Finite Group Databasebeta

← #38745489

Subgroups — #38745489

Order 25216.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263319C211yesmaximal, characteristic
2#27160977C211yesmaximal, characteristic, complemented
3#27160976C211yesmaximal, characteristic, complemented
4#27160973C211yesmaximal, characteristic
5#27160972C211yesmaximal, characteristic
6#27160967C211yesmaximal, characteristic
7#27160964C211yesmaximal, characteristic, complemented
8#27160963C211yesmaximal, characteristic, complemented
9#27160961C211yesmaximal, characteristic, complemented
10#27160952C211yesmaximal, characteristic
11#27160952C211yesmaximal, characteristic
12#27160943C211yesmaximal, characteristic, complemented
13C4×#15691337C211yesmaximal, characteristic, complemented
14C4×#15691337C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C197C211yesmaximal, characteristic
16#20263319C418yescomplemented
17C2×C4⋊Dic197C2211yescharacteristic
18C2×C4⋊Dic197C2211yescharacteristic
19C2×#15691337C2211yescharacteristic
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
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
45C23×C4×C197C2211yescharacteristic
46C2×C22⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48C2×C42×C197C2211yescharacteristic
49C2×C42×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C22×Dic197C2311yescharacteristic
53C22×Dic197C2311yescharacteristic
54C22×Dic197C2311yescharacteristic
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C4⋊Dic197—41no—
61C4⋊Dic197C2×C414yescomplemented
62#15691337—22no—
63#15691337—22no—
64#15691337—22no—
65#15691337—41no—
66#15691337—41no—
67#15691337—41no—
68#15691337C2×C414yescomplemented
69#15691337C2×C414yescomplemented
70C22⋊Dic197—41no—
71C22⋊Dic197—41no—
72C22⋊Dic197C2×C414yes—
73C22⋊Dic197C2×C414yes—
74C4×Dic197—22no—
75C4×Dic197—41no—
76C4×Dic197—41no—
77C4×Dic197C2×C414yescomplemented
78C24×C197C2311yescharacteristic
79C22×C4×C197—22no—
80C22×C4×C197—41no—
81C22×C4×C197—41no—
82C22×C4×C197C2311yescharacteristic
83C22×C4×C197C2311yescharacteristic
84C22×C4×C197C2311yescharacteristic
85C22×C4×C197C2311yescharacteristic
86C22×C4×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C22⋊C4×C197—22no—
89C22⋊C4×C197C2×C414yes—
90C42×C197—41no—
91C42×C197—41no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—21no—
96C2×Dic197—21no—
97C2×Dic197—21no—
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×Dic197C22×C412yes—
113C2×Dic197C22×C412yes—
114C2×Dic197C22×C412yes—
115C2×Dic197C22×C412yes—
116C23×C197—21no—
117C23×C197—21no—
118C23×C197—21no—
119C23×C197—21no—
120C23×C197—21no—
121C23×C197—21no—
122C23×C197C22×C412yes—
123C23×C197C2411yescharacteristic
124C2×C4×C197—21no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—44no—
139C2×C4×C197C22×C412yes—
140C2×C4×C197C22×C412yes—
141C2×C4×C197#4112yes—
142C2×C4×C197#4112yes—
143C22×C197—22no—
144C22×C197—22no—
145C22×C197—22no—
146C22×C197—41no—
147C22×C197—41no—
148C22×C197—41no—
149C22×C197—41no—
150C22×C197C23×C411yescharacteristic
151C22×C197C2×#4111yescharacteristic
152C22×C197C2×#4111yescharacteristic
153C22×C197H(5,𝔽2)11yescharacteristic
154C22×C197H(5,𝔽2)11yescharacteristic
155C22×C1972−1+411yescharacteristic
156C22×C1972−1+411yescharacteristic
157C4×C197—22no—
158C4×C197—22no—
159C4×C197—22no—
160C4×C197—22no—
161C4×C197—41no—
162C4×C197—41no—
163Dic197—22no—
164Dic197—22no—
165Dic197—22no—
166Dic197—22no—
167Dic197—41no—
168Dic197—41no—
169Dic197—41no—
170Dic197—41no—
171C2×C197—42no—
172C2×C197C4×#4111yescharacteristic
173C2×C197#52011yescharacteristic
174C2×C197#52011yescharacteristic
175C2×C197#53811yescharacteristic
176C2×C197#53911yescharacteristic
177C2×C197#54111yescharacteristic
178C2×C197#54211yescharacteristic
179C197#233411yescharacteristic, complemented, Sylow
180#2334—1971nomaximal, Sylow
181C2×#126—1971no—
182#389—1971no—
183#389—1971no—
184#388—1971no—
