Finite Group Databasebeta

← #38754559

Subgroups — #38754559

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161782C211yesmaximal, characteristic, complemented
2#27161724C211yesmaximal, characteristic, complemented
3#27161436C211yesmaximal, characteristic, complemented
4#27161416C211yesmaximal, characteristic, complemented
5#27161405C211yesmaximal, characteristic, complemented
6#27161316C211yesmaximal, characteristic, complemented
7#27161309C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161225C211yesmaximal, characteristic, complemented
10#27161214C211yesmaximal, characteristic, complemented
11C4×#15691343C211yesmaximal, characteristic, complemented
12#27161144C211yesmaximal, characteristic, complemented
13#27161072C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#478×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C2×#15691347C2211yescharacteristic, complemented
19C2×D1576C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C2×C8×D197C2211yescharacteristic
22C2×C8×D197C2211yescharacteristic, complemented
23#20263321—21no—
24#20263314C2211yescharacteristic
25#20263311—21no—
26#20263310—21no—
27#20263306C2211yescharacteristic
28#20263301—21no—
29#20263301C2211yescharacteristic
30#20263301C2211yescharacteristic
31#20263301C2211yescharacteristic, complemented
32#20263300—21no—
33#20263296—21no—
34C4×D788C2211yescharacteristic
35C4×D788C2211yescharacteristic, complemented
36#20263288—21no—
37#20263284C2211yescharacteristic, complemented
38C4⋊C4×D197C2211yescharacteristic
39C2×#15691335—21no—
40#20263259C2211yescharacteristic, complemented
41#20263251C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263248C2211yescharacteristic
45#20263247C2211yescharacteristic
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic, complemented
49#20263238C2211yescharacteristic
50#20263238C2211yescharacteristic, complemented
51C4×#13096802C2211yescharacteristic, complemented
52#125×C197C2211yescharacteristic, complemented
53C4×D4×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56C4⋊C8×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59C23×D197—21no—
60#15691357—22no—
61#15691357—41no—
62C2×Dic394C2311yescharacteristic
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788D411yescharacteristic
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71D4×D197—22no—
72D4×D197—41no—
73C22×Dic197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197C2311yescharacteristic
77C2×C4×D197C2311yescharacteristic
78#15691347—41no—
79D1576—41no—
80#15691343—22no—
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C8×D197—41no—
84C8×D197—41no—
85C4⋊Dic197—21no—
86C4⋊Dic197D411yescharacteristic
87C4⋊Dic197C2311yescharacteristic
88#15691337—21no—
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#15691335D411yescharacteristic
98#15691335D411yescharacteristic
99C4×Dic197—21no—
100C2×D4×C197C2311yescharacteristic
101C22×C4×C197—21no—
102C2×C8×C197C2311yescharacteristic
103C2×C8×C197C2311yescharacteristic
104C4⋊C4×C197—21no—
105C4⋊C4×C197—21no—
106C4⋊C4×C197C2311yescharacteristic
107C4⋊C4×C197C2311yescharacteristic
108C4⋊C4×C197C2311yescharacteristic
109C22⋊C4×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116C22×D197C2×D411yescharacteristic
117C22×D197C2×D411yescharacteristic
118Dic394—21no—
119Dic394—21no—
120D788—22no—
121D788—21no—
122D788—21no—
123D788—21no—
124D788—21no—
125D788—42no—
126D788—81no—
127D788—81no—
128C22⋊D197—42no—
129C22⋊D197—42no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197—42no—
133C2×Dic197C2×D411yescharacteristic
134C2×Dic197C2×D411yescharacteristic
135C4×D197—21no—
136C4×D197—21no—
137C4×D197—21no—
138C4×D197—21no—
139C4×D197—81no—
140C4×D197—81no—
141#13096802—21no—
142#13096802—21no—
143C23×C197—21no—
144D4×C197—21no—
145D4×C197#4112yes—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—81no—
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2411yescharacteristic
153C8×C197—21no—
154C8×C197—21no—
155C22×C197—22no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197C22×D411yescharacteristic
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—21no—
162C2×D197—21no—
163C2×D197—21no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C2×D197—41no—
169C2×D197—41no—
170C2×D197—84no—
171C2×D197#13512yes—
172C4×C197—21no—
173C4×C197—21no—
174C4×C197—41no—
175C4×C197—41no—
176C4×C197—41no—
177C4×C197C2×#4111yescharacteristic
178C4×C197H(5,𝔽2)11yescharacteristic
179Dic197—21no—
180Dic197—21no—
181Dic197—41no—
182Dic197—41no—
183C2×C197—42no—
184C2×C197#54611yescharacteristic
185C2×C197C2×#13511yescharacteristic
186C2×C197#57611yescharacteristic
187D197—22no—
188D197—41no—
189D197—81no—
190D197—81no—
191C197#329711yescharacteristic, complemented, Sylow
192#3297—1971nomaximal, Sylow
