Finite Group Databasebeta

← #38745499

Subgroups — #38745499

Order 25216.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263307C211yesmaximal, characteristic
2#27160974C211yesmaximal, characteristic, complemented
3#27160973C211yesmaximal, characteristic, complemented
4#27160967C211yesmaximal, characteristic, complemented
5#27160967C211yesmaximal, characteristic, complemented
6#27160967C211yesmaximal, characteristic, complemented
7#27160963C211yesmaximal, characteristic, complemented
8#27160960C211yesmaximal, characteristic, complemented
9#27160959C211yesmaximal, characteristic
10#27160951C211yesmaximal, characteristic, complemented
11#27160937C211yesmaximal, characteristic, complemented
12C4×#15691337C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
15#390×C197C211yesmaximal, characteristic
16C23×Dic197C2211yescharacteristic
17C23×Dic197C2211yescharacteristic
18#20263307C418yescomplemented
19C2×#15691337C2211yescharacteristic
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×C22⋊Dic197C2211yescharacteristic
26C2×C22⋊Dic197C2211yescharacteristic
27C2×C22⋊Dic197C2211yescharacteristic
28C2×C22⋊Dic197C2211yescharacteristic
29C2×C22⋊Dic197C2211yescharacteristic
30C2×C22⋊Dic197C2211yescharacteristic
31C2×C4×Dic197C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C22×D4×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C22⋊C4×C197C2211yescharacteristic
50C2×C22⋊C4×C197C2211yescharacteristic
51C2×C42×C197C2211yescharacteristic
52C22×Dic197—22no—
53C22×Dic197—41no—
54C22×Dic197—41no—
55C22×Dic197—41no—
56C22×Dic197—41no—
57C22×Dic197C2×C414yescomplemented
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66#15691337—22no—
67#15691337—41no—
68#15691337—41no—
69#15691337—41no—
70#15691337C2×C414yescomplemented
71#15691337C2×C414yescomplemented
72C22⋊Dic197—22no—
73C22⋊Dic197—41no—
74C22⋊Dic197—41no—
75C22⋊Dic197C2×C414yes—
76C22⋊Dic197C2×C414yes—
77C22⋊Dic197C2×C414yescomplemented
78C4×Dic197—22no—
79C4×Dic197—41no—
80C4×Dic197—41no—
81C4×Dic197—41no—
82C24×C197C2311yescharacteristic
83C24×C197C2311yescharacteristic
84C2×D4×C197—41no—
85C2×D4×C197—41no—
86C2×D4×C197C2×C414yes—
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C4⋊C4×C197—41no—
93C22⋊C4×C197—22no—
94C22⋊C4×C197—22no—
95C22⋊C4×C197—41no—
96C22⋊C4×C197—41no—
97C42×C197—41no—
98C2×Dic197—21no—
99C2×Dic197—21no—
100C2×Dic197—21no—
101C2×Dic197—21no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197—21no—
105C2×Dic197—21no—
106C2×Dic197—21no—
107C2×Dic197—21no—
108C2×Dic197—21no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—44no—
118C2×Dic197—44no—
119C2×Dic197C22×C412yes—
120C2×Dic197C22×C412yes—
121C2×Dic197C22×C412yes—
122C2×Dic197C22×C412yes—
123C2×Dic197#4112yes—
124C23×C197—21no—
125C23×C197—21no—
126C23×C197—21no—
127C23×C197—21no—
128C23×C197—21no—
129C23×C197—21no—
130C23×C197—21no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—21no—
134C23×C197—21no—
135C23×C197—21no—
136C23×C197C22×C412yes—
137C23×C197C22×C412yes—
138C23×C197C2411yescharacteristic
139D4×C197—44no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197C22×C412yes—
154C2×C4×C197#4112yes—
155C22×C197—22no—
156C22×C197—22no—
157C22×C197—22no—
158C22×C197—22no—
159C22×C197—22no—
160C22×C197—22no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197C23×C411yescharacteristic
170C22×C197C2×#4111yescharacteristic
171C22×C197C2×#4111yescharacteristic
172C22×C197H(5,𝔽2)11yescharacteristic
173C22×C197H(5,𝔽2)11yescharacteristic
174C22×C197H(5,𝔽2)11yescharacteristic
175C22×C1972−1+411yescharacteristic
176C4×C197—22no—
177C4×C197—22no—
178C4×C197—41no—
179C4×C197—41no—
180C4×C197—41no—
181Dic197—22no—
182Dic197—22no—
183Dic197—22no—
184Dic197—22no—
185Dic197—22no—
186Dic197—41no—
187Dic197—41no—
188Dic197—41no—
189C2×C197—42no—
190C2×C197—42no—
191C2×C197C4×#4111yescharacteristic
192C2×C197#51811yescharacteristic
193C2×C197#52011yescharacteristic
194C2×C197#53811yescharacteristic
195C2×C197#53911yescharacteristic
196C2×C197#54011yescharacteristic
197C2×C197#54211yescharacteristic
198C197#233511yescharacteristic, complemented, Sylow
199#2335—1971nomaximal, Sylow
200C2×#123—1971no—
