Finite Group Databasebeta

← #38754688

Subgroups — #38754688

Order 25216.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161775C211yesmaximal, characteristic, complemented
2#27161447C211yesmaximal, characteristic, complemented
3#27161417C211yesmaximal, characteristic, complemented
4#27161407C211yesmaximal, characteristic, complemented
5#27161379C211yesmaximal, characteristic, complemented
6#27161327C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161281C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic, complemented
10#27161224C211yesmaximal, characteristic, complemented
11#27161189C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161080C211yesmaximal, characteristic, complemented
14#27161075C211yesmaximal, characteristic, complemented
15#547×C197C211yesmaximal, characteristic, complemented
16C2×#15691345C2211yescharacteristic
17C2×#15691345C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691342C2211yescharacteristic
20C22×#13096802C2211yescharacteristic
21#20263321—21no—
22#20263318C2211yescharacteristic
23#20263308—21no—
24#20263300C2211yescharacteristic
25#20263300C2211yescharacteristic
26#20263298—21no—
27C4×D788C2211yescharacteristic
28#20263286—21no—
29#20263285C2211yescharacteristic
30C2×C4⋊Dic197C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263250C2211yescharacteristic
39#20263250C2211yescharacteristic
40#20263250C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48C2×#41×C197C2211yescharacteristic
49C2×#41×C197C2211yescharacteristic
50C4⋊Q8×C197C2211yescharacteristic
51#123×C197—21no—
52#123×C197—21no—
53C22⋊Q8×C197C2211yescharacteristic
54C22⋊Q8×C197C2211yescharacteristic
55#121×C197—21no—
56C4×D4×C197C2211yescharacteristic
57C4×D4×C197C2211yescharacteristic
58C2×C4⋊C4×C197—21no—
59C2×D788C2311yescharacteristic
60C2×C22⋊D197—21no—
61C2×C22⋊D197—21no—
62C22×Dic197—21no—
63C2×C4×D197—21no—
64#15691345—41no—
65#15691345—41no—
66#15691343—22no—
67#15691342—41no—
68C2×#13096802—41no—
69C2×#13096802C2311yescharacteristic
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C4⋊Dic197—21no—
74C4⋊Dic197C2311yescharacteristic
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83C4×Dic197—21no—
84#41×C197—22no—
85#41×C197—22no—
86#41×C197—41no—
87#41×C197—41no—
88C2×Q8×C197C2311yescharacteristic
89C2×Q8×C197C2311yescharacteristic
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197D411yescharacteristic
93C2×D4×C197C2311yescharacteristic
94C22×C4×C197—21no—
95C22×C4×C197—21no—
96C22×C4×C197—21no—
97C22×C4×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197—21no—
102C4⋊C4×C197—21no—
103C4⋊C4×C197D411yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112D788—21no—
113D788—21no—
114C22⋊D197—42no—
115C22⋊D197—81no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—81no—
120C4×D197—81no—
121#13096802—21no—
122#13096802—21no—
123#13096802—21no—
124#13096802—21no—
125C23×C197—21no—
126C23×C197C2×D411yescharacteristic
127C23×C197C2×D411yescharacteristic
128Q8×C197—21no—
129Q8×C197—21no—
130Q8×C197—21no—
131Q8×C197—21no—
132D4×C197—21no—
133D4×C197—41no—
134D4×C197—42no—
135D4×C197—42no—
136D4×C197#4112yes—
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—21no—
144C2×C4×C197—42no—
145C2×C4×C197—42no—
146C2×C4×C197—42no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2411yescharacteristic
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—22no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197C22×D411yescharacteristic
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197—21no—
168C4×C197—21no—
169C4×C197—41no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197C2×#4111yescharacteristic
173C4×C197H(5,𝔽2)11yescharacteristic
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177C2×C197—41no—
178C2×C197—41no—
179C2×C197—42no—
180C2×C197#54611yescharacteristic
181C2×C197#57511yescharacteristic
182C2×C197#57711yescharacteristic
183D197—81no—
184C197#330711yescharacteristic, complemented, Sylow
