Finite Group Databasebeta

← #38752890

Subgroups — #38752890

Order 25216.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161553C211yesmaximal, characteristic, complemented
2#27161417C211yesmaximal, characteristic, complemented
3#27161400C211yesmaximal, characteristic, complemented
4#27161366C211yesmaximal, characteristic, complemented
5#27161356C211yesmaximal, characteristic, complemented
6#27161291C211yesmaximal, characteristic, complemented
7#27161291C211yesmaximal, characteristic, complemented
8#27161229C211yesmaximal, characteristic, complemented
9#27161212C211yesmaximal, characteristic
10#27161189C211yesmaximal, characteristic, complemented
11#27161177C211yesmaximal, characteristic, complemented
12C4×#15691345C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14#27161140C211yesmaximal, characteristic, complemented
15#542×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C2×#15691345C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C4×D788C2211yescharacteristic
22C4×D788C2211yescharacteristic
23C4×D788C2211yescharacteristic
24C4×D788C2211yescharacteristic
25#20263288—21no—
26#20263288—21no—
27#20263285C2211yescharacteristic
28C22⋊C4×D197—21no—
29C22⋊C4×D197—21no—
30#20263259C2211yescharacteristic, complemented
31#20263259C2211yescharacteristic, complemented
32#20263257C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35C4⋊#13096802C2211yescharacteristic, complemented
36#20263250C2211yescharacteristic, complemented
37#20263250C2211yescharacteristic, complemented
38#20263247C2211yescharacteristic, complemented
39#20263247C2211yescharacteristic, complemented
40#20263246C2211yescharacteristic
41#20263246C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44C22⋊#13096802C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46#20263237C2211yescharacteristic, complemented
47C4×#13096802C2211yescharacteristic, complemented
48C4⋊Q8×C197C2211yescharacteristic, complemented
49#126×C197—21no—
50#124×C197—21no—
51#124×C197C2211yescharacteristic, complemented
52#123×C197—21no—
53C22⋊Q8×C197—21no—
54C22⋊Q8×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic
56C4×Q8×C197C2211yescharacteristic, complemented
57C4×D4×C197C2211yescharacteristic, complemented
58#117×C197C2211yescharacteristic
59C23×D197—21no—
60C2×D788—22no—
61C2×D788—22no—
62C2×D788—21no—
63C2×D788C2311yescharacteristic
64C2×D788C2311yescharacteristic
65C2×C4×D197—21no—
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69#15691345—22no—
70#15691345—22no—
71#15691343—22no—
72#15691343—22no—
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C2×#13096802C2311yescharacteristic
76C2×#13096802C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79C22⋊Dic197—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83#15691335—21no—
84C4×Dic197—21no—
85C2×Q8×C197—21no—
86C2×Q8×C197C2311yescharacteristic
87C2×D4×C197—21no—
88C2×D4×C197C2311yescharacteristic
89C22×C4×C197—21no—
90C22×C4×C197C2311yescharacteristic
91C4⋊C4×C197—21no—
92C4⋊C4×C197—21no—
93C4⋊C4×C197—21no—
94C4⋊C4×C197D411yescharacteristic
95C4⋊C4×C197D411yescharacteristic
96C4⋊C4×C197C2311yescharacteristic
97C4⋊C4×C197C2311yescharacteristic
98C22⋊C4×C197—21no—
99C22⋊C4×C197—21no—
100C22⋊C4×C197—21no—
101C22⋊C4×C197—21no—
102C22⋊C4×C197—21no—
103C22⋊C4×C197D411yescharacteristic
104C22⋊C4×C197D411yescharacteristic
105C42×C197—21no—
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—21no—
110C22×D197—24no—
111C22×D197—81no—
112C22×D197—81no—
113D788—24no—
114D788—21no—
115D788—21no—
116D788#4112yes—
117D788#4112yes—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C4×D197—81no—
121C4×D197—81no—
122C4×D197—81no—
123C4×D197—81no—
124#13096802—21no—
125#13096802—21no—
126#13096802—21no—
127#13096802—21no—
128C23×C197—21no—
129C23×C197C2×D411yescharacteristic
130Q8×C197—21no—
131Q8×C197—21no—
132Q8×C197—81no—
133D4×C197—21no—
134D4×C197—21no—
135D4×C197—81no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—81no—
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C22×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197C22×D411yescharacteristic
155C2×D197—22no—
156C2×D197—22no—
157C2×D197—41no—
158C2×D197—41no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—84no—
162C4×C197—21no—
163C4×C197—21no—
164C4×C197—21no—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197C2×#4111yescharacteristic
171C4×C197C2×#4111yescharacteristic
172Dic197—41no—
173Dic197—41no—
174C2×C197—41no—
175C2×C197—81no—
176C2×C197#52511yescharacteristic
177C2×C197#57611yescharacteristic
178C2×C197#57711yescharacteristic
