Finite Group Databasebeta

← #38751683

Subgroups — #38751683

Order 25216.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161607C211yesmaximal, characteristic
2#27161207C211yesmaximal, characteristic, complemented
3C8×Dic394C211yesmaximal, characteristic, complemented
4#27161155C211yesmaximal, characteristic, complemented
5#27161152C211yesmaximal, characteristic, complemented
6#27161148C211yesmaximal, characteristic, complemented
7C8×D788C211yesmaximal, characteristic, complemented
8C8×C22⋊D197C211yesmaximal, characteristic, complemented
9C8×C22⋊D197C211yesmaximal, characteristic, complemented
10#27161141C211yesmaximal, characteristic, complemented
11#27161140C211yesmaximal, characteristic, complemented
12#27161134C211yesmaximal, characteristic, complemented
13#27161033C211yesmaximal, characteristic, complemented
14C4×C8×D197C211yesmaximal, characteristic, complemented
15#405×C197C211yesmaximal, characteristic, complemented
16C2×#15691341C2211yescharacteristic
17C2×C8×D197C2211yescharacteristic
18C2×C8×D197C2211yescharacteristic
19C2×C8×D197C2211yescharacteristic
20#20263321C411yescharacteristic
21#20263319C411yescharacteristic
22#20263316C411yescharacteristic
23#20263312C411yescharacteristic
24#20263311C411yescharacteristic
25#20263308C411yescharacteristic
26#20263304C411yescharacteristic
27#20263299C411yescharacteristic
28C4×Dic394C2211yescharacteristic
29C4×D788C2211yescharacteristic
30C4×C22⋊D197C2211yescharacteristic
31C4×C22⋊D197C2211yescharacteristic
32#20263283C2211yescharacteristic
33C42×D197C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35C4⋊#13096802C2211yescharacteristic
36#20263255C2211yescharacteristic
37#20263255C2211yescharacteristic
38#20263255C2211yescharacteristic
39#20263255C2211yescharacteristic
40C22⋊#13096802C2211yescharacteristic
41C22⋊#13096802C2211yescharacteristic
42#20263238C2211yescharacteristic
43#20263238C2211yescharacteristic
44#20263238C2211yescharacteristic
45#20263238C2211yescharacteristic
46#20263237C2211yescharacteristic
47#20263236C2211yescharacteristic
48C4×#13096802C2211yescharacteristic
49C8×Dic197C2211yescharacteristic
50C8×Dic197C2211yescharacteristic
51C8×Dic197C2211yescharacteristic
52C2×M16×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic
54#117×C197C2211yescharacteristic
55#97×C197C2211yescharacteristic
56#97×C197C2211yescharacteristic
57C4×C8×C197C2211yescharacteristic
58C4×C8×C197C2211yescharacteristic
59C2×Dic394C2×C411yescharacteristic
60C2×D788C2×C411yescharacteristic
61C2×C22⋊D197C2×C411yescharacteristic
62C2×C22⋊D197C2×C411yescharacteristic
63C2×C4×D197C2×C411yescharacteristic
64C2×C4×D197C2×C411yescharacteristic
65C2×C4×D197C2311yescharacteristic
66C2×C4×D197C2311yescharacteristic
67#15691341—22no—
68C2×#13096802C2311yescharacteristic
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C8×D197—22no—
73C8×D197—22no—
74C8×D197—22no—
75C4⋊Dic197C2×C411yescharacteristic
76C4⋊Dic197C2×C411yescharacteristic
77#15691337C2×C411yescharacteristic
78#15691337C2×C411yescharacteristic
79#15691337C2×C411yescharacteristic
80#15691337C2×C411yescharacteristic
81#15691337C2×C411yescharacteristic
82#15691337C2×C411yescharacteristic
83C22⋊Dic197C2×C411yescharacteristic
84C22⋊Dic197C2×C411yescharacteristic
85#15691335C2×C411yescharacteristic
86#15691335C2×C411yescharacteristic
