Finite Group Databasebeta

← #38754697

Subgroups — #38754697

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161738C211yesmaximal, characteristic, complemented
2#27161441C211yesmaximal, characteristic, complemented
3#27161414C211yesmaximal, characteristic, complemented
4#27161392C211yesmaximal, characteristic, complemented
5#27161371C211yesmaximal, characteristic, complemented
6#27161323C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic, complemented
8#27161294C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic, complemented
10#27161217C211yesmaximal, characteristic, complemented
11#27161179C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161078C211yesmaximal, characteristic, complemented
14C2×#20263249C211yesmaximal, characteristic, complemented
15#547×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×Dic394C2211yescharacteristic
18C2×#15691346C2211yescharacteristic
19C2×#15691346C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C2×#15691342C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263314C2211yescharacteristic
24#20263312—21no—
25#20263305C2211yescharacteristic
26#20263304—21no—
27#20263300C2211yescharacteristic
28#20263298—21no—
29C4×Dic394C2211yescharacteristic
30#20263286—21no—
31#20263259C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263259C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263249C2211yescharacteristic
36#20263249C2211yescharacteristic
37#20263249C2211yescharacteristic
38#20263249C2211yescharacteristic
39#20263249C2211yescharacteristic
40#20263249C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48C2×#41×C197—21no—
49C4⋊Q8×C197C2211yescharacteristic
50#123×C197—21no—
51#123×C197—21no—
52C22⋊Q8×C197—21no—
53#121×C197C2211yescharacteristic
54#121×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic
56C4×D4×C197C2211yescharacteristic
57C2×C4⋊C4×C197C2211yescharacteristic
58C2×C4⋊C4×C197C2211yescharacteristic
59#15691358—22no—
60#15691358—41no—
61C2×Dic394—22no—
62C2×Dic394—21no—
63C2×Dic394—41no—
64C2×Dic394C2311yescharacteristic
65C2×Dic394C2311yescharacteristic
66C2×D788C2311yescharacteristic
67C2×C22⋊D197—21no—
68C2×C22⋊D197—21no—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71#15691346—22no—
72#15691346—41no—
73#15691343—41no—
74#15691342—41no—
75C2×#13096802—41no—
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81#15691337—21no—
82#15691337—21no—
83C22⋊Dic197—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87C4×Dic197—21no—
88#41×C197—44no—
89C2×Q8×C197—21no—
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×C197D411yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C4⋊C4×C197C2311yescharacteristic
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197—21no—
109C22⋊C4×C197D411yescharacteristic
110C22⋊C4×C197D411yescharacteristic
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—21no—
116Dic394—42no—
117Dic394#4112yescomplemented
118D788—21no—
119D788—21no—
120C22⋊D197—42no—
121C22⋊D197—42no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—42no—
126C4×D197—42no—
127#13096802—21no—
128#13096802—21no—
129#13096802—21no—
130#13096802—21no—
131C23×C197—21no—
132C23×C197C2×D411yescharacteristic
133C23×C197C2×D411yescharacteristic
134Q8×C197—42no—
135D4×C197—22no—
136D4×C197—21no—
137D4×C197—21no—
138D4×C197—42no—
139D4×C197—42no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—42no—
148C2×C4×C197—42no—
149C2×C4×C197—42no—
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2×D411yescharacteristic
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2411yescharacteristic
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197AGL(1,ℤ/8)12yes—
164C22×C197C22×D411yescharacteristic
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—21no—
171C4×C197—21no—
172C4×C197—41no—
173C4×C197—41no—
174C4×C197—41no—
175C4×C197C2×#4111yescharacteristic
176C4×C197H(5,𝔽2)11yescharacteristic
177Dic197—22no—
178Dic197—41no—
179Dic197—41no—
180C2×C197—22no—
181C2×C197—41no—
182C2×C197—81no—
183C2×C197#54611yescharacteristic
184C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C197#57811yescharacteristic
186D197—81no—
