Finite Group Databasebeta

← #38754497

Subgroups — #38754497

Order 25216.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#27161892C211yesmaximal, characteristic, complemented
2#27161885C212yesmaximal, complemented
3#27161880C211yesmaximal, characteristic, complemented
4#27161351C211yesmaximal, characteristic, complemented
5#27161346C211yesmaximal, characteristic, complemented
6#27161339C212yesmaximal, complemented
7#27161256C212yesmaximal, complemented
8#27161251C212yesmaximal, complemented
9#27161197C211yesmaximal, characteristic, complemented
10#27161197C211yesmaximal, characteristic, complemented
11#495×C197C211yesmaximal, characteristic, complemented
12#20263374C2212yescomplemented
13#20263371C2212yes—
14#20263358—22no—
15#20263357C2212yes—
16#20263357C2212yescomplemented
17#20263355—22no—
18#20263353—22no—
19#20263352C2212yes—
20#20263352C2212yescomplemented
21#20263350—22no—
22#20263349—21no—
23#20263348C2211yescharacteristic
24#20263348C2211yescharacteristic
25#20263347—22no—
26#20263345—21no—
27Q16×D197—21no—
28SD16×D197—22no—
29D8×D197—21no—
30#20263332C2211yescharacteristic
31#20263332C2211yescharacteristic, complemented
32#20263332C2212yescomplemented
33#20263262C2211yescharacteristic, complemented
34#20263262C2211yescharacteristic, complemented
35#20263254C2211yescharacteristic
36#20263254C2211yescharacteristic, complemented
37#20263254C2212yescomplemented
38#20263245C2212yescomplemented
39#20263243C2212yescomplemented
40C2×Q16×C197C2211yescharacteristic, complemented
41C2×SD16×C197C2212yescomplemented
42C2×D8×C197C2211yescharacteristic
43#124×C197C2212yescomplemented
44C4×C8×C197C2211yescharacteristic, complemented
45#15691359—22no—
46#15691359—22no—
47#15691358—42no—
48#15691358C2311yescharacteristic
49#15691358C2312yes—
50#15691358C2311yescharacteristic
51#15691357—22no—
52#15691357—22no—
53Q8×D197—22no—
54Q8×D197—22no—
55C2×C22⋊D197—42no—
56D4×D197—22no—
57D4×D197—22no—
58Dic788D411yescharacteristic
59Dic788D411yescharacteristic
60#15691348—22no—
61#15691348—22no—
62#15691347D411yescharacteristic
63#15691347D411yescharacteristic
64#15691347D412yes—
65#15691346—22no—
66#15691346—22no—
67#15691345—22no—
68#15691345—22no—
69D1576D411yescharacteristic
70D1576D411yescharacteristic
71#15691343—22no—
72#15691343—22no—
73#15691342C2311yescharacteristic
74#15691342C2312yes—
75#15691342C2311yescharacteristic
76#15691341—21no—
77#15691341—22no—
78#15691341—21no—
79C8×D197—21no—
80C8×D197—22no—
81C8×D197—21no—
82C2×Q8×C197C2312yes—
83C2×D4×C197C2312yes—
84Q16×C197—21no—
85Q16×C197—21no—
86SD16×C197—22no—
87SD16×C197—22no—
88D8×C197—21no—
89D8×C197—21no—
90C2×C8×C197C2311yescharacteristic
91C2×C8×C197C2311yescharacteristic
92C22⋊C4×C197—42no—
93C42×C197C2311yescharacteristic
94C22×D197—42no—
95Dic394—42no—
96Dic394C2×D411yescharacteristic
97Dic394C2×D412yes—
98Dic394C2×D411yescharacteristic
99D788—42no—
100D788C2×D411yescharacteristic
101D788C2×D412yes—
102D788C2×D411yescharacteristic
103C22⋊D197—21no—
104C22⋊D197—22no—
105C22⋊D197—21no—
106C22⋊D197—42no—
107C22⋊D197—42no—
108C2×Dic197—42no—
109C4×D197—21no—
110C4×D197—22no—
111C4×D197—21no—
112C4×D197—42no—
113C4×D197—42no—
114#13096802—21no—
115#13096802—22no—
116#13096802—21no—
117C23×C197—22no—
118Q8×C197—22no—
119Q8×C197—22no—
120D4×C197—22no—
121D4×C197—22no—
122C2×C4×C197—21no—
123C2×C4×C197—22no—
124C2×C4×C197C2411yescharacteristic
125C8×C197C2×D411yescharacteristic
126C8×C197C2×D411yescharacteristic
127C8×C197C2×D411yescharacteristic
128C8×C197C2×D411yescharacteristic
129C22×C197—42no—
130C22×C197—82no—
131C22×C197H(5,𝔽2)11yescharacteristic
132C2×D197—21no—
133C2×D197—22no—
134C2×D197—21no—
135C2×D197—162no—
136C4×C197—41no—
137C4×C197—42no—
138C4×C197C22×D411yescharacteristic
139C4×C197C22×D411yescharacteristic
140Dic197—21no—
141Dic197—22no—
142Dic197—21no—
143C2×C197—21no—
144C2×C197—82no—
145C2×C197D4211yescharacteristic
146D197—41no—
147D197—42no—
148D197—41no—