185#388—1971no—
186#388—1971no—
187#387—1971no—
188#387—1971no—
189#387—1971no—
190#385—1971no—
191#385—1971no—
192#382—1971no—
193C4×C4⋊C4—1971no—
194C4×C4⋊C4—1971no—
195C4×C22⋊C4D19711yescharacteristic
196C23×C4C2×D19711yescharacteristic
197#126—1978no—
198C2×C4⋊C4—1971no—
199C2×C4⋊C4—1971no—
200C2×C4⋊C4—1971no—
201C2×C4⋊C4—1971no—
202C2×C4⋊C4—1971no—
203C2×C4⋊C4—1971no—
204C2×C4⋊C4—1971no—
205C2×C4⋊C4—1971no—
206C2×C4⋊C4—1971no—
207C2×C4⋊C4—1971no—
208C2×C22⋊C4—1971no—
209C2×C22⋊C4—1971no—
210C2×C22⋊C4—1971no—
211C2×C22⋊C4—1971no—
212C2×C22⋊C4C2×D19711yescharacteristic
213C2×C22⋊C4C2×D19711yescharacteristic
214C2×C42—1971no—
215C2×C42—1971no—
216C2×C42—1971no—
217C2×C42—1971no—
218C2×C42C2×D19711yescharacteristic
219C2×C42C2×D19711yescharacteristic
220#95—1971no—
221#95—1971no—
222#95—1971no—
223#95—1971no—
224#95—1971no—
225#95—1971no—
226#95—1971no—
227#95—1971no—
228#95—1971no—
229#95—1971no—
230#95C2×D19711yescharacteristic
231#95C2×D19711yescharacteristic
232C24C22×D19711yescharacteristic
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—1971no—
237C22×C4—1971no—
238C22×C4—1971no—
239C22×C4—1971no—
240C22×C4—1971no—
241C22×C4—1971no—
242C22×C4—1971no—
243C22×C4—1971no—
244C22×C4C22×D19711yescharacteristic
245C22×C4C22×D19711yescharacteristic
246C22×C4C22×D19711yescharacteristic
247C22×C4C22×D19711yescharacteristic
248C22×C4C22×D19711yescharacteristic
249C22×C4C22×D19711yescharacteristic
250C4⋊C4—7881no—
251C4⋊C4—7881no—
252C4⋊C4—7881no—
253C4⋊C4—7881no—
254C4⋊C4—3942no—
255C4⋊C4—3942no—
256C4⋊C4—3942no—
257C4⋊C4—1974no—
258C4⋊C4—1974no—
259C4⋊C4—1974no—
260C22⋊C4—22no—
261C22⋊C4—7881no—
262C22⋊C4—7881no—
263C22⋊C4—1974no—
264C22⋊C4—1974no—
265C22⋊C4C4×D19714yes—
266C42—41no—
267C42—41no—
268C42—7881no—
269C42—7881no—
270C42—3942no—
271C42—1974no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D19712yes—
279C23C23×D19711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—3941no—
296C2×C4—3941no—
297C2×C4—3941no—
298C2×C4—3941no—
299C2×C4—3941no—
300C2×C4—3941no—
301C2×C4—3941no—
302C2×C4—3941no—
303C2×C4—3941no—
304C2×C4—3941no—
305C2×C4—3941no—
306C2×C4—3941no—
307C2×C4—3941no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—1972no—
316C2×C4—1972no—
317C2×C4—1972no—
318C2×C4—1972no—
319C2×C4C2×C4×D19712yes—
320C2×C4C2×C4×D19712yes—
321C2×C4#1569135812yes—
322C2×C4#1569135812yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D19711yescharacteristic
331C22C2×#1569135811yescharacteristic
332C22C2×#1569135811yescharacteristic
333C22#2026337111yescharacteristic
334C22#2026337111yescharacteristic
335C22#2026337311yescharacteristic
336C22#2026337311yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—7881no—
344C4—7881no—
345C4—7881no—
346C4—7881no—
347C4—3942no—
348C4—3942no—
349C4—3942no—
350C4—3942no—
351C2—42no—
352C2C4×#1569135811yescharacteristic
353C2#2716155811yescharacteristic
354C2#2716155811yescharacteristic
355C2#2716166711yescharacteristic
356C2#2716167111yescharacteristic
357C2#2716169111yescharacteristic
358C2#2716169411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
FittingC4×C22⋊C4×C197C2
socle, cosocleC23×C197C24
prime radicalC24×C197C23
hypercenterC4×C22⋊C4D197
hypocenterC197#2334
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #27160977C2semidirect
3. #27160976C2semidirect
7. #27160964C2semidirect
8. #27160963C2semidirect
9. #27160961C2semidirect
12. #27160943C2semidirect
13. C4×#15691337C2semidirect
14. C4×#15691337C2semidirect
16. #20263319C4semidirect
61. C4⋊Dic197C2×C4semidirect
68. #15691337C2×C4semidirect
69. #15691337C2×C4semidirect
77. C4×Dic197C2×C4semidirect
179. C197#2334semidirect

Maximal subgroups

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

subgroupindex
C2×#202633192
#271609772
#271609762
#271609732
#271609722
#271609672
#271609642
#271609632
#271609612
#271609522
#271609432
C4×#156913372
C4×C22⋊C4×C1972
#2334197