Finite Group Databasebeta

← #385×D197 #38745120

Subgroups — #385×D197 #38745120

Order 25216.

1232 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D197C211yesmaximal, characteristic
2C2×C4⋊C4×D197C212yesmaximal, complemented
3C2×C22⋊C4×D197C211yesmaximal, characteristic
4C2×C22⋊C4×D197C211yesmaximal, characteristic
5#27160954C211yesmaximal, characteristic, complemented
6#27160953C212yesmaximal, complemented
7#27160952C211yesmaximal, characteristic, complemented
8#27160951C211yesmaximal, characteristic, complemented
9#27160950C212yesmaximal, complemented
10#95×D197C212yesmaximal, complemented
11#385×C197C211yesmaximal, characteristic, complemented
12C24×D197C2211yescharacteristic
13C23×Dic197C2211yescharacteristic
14C22×C4×D197—24no—
15C22×C4×D197—28no—
16C22×C4×D197C2212yes—
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C22×C4×D197C2212yes—
20C22×C4×D197C2212yes—
21C2×C4⋊Dic197C2212yescomplemented
22C2×#15691337C2212yescomplemented
23C2×#15691337C2212yescomplemented
24C4⋊C4×D197—28no—
25C2×C22⋊Dic197C2211yescharacteristic
26C2×C22⋊Dic197C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2212yes—
30C22⋊C4×D197—24no—
31C22⋊C4×D197C418yescomplemented
32#20263233C2212yescomplemented
33#20263233C2212yescomplemented
34#20263233C2212yescomplemented
35C23×C4×C197C2211yescharacteristic
36C2×C4⋊C4×C197C2212yescomplemented
37C2×C22⋊C4×C197C2211yescharacteristic
38C2×C22⋊C4×C197C2211yescharacteristic
39#95×C197C2212yescomplemented
40C23×D197—24no—
41C23×D197—24no—
42C23×D197C2×C414yes—
43C23×D197D414yes—
44C23×D197Q814yes—
45C23×D197C2311yescharacteristic
46C23×D197C2311yescharacteristic
47C22×Dic197—24no—
48C22×Dic197—22no—
49C22×Dic197C2311yescharacteristic
50C22×Dic197C2311yescharacteristic
51C22×Dic197C2312yes—
52C22×Dic197C2312yes—
53C2×C4×D197—232no—
54C2×C4×D197—24no—
55C2×C4×D197—24no—
56C2×C4×D197—24no—
57C2×C4×D197—24no—
58C2×C4×D197—24no—
59C2×C4×D197—24no—
60C2×C4×D197—22no—
61C2×C4×D197—22no—
62C2×C4×D197—24no—
63C2×C4×D197—24no—
64C2×C4×D197—24no—
65C2×C4×D197C2×C418yescomplemented
66C2×C4×D197D418yes—
67C4⋊Dic197—24no—
68#15691337—24no—
69#15691337—24no—
70C22⋊Dic197—22no—
71C22⋊Dic197C2×C414yescomplemented
72#15691335—22no—
73#15691335—22no—
74#15691335C2×C418yescomplemented
75C24×C197C2311yescharacteristic
76C22×C4×C197—22no—
77C22×C4×C197—24no—
78C22×C4×C197C2311yescharacteristic
79C22×C4×C197C2311yescharacteristic
80C22×C4×C197C2312yes—
81C22×C4×C197C2312yes—
82C4⋊C4×C197—24no—
83C22⋊C4×C197—22no—
84C22⋊C4×C197C2×C414yescomplemented
85C22×D197—22no—
86C22×D197—22no—
87C22×D197—216no—
88C22×D197—28no—
89C22×D197—216no—
90C22×D197—28no—
91C22×D197C4⋊C4116yescomplemented
92C22×D197C22×C412yes—
93C22×D197C22×C412yes—
94C22×D197C2×D412yes—
95C22×D197C2×D412yes—
96C22×D197C2×D414yes—
97C22×D197C2×Q812yes—
98C22×D197C2×Q812yes—
99C22×D197#4114yes—
100C2×Dic197—28no—
101C2×Dic197—22no—
102C2×Dic197—22no—
103C2×Dic197—22no—
104C2×Dic197—22no—
105C2×Dic197—21no—
106C2×Dic197—21no—
107C2×Dic197—22no—
108C2×Dic197—22no—