179D197—42no—
180D197—42no—
181C197#311811yescharacteristic, complemented, Sylow
182#3118—1971nomaximal, Sylow
183#542D19711yescharacteristic, complemented
184#518—1971no—
185#485—1971no—
186#482—1971no—
187#476—1971no—
188#474—1971no—
189#463—1971no—
190#463—1971no—
191#450—1971no—
192#447—1971no—
193#442—1971no—
194#440—1971no—
195C4×SD16—1971no—
196C4×D8—1971no—
197#433—1971no—
198C22×D4—1971no—
199C2×SD16—1971no—
200C2×SD16—1971no—
201C2×D8—1971no—
202C2×D8—1971no—
203C4⋊Q8C2×D19711yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D19711yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D19711yescharacteristic
210#121C2×D19711yescharacteristic
211C4×Q8C2×D19711yescharacteristic, complemented
212C4×D4—3941no—
213C4×D4—3941no—
214C4×D4—1971no—
215C4×D4—1971no—
216C4×D4—1971no—
217C4×D4—1971no—
218C4×D4C2×D19711yescharacteristic, complemented
219#117—1971no—
220#117C2×D19711yescharacteristic
221C2×C22⋊C4—3941no—
222C2×C22⋊C4—3941no—
223#107—1971no—
224#107—1971no—
225#106—1971no—
226#106—1971no—
227C4⋊C8—1971no—
228C4⋊C8—1971no—
229#103—1971no—
230#103—1971no—
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102—1971no—
236#102—1971no—
237C22⋊C8—1971no—
238C22⋊C8—1971no—
239#97—1971no—
240C4×C8—1971no—
241C24—3941no—
242C2×Q8—21no—
243C2×Q8C22×D19711yescharacteristic
244C2×D4—21no—
245C2×D4—3942no—
246C2×D4—3942no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4—1971no—
250C2×D4C22×D19711yescharacteristic
251C22×C4—21no—
252C22×C4—3941no—
253C22×C4—3941no—
254C22×C4—3941no—
255C22×C4—3941no—
256C22×C4C22×D19711yescharacteristic
257SD16—3942no—
258SD16—3942no—
259D8—3942no—
260D8—3942no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8—1971no—
264C2×C8—1971no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4C22⋊D19711yescharacteristic
271C4⋊C4C22⋊D19711yescharacteristic
272C4⋊C4C22×D19711yescharacteristic
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4C22⋊D19711yescharacteristic
285C22⋊C4C22⋊D19711yescharacteristic
286C42—21no—
287C42—3941no—
288C42C22×D19711yescharacteristic
289C42C22×D19711yescharacteristic
290C23—21no—
291C23—15761no—
292C23—15761no—
293C23—3944no—
294C23—3941no—
295C23—3941no—
296C23C2×C22⋊D19711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—3944no—
304D4—3941no—
305D4—3941no—
306D4—1972no—
307D4—1972no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—15761no—
315C2×C4—15761no—
316C2×C4—15761no—
317C2×C4—15761no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4C2×C22⋊D19711yescharacteristic
321C2×C4C2×C22⋊D19711yescharacteristic
322C2×C4C2×C22⋊D19711yescharacteristic
323C2×C4C2×C22⋊D19711yescharacteristic
324C2×C4C2×C22⋊D19711yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—3941no—
327C8—3941no—
328C8—3941no—
329C8—3941no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—7881no—
337C22—7881no—
338C22—7881no—
339C22—7881no—
340C22—15764no—
341C22—3942no—
342C22—3942no—
343C22C22×C22⋊D19711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—7881no—
353C4—7881no—
354C4#41×D19711yescharacteristic
355C4#41×D19711yescharacteristic
356C2—41no—
357C2—81no—
358C2—7882no—
359C2—7882no—
360C2#2716158611yescharacteristic
361C2#2716188211yescharacteristic
362C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#542×C197C2
socleC22×C197C22×D4
prime radical#27161212C2
hypercenter#542D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #27161553C2semidirect
2. #27161417C2semidirect
3. #27161400C2semidirect
4. #27161366C2semidirect
5. #27161356C2semidirect
6. #27161291C2semidirect
7. #27161291C2semidirect
8. #27161229C2semidirect
10. #27161189C2semidirect
11. #27161177C2semidirect
12. C4×#15691345C2semidirect
13. C4×#15691343C2semidirect
14. #27161140C2semidirect
15. #542×C197C2semidirect
30. #20263259C22semidirect
31. #20263259C22semidirect
32. #20263257C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. C4⋊#13096802C22semidirect
36. #20263250C22semidirect
37. #20263250C22semidirect
38. #20263247C22semidirect
39. #20263247C22semidirect
46. #20263237C22semidirect
47. C4×#13096802C22semidirect
48. C4⋊Q8×C197C22semidirect
51. #124×C197C22semidirect
56. C4×Q8×C197C22semidirect
57. C4×D4×C197C22semidirect
181. C197#3118semidirect
183. #542D197semidirect
203. C4⋊Q8C2×D197semidirect
206. #124C2×D197semidirect
211. C4×Q8C2×D197semidirect
218. C4×D4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615532
#271614172
#271614002
#271613662
#271613562
#271612912
#271612292
#271612122
#271611892
#271611772
C4×#156913452
C4×#156913432
#271611402
#542×C1972
#3118197