193#553—1971no—
194#546—1971no—
195#488—1971no—
196#484—1971no—
197#482—1971no—
198#478D19711yescharacteristic, complemented
199#466—1971no—
200#465—1971no—
201#463—1971no—
202#449—1971no—
203#447—1971no—
204C4×D8—1971no—
205C8×D4—1971no—
206#416—1971no—
207C2×#102—1971no—
208C2×#41—1971no—
209C22×D4—1971no—
210C2×SD16—1971no—
211C2×D8—1971no—
212C2×D8—1971no—
213C22×C8—1971no—
214C22×C8—1971no—
215#126—3941no—
216#125C2×D19711yescharacteristic, complemented
217#124—1971no—
218#123—3941no—
219#123—3941no—
220C22⋊Q8—1971no—
221#121—3941no—
222#121—3941no—
223#121—1971no—
224#121—1971no—
225#121—1971no—
226#120—3941no—
227C4×D4—3941no—
228C4×D4—1971no—
229C4×D4—1971no—
230C4×D4C2×D19711yescharacteristic
231#117—1971no—
232C2×C4⋊C4—1971no—
233C2×C22⋊C4—3941no—
234#107—1971no—
235#107C2×D19711yescharacteristic, complemented
236#106C2×D19711yescharacteristic, complemented
237C4⋊C8C2×D19711yescharacteristic, complemented
238#103—1971no—
239#103—1971no—
240#103—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102C2×D19711yescharacteristic
247#102C2×D19711yescharacteristic
248C22⋊C8—1971no—
249C22⋊C8—1971no—
250C4×C8—1971no—
251C24—3941no—
252#41—7881no—
253#41—3942no—
254C2×Q8—1971no—
255C2×D4—7881no—
256C2×D4—3941no—
257C2×D4—3941no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—3942no—
261C2×D4—3941no—
262C2×D4—1971no—
263C2×D4—1971no—
264C2×D4—1971no—
265C2×D4C22×D19711yescharacteristic
266C22×C4—21no—
267C22×C4—3941no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4—1971no—
271C22×C4—1971no—
272SD16—7881no—
273D8—7881no—
274D8—3942no—
275C2×C8—7881no—
276C2×C8—7881no—
277C2×C8—1971no—
278C2×C8—1971no—
279C2×C8C22×D19711yescharacteristic
280C2×C8C22×D19711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3941no—
284C4⋊C4—3941no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4C22×D19711yescharacteristic
288C4⋊C4C22×D19711yescharacteristic
289C4⋊C4C22×D19711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3941no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C22⋊C4—1971no—
300C22⋊C4—1971no—
301C42—3941no—
302C42C22×D19711yescharacteristic
303C23—21no—
304C23—7882no—
305C23—7882no—
306C23—15761no—
307C23—3941no—
308C23—3941no—
309C23—1971no—
310C23—1971no—
311Q8—3941no—
312Q8—3941no—
313D4—21no—
314D4—7882no—
315D4—7882no—
316D4—7882no—
317D4—15761no—
318D4—15761no—
319D4—3942no—
320D4—3941no—
321D4—3941no—
322D4—3941no—
323D4—3941no—
324D4#1569135912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—81no—
329C2×C4—7882no—
330C2×C4—15761no—
331C2×C4—15761no—
332C2×C4—3941no—
333C2×C4—3941no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—1971no—
339C2×C4—1971no—
340C2×C4D4×D19711yescharacteristic
341C2×C4D4×D19711yescharacteristic
342C2×C4C23×D19711yescharacteristic
343C8—21no—
344C8—21no—
345C8—3941no—
346C8—3941no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—7881no—
351C22—7881no—
352C22—7881no—
353C22—7881no—
354C22—7881no—
355C22—7881no—
356C22—15764no—
357C22—3941no—
358C22—3941no—
359C22—3941no—
360C22—3941no—
361C22—3941no—
362C22—1972no—
363C22C2×D4×D19711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—7881no—
370C4—7881no—
371C4—3941no—
372C4—3941no—
373C4C2×#1569135911yescharacteristic
374C4#2026337211yescharacteristic
375C2—42no—
376C2—7881no—
377C2—15761no—
378C2—15761no—
379C2—3942no—
380C2#2716172011yescharacteristic
381C2#135×D19711yescharacteristic
382C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#478×C197C2
socleC22×C197C22×D4
hypercenter#478D197
hypocenterC197#3297
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #27161782C2semidirect
2. #27161724C2semidirect
3. #27161436C2semidirect
4. #27161416C2semidirect
5. #27161405C2semidirect
6. #27161316C2semidirect
7. #27161309C2semidirect
8. #27161295C2semidirect
9. #27161225C2semidirect
10. #27161214C2semidirect
11. C4×#15691343C2semidirect
12. #27161144C2semidirect
13. #27161072C2semidirect
14. #102×D197C2semidirect
15. #478×C197C2semidirect
18. C2×#15691347C22semidirect
22. C2×C8×D197C22semidirect
31. #20263301C22semidirect
35. C4×D788C22semidirect
37. #20263284C22semidirect
40. #20263259C22semidirect
41. #20263251C22semidirect
42. #20263249C22semidirect
43. #20263249C22semidirect
48. #20263246C22semidirect
50. #20263238C22semidirect
51. C4×#13096802C22semidirect
52. #125×C197C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. C4⋊C8×C197C22semidirect
191. C197#3297semidirect
198. #478D197semidirect
216. #125C2×D197semidirect
235. #107C2×D197semidirect
236. #106C2×D197semidirect
237. C4⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617822
#271617242
#271614362
#271614162
#271614052
#271613162
#271613092
#271612952
#271612252
#271612142
C4×#156913432
#271611442
#271610722
#102×D1972
#478×C1972
#3297197