Finite Group Databasebeta

← #38752894

Subgroups — #38752894

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#27161694C211yesmaximal, characteristic, complemented
2#27161551C211yesmaximal, characteristic, complemented
3#27161423C211yesmaximal, characteristic, complemented
4#27161401C211yesmaximal, characteristic, complemented
5#27161368C211yesmaximal, characteristic, complemented
6#27161357C211yesmaximal, characteristic, complemented
7#27161293C211yesmaximal, characteristic, complemented
8#27161292C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic, complemented
10#27161211C211yesmaximal, characteristic
11#27161181C211yesmaximal, characteristic, complemented
12C4×#15691346C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14#27161141C211yesmaximal, characteristic, complemented
15#442×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C2211yescharacteristic
17C2×#15691347C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20#20263325C2211yescharacteristic, complemented
21#20263319—21no—
22#20263313C2211yescharacteristic, complemented
23#20263312—21no—
24#20263309—21no—
25#20263306C2211yescharacteristic
26#20263305—21no—
27#20263297C2211yescharacteristic
28C4×Dic394C2211yescharacteristic, complemented
29C4×D788C2211yescharacteristic
30D4×Dic197C2211yescharacteristic
31#20263288C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263284C2211yescharacteristic
35#20263283C2211yescharacteristic
36C2×#15691335—21no—
37C22⋊C4×D197—21no—
38#20263260C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263249C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263247C2211yescharacteristic
47#20263247C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic
51C4×#13096802C2211yescharacteristic, complemented
52C2×D8×C197C2211yescharacteristic
53C4×D4×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic, complemented
58#97×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C2×Dic394—21no—
61C2×Dic394C2311yescharacteristic
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64C2×C22⋊D197—21no—
65D4×D197—22no—
66D4×D197—22no—
67C22×Dic197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197C2311yescharacteristic
71#15691347—22no—
72#15691346—22no—
73#15691343—22no—
74C2×#13096802C2311yescharacteristic
75C2×#13096802C2311yescharacteristic
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79#15691337—21no—
80#15691337—21no—
81#15691337D411yescharacteristic
82#15691337D411yescharacteristic
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87#15691335—21no—
88#15691335—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335D411yescharacteristic
92#15691335D411yescharacteristic
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197C2311yescharacteristic
96C2×D4×C197C2311yescharacteristic
97C2×D4×C197C2311yescharacteristic
98C22×C4×C197—21no—
99C22×C4×C197—21no—
100D8×C197—22no—
101C2×C8×C197C2311yescharacteristic
102C2×C8×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—24no—
110C22×D197—81no—
111C22×D197—81no—
112C22×D197C2×D411yescharacteristic
113Dic394—21no—
114Dic394—21no—
115Dic394—81no—
116D788—21no—
117D788#4112yes—
118C22⋊D197—24no—
119C22⋊D197—81no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—81no—
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C4×D197—21no—
128C4×D197—21no—
129C4×D197—81no—
130C4×D197—81no—
131#13096802—21no—
132#13096802—21no—
133C23×C197—21no—
134C23×C197—21no—
135D4×C197—21no—
136D4×C197—21no—
137D4×C197—21no—
138D4×C197#4112yes—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—81no—
142C2×C4×C197—81no—
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—21no—
147C8×C197—21no—
148C22×C197—22no—
149C22×C197—41no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197C22×D411yescharacteristic
155C2×D197—21no—
156C2×D197—22no—
157C2×D197—21no—
158C2×D197—21no—
159C2×D197—41no—
160C2×D197—41no—
161C2×D197—84no—
162C4×C197—21no—
163C4×C197—21no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197C2×#4111yescharacteristic
167C4×C197C2×#4111yescharacteristic
168Dic197—21no—
169Dic197—21no—
170Dic197—21no—
171Dic197—41no—
172Dic197—41no—
173Dic197—41no—
174C2×C197—42no—
175C2×C197—81no—
176C2×C197#52511yescharacteristic
177C2×C197#57611yescharacteristic
178C2×C197#57711yescharacteristic
179D197—41no—
