Finite Group Databasebeta

← #38754593

Subgroups — #38754593

Order 25216.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161738C211yesmaximal, characteristic
2#27161436C211yesmaximal, characteristic, complemented
3#27161420C211yesmaximal, characteristic, complemented
4#27161392C211yesmaximal, characteristic, complemented
5#27161383C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161301C211yesmaximal, characteristic, complemented
9#27161239C211yesmaximal, characteristic, complemented
10#27161234C211yesmaximal, characteristic, complemented
11C4×#15691348C211yesmaximal, characteristic, complemented
12D4×#13096802C211yesmaximal, characteristic, complemented
13#27161073C211yesmaximal, characteristic, complemented
14C2×#20263249C211yesmaximal, characteristic, complemented
15#552×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×Dic394C2211yescharacteristic
18C2×#15691348C2211yescharacteristic
19C2×#15691348C2211yescharacteristic
20C2×#15691345C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263314C2211yescharacteristic
24#20263312—21no—
25#20263305C2211yescharacteristic
26#20263304—21no—
27#20263300C2211yescharacteristic
28#20263298—21no—
29C4×Dic394C2211yescharacteristic
30#20263286—21no—
31#20263259C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263249C2211yescharacteristic
37#20263249C2211yescharacteristic
38#20263249C2211yescharacteristic
39#20263249C2211yescharacteristic
40#20263249C2211yescharacteristic
41#20263249C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
48#126×C197—21no—
49#125×C197—21no—
50#125×C197C2211yescharacteristic
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic
53C22⋊Q8×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic
56#117×C197—21no—
57#117×C197C2211yescharacteristic
58C2×C4⋊C4×C197C2211yescharacteristic
59#15691358—22no—
60#15691358—41no—
61C2×Dic394—22no—
62C2×Dic394—21no—
63C2×Dic394—41no—
64C2×Dic394C2311yescharacteristic
65C2×Dic394C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71#15691348—22no—
72#15691348—41no—
73#15691345—41no—
74C2×#13096802—41no—
75C2×#13096802—41no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81#15691337—21no—
82#15691337—21no—
83C22⋊Dic197—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87C4×Dic197—21no—
88C2×Q8×C197C2311yescharacteristic
89C2×D4×C197D411yescharacteristic
90C22×C4×C197—21no—
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—21no—
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197D411yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197D411yescharacteristic
107C22⋊C4×C197D411yescharacteristic
108C42×C197—21no—
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112Dic394—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—42no—
116Dic394#4112yes—
117D788—21no—
118D788—21no—
119C22⋊D197—42no—
120C22⋊D197—42no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—42no—
125C4×D197—42no—
126#13096802—21no—
127#13096802—21no—
128#13096802—21no—
129#13096802—21no—
130C23×C197C2×D411yescharacteristic
131C23×C197C2×D411yescharacteristic
132Q8×C197—21no—
133Q8×C197—21no—
134D4×C197—22no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—42no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C22×C197—21no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197—21no—
153C22×C197—21no—
154C22×C197#13512yes—
155C22×C197C22×D411yescharacteristic
156C2×D197—41no—
157C2×D197—41no—
158C2×D197—41no—
159C4×C197—21no—
160C4×C197—21no—
161C4×C197—21no—
162C4×C197—21no—
163C4×C197—41no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197—41no—
167C4×C197C2×#4111yescharacteristic
168C4×C197H(5,𝔽2)11yescharacteristic
169Dic197—22no—
170Dic197—41no—
171Dic197—41no—
172C2×C197—22no—
173C2×C197—41no—
174C2×C197#54611yescharacteristic
175C2×C197C2×#13511yescharacteristic
176C2×C197#57811yescharacteristic
177D197—81no—
178C197#330011yescharacteristic, complemented, Sylow
179#3300—1971nomaximal, Sylow
180#552D19711yescharacteristic, complemented
