Finite Group Databasebeta

← #38745291

Subgroups — #38745291

Order 25216.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D197C211yesmaximal, characteristic
2C2×#20263284C211yesmaximal, characteristic, complemented
3C2×#20263283C211yesmaximal, characteristic, complemented
4#27160982C211yesmaximal, characteristic, complemented
5#27160981C211yesmaximal, characteristic, complemented
6#27160980C211yesmaximal, characteristic, complemented
7#27160979C214yesmaximal, complemented
8#27160934C214yesmaximal, complemented
9#390×C197C211yesmaximal, characteristic, complemented
10C24×D197C2212yes—
11C22×D788C2211yescharacteristic
12C22×C22⋊D197C2212yes—
13C2×D4×D197—24no—
14C2×D4×D197—24no—
15C2×D4×D197C418yescomplemented
16C22×C4×D197C2211yescharacteristic
17#20263284—24no—
18#20263283—24no—
19C2×C4⋊Dic197C2211yescharacteristic, complemented
20C2×#15691337C2212yescomplemented
21C2×C22⋊Dic197C2214yescomplemented
22C2×#15691335C2212yes—
23C2×#15691335C2212yes—
24C2×#15691335C2214yescomplemented
25C2×#15691335C2214yescomplemented
26C2×C4×Dic197C2212yescomplemented
27C2×C4×Dic197C2211yescharacteristic, complemented
28C22×D4×C197C2211yescharacteristic
29C2×C4⋊C4×C197C2211yescharacteristic, complemented
30C2×C22⋊C4×C197C2214yescomplemented
31C2×C42×C197C2211yescharacteristic, complemented
32C23×D197—24no—
33C23×D197—24no—
34C23×D197—44no—
35C23×D197—44no—
36C23×D197C2×C418yes—
37C23×D197C2312yes—
38C23×D197C2311yescharacteristic
39C2×D788—22no—
40C2×D788—22no—
41C2×D788C2×C414yes—
42C2×C22⋊D197—24no—
43C2×C22⋊D197—24no—
44C2×C22⋊D197C2×C418yes—
45D4×D197—232no—
46C22×Dic197C2312yes—
47C22×Dic197C2312yes—
48C22×Dic197C2311yescharacteristic
49C2×C4×D197C2×C414yes—
50C2×C4×D197D414yes—
51C2×C4×D197D414yes—
52C4⋊Dic197—22no—
53#15691337—24no—
54C22⋊Dic197—44no—
55#15691335—24no—
56#15691335—24no—
57#15691335—44no—
58#15691335—44no—
59C4×Dic197—24no—
60C4×Dic197—22no—
61C24×C197C2312yes—
62C2×D4×C197—22no—
63C2×D4×C197—22no—
64C2×D4×C197C2×C414yes—
65C22×C4×C197C2312yes—
66C22×C4×C197C2312yes—
67C22×C4×C197C2311yescharacteristic
68C4⋊C4×C197—22no—
69C22⋊C4×C197—44no—
70C42×C197—22no—
71C22×D197—22no—
72C22×D197—22no—
73C22×D197—22no—
74C22×D197—22no—
75C22×D197—24no—
76C22×D197—24no—
77C22×D197—216no—
78C22×D197—416no—
79C22×D197—48no—
80C22×D197—416no—
81C22×D197—48no—
82C22×D197C22×C414yes—
83C22×D197C22×C412yes—
84C22×D197C2×D412yes—
85C22×D197C2×D412yes—
86D788—28no—
87C22⋊D197—216no—
88C2×Dic197—22no—
89C2×Dic197—22no—
90C2×Dic197—22no—
91C2×Dic197—24no—
92C2×Dic197—22no—
93C2×Dic197C22×C412yes—
94C2×Dic197C2×D412yes—
95C2×Dic197C2×D412yes—
96C4×D197C22⋊C4116yescomplemented
97C23×C197—22no—
98C23×C197—22no—
99C23×C197—24no—
100C23×C197—24no—
101C23×C197C22×C414yes—
102C23×C197C2411yescharacteristic
103D4×C197—28no—
104C2×C4×C197—22no—
105C2×C4×C197—22no—
106C2×C4×C197—22no—
107C2×C4×C197—24no—
108C2×C4×C197—22no—
109C2×C4×C197C22×C412yes—
110C2×C4×C197C2×D412yes—
111C2×C4×C197C2×D412yes—
112C22×C197—24no—
113C22×C197—28no—
114C22×C197—42no—
115C22×C197—44no—
116C22×C197—42no—
117C22×C197C23×C411yescharacteristic
118C22×C197C22×D411yescharacteristic
119C22×C197C22×D411yescharacteristic
120C22×C197H(5,𝔽2)12yes—
121C22×C197H(5,𝔽2)12yes—
122C2×D197—24no—
123C2×D197—22no—
124C2×D197—24no—
125C2×D197—22no—
126C2×D197—24no—
127C2×D197—28no—
128C2×D197—432no—
129C2×D197—42no—
130C2×D197—44no—
131C2×D197—42no—
132C2×D197C2×C22⋊C414yes—
133C4×C197—42no—
134C4×C197—42no—
135C4×C197C2×C22⋊C414yes—
136Dic197—42no—
137Dic197—42no—
138Dic197C2×C22⋊C414yes—
139C2×C197—44no—
140C2×C197C22×C22⋊C411yescharacteristic
141C2×C197#51812yes—
142C2×C197#53511yescharacteristic
143C2×C197#53512yes—
144C2×C197#53511yescharacteristic
145D197—24no—
146D197—44no—
147C197#230911yescharacteristic, complemented, Sylow