87#15691335C2×C411yescharacteristic
88#15691335C2×C411yescharacteristic
89#15691335C2×C411yescharacteristic
90#15691335C2×C411yescharacteristic
91C4×Dic197C2×C411yescharacteristic
92C4×Dic197C2×C411yescharacteristic
93C4×Dic197C2311yescharacteristic
94C4×Dic197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96M16×C197—22no—
97C2×C8×C197—22no—
98C2×C8×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C2×C8×C197C2311yescharacteristic
102C4⋊C4×C197C2×C411yescharacteristic
103C4⋊C4×C197C2×C411yescharacteristic
104C22⋊C4×C197C2×C411yescharacteristic
105C22⋊C4×C197C2×C411yescharacteristic
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197C22×C411yescharacteristic
109C22×D197C22×C411yescharacteristic
110Dic394—22no—
111D788—22no—
112C22⋊D197—22no—
113C22⋊D197—22no—
114C2×Dic197C22×C411yescharacteristic
115C2×Dic197C22×C411yescharacteristic
116C2×Dic197C22×C411yescharacteristic
117C2×Dic197C22×C411yescharacteristic
118C2×Dic197C22×C411yescharacteristic
119C2×Dic197C22×C411yescharacteristic
120C4×D197—22no—
121C4×D197—21no—
122C4×D197—21no—
123C4×D197—21no—
124C4×D197—21no—
125C4×D197—22no—
126#13096802—21no—
127#13096802—21no—
128#13096802—21no—
129#13096802—21no—
130C23×C197C22×C411yescharacteristic
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197C22×C411yescharacteristic
134C2×C4×C197C22×C411yescharacteristic
135C2×C4×C197C22×C411yescharacteristic
136C2×C4×C197C22×C411yescharacteristic
137C2×C4×C197C22×C411yescharacteristic
138C2×C4×C197C2411yescharacteristic
139C8×C197—21no—
140C8×C197—21no—
141C8×C197#4112yes—
142C8×C197#4112yes—
143C22×C197—21no—
144C22×C197—21no—
145C22×C197—21no—
146C22×C197C23×C411yescharacteristic
147C2×D197—21no—
148C2×D197—21no—
149C2×D197—21no—
150C2×D197—21no—
151C2×D197—21no—
152C2×D197#13112yes—
153C4×C197—21no—
154C4×C197—21no—
155C4×C197—21no—
156C4×C197—21no—
157C4×C197—21no—
158C4×C197C2×#4111yescharacteristic
159C4×C197C2×#4111yescharacteristic
160Dic197—21no—
161Dic197—21no—
162Dic197—21no—
163Dic197—21no—
164Dic197—21no—
165Dic197#13112yes—
166C2×C197—41no—
167C2×C197C4×#4111yescharacteristic
168C2×C197C2×#13111yescharacteristic
169C2×C197C2×#13111yescharacteristic
170D197—22no—
171D197—41no—
172C197#296611yescharacteristic, complemented, Sylow
173#2966—1971nomaximal, Sylow
174#529—1971no—
175#446—1971no—
176C8×Q8—1971no—
177#436—1971no—
178#435—1971no—
179#435—1971no—
180C8×D4—1971no—
181C8×D4—1971no—
182C8×D4—1971no—
183#433—1971no—
184#433—1971no—
185#432—1971no—
186#405—1971no—
187#405D19711yescharacteristic, complemented
188C2×C4×C8—1971no—
189C2×M16—1971no—
190C2×M16C2×D19711yescharacteristic
191C22×C8—1971no—
192C22×C8—1971no—
193C22×C8—1971no—
194C22×C8C2×D19711yescharacteristic
195#126—1971no—
196#126—1971no—
197#125—1971no—
198#124—1971no—
199#123—1971no—
200#123—1971no—
201C22⋊Q8—1971no—
202#121—1971no—
203C4×Q8—1971no—
204C4×D4—1971no—
205C4×D4—1971no—
206C4×D4—1971no—
207#117—1971no—
208#117C2×D19711yescharacteristic
209C2×C42—1971no—
210C4⋊C8—1971no—
211C4⋊C8—1971no—
212C4⋊C8—1971no—
213C4⋊C8—1971no—
214C4⋊C8—1971no—
215C4⋊C8—1971no—
216C22⋊C8—1971no—
217C22⋊C8—1971no—
218C22⋊C8—1971no—
219C22⋊C8—1971no—
220C22⋊C8—1971no—
221C22⋊C8—1971no—
222#97—1971no—
223#97—1971no—
224#97C2×D19711yescharacteristic
225#97C2×D19711yescharacteristic
226C4×C8—1971no—
227C4×C8—1971no—
228C4×C8—1971no—
229C4×C8—1971no—