181#548—1971no—
182#488—1971no—
183#485—1971no—
184#480—1971no—
185#479—1971no—
186#467—1971no—
187#465—1971no—
188#464—1971no—
189#452—1971no—
190#451—1971no—
191C4×Q16—1971no—
192C8×D4—1971no—
193#416—1971no—
194C2×#103—1971no—
195C2×#41—1971no—
196C22×Q8—1971no—
197C2×Q16—1971no—
198C2×Q16—1971no—
199C2×SD16—1971no—
200C22×C8—1971no—
201C22×C8—1971no—
202#126—21no—
203#125—21no—
204#125C2×D19711yescharacteristic
205#124—3941no—
206#124—1971no—
207#123—21no—
208C22⋊Q8—3941no—
209C22⋊Q8—1971no—
210C22⋊Q8C2×D19711yescharacteristic
211C22⋊Q8C2×D19711yescharacteristic
212#121—3941no—
213#121—1971no—
214C4×Q8—1971no—
215C4×Q8C2×D19711yescharacteristic
216C4×D4—3941no—
217C4×D4C2×D19711yescharacteristic
218#117—21no—
219#117C2×D19711yescharacteristic
220C2×C4⋊C4C2×D19711yescharacteristic
221#107—1971no—
222#107—1971no—
223#106—1971no—
224C4⋊C8—1971no—
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103—1971no—
230#103—1971no—
231#103—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235C22⋊C8—1971no—
236C22⋊C8—1971no—
237C4×C8—1971no—
238#41—7881no—
239#41—3942no—
240C2×Q8—7881no—
241C2×Q8—3942no—
242C2×Q8—3941no—
243C2×Q8—1971no—
244C2×Q8—1971no—
245C2×Q8C22×D19711yescharacteristic
246C2×D4—3941no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4C22⋊D19711yescharacteristic
250C22×C4—21no—
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4C22×D19711yescharacteristic
254C22×C4C22×D19711yescharacteristic
255Q16—7881no—
256Q16—3942no—
257SD16—7881no—
258C2×C8—7881no—
259C2×C8—7881no—
260C2×C8—1971no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8—1971no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—3941no—
271C4⋊C4—3941no—
272C4⋊C4—1971no—
273C4⋊C4C22⋊D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C4⋊C4C22×D19711yescharacteristic
276C4⋊C4C22×D19711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4C22⋊D19711yescharacteristic
285C22⋊C4C22⋊D19711yescharacteristic
286C42—21no—
287C42—21no—
288C42—3941no—
289C42C22×D19711yescharacteristic
290C23—3941no—
291C23C2×C22⋊D19711yescharacteristic
292C23C2×C22⋊D19711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—7882no—
296Q8—3941no—
297Q8—3941no—
298Q8—3941no—
299Q8—1972no—
300D4—22no—
301D4—7882no—
302D4—7882no—
303D4—3941no—
304D4—3941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—7882no—
315C2×C4—7882no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4C2×C22⋊D19711yescharacteristic
320C2×C4C2×C22⋊D19711yescharacteristic
321C2×C4C2×C22⋊D19711yescharacteristic
322C2×C4C2×C22⋊D19711yescharacteristic
323C2×C4C23×D19711yescharacteristic
324C8—3941no—
325C8—3941no—
326C8—3941no—
327C8—3941no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—7881no—
334C22—7881no—
335C22—7881no—
336C22#2026334612yes—
337C22C22×C22⋊D19711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—7881no—
347C4—7881no—
348C4—3942no—
349C4#41×D19711yescharacteristic
350C4#2026337211yescharacteristic
351C2—22no—
352C2—41no—
353C2—15761no—
354C2#2716171711yescharacteristic
355C2C2×#2026334611yescharacteristic
356C2#2716189411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#552×C197C2
socleC22×C197C22×D4
prime radical#27161738C2
hypercenter#552D197
hypocenterC197#3300
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #27161436C2semidirect
3. #27161420C2semidirect
4. #27161392C2semidirect
5. #27161383C2semidirect
6. #27161318C2semidirect
7. #27161310C2semidirect
8. #27161301C2semidirect
9. #27161239C2semidirect
10. #27161234C2semidirect
11. C4×#15691348C2semidirect
12. D4×#13096802C2semidirect
13. #27161073C2semidirect
14. C2×#20263249C2semidirect
15. #552×C197C2semidirect
178. C197#3300semidirect
180. #552D197semidirect

Maximal subgroups

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

subgroupindex
#271617382
#271614362
#271614202
#271613922
#271613832
#271613182
#271613102
#271613012
#271612392
#271612342
C4×#156913482
D4×#130968022
#271610732
C2×#202632492
#552×C1972
#3300197