109C2×Dic197C22×C414yescomplemented
110C2×Dic197C2×D414yes—
111C4×D197—216no—
112C4×D197—216no—
113C4×D197—216no—
114C4×D197—44no—
115C4×D197—44no—
116C23×C197—22no—
117C23×C197—22no—
118C23×C197C22×C412yes—
119C23×C197C2×D412yes—
120C23×C197C2×Q812yes—
121C23×C197C2411yescharacteristic
122C2×C4×C197—28no—
123C2×C4×C197—22no—
124C2×C4×C197—22no—
125C2×C4×C197—22no—
126C2×C4×C197—22no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—22no—
130C2×C4×C197—22no—
131C2×C4×C197C22×C414yescomplemented
132C2×C4×C197C2×D414yes—
133C22×C197—24no—
134C22×C197—22no—
135C22×C197—24no—
136C22×C197—22no—
137C22×C197C2×C4⋊C414yescomplemented
138C22×C197C23×C411yescharacteristic
139C22×C197C22×D411yescharacteristic
140C22×C197C22×D412yes—
141C22×C197C22×Q811yescharacteristic
142C22×C197C2×#4112yes—
143C2×D197—24no—
144C2×D197—22no—
145C2×D197—24no—
146C2×D197—22no—
147C2×D197—232no—
148C2×D197C2×C4⋊C414yes—
149C2×D197C2×C4⋊C414yescomplemented
150C2×D197C4×D418yes—
151C2×D197#12014yes—
152C2×D197C22⋊Q818yes—
153C2×D197#12314yes—
154C4×C197—24no—
155C4×C197—24no—
156C4×C197—41no—
157C4×C197—41no—
158Dic197—24no—
159Dic197—24no—
160Dic197—41no—
161Dic197—41no—
162C2×C197—24no—
163C2×C197C22×C4⋊C411yescharacteristic
164C2×C197C2×C4×D412yes—
165C2×C197C2×#12011yescharacteristic
166C2×C197C2×C22⋊Q812yes—
167C2×C197C2×#12311yescharacteristic
168D197—24no—
169D197#38518yesdirect product factor
170C197C2×#38511yescharacteristic, complemented, Sylow
171C2×#385—1971nomaximal, Sylow
172C24×C4—1971no—
173C22×C4⋊C4—1972no—
174C22×C22⋊C4—1971no—
175C22×C22⋊C4—1971no—
176#385—1971no—
177#385—1972no—
178#385—1971no—
179#385—1971no—
180#385—1972no—
181#385D19711yescharacteristic, direct product factor
182C2×#95—1972no—
183C25—1971no—
184C23×C4—3944no—
185C23×C4—3948no—
186C23×C4—1971no—
187C23×C4—1971no—
188C23×C4—1971no—
189C23×C4—1972no—
190C23×C4—1972no—
191C23×C4—1972no—
192C23×C4C2×D19711yescharacteristic
193C2×C4⋊C4—3948no—
194C2×C4⋊C4—1972no—
195C2×C4⋊C4—1972no—
196C2×C4⋊C4—1972no—
197C2×C4⋊C4C2×D19712yescomplemented
198C2×C22⋊C4—3944no—
199C2×C22⋊C4—1971no—
200C2×C22⋊C4—1971no—
201C2×C22⋊C4—1971no—
202C2×C22⋊C4—1971no—
203C2×C22⋊C4—1972no—
204C2×C22⋊C4—1978no—
205C2×C22⋊C4C2×D19711yescharacteristic
206C2×C22⋊C4C2×D19711yescharacteristic
207#95—1972no—
208#95—1972no—
209#95—1972no—
210#95C2×D19712yescomplemented
211C24—3944no—
212C24—3944no—
213C24—1974no—
214C24—1974no—
215C24—1974no—
216C24—1971no—
217C24—1971no—
218C24C22×D19711yescharacteristic
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—3942no—
222C22×C4—3942no—
223C22×C4—3944no—
224C22×C4—3944no—
225C22×C4—39432no—
226C22×C4—3944no—
227C22×C4—3944no—
228C22×C4—3944no—
229C22×C4—3944no—
230C22×C4—3944no—
231C22×C4—3944no—
232C22×C4—3942no—
233C22×C4—3944no—
234C22×C4—3944no—
235C22×C4—1971no—
236C22×C4—1971no—
237C22×C4—1972no—
238C22×C4—1972no—
239C22×C4—1978no—
240C22×C4—1978no—
241C22×C4C22×D19711yescharacteristic
242C22×C4C22×D19711yescharacteristic
243C22×C4C22×D19712yes—
