Finite Group Databasebeta

← #38752060

Subgroups — #38752060

Order 25216.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#27161915C211yesmaximal, characteristic, complemented
2#27161877C212yesmaximal, complemented
3#27161261C212yesmaximal, complemented
4#27161256C212yesmaximal, complemented
5#27161251C212yesmaximal, complemented
6#27161246C212yesmaximal, complemented
7C2×#20263254C212yesmaximal, complemented
8#27161063C211yesmaximal, characteristic, complemented
9#532×C197C211yesmaximal, characteristic, complemented
10#20263374C2212yes—
11#20263373—22no—
12#20263372—22no—
13#20263371—22no—
14#20263371C2212yes—
15C2×#15691358—22no—
16C2×#15691358C2212yes—
17#41×D197—24no—
18#20263357C2214yescomplemented
19#20263356—22no—
20#20263352C2214yescomplemented
21#20263351—22no—
22#20263346—24no—
23C2×#15691342C2212yescomplemented
24#20263254C2214yescomplemented
25#20263254C2214yescomplemented
26#20263245C2212yescomplemented
27#20263243C2212yescomplemented
28C2×#41×C197C2211yescharacteristic
29C2×#41×C197C2211yescharacteristic, complemented
30C4⋊Q8×C197C2211yescharacteristic, complemented
31#127×C197C2211yescharacteristic, complemented
32#124×C197C2211yescharacteristic, complemented
33#124×C197C2211yescharacteristic, complemented
34#121×C197—22no—
35C4×D4×C197—22no—
36C2×C42×C197C2211yescharacteristic, complemented
37#15691359—24no—
38#15691358—24no—
39#15691358—24no—
40#15691358—22no—
41#15691358—24no—
42#15691358—22no—
43#15691358D414yescomplemented
44#15691358C2314yes—
45#15691357—28no—
46#15691357—24no—
47C2×Dic394—22no—
48C2×Dic394D412yescomplemented
49Q8×D197—24no—
50C2×D788—22no—
51C2×D788D412yescomplemented
52C2×C22⋊D197—24no—
53C2×C22⋊D197—22no—
54D4×D197—28no—
55D4×D197—24no—
56C2×C4×D197—24no—
57C2×C4×D197—22no—
58#15691348—24no—
59#15691346—24no—
60#15691345—24no—
61#15691343—24no—
62#15691342—22no—
63#15691342C2314yes—
64C2×#13096802—22no—
65#41×C197—22no—
66#41×C197—22no—
67#41×C197—22no—
68#41×C197—22no—
69C2×Q8×C197C2311yescharacteristic
70C2×Q8×C197C2311yescharacteristic
71C2×D4×C197—22no—
72C2×D4×C197—21no—
73C2×D4×C197D412yes—
74C2×D4×C197C2311yescharacteristic
75C2×D4×C197C2311yescharacteristic
76C22×C4×C197—21no—
77C22×C4×C197—21no—
78C22×C4×C197D412yes—
79C22×C4×C197C2311yescharacteristic
80C4⋊C4×C197—22no—
81C22⋊C4×C197—22no—
82C22⋊C4×C197—22no—
83C42×C197—21no—
84C42×C197C2311yescharacteristic
85C42×C197C2311yescharacteristic
86C22×D197—24no—
87C22×D197—22no—
88Dic394—24no—
89Dic394—22no—
90Dic394C2×D414yes—
91D788—24no—
92D788—22no—
93D788C2×D414yes—
94C22⋊D197—28no—
95C22⋊D197—24no—
96C22⋊D197—24no—
97C2×Dic197—24no—
98C2×Dic197—22no—
99C4×D197—28no—
100C4×D197—24no—
101C4×D197—24no—
102#13096802—24no—
103C23×C197—21no—
104C23×C197C2×D412yes—
105C23×C197C2×D411yescharacteristic
106Q8×C197—22no—
107Q8×C197—22no—
108D4×C197—24no—
109D4×C197—22no—
110D4×C197—22no—
111D4×C197—44no—
112D4×C197—42no—
113C2×C4×C197—21no—
114C2×C4×C197—22no—
115C2×C4×C197—22no—
116C2×C4×C197—21no—
117C2×C4×C197—21no—
118C2×C4×C197—42no—
119C2×C4×C197—41no—
120C2×C4×C197—41no—
121C2×C4×C197C2×D412yes—
122C2×C4×C197C2×D412yes—
123C2×C4×C197C2×D412yes—
124C2×C4×C197C2×D411yescharacteristic
125C2×C4×C197C2411yescharacteristic
126C22×C197—22no—
127C22×C197—41no—
128C22×C197—42no—
129C22×C197—42no—
130C22×C197—41no—
131C22×C197#12012yes—
132C22×C197C22×D411yescharacteristic
133C2×D197—24no—
134C2×D197—82no—
135C2×D197—84no—
136C4×C197—22no—
137C4×C197—22no—
138C4×C197—41no—
139C4×C197—41no—
140C4×C197#12011yescharacteristic
141C4×C197#12011yescharacteristic
142C4×C197C22×D412yes—
143Dic197—24no—
144C2×C197—21no—
145C2×C197—22no—
146C2×C197—42no—
147C2×C197—81no—
148C2×C197C2×#12011yescharacteristic
149D197—44no—
150C197#301811yescharacteristic, complemented, Sylow
151#3018—1971nomaximal, Sylow
152#585—1971no—
153#575—1972no—
154#532D19711yescharacteristic, complemented