180D197—42no—
181C197#311811yescharacteristic, complemented, Sylow
182#3118—1971nomaximal, Sylow
183#542—1971no—
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#442D19711yescharacteristic, complemented
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×D8C2×D19711yescharacteristic
203C4⋊Q8—1971no—
204#126—3941no—
205#124—3941no—
206#124—1971no—
207#123—3941no—
208C22⋊Q8—3941no—
209C22⋊Q8—1971no—
210#121—1971no—
211C4×Q8—1971no—
212C4×D4—3941no—
213C4×D4—3941no—
214C4×D4—1971no—
215C4×D4—1971no—
216C4×D4—1971no—
217C4×D4C2×D19711yescharacteristic
218C4×D4C2×D19711yescharacteristic, complemented
219#117—1971no—
220#117—1971no—
221C2×C22⋊C4—3941no—
222C2×C22⋊C4—3941no—
223#107—1971no—
224#107—1971no—
225#106—1971no—
226#106C2×D19711yescharacteristic, complemented
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#102C2×D19711yescharacteristic
236#102C2×D19711yescharacteristic, complemented
237C22⋊C8—1971no—
238C22⋊C8—1971no—
239#97C2×D19711yescharacteristic, complemented
240C4×C8—1971no—
241C24—3941no—
242C2×Q8—3941no—
243C2×Q8—1971no—
244C2×D4—3942no—
245C2×D4—3941no—
246C2×D4—3942no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4C22×D19711yescharacteristic
250C2×D4C22×D19711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—3941no—
254C22×C4—3941no—
255C22×C4—3941no—
256C22×C4—1971no—
257SD16—3942no—
258SD16—3942no—
259D8—22no—
260D8—3942no—
261C2×C8—1971no—
262C2×C8—1971no—
263C2×C8C22×D19711yescharacteristic
264C2×C8C22×D19711yescharacteristic
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—1971no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4C22×D19711yescharacteristic
273C4⋊C4C22×D19711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C22⋊C4—1971no—
285C22⋊C4—1971no—
286C42—3941no—
287C42—3941no—
288C42—1971no—
289C42C22×D19711yescharacteristic
290C23—21no—
291C23—21no—
292C23—15761no—
293C23—15761no—
294C23—3944no—
295C23—3941no—
296C23—1971no—
297Q8—15761no—
298Q8—3941no—
299Q8—3941no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—15761no—
304D4—3944no—
305D4—3941no—
306D4—1972no—
307D4#1569135812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—15761no—
313C2×C4—15761no—
314C2×C4—15761no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—1971no—
321C2×C4—1971no—
322C2×C4—1971no—
323C2×C4D4×D19711yescharacteristic
324C2×C4D4×D19711yescharacteristic
325C2×C4C23×D19711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3941no—
329C8—3941no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—7881no—
337C22—7881no—
338C22—15764no—
339C22—3941no—
340C22—3941no—
341C22—3941no—
342C22—3942no—
343C22C2×D4×D19711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—7881no—
349C4—7881no—
350C4—7881no—
351C4—3941no—
352C4—3941no—
353C4—3941no—
354C4#41×D19711yescharacteristic
355C4C2×#1569135811yescharacteristic
356C2—42no—
357C2—81no—
358C2—7882no—
359C2—7881no—
360C2#2716158811yescharacteristic
361C2#2716188311yescharacteristic
362C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#442×C197C2
socleC22×C197C22×D4
prime radical#27161211C2
hypercenter#442D197
hypocenterC197#3118
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #27161694C2semidirect
2. #27161551C2semidirect
3. #27161423C2semidirect
4. #27161401C2semidirect
5. #27161368C2semidirect
6. #27161357C2semidirect
7. #27161293C2semidirect
8. #27161292C2semidirect
9. #27161231C2semidirect
11. #27161181C2semidirect
12. C4×#15691346C2semidirect
13. C4×#15691343C2semidirect
14. #27161141C2semidirect
15. #442×C197C2semidirect
20. #20263325C22semidirect
22. #20263313C22semidirect
28. C4×Dic394C22semidirect
38. #20263260C22semidirect
39. #20263259C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263251C22semidirect
44. #20263249C22semidirect
47. #20263247C22semidirect
51. C4×#13096802C22semidirect
54. C4×D4×C197C22semidirect
55. #106×C197C22semidirect
57. #102×C197C22semidirect
58. #97×C197C22semidirect
181. C197#3118semidirect
193. #442D197semidirect
218. C4×D4C2×D197semidirect
226. #106C2×D197semidirect
236. #102C2×D197semidirect
239. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616942
#271615512
#271614232
#271614012
#271613682
#271613572
#271612932
#271612922
#271612312
#271612112
#271611812
C4×#156913462
C4×#156913432
#271611412
#442×C1972
#3118197