244C22×C4C22×D19712yes—
245C4⋊C4—24no—
246C4⋊C4—3944no—
247C4⋊C4—3944no—
248C4⋊C4—3944no—
249C22⋊C4—22no—
250C22⋊C4—3942no—
251C22⋊C4—3942no—
252C22⋊C4—3942no—
253C22⋊C4—1974no—
254C22⋊C4—1978no—
255C22⋊C4C4×D19714yescomplemented
256C23—22no—
257C23—22no—
258C23—3948no—
259C23—39416no—
260C23—39416no—
261C23—3948no—
262C23—3942no—
263C23—3942no—
264C23—19716no—
265C23—1972no—
266C23—1972no—
267C23—1972no—
268C23—1972no—
269C23—1972no—
270C23—1974no—
271C23—1972no—
272C23—1974no—
273C23C2×C4×D19712yes—
274C23D4×D19712yes—
275C23Q8×D19712yes—
276C23C23×D19711yescharacteristic
277C2×C4—28no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—7884no—
287C2×C4—7884no—
288C2×C4—39416no—
289C2×C4—39416no—
290C2×C4—39416no—
291C2×C4—3941no—
292C2×C4—3941no—
293C2×C4—3942no—
294C2×C4—3942no—
295C2×C4—3948no—
296C2×C4—3942no—
297C2×C4—3942no—
298C2×C4—3942no—
299C2×C4—3942no—
300C2×C4—1974no—
301C2×C4—1974no—
302C2×C4C2×C4×D19714yescomplemented
303C2×C4D4×D19714yes—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—22no—
308C22—39432no—
309C22—3944no—
310C22—3942no—
311C22—3944no—
312C22—3942no—
313C22—1974no—
314C22—1974no—
315C22—1978no—
316C22—1974no—
317C22—1978no—
318C22—1974no—
319C22C4⋊C4×D19714yescomplemented
320C22C22×C4×D19711yescharacteristic
321C22C2×D4×D19711yescharacteristic
322C22C2×D4×D19712yes—
323C22C2×Q8×D19711yescharacteristic
324C22#41×D19712yes—
325C4—24no—
326C4—24no—
327C4—41no—
328C4—41no—
329C4—7881no—
330C4—7881no—
331C4—3944no—
332C4—3944no—
333C2—24no—
334C2—3944no—
335C2—1978no—
336C2C2×C4⋊C4×D19711yescharacteristic
337C2C4×D4×D19712yes—
338C2#120×D19711yescharacteristic
339C2C22⋊Q8×D19712yes—
340C2#123×D19711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#385×C197C2
socle, cosocleC23×C197C24
prime radicalC24×D197C22
hypercenter#385D197
hypocenterC197C2×#385
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D197C2semidirect
5. #27160954C2semidirect
6. #27160953C2semidirect
7. #27160952C2semidirect
8. #27160951C2semidirect
9. #27160950C2semidirect
10. #95×D197C2semidirect
11. #385×C197C2semidirect
21. C2×C4⋊Dic197C22semidirect
22. C2×#15691337C22semidirect
23. C2×#15691337C22semidirect
31. C22⋊C4×D197C4semidirect
32. #20263233C22semidirect
33. #20263233C22semidirect
34. #20263233C22semidirect
36. C2×C4⋊C4×C197C22semidirect
39. #95×C197C22semidirect
65. C2×C4×D197C2×C4semidirect
71. C22⋊Dic197C2×C4semidirect
74. #15691335C2×C4semidirect
84. C22⋊C4×C197C2×C4semidirect
91. C22×D197C4⋊C4semidirect
109. C2×Dic197C22×C4semidirect
131. C2×C4×C197C22×C4semidirect
137. C22×C197C2×C4⋊C4semidirect
149. C2×D197C2×C4⋊C4semidirect
169. D197#385direct (N × H)
170. C197C2×#385semidirect
181. #385D197direct (N × H)
197. C2×C4⋊C4C2×D197semidirect
210. #95C2×D197semidirect
255. C22⋊C4C4×D197semidirect
302. C2×C4C2×C4×D197semidirect
319. C22C4⋊C4×D197semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1972
C2×C4⋊C4×D1972
C2×C22⋊C4×D1972
#271609542
#271609532
#271609522
#271609512
#271609502
#95×D1972
#385×C1972
C2×#385197