155#456—1972no—
156#455—1972no—
157#454—1972no—
158#453—1972no—
159C2×#104—1972no—
160#413—1971no—
1612−1+4—3942no—
1622−1+4—1972no—
163H(5,𝔽2)—3942no—
164H(5,𝔽2)—3942no—
165H(5,𝔽2)—1972no—
166C2×#41—3944no—
167C2×#41—3942no—
168C2×#41—1972no—
169C2×#41C2×D19711yescharacteristic
170C2×#41C2×D19711yescharacteristic, complemented
171#137—3942no—
172#137—1974no—
173AGL(1,ℤ/8)—3942no—
174AGL(1,ℤ/8)—1974no—
175#135—3944no—
176C2×M16—1972no—
177C4⋊Q8C2×D19711yescharacteristic, complemented
178#127C2×D19711yescharacteristic, complemented
179#124C2×D19711yescharacteristic, complemented
180#124C2×D19711yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D19711yescharacteristic, complemented
184#104—1974no—
185#104—1974no—
186#101—1972no—
187#100—1972no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—3944no—
193#41—3944no—
194#41—3948no—
195#41—3944no—
196#41—3944no—
197#41—3942no—
198#41—3942no—
199#41—3944no—
200#41—1974no—
201#41—1974no—
202C2×Q8—3942no—
203C2×Q8—3944no—
204C2×Q8—1972no—
205C2×Q8C22×D19711yescharacteristic
206C2×Q8C22×D19711yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—3942no—
210C2×D4—3948no—
211C2×D4—3944no—
212C2×D4—3944no—
213C2×D4—3942no—
214C2×D4—1972no—
215C2×D4C22⋊D19712yes—
216C2×D4C22×D19711yescharacteristic
217C2×D4C22×D19711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—3944no—
221C22×C4—3942no—
222C22×C4C22⋊D19712yes—
223C22×C4C22×D19711yescharacteristic
224Q16—3944no—
225SD16—3944no—
226SD16—3944no—
227D8—3944no—
228M16—3942no—
229M16—1974no—
230C2×C8—3942no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D19711yescharacteristic
236C42C22×D19711yescharacteristic
237C23—21no—
238C23—3944no—
239C23—3942no—
240C23C2×C22⋊D19712yes—
241C23C2×C22⋊D19711yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—3944no—
245Q8—3942no—
246Q8—1974no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—3944no—
253D4—3948no—
254D4—3944no—
255D4—3944no—
256D4—3942no—
257D4—1974no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3948no—
267C2×C4—3944no—
268C2×C4—3942no—
269C2×C4—3944no—
270C2×C4—3944no—
271C2×C4D4×D19712yes—
272C2×C4D4×D19712yes—
273C2×C4C2×C22⋊D19712yes—
274C2×C4C2×C22⋊D19711yescharacteristic
275C2×C4C23×D19711yescharacteristic
276C8—3944no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—15762no—
283C22—15764no—
284C22—3944no—
285C22#2026329412yes—
286C22C22×C22⋊D19711yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—3944no—
292C4#2026329411yescharacteristic
293C4#2026329411yescharacteristic
294C4C2×D4×D19712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—7884no—
300C2C2×#2026329411yescharacteristic
subgroupG/H
centerC4#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#532×C197C2
socleC2×C197C2×#120
hypercenter#532D197
hypocenterC197#3018
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #27161915C2semidirect
2. #27161877C2semidirect
3. #27161261C2semidirect
4. #27161256C2semidirect
5. #27161251C2semidirect
6. #27161246C2semidirect
7. C2×#20263254C2semidirect
8. #27161063C2semidirect
9. #532×C197C2semidirect
18. #20263357C22semidirect
20. #20263352C22semidirect
23. C2×#15691342C22semidirect
24. #20263254C22semidirect
25. #20263254C22semidirect
26. #20263245C22semidirect
27. #20263243C22semidirect
29. C2×#41×C197C22semidirect
30. C4⋊Q8×C197C22semidirect
31. #127×C197C22semidirect
32. #124×C197C22semidirect
33. #124×C197C22semidirect
36. C2×C42×C197C22semidirect
43. #15691358D4semidirect
48. C2×Dic394D4semidirect
51. C2×D788D4semidirect
150. C197#3018semidirect
154. #532D197semidirect
170. C2×#41C2×D197semidirect
177. C4⋊Q8C2×D197semidirect
178. #127C2×D197semidirect
179. #124C2×D197semidirect
180. #124C2×D197semidirect
183. C2×C42C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271618772
#271612612
#271612562
#271612512
#271612462
C2×#202632542
#271610632
#532×C1972
#3018197