Finite Group Databasebeta

← #38752958

Subgroups — #38752958

Order 25216.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161689C211yesmaximal, characteristic, complemented
2#27161554C211yesmaximal, characteristic
3#27161421C211yesmaximal, characteristic, complemented
4#27161417C211yesmaximal, characteristic, complemented
5#27161381C211yesmaximal, characteristic, complemented
6#27161377C211yesmaximal, characteristic, complemented
7#27161282C211yesmaximal, characteristic, complemented
8#27161282C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic
10#27161180C211yesmaximal, characteristic, complemented
11#27161176C211yesmaximal, characteristic, complemented
12#27161170C211yesmaximal, characteristic, complemented
13SD16×Dic197C211yesmaximal, characteristic, complemented
14#27161137C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic
16C2×#15691346C2211yescharacteristic
17C2×#15691346C2211yescharacteristic
18#20263324C2211yescharacteristic, complemented
19#20263319—21no—
20#20263319—21no—
21#20263317—21no—
22#20263317C2211yescharacteristic, complemented
23#20263316—21no—
24#20263305C2211yescharacteristic
25Q8×Dic197C2211yescharacteristic, complemented
26#20263291C2211yescharacteristic, complemented
27D4×Dic197—21no—
28D4×Dic197—21no—
29D4×Dic197C2211yescharacteristic
30D4×Dic197C2211yescharacteristic
31D4×Dic197C2211yescharacteristic
32D4×Dic197C2211yescharacteristic
33#20263285C2211yescharacteristic
34#20263285C2211yescharacteristic
35C2×C22⋊Dic197—21no—
36C2×C22⋊Dic197—21no—
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263258C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263255C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263250C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263236C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C22×D4×C197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic
54C2×SD16×C197C2211yescharacteristic
55C22⋊Q8×C197C2211yescharacteristic
56#102×C197C2211yescharacteristic
57#102×C197C2211yescharacteristic
58C22⋊C8×C197C2211yescharacteristic
59C2×Dic394C2311yescharacteristic
60C22×Dic197—21no—
61C22×Dic197—21no—
62C22×Dic197—21no—
63C22×Dic197—21no—
64#15691346—22no—
65#15691346—22no—
66C2×#13096802C2311yescharacteristic
67C2×#13096802C2311yescharacteristic
68C4⋊Dic197—21no—
69C4⋊Dic197—21no—
70C4⋊Dic197D411yescharacteristic
71C4⋊Dic197D411yescharacteristic
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74C4⋊Dic197C2311yescharacteristic
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79#15691337—21no—
80C22⋊Dic197—21no—
81C22⋊Dic197—21no—
82C22⋊Dic197—21no—
83C22⋊Dic197—21no—
84C22⋊Dic197—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197D411yescharacteristic
87C22⋊Dic197D411yescharacteristic
88C4×Dic197—21no—
89C4×Dic197—21no—
90C4×Dic197—21no—
91C4×Dic197C2311yescharacteristic
92C4×Dic197C2311yescharacteristic
93C24×C197—21no—
94C2×Q8×C197C2311yescharacteristic
95C2×D4×C197—22no—
96C2×D4×C197—22no—
97C2×D4×C197—21no—
98C2×D4×C197C2311yescharacteristic
99C2×D4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101SD16×C197—22no—
102SD16×C197—22no—
103C2×C8×C197C2311yescharacteristic
104C2×C8×C197C2311yescharacteristic
105C4⋊C4×C197—21no—
106C4⋊C4×C197C2311yescharacteristic
107C22⋊C4×C197—21no—
108Dic394—21no—
109Dic394—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—81no—
115C2×Dic197—81no—
116C2×Dic197—81no—
117C2×Dic197—81no—
118C2×Dic197C2×D411yescharacteristic
119C2×Dic197C2×D411yescharacteristic
120C2×Dic197C2×D411yescharacteristic
121C2×Dic197C2×D411yescharacteristic
122#13096802—21no—
123#13096802—21no—
124C23×C197—24no—
125C23×C197—21no—
126C23×C197—21no—
127C23×C197—81no—
128C23×C197—81no—
129C23×C197C2×D411yescharacteristic
130Q8×C197—21no—
131Q8×C197—21no—
132D4×C197—24no—
133D4×C197—21no—
134D4×C197—21no—
135D4×C197#4112yes—
136D4×C197#4112yes—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2411yescharacteristic
143C8×C197—21no—
144C8×C197—21no—
145C22×C197—21no—
146C22×C197—22no—
147C22×C197—22no—
148C22×C197—21no—
149C22×C197—21no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197—84no—
155C22×C197C22×D411yescharacteristic
156C4×C197—21no—
157C4×C197—41no—
158C4×C197—41no—
159C4×C197C2×#4111yescharacteristic
160C4×C197C2×#4111yescharacteristic
161Dic197—21no—
162Dic197—21no—
163Dic197—21no—
164Dic197—21no—
165Dic197—41no—
166Dic197—41no—
167Dic197—41no—
168Dic197—41no—
169C2×C197—41no—
