#117C2×D28311yescharacteristic, complemented
178#99—2834no—
179#99—2838no—
180C24C22×D28311yescharacteristic
181#41—22no—
182#41—22no—
183#41—5664no—
184#41—5664no—
185#41—5668no—
186#41—5664no—
187#41—5664no—
188#41—5662no—
189#41—5662no—
190#41—5662no—
191#41—5662no—
192#41—5664no—
193C2×Q8—2832no—
194C2×Q8—5662no—
195C2×Q8—5664no—
196C2×Q8C22⋊D28311yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—2834no—
199C2×D4—2832no—
200C2×D4—5662no—
201C2×D4—5668no—
202C2×D4—5664no—
203C2×D4—5664no—
204C2×D4—5664no—
205C2×D4C22⋊D28311yescharacteristic, complemented
206C2×D4C22×D28312yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—2834no—
212C22×C4—5664no—
213C22×C4C22⋊D28312yescomplemented
214C22×C4C22×D28311yescharacteristic
215C22×C4C22×D28311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—5664no—
219C4⋊C4—5662no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—2834no—
223C22⋊C4—5664no—
224C22⋊C4—5664no—
225C22⋊C4C22×D28312yes—
226C42—21no—
227C42—5662no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—2834no—
232C23—5664no—
233C23C2×C22⋊D28312yes—
234C23C23×D28311yescharacteristic
235Q8—22no—
236Q8—5664no—
237Q8—5662no—
238Q8—5662no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—5664no—
243D4—5668no—
244D4—5664no—
245D4—5664no—
246D4—5662no—
247D4—5662no—
248D4—11328no—
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—2834no—
258C2×C4—5668no—
259C2×C4—5664no—
260C2×C4—5664no—
261C2×C4—5664no—
262C2×C4—5664no—
263C2×C4D4×D28312yes—
264C2×C4D4×D28312yes—
265C2×C4C2×C22⋊D28312yes—
266C2×C4C2×C22⋊D28311yescharacteristic
267C2×C4C2×C22⋊D28311yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—5664no—
277C22—11324no—
278C22—22644no—
279C22C2×D4×D28312yes—
280C22C22×C22⋊D28311yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—5664no—
285C4—11324no—
286C4#2258357811yescharacteristic
287C4#2258357812yes—
288C4#2258357811yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—11324no—
294C2C2×#2258357811yescharacteristic
subgroupG/H
centerC4#22583578
commutator, cosocleC23×C283C24
FrattiniC23C23×D283
Fitting#525×C283C2
socleC2×C283C2×#120
hypercenter#525D283
hypocenterC283#3025
LCSG ▷ C23×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #31827301C2semidirect
2. #31826976C2semidirect
3. #31826654C2semidirect
4. #31826651C2semidirect
5. #31826439C2semidirect
6. #31826438C2semidirect
7. #525×C283C2semidirect
16. #22583593C22semidirect
19. #22583579C22semidirect
21. #22583569C22semidirect
22. #22583525C22semidirect
23. #22583524C22semidirect
25. C23×C4×C283C22semidirect
26. #123×C283C22semidirect
29. #120×C283C22semidirect
31. #117×C283C22semidirect
43. C2×Dic566D4semidirect
46. C2×D1132D4semidirect
53. C2×C4×D283D4semidirect
63. C2×Q8×C283D4semidirect
65. C2×D4×C283D4semidirect
71. C22×C4×C283D4semidirect
147. C283#3025semidirect
151. #525D283semidirect
165. C23×C4C2×D283semidirect
167. #123C2×D283semidirect
173. #120C2×D283semidirect
177. #117C2×D283semidirect
196. C2×Q8C22⋊D283semidirect
205. C2×D4C22⋊D283semidirect
213. C22×C4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318269762
#318266542
#318266512
#318264392
#318264382
#525×C2832
#3025283