187C197#330811yescharacteristic, complemented, Sylow
188#3308—1971nomaximal, Sylow
189#548—1971no—
190#547D19711yescharacteristic, complemented
191#489—1971no—
192#484—1971no—
193#480—1971no—
194#477—1971no—
195#468—1971no—
196#466—1971no—
197#463—1971no—
198#449—1971no—
199#448—1971no—
200#440—1971no—
201#435—1971no—
202#417—1971no—
203C2×#103—1971no—
204C2×#41—21no—
205C2×#41—1971no—
206C22×Q8—1971no—
207C2×SD16—1971no—
208C2×SD16—1971no—
209C2×D8—1971no—
210C2×M16—1971no—
211C22×C8—1971no—
212C4⋊Q8C2×D19711yescharacteristic
213#124—3941no—
214#124—1971no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—3941no—
219C22⋊Q8—1971no—
220#121—3941no—
221#121—1971no—
222#121C2×D19711yescharacteristic
223#121C2×D19711yescharacteristic
224C4×Q8—1971no—
225C4×D4—3941no—
226C4×D4C2×D19711yescharacteristic
227C4×D4C2×D19711yescharacteristic
228C2×C4⋊C4C2×D19711yescharacteristic
229C2×C4⋊C4C2×D19711yescharacteristic
230#107—1971no—
231#107—1971no—
232#107—1971no—
233C4⋊C8—1971no—
234#103—1971no—
235#103—1971no—
236#103—1971no—
237#103—1971no—
238#103—1971no—
239#103—1971no—
240#102—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244C22⋊C8—1971no—
245C22⋊C8—1971no—
246#97—1971no—
247#41—44no—
248#41—7881no—
249#41—3942no—
250C2×Q8—21no—
251C2×Q8—7881no—
252C2×Q8—3942no—
253C2×Q8—3941no—
254C2×Q8—1971no—
255C2×Q8—1971no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—1971no—
261C2×D4C22⋊D19711yescharacteristic
262C2×D4C22×D19711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4C22×D19711yescharacteristic
269C22×C4C22×D19711yescharacteristic
270SD16—7881no—
271SD16—3942no—
272D8—7881no—
273M16—7881no—
274C2×C8—7881no—
275C2×C8—1971no—
276C2×C8—1971no—
277C2×C8—1971no—
278C2×C8—1971no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—3941no—
283C4⋊C4—3941no—
284C4⋊C4—1971no—
285C4⋊C4C22⋊D19711yescharacteristic
286C4⋊C4C22×D19711yescharacteristic
287C4⋊C4C22×D19711yescharacteristic
288C4⋊C4C22×D19711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4C22⋊D19711yescharacteristic
297C22⋊C4C22⋊D19711yescharacteristic
298C42—3941no—
299C42C22×D19711yescharacteristic
300C23—21no—
301C23—3941no—
302C23C2×C22⋊D19711yescharacteristic
303C23C2×C22⋊D19711yescharacteristic
304Q8—42no—
305Q8—7882no—
306Q8—3941no—
307Q8—3941no—
308Q8—3941no—
309Q8—1972no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—7882no—
316D4—7882no—
317D4—3941no—
318D4—3941no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—7882no—
330C2×C4—7882no—
331C2×C4—3941no—
332C2×C4—3941no—
333C2×C4—3941no—
334C2×C4C2×C22⋊D19711yescharacteristic
335C2×C4C2×C22⋊D19711yescharacteristic
336C2×C4C2×C22⋊D19711yescharacteristic
337C2×C4C2×C22⋊D19711yescharacteristic
338C2×C4C23×D19711yescharacteristic
339C8—3941no—
340C8—3941no—
341C8—3941no—
342C8—3941no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—7881no—
352C22—7881no—
353C22—7881no—
354C22#2026335212yes—
355C22C22×C22⋊D19711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—7881no—
364C4—7881no—
365C4—3942no—
366C4#41×D19711yescharacteristic
367C4#2026337211yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—15761no—
372C2#2716171711yescharacteristic
373C2C2×#2026335211yescharacteristic
374C2#2716189411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#547×C197C2
socleC22×C197C22×D4
hypercenter#547D197
hypocenterC197#3308
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. #27161738C2semidirect
2. #27161441C2semidirect
3. #27161414C2semidirect
4. #27161392C2semidirect
5. #27161371C2semidirect
6. #27161323C2semidirect
7. #27161313C2semidirect
8. #27161294C2semidirect
9. #27161222C2semidirect
10. #27161217C2semidirect
11. #27161179C2semidirect
12. #27161151C2semidirect
13. #27161078C2semidirect
14. C2×#20263249C2semidirect
15. #547×C197C2semidirect
117. Dic394#41semidirect
187. C197#3308semidirect
190. #547D197semidirect

Maximal subgroups

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

subgroupindex
#271617382
#271614412
#271614142
#271613922
#271613712
#271613232
#271613132
#271612942
#271612222
#271612172
#271611792
#271611512
#271610782
C2×#202632492
#547×C1972
#3308197