170C2×C197—42no—
171C2×C197—42no—
172C2×C197#52511yescharacteristic
173C2×C197#57711yescharacteristic
174C2×C197#57711yescharacteristic
175C197#312311yescharacteristic, complemented, Sylow
176#3123—1971nomaximal, Sylow
177#541—1971no—
178#518—1971no—
179#485—1971no—
180#485—1971no—
181#478—1971no—
182#478—1971no—
183#461—1971no—
184#461—1971no—
185#450—1971no—
186#450D19711yescharacteristic
187#440—1971no—
188#440—1971no—
189C4×SD16—1971no—
190C4×SD16—1971no—
191#433—1971no—
192C22×D4C2×D19711yescharacteristic
193C2×SD16—1971no—
194C2×SD16—1971no—
195C2×SD16C2×D19711yescharacteristic
196C2×SD16C2×D19711yescharacteristic
197C4⋊Q8—1971no—
198#126—3941no—
199#126—3941no—
200#125—3941no—
201#125—3941no—
202#125—1971no—
203C22⋊Q8—1971no—
204C22⋊Q8C2×D19711yescharacteristic
205C4×Q8—1971no—
206C4×Q8—1971no—
207C4×D4—3941no—
208C4×D4—3941no—
209C4×D4—1971no—
210C4×D4—1971no—
211C4×D4—1971no—
212C4×D4—1971no—
213#117—1971no—
214#117—1971no—
215C2×C22⋊C4—3941no—
216C2×C22⋊C4—3941no—
217#107—1971no—
218#107—1971no—
219#106—1971no—
220#106—1971no—
221C4⋊C8—1971no—
222C4⋊C8—1971no—
223#103—1971no—
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#102—1971no—
228#102—1971no—
229#102C2×D19711yescharacteristic
230#102C2×D19711yescharacteristic
231C22⋊C8—1971no—
232C22⋊C8C2×D19711yescharacteristic
233#97—1971no—
234C4×C8—1971no—
235C24—21no—
236C2×Q8—1971no—
237C2×Q8C22×D19711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D19711yescharacteristic
242C2×D4C22×D19711yescharacteristic
243C22×C4—3941no—
244C22×C4—3941no—
245C22×C4—3941no—
246C22×C4—3941no—
247C22×C4C22×D19711yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—3942no—
251SD16—3942no—
252C2×C8—1971no—
253C2×C8—1971no—
254C2×C8C22×D19711yescharacteristic
255C2×C8C22×D19711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3941no—
258C4⋊C4—3941no—
259C4⋊C4—3941no—
260C4⋊C4—3941no—
261C4⋊C4—3941no—
262C4⋊C4—3941no—
263C4⋊C4—3941no—
264C4⋊C4—1971no—
265C4⋊C4—1971no—
266C4⋊C4—1971no—
267C4⋊C4—1971no—
268C4⋊C4—1971no—
269C4⋊C4C22×D19711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3941no—
272C22⋊C4—3941no—
273C22⋊C4—3941no—
274C22⋊C4—3941no—
275C22⋊C4—3941no—
276C22⋊C4—3941no—
277C22⋊C4—1971no—
278C22⋊C4—1971no—
279C42—3941no—
280C42—3941no—
281C42—3941no—
282C42—1971no—
283C42—1971no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D19711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3941no—
293Q8—3941no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1569135712yes—
298D4#1569135712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—15761no—
304C2×C4—15761no—
305C2×C4—15761no—
306C2×C4—15761no—
307C2×C4—3941no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—1971no—
312C2×C4—1971no—
313C2×C4—1971no—
314C2×C4—1971no—
315C2×C4D4×D19711yescharacteristic
316C2×C4C23×D19711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3941no—
320C8—3941no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D19711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—7881no—
336C4—7881no—
337C4—7881no—
338C4—7881no—
339C4—3941no—
340C4—3941no—
341C4—3941no—
342C4—3941no—
343C4C2×#1569135711yescharacteristic
344C4C2×#1569135711yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2716158511yescharacteristic
349C2#2716188511yescharacteristic
350C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
prime radicalC22×D4×C197C22
hypercenter#450D197
hypocenterC197#3123
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27161689C2semidirect
3. #27161421C2semidirect
4. #27161417C2semidirect
5. #27161381C2semidirect
6. #27161377C2semidirect
7. #27161282C2semidirect
8. #27161282C2semidirect
10. #27161180C2semidirect
11. #27161176C2semidirect
12. #27161170C2semidirect
13. SD16×Dic197C2semidirect
14. #27161137C2semidirect
18. #20263324C22semidirect
22. #20263317C22semidirect
25. Q8×Dic197C22semidirect
26. #20263291C22semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263258C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263255C22semidirect
43. #20263251C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263250C22semidirect
50. #20263236C22semidirect
51. C8×Dic197C22semidirect
175. C197#3123semidirect

Maximal subgroups

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

subgroupindex
#271616892
#271615542
#271614212
#271614172
#271613812
#271613772
#271612822
#271612302
#271611802
#271611762
#271611702
SD16×Dic1972
#271611372
#450×C1972
#3123197