Finite Group Databasebeta

← #38752247

Subgroups — #38752247

Order 25216.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263352C211yesmaximal, characteristic, complemented
2C2×#20263346C211yesmaximal, characteristic, complemented
3#27161559C211yesmaximal, characteristic, complemented
4#27161298C212yesmaximal, complemented
5#27161291C212yesmaximal, complemented
6#27161276C212yesmaximal, complemented
7#27161271C212yesmaximal, complemented
8#27161106C211yesmaximal, characteristic, complemented
9#27161080C211yesmaximal, characteristic, complemented
10C2×#20263247C211yesmaximal, characteristic, complemented
11#535×C197C211yesmaximal, characteristic, complemented
12C2×#15691358C2211yescharacteristic
13#20263352—24no—
14#20263346—24no—
15C2×#15691348C2211yescharacteristic, complemented
16C2×#15691346C2212yes—
17C2×#15691346C2211yescharacteristic, complemented
18C2×#15691345C2211yescharacteristic, complemented
19C2×#15691343C2212yes—
20C2×#15691343C2211yescharacteristic, complemented
21C2×#15691342C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23C4×Dic394C2212yescomplemented
24C4×D788C2212yescomplemented
25#20263286—22no—
26#20263284—21no—
27C2×C4⋊Dic197C2211yescharacteristic
28C4⋊C4×D197—21no—
29C4⋊#13096802C2212yescomplemented
30C4⋊#13096802C2212yescomplemented
31#20263250C2212yescomplemented
32#20263247C2212yes—
33#20263247C2212yes—
34#20263247C2212yescomplemented
35C2×#41×C197C2211yescharacteristic
36C22×D4×C197C2211yescharacteristic
37#127×C197C2212yes—
38#124×C197C2212yes—
39#121×C197—21no—
40#121×C197—21no—
41#120×C197—22no—
42#117×C197C2211yescharacteristic
43#15691358—22no—
44#15691358D414yescomplemented
45C2×Dic394C2311yescharacteristic
46C2×D788C2311yescharacteristic
47C2×C22⋊D197—21no—
48C22×Dic197—21no—
49C2×C4×D197—21no—
50C2×C4×D197—22no—
51#15691348—22no—
52#15691346—24no—
53#15691346—22no—
54#15691345—22no—
55#15691343—24no—
56#15691343—22no—
57#15691342—22no—
58C2×#13096802—22no—
59C2×#13096802C2311yescharacteristic
60C2×#13096802C2311yescharacteristic
61C2×#13096802C2312yes—
62C4⋊Dic197—21no—
63C4⋊Dic197C2312yes—
64#15691337—22no—
65#15691335—22no—
66C4×Dic197—22no—
67C24×C197—21no—
68#41×C197—22no—
69#41×C197—22no—
70C2×Q8×C197C2311yescharacteristic
71C2×D4×C197—21no—
72C2×D4×C197—22no—
73C2×D4×C197—21no—
74C2×D4×C197—22no—
75C2×D4×C197—22no—
76C2×D4×C197—22no—
77C2×D4×C197C2312yes—
78C2×D4×C197C2311yescharacteristic
79C22×C4×C197—21no—
80C22×C4×C197C2311yescharacteristic
81C4⋊C4×C197D411yescharacteristic
82C4⋊C4×C197D411yescharacteristic
83C22⋊C4×C197—22no—
84C22⋊C4×C197—22no—
85C22⋊C4×C197D412yes—
86C42×C197C2312yes—
87C22×D197—21no—
88Dic394—21no—
89Dic394C2×D412yes—
90D788—21no—
91D788C2×D412yes—
92C22⋊D197—24no—
93C2×Dic197—21no—
94C2×Dic197—22no—
95C2×Dic197—81no—
96C4×D197—24no—
97C4×D197—82no—
98#13096802—21no—
99#13096802—21no—
100#13096802—22no—
101C23×C197—22no—
102C23×C197—21no—
103C23×C197—42no—
104C23×C197—81no—
105C23×C197—81no—
106C23×C197C2×D411yescharacteristic
107Q8×C197—21no—
108Q8×C197—21no—
109D4×C197—22no—
110D4×C197—21no—
111D4×C197—22no—
112D4×C197—21no—
113D4×C197—42no—
114D4×C197—42no—
115D4×C197—82no—
116D4×C197—82no—
117C2×C4×C197—21no—
118C2×C4×C197—42no—
119C2×C4×C197C2×D412yes—
120C2×C4×C197C2×D412yes—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D411yescharacteristic
123C2×C4×C197#4112yes—
124C2×C4×C197C2411yescharacteristic
125C22×C197—21no—
126C22×C197—21no—
127C22×C197—41no—
128C22×C197—42no—
129C22×C197—41no—
130C22×C197—42no—
131C22×C197—41no—
132C22×C197—42no—
133C22×C197—84no—
134C22×C197#12112yes—
135C22×C197C22×D411yescharacteristic
136C2×D197—22no—
137C2×D197—41no—
138C2×D197—41no—
139C4×C197—22no—
140C4×C197—22no—
141C4×C197—41no—
142C4×C197#12112yes—
143C4×C197C22×D411yescharacteristic
144C4×C197C2×#4111yescharacteristic
145Dic197—22no—
146Dic197—42no—
147C2×C197—22no—
148C2×C197—81no—