185#3307—1971nomaximal, Sylow
186#553—1971no—
187#547D19711yescharacteristic, complemented
188#491—1971no—
189#485—1971no—
190#483—1971no—
191#478—1971no—
192#468—1971no—
193#465—1971no—
194#461—1971no—
195#449—1971no—
196#449—1971no—
197#442—1971no—
198#435—1971no—
199#417—1971no—
200#416—1971no—
201C2×#41C2×D19711yescharacteristic
202C2×#41C2×D19711yescharacteristic
203C2×SD16—1971no—
204C2×SD16—1971no—
205C2×D8—1971no—
206C2×M16—1971no—
207C22×C8—1971no—
208C4⋊Q8C2×D19711yescharacteristic
209#126—3941no—
210#125—1971no—
211#123—21no—
212#123—21no—
213#123—3941no—
214C22⋊Q8C2×D19711yescharacteristic
215C22⋊Q8C2×D19711yescharacteristic
216#121—21no—
217#121—3941no—
218#121—1971no—
219#121—1971no—
220C4×D4—3941no—
221C4×D4—1971no—
222C4×D4C2×D19711yescharacteristic
223C4×D4C2×D19711yescharacteristic
224#117—1971no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1971no—
227#107—1971no—
228#106—1971no—
229#106—1971no—
230C4⋊C8—1971no—
231#103—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#103—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241C22⋊C8—1971no—
242C22⋊C8—1971no—
243#97—1971no—
244#41—22no—
245#41—22no—
246#41—41no—
247#41—41no—
248C2×Q8C22×D19711yescharacteristic
249C2×Q8C22×D19711yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3941no—
253C2×D4—3941no—
254C2×D4—1971no—
255C2×D4C22⋊D19711yescharacteristic
256C2×D4C22×D19711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3941no—
261C22×C4—3941no—
262C22×C4C22×D19711yescharacteristic
263C22×C4C22×D19711yescharacteristic
264SD16—7881no—
265SD16—7881no—
266D8—3942no—
267M16—7881no—
268C2×C8—7881no—
269C2×C8—1971no—
270C2×C8—1971no—
271C2×C8—1971no—
272C2×C8—1971no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—3941no—
278C4⋊C4—3941no—
279C4⋊C4—3941no—
280C4⋊C4—1971no—
281C4⋊C4—1971no—
282C4⋊C4—1971no—
283C4⋊C4C22⋊D19711yescharacteristic
284C4⋊C4C22×D19711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4C22⋊D19711yescharacteristic
293C22⋊C4C22⋊D19711yescharacteristic
294C42—3941no—
295C42C22×D19711yescharacteristic
296C23—21no—
297C23—3941no—
298C23C2×C22⋊D19711yescharacteristic
299C23C2×C22⋊D19711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—41no—
306D4—42no—
307D4—42no—
308D4—7882no—
309D4—15761no—
310D4—3941no—
311D4—3941no—
312D4#1569135712yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—15761no—
324C2×C4—15761no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4C2×C22⋊D19711yescharacteristic
329C2×C4C2×C22⋊D19711yescharacteristic
330C2×C4C2×C22⋊D19711yescharacteristic
331C2×C4C2×C22⋊D19711yescharacteristic
332C2×C4C23×D19711yescharacteristic
333C8—3941no—
334C8—3941no—
335C8—3941no—
336C8—3941no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—7881no—
347C22—7881no—
348C22—7881no—
349C22C22×C22⋊D19711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—7881no—
358C4—7881no—
359C4—7881no—
360C4C2×#1569135711yescharacteristic
361C4#2026337111yescharacteristic
362C2—41no—
363C2—41no—
364C2—42no—
365C2—15761no—
366C2#2716172311yescharacteristic
367C2#2716187711yescharacteristic
368C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#547×C197C2
socleC22×C197C22×D4
hypercenter#547D197
hypocenterC197#3307
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #27161775C2semidirect
2. #27161447C2semidirect
3. #27161417C2semidirect
4. #27161407C2semidirect
5. #27161379C2semidirect
6. #27161327C2semidirect
7. #27161310C2semidirect
8. #27161281C2semidirect
9. #27161224C2semidirect
10. #27161224C2semidirect
11. #27161189C2semidirect
12. #27161151C2semidirect
13. #27161080C2semidirect
14. #27161075C2semidirect
15. #547×C197C2semidirect
184. C197#3307semidirect
187. #547D197semidirect

Maximal subgroups

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

subgroupindex
#271617752
#271614472
#271614172
#271614072
#271613792
#271613272
#271613102
#271612812
#271612242
#271611892
#271611512
#271610802
#271610752
#547×C1972
#3307197