148#2309—1971nomaximal, Sylow
149C23×D4—1971no—
150C2×#117—1971no—
151C2×#117—1971no—
152#390—1971no—
153#390—1971no—
154#390—1974no—
155#390—1971no—
156#390D19711yescharacteristic, complemented
157#379—1974no—
158C25—1972no—
159C22×D4—3944no—
160C22×D4—3944no—
161C22×D4—1972no—
162C22×D4—1978no—
163C22×D4—1971no—
164C22×D4C2×D19711yescharacteristic
165C23×C4—1971no—
166#117—3944no—
167#117—3944no—
168C2×C4⋊C4—1972no—
169C2×C4⋊C4—1971no—
170C2×C4⋊C4C2×D19711yescharacteristic, complemented
171C2×C22⋊C4—1974no—
172C2×C22⋊C4—1974no—
173C2×C22⋊C4—1974no—
174C2×C22⋊C4—1972no—
175C2×C22⋊C4—1972no—
176C2×C22⋊C4C2×D19714yescomplemented
177C2×C42—1972no—
178C2×C42—1971no—
179C2×C42C2×D19711yescharacteristic, complemented
180C24—7884no—
181C24—7884no—
182C24—3944no—
183C24—3944no—
184C24—1978no—
185C24—1972no—
186C24—1971no—
187C24C22×D19712yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—39432no—
191C2×D4—3944no—
192C2×D4—3944no—
193C2×D4—3942no—
194C2×D4—3942no—
195C2×D4—1978no—
196C2×D4—1974no—
197C2×D4C4×D19714yes—
198C22×C4—1972no—
199C22×C4—1972no—
200C22×C4—1971no—
201C22×C4—1974no—
202C22×C4—1974no—
203C22×C4—1974no—
204C22×C4C22×D19712yes—
205C22×C4C22×D19712yes—
206C22×C4C22×D19711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—3944no—
209C4⋊C4—3942no—
210C22⋊C4—44no—
211C22⋊C4—7884no—
212C22⋊C4—7884no—
213C22⋊C4—7884no—
214C22⋊C4—3944no—
215C22⋊C4—3944no—
216C42—22no—
217C42—3944no—
218C42—3942no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—78816no—
224C23—7888no—
225C23—78816no—
226C23—7888no—
227C23—39416no—
228C23—3942no—
229C23—3942no—
230C23—3944no—
231C23—3944no—
232C23—3942no—
233C23—3942no—
234C23—1974no—
235C23—1972no—
236C23—1972no—
237C23—1972no—
238C23C2×C4×D19714yes—
239C23C23×D19711yescharacteristic
240D4—28no—
241D4—39416no—
242D4—3948no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—3942no—
249C2×C4—3942no—
250C2×C4—3942no—
251C2×C4—3942no—
252C2×C4—3944no—
253C2×C4—1972no—
254C2×C4—1972no—
255C2×C4—1972no—
256C2×C4—19716no—
257C2×C4C2×C4×D19712yes—
258C2×C4D4×D19712yes—
259C2×C4D4×D19712yes—
260C22—24no—
261C22—28no—
262C22—42no—
263C22—44no—
264C22—42no—
265C22—78832no—
266C22—7882no—
267C22—7884no—
268C22—7882no—
269C22—3944no—
270C22—3948no—
271C22—3942no—
272C22—3944no—
273C22—3942no—
274C22—3944no—
275C22—1974no—
276C22C22×C4×D19711yescharacteristic
277C22C2×D4×D19711yescharacteristic
278C22C2×D4×D19711yescharacteristic
279C22#2026337112yes—
280C22#2026337212yes—
281C4—42no—
282C4—42no—
283C4—7882no—
284C4—7882no—
285C4—1974no—
286C4C22⋊C4×D19714yes—
287C2—44no—
288C2—7884no—
289C2—3944no—
290C2C2×C22⋊C4×D19711yescharacteristic
291C2#2716155112yes—
292C2#2716164211yescharacteristic
293C2#2716164512yes—
294C2#2716164711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#390×C197C2
socle, cosocleC23×C197C24
prime radicalC22×D4×D197C2
hypercenter#390D197
hypocenterC197#2309
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#20263284C2semidirect
3. C2×#20263283C2semidirect
4. #27160982C2semidirect
5. #27160981C2semidirect
6. #27160980C2semidirect
7. #27160979C2semidirect
8. #27160934C2semidirect
9. #390×C197C2semidirect
15. C2×D4×D197C4semidirect
19. C2×C4⋊Dic197C22semidirect
20. C2×#15691337C22semidirect
21. C2×C22⋊Dic197C22semidirect
24. C2×#15691335C22semidirect
25. C2×#15691335C22semidirect
26. C2×C4×Dic197C22semidirect
27. C2×C4×Dic197C22semidirect
29. C2×C4⋊C4×C197C22semidirect
30. C2×C22⋊C4×C197C22semidirect
31. C2×C42×C197C22semidirect
96. C4×D197C22⋊C4semidirect
147. C197#2309semidirect
156. #390D197semidirect
170. C2×C4⋊C4C2×D197semidirect
176. C2×C22⋊C4C2×D197semidirect
179. C2×C42C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C2×#202632842
C2×#202632832
#271609822
#271609812
#271609802
#271609792
#271609342
#390×C1972
#2309197