149C2×C197—82no—
150C2×C197C2×#12111yescharacteristic
151C2×C197#57611yescharacteristic
152C2×C197#57711yescharacteristic
153D197—42no—
154C197#304511yescharacteristic, complemented, Sylow
155#3045—1971nomaximal, Sylow
156C2×AGL(1,ℤ/8)—1971no—
157C2×#135—1971no—
158#535D19711yescharacteristic, complemented
159#520—1971no—
160#464—1972no—
161#463—1972no—
162#460—1972no—
163#459—1972no—
164#424—1971no—
165#417—1971no—
166C2×#102—1971no—
167C2×#41—1971no—
168C2×#41C2×D19711yescharacteristic
169C22×D4C2×D19711yescharacteristic
170AGL(1,ℤ/8)—3944no—
171#135—3944no—
172C2×Q16—1971no—
173C2×SD16—1971no—
174C2×SD16—1971no—
175C2×SD16—1972no—
176C2×D8—1971no—
177C2×D8—1972no—
178C2×M16—1971no—
179C22×C8—1971no—
180#127C2×D19712yes—
181#124C2×D19712yes—
182#121—21no—
183#121—21no—
184#120—22no—
185C4×Q8—1972no—
186C4×D4—3942no—
187C4×D4—1972no—
188#117—3941no—
189#117C2×D19711yescharacteristic
190C2×C4⋊C4—3941no—
191C2×C4⋊C4—1971no—
192C4⋊C8—1972no—
193C4⋊C8—1972no—
194#103—1972no—
195#102—1972no—
196#102—1972no—
197#102—1972no—
198C24—21no—
199#41—22no—
200#41—22no—
201#41—3942no—
202#41—1974no—
203C2×Q8—1971no—
204C2×Q8C22×D19711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—3941no—
212C2×D4—1971no—
213C2×D4C22×D19712yes—
214C2×D4C22×D19711yescharacteristic
215C22×C4—21no—
216C22×C4—3941no—
217C22×C4—3942no—
218C22×C4—3941no—
219C22×C4C22×D19711yescharacteristic
220Q16—3942no—
221SD16—3942no—
222SD16—3944no—
223SD16—3942no—
224D8—3942no—
225D8—3944no—
226M16—3942no—
227C2×C8—3942no—
228C2×C8—1971no—
229C2×C8—1971no—
230C2×C8—1972no—
231C4⋊C4—3942no—
232C4⋊C4—3941no—
233C4⋊C4—1972no—
234C4⋊C4C22⋊D19711yescharacteristic
235C4⋊C4C22⋊D19711yescharacteristic
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—3942no—
239C22⋊C4C22⋊D19712yes—
240C42—3942no—
241C42C22×D19712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—3941no—
248C23C2×C22⋊D19711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—3941no—
252Q8—1972no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—42no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—3944no—
262D4—3941no—
263D4—1972no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—15762no—
267C2×C4—15761no—
268C2×C4—3942no—
269C2×C4—3941no—
270C2×C4—3944no—
271C2×C4D4×D19712yes—
272C2×C4C2×C22⋊D19712yes—
273C2×C4C2×C22⋊D19712yes—
274C2×C4C2×C22⋊D19711yescharacteristic
275C2×C4#1569135712yes—
276C2×C4C23×D19711yescharacteristic
277C8—3941no—
278C8—3941no—
279C8—3942no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—84no—
289C22—7881no—
290C22—7881no—
291C22—3942no—
292C22#2026329712yes—
293C22C22×C22⋊D19711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—7882no—
298C4—3942no—
299C4#2026329712yes—
300C4C2×D4×D19711yescharacteristic
301C4C2×#1569135711yescharacteristic
302C2—22no—
303C2—82no—
304C2—81no—
305C2—7882no—
306C2C2×#2026329711yescharacteristic
307C2#2716188211yescharacteristic
308C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC22×C197C22×D4
hypercenter#535D197
hypocenterC197#3045
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. C2×#20263346C2semidirect
3. #27161559C2semidirect
4. #27161298C2semidirect
5. #27161291C2semidirect
6. #27161276C2semidirect
7. #27161271C2semidirect
8. #27161106C2semidirect
9. #27161080C2semidirect
10. C2×#20263247C2semidirect
11. #535×C197C2semidirect
15. C2×#15691348C22semidirect
17. C2×#15691346C22semidirect
18. C2×#15691345C22semidirect
20. C2×#15691343C22semidirect
23. C4×Dic394C22semidirect
24. C4×D788C22semidirect
29. C4⋊#13096802C22semidirect
30. C4⋊#13096802C22semidirect
31. #20263250C22semidirect
34. #20263247C22semidirect
44. #15691358D4semidirect
154. C197#3045semidirect
158. #535D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
C2×#202633462
#271615592
#271612982
#271612912
#271612762
#271612712
#271611062
#271610802
C2×#202632472
#535×C1972
#3045197