230C4×C8C2×D19711yescharacteristic
231C4×C8C2×D19711yescharacteristic
232C2×Q8—1971no—
233C2×D4—1971no—
234C2×D4—1971no—
235C2×D4—1971no—
236C22×C4—1971no—
237C22×C4—1971no—
238C22×C4—1971no—
239C22×C4—1971no—
240C22×C4C22×D19711yescharacteristic
241M16—22no—
242M16—3942no—
243C2×C8—22no—
244C2×C8—3942no—
245C2×C8—3942no—
246C2×C8—3942no—
247C2×C8—1971no—
248C2×C8—1971no—
249C2×C8—1971no—
250C2×C8—1971no—
251C2×C8C22×D19711yescharacteristic
252C2×C8C22×D19711yescharacteristic
253C2×C8C22×D19711yescharacteristic
254C2×C8C22×D19711yescharacteristic
255C4⋊C4—1971no—
256C4⋊C4—1971no—
257C4⋊C4—1971no—
258C4⋊C4—1971no—
259C4⋊C4—1971no—
260C4⋊C4—1971no—
261C4⋊C4—1971no—
262C4⋊C4—1971no—
263C4⋊C4C4×D19711yescharacteristic
264C4⋊C4C4×D19711yescharacteristic
265C22⋊C4—1971no—
266C22⋊C4—1971no—
267C22⋊C4—1971no—
268C22⋊C4—1971no—
269C22⋊C4—1971no—
270C22⋊C4—1971no—
271C22⋊C4—1971no—
272C22⋊C4—1971no—
273C22⋊C4C4×D19711yescharacteristic
274C22⋊C4C4×D19711yescharacteristic
275C42—1971no—
276C42—1971no—
277C42—1971no—
278C42—1971no—
279C42C22×D19711yescharacteristic
280C42C22×D19711yescharacteristic
281C23—1971no—
282C23—1971no—
283C23C2×C4×D19711yescharacteristic
284Q8—3942no—
285D4—3942no—
286D4—3942no—
287D4—3942no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—3942no—
291C2×C4—3942no—
292C2×C4—3941no—
293C2×C4—3941no—
294C2×C4—3941no—
295C2×C4—3941no—
296C2×C4—1971no—
297C2×C4—1971no—
298C2×C4—1971no—
299C2×C4—1971no—
300C2×C4—1971no—
301C2×C4—1971no—
302C2×C4C2×C4×D19711yescharacteristic
303C2×C4C2×C4×D19711yescharacteristic
304C2×C4C2×C4×D19711yescharacteristic
305C2×C4C2×C4×D19711yescharacteristic
306C2×C4C2×C4×D19711yescharacteristic
307C2×C4C23×D19711yescharacteristic
308C8—21no—
309C8—21no—
310C8—3941no—
311C8—3941no—
312C8—3941no—
313C8—3941no—
314C8#1569135812yes—
315C8#1569135812yes—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—3941no—
320C22—3941no—
321C22—3941no—
322C22—3941no—
323C22—3941no—
324C22—1972no—
325C22C22×C4×D19711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—3941no—
332C4—3941no—
333C4—3941no—
334C4—3941no—
335C4—3941no—
336C4—1972no—
337C4C2×#1569135811yescharacteristic
338C4C2×#1569135811yescharacteristic
339C2—41no—
340C2—7881no—
341C2—3942no—
342C2C4×#1569135811yescharacteristic
343C2#131×D19711yescharacteristic
344C2#131×D19711yescharacteristic
subgroupG/H
centerC2×C8C22×D197
commutator, socleC22×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#405×C197C2
prime radical#20263299C4
cosocleC2×C4×C197C24
hypercenter#405D197
hypocenterC197#2966
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C2×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
2. #27161207C2semidirect
3. C8×Dic394C2semidirect
4. #27161155C2semidirect
5. #27161152C2semidirect
6. #27161148C2semidirect
7. C8×D788C2semidirect
8. C8×C22⋊D197C2semidirect
9. C8×C22⋊D197C2semidirect
10. #27161141C2semidirect
11. #27161140C2semidirect
12. #27161134C2semidirect
13. #27161033C2semidirect
14. C4×C8×D197C2semidirect
15. #405×C197C2semidirect
172. C197#2966semidirect
187. #405D197semidirect

Maximal subgroups

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

subgroupindex
#271616072
#271612072
C8×Dic3942
#271611552
#271611522
#271611482
C8×D7882
C8×C22⋊D1972
#271611412
#271611402
#271611342
#271610332
C4×C8×D1972
#405×C1972
#2966197