149C197#329011yescharacteristic, complemented, Sylow
150#3290—1971nomaximal, Sylow
151#578—1971no—
152#577—1972no—
153#576—1971no—
154#495D19711yescharacteristic, complemented
155#473—1971no—
156#472—1971no—
157#471—1972no—
158#455—1972no—
159#454—1972no—
160#443—1971no—
161#443—1971no—
1622−1+4—1972no—
163H(5,𝔽2)—1972no—
164#137—3942no—
165#137—3942no—
166#137—1972no—
167#137—1972no—
168AGL(1,ℤ/8)—3942no—
169AGL(1,ℤ/8)—3942no—
170AGL(1,ℤ/8)—1972no—
171AGL(1,ℤ/8)—1972no—
172#135—3941no—
173#135—3942no—
174#135—3941no—
175#135—1971no—
176#135—1971no—
177C2×Q16—3941no—
178C2×Q16C2×D19711yescharacteristic, complemented
179C2×SD16—3942no—
180C2×SD16C2×D19712yescomplemented
181C2×D8—3941no—
182C2×D8C2×D19711yescharacteristic
183#131—1971no—
184#131—1972no—
185#131—1971no—
186#124C2×D19712yescomplemented
187#108—1971no—
188#108—1971no—
189#104—1971no—
190#104—1971no—
191#104—1972no—
192#101—1972no—
193#100—1972no—
194C4×C8C2×D19711yescharacteristic, complemented
195#41—7882no—
196#41—3942no—
197#41—3942no—
198#41—3942no—
199#41—3942no—
200#41—1971no—
201#41—1972no—
202#41—1971no—
203C2×Q8—3942no—
204C2×Q8—3942no—
205C2×Q8C22×D19712yes—
206C2×D4—7882no—
207C2×D4—3942no—
208C2×D4—3942no—
209C2×D4C22×D19712yes—
210Q16—21no—
211Q16—21no—
212Q16—3942no—
213Q16—3942no—
214Q16—1971no—
215Q16—1971no—
216SD16—22no—
217SD16—22no—
218SD16—3942no—
219SD16—3942no—
220SD16—3942no—
221SD16—3942no—
222SD16—1971no—
223SD16—1971no—
224SD16—1972no—
225D8—21no—
226D8—21no—
227D8—3942no—
228D8—3942no—
229D8—1971no—
230D8—1971no—
231M16—3941no—
232M16—3942no—
233M16—3941no—
234M16—1971no—
235M16—1972no—
236M16—1971no—
237C2×C8—3941no—
238C2×C8—3942no—
239C2×C8—3941no—
240C2×C8C22×D19711yescharacteristic
241C2×C8C22×D19711yescharacteristic
242C22⋊C4—42no—
243C42C22×D19711yescharacteristic
244C23—22no—
245C23—7882no—
246Q8—22no—
247Q8—22no—
248Q8—7882no—
249Q8—1971no—
250Q8—1972no—
251Q8—1971no—
252D4—22no—
253D4—22no—
254D4—7882no—
255D4—7882no—
256D4—7882no—
257D4—3941no—
258D4—3942no—
259D4—3941no—
260D4—1971no—
261D4—1972no—
262D4—1971no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—7882no—
266C2×C4—7882no—
267C2×C4—7882no—
268C2×C4—3941no—
269C2×C4—3942no—
270C2×C4—3941no—
271C2×C4C23×D19711yescharacteristic
272C8—3941no—
273C8—3942no—
274C8—3941no—
275C8D4×D19711yescharacteristic
276C8D4×D19711yescharacteristic
277C8D4×D19711yescharacteristic
278C8D4×D19711yescharacteristic
279C22—42no—
280C22—82no—
281C22—31522no—
282C22—3941no—
283C22—3942no—
284C22—3941no—
285C22#2026337111yescharacteristic
286C4—41no—
287C4—42no—
288C4—3941no—
289C4—3942no—
290C4—3941no—
291C4C2×D4×D19711yescharacteristic
292C4C2×D4×D19711yescharacteristic
293C2—21no—
294C2—82no—
295C2—7881no—
296C2—7882no—
297C2—7881no—
298C2#2716171411yescharacteristic
subgroupG/H
centerC2#27161714
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#495×C197C2
socleC2×C197D42
hypercenter#495D197
hypocenterC197#3290
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #27161892C2semidirect
2. #27161885C2semidirect
3. #27161880C2semidirect
4. #27161351C2semidirect
5. #27161346C2semidirect
6. #27161339C2semidirect
7. #27161256C2semidirect
8. #27161251C2semidirect
9. #27161197C2semidirect
10. #27161197C2semidirect
11. #495×C197C2semidirect
12. #20263374C22semidirect
16. #20263357C22semidirect
20. #20263352C22semidirect
31. #20263332C22semidirect
32. #20263332C22semidirect
33. #20263262C22semidirect
34. #20263262C22semidirect
36. #20263254C22semidirect
37. #20263254C22semidirect
38. #20263245C22semidirect
39. #20263243C22semidirect
40. C2×Q16×C197C22semidirect
41. C2×SD16×C197C22semidirect
43. #124×C197C22semidirect
44. C4×C8×C197C22semidirect
149. C197#3290semidirect
154. #495D197semidirect
178. C2×Q16C2×D197semidirect
180. C2×SD16C2×D197semidirect
186. #124C2×D197semidirect
194. C4×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618922
#271618852
#271618802
#271613512
#271613462
#271613392
#271612562
#271612512
#271611972
#495×C1972
#3290197