201#390D19711yescharacteristic
202#389—1971no—
203#388—1971no—
204#388—1971no—
205#388—1971no—
206#388—1971no—
207#387—1971no—
208#387—1971no—
209#386—1971no—
210#385—1971no—
211#380—1971no—
212C4×C4⋊C4—1971no—
213C4×C22⋊C4—1971no—
214C4×C22⋊C4—1971no—
215C22×D4C2×D19711yescharacteristic
216C23×C4—1971no—
217C23×C4—1971no—
218#123—1978no—
219C2×C4⋊C4—1971no—
220C2×C4⋊C4—1971no—
221C2×C4⋊C4—1971no—
222C2×C4⋊C4—1971no—
223C2×C4⋊C4—1971no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4C2×D19711yescharacteristic
226C2×C22⋊C4—1971no—
227C2×C22⋊C4—1971no—
228C2×C22⋊C4—1971no—
229C2×C22⋊C4—1971no—
230C2×C22⋊C4—1971no—
231C2×C22⋊C4—1971no—
232C2×C22⋊C4C2×D19711yescharacteristic
233C2×C22⋊C4C2×D19711yescharacteristic
234C2×C22⋊C4C2×D19711yescharacteristic
235C2×C22⋊C4C2×D19711yescharacteristic
236C2×C42—1971no—
237C2×C42—1971no—
238C2×C42—1971no—
239C2×C42—1971no—
240C2×C42C2×D19711yescharacteristic
241#95—1971no—
242#95—1971no—
243#95—1971no—
244#95—1971no—
245#95—1971no—
246#95—1971no—
247#95—1971no—
248#95—1971no—
249#95—1971no—
250#95—1971no—
251C24C22×D19711yescharacteristic
252C24C22×D19711yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4C4×D19714yes—
256C22×C4—7881no—
257C22×C4—7881no—
258C22×C4—7881no—
259C22×C4—7881no—
260C22×C4—3942no—
261C22×C4—1974no—
262C22×C4—1971no—
263C22×C4—1971no—
264C22×C4—1971no—
265C22×C4—1971no—
266C22×C4—1971no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4—1971no—
270C22×C4C22×D19711yescharacteristic
271C22×C4C22×D19711yescharacteristic
272C22×C4C22×D19711yescharacteristic
273C22×C4C22×D19711yescharacteristic
274C22×C4C22×D19711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—7881no—
277C4⋊C4—7881no—
278C4⋊C4—7881no—
279C4⋊C4—3942no—
280C4⋊C4—1974no—
281C4⋊C4—1974no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—7881no—
287C22⋊C4—7881no—
288C22⋊C4—3942no—
289C22⋊C4—1974no—
290C22⋊C4—1974no—
291C22⋊C4—1974no—
292C42—41no—
293C42—7881no—
294C42—7881no—
295C42—7881no—
296C42—3942no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C4×D19712yes—
310C23C2×C4×D19712yes—
311C23C23×D19711yescharacteristic
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—7884no—
327C2×C4—7884no—
328C2×C4—3941no—
329C2×C4—3941no—
330C2×C4—3941no—
331C2×C4—3941no—
332C2×C4—3941no—
333C2×C4—3941no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—3941no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—1972no—
348C2×C4—1972no—
349C2×C4—1972no—
350C2×C4—1972no—
351C2×C4—1972no—
352C2×C4C2×C4×D19712yes—
353C2×C4#1569135712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C22×C4×D19711yescharacteristic
369C22#41×D19711yescharacteristic
370C22C2×#1569135711yescharacteristic
371C22#2026337111yescharacteristic
372C22#2026337111yescharacteristic
373C22#2026337111yescharacteristic
374C22#2026337311yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—7881no—
381C4—7881no—
382C4—7881no—
383C4—3942no—
384C4—3942no—
385C4—3942no—
386C4—3942no—
387C4—3942no—
388C2—42no—
389C2—42no—
390C2C4×#1569135711yescharacteristic
391C2#2716155211yescharacteristic
392C2#2716155811yescharacteristic
393C2#2716166111yescharacteristic
394C2#2716167411yescharacteristic
395C2#2716168311yescharacteristic
396C2#2716169311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#390×C197C2
socle, cosocleC23×C197C24
prime radicalC22×D4×C197C22
hypercenter#390D197
hypocenterC197#2335
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #27160974C2semidirect
3. #27160973C2semidirect
4. #27160967C2semidirect
5. #27160967C2semidirect
6. #27160967C2semidirect
7. #27160963C2semidirect
8. #27160960C2semidirect
10. #27160951C2semidirect
11. #27160937C2semidirect
12. C4×#15691337C2semidirect
13. C22⋊C4×Dic197C2semidirect
14. C22⋊C4×Dic197C2semidirect
18. #20263307C4semidirect
57. C22×Dic197C2×C4semidirect
70. #15691337C2×C4semidirect
71. #15691337C2×C4semidirect
77. C22⋊Dic197C2×C4semidirect
198. C197#2335semidirect

Maximal subgroups

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

subgroupindex
C2×#202633072
#271609742
#271609732
#271609672
#271609632
#271609602
#271609592
#271609512
#271609372
C4×#156913372
C22⋊C4×Dic1972
#390×C1972
#2335197