Finite Group Databasebeta

← #38752067

Subgroups — #38752067

Order 25216.

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

normal subgroups only

This group has 762 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
1C2×#20263372C211yesmaximal, characteristic, complemented
2#27161877C211yesmaximal, characteristic, complemented
3#27161643C211yesmaximal, characteristic, complemented
4#27161255C214yesmaximal, complemented
5#27161245C214yesmaximal, complemented
6#27161096C211yesmaximal, characteristic, complemented
7#27161095C211yesmaximal, characteristic, complemented
8C2×#20263242C211yesmaximal, characteristic, complemented
9#575×C197C211yesmaximal, characteristic, complemented
10#20263372—22no—
11#20263372—22no—
12#20263372—22no—
13#20263372C2214yes—
14C2×#15691359—21no—
15C2×#15691358—21no—
16C2×#15691358C2211yescharacteristic
17C22×D788—21no—
18C22×D788C2211yescharacteristic
19C2×D4×D197—22no—
20C2×D4×D197—21no—
21#20263357C2212yescomplemented
22#20263356—21no—
23#20263352C2212yescomplemented
24#20263351—21no—
25#20263346—22no—
26C2×#15691342C2211yescharacteristic, complemented
27#20263322C2212yescomplemented
28#20263315C2212yescomplemented
29#20263300—22no—
30#20263294—22no—
31#20263285C2211yescharacteristic, complemented
32#20263253C2214yescomplemented
33#20263252C2214yescomplemented
34#20263242C2214yescomplemented
35C2×#41×C197C2211yescharacteristic
36C2×#41×C197C2211yescharacteristic, complemented
37#137×C197—21no—
38#137×C197C2212yescomplemented
39AGL(1,ℤ/8)×C197—21no—
40AGL(1,ℤ/8)×C197C2212yescomplemented
41#135×C197—22no—
42C2×M16×C197C2211yescharacteristic, complemented
43C23×D197—22no—
44C23×D197D412yes—
45#15691359—24no—
46#15691359—24no—
47#15691358—24no—
48#15691358—24no—
49#15691358—21no—
50#15691358—21no—
51#15691358D412yescomplemented
52#15691358C2312yes—
53C2×Dic394D411yescharacteristic, complemented
54C2×D788—22no—
55C2×D788—24no—
56C2×D788—24no—
57C2×D788—21no—
58C2×D788—21no—
59C2×D788—21no—
60C2×D788—21no—
61C2×D788D412yes—
62C2×D788D411yescharacteristic, complemented
63C2×D788C2312yes—
64C2×C22⋊D197—22no—
65C2×C22⋊D197—24no—
66C2×C22⋊D197—21no—
67D4×D197—28no—
68D4×D197—24no—
69D4×D197—28no—
70D4×D197—24no—
71C2×C4×D197—22no—
72C2×C4×D197—21no—
73#15691348—22no—
74#15691346—22no—
75#15691345—22no—
76#15691343—22no—
77#15691342—21no—
78#15691342C2312yescomplemented
79C2×#13096802—21no—
80C4⋊Dic197—21no—
81C22⋊Dic197—21no—
82#15691335—24no—
83C4×Dic197C2312yescomplemented
84#41×C197—21no—
85#41×C197—21no—
86#41×C197—21no—
87#41×C197—21no—
88#41×C197—21no—
89#41×C197—21no—
90#41×C197D412yescomplemented
91#41×C197C2312yes—
92C2×Q8×C197D411yescharacteristic, complemented
93C2×Q8×C197C2311yescharacteristic, complemented
94C2×D4×C197—21no—
95C2×D4×C197—21no—
96C2×D4×C197D411yescharacteristic, complemented
97C2×D4×C197C2311yescharacteristic, complemented
98C22×C4×C197—21no—
99C22×C4×C197—21no—
100C22×C4×C197C2311yescharacteristic
101Q16×C197—22no—
102SD16×C197—22no—
103SD16×C197—22no—
104D8×C197—22no—
105M16×C197—21no—
106M16×C197C2312yescomplemented
107C2×C8×C197—21no—
108C22×D197—22no—
109C22×D197—21no—
110C22×D197—28no—
111C22×D197—22no—
112C22×D197—82no—
113C22×D197—84no—
114C22×D197C2×D414yes—
115Dic394—21no—
116Dic394C2×D412yes—
117D788—28no—
118D788—22no—
119D788—22no—
120D788—21no—
121D788—22no—
122D788—22no—
123D788C2×D412yes—
124C22⋊D197—28no—
125C22⋊D197—22no—
126C22⋊D197—22no—
127C22⋊D197—84no—
128C2×Dic197—21no—
129C2×Dic197—22no—
130C4×D197—28no—
131C4×D197—22no—
132C4×D197—22no—
133#13096802—22no—
134C23×C197—21no—
135C23×C197—21no—
136C23×C197C2×D411yescharacteristic
137Q8×C197—21no—
138Q8×C197—21no—
139Q8×C197—21no—
140Q8×C197C2×D412yes—
141D4×C197—22no—
142D4×C197—22no—
143D4×C197—21no—
144D4×C197—21no—
145D4×C197—21no—
146D4×C197—41no—
147D4×C197—41no—
148D4×C197C2×D412yes—
149C2×C4×C197—21no—
150C2×C4×C197—22no—
151C2×C4×C197—22no—
152C2×C4×C197—21no—
153C2×C4×C197—41no—
154C2×C4×C197—41no—
155C2×C4×C197C2×D411yescharacteristic
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2×D411yescharacteristic
158C2×C4×C197C2×D411yescharacteristic
159C2×C4×C197C2×D411yescharacteristic
160C2×C4×C197C2411yescharacteristic
161C8×C197—22no—
162C22×C197—22no—
163C22×C197—41no—
164C22×C197—42no—
165C22×C197—42no—
166C22×C197—41no—
167C22×C197#12011yescharacteristic
168C22×C197#12011yescharacteristic
169C22×C197C22×D411yescharacteristic
170C2×D197—24no—
171C2×D197—22no—
172C2×D197—44no—
173C2×D197—42no—
174C2×D197—44no—
175C2×D197—82no—
176C2×D197—88no—
177C4×C197—22no—
178C4×C197—41no—
179C4×C197#12011yescharacteristic
180C4×C197#12011yescharacteristic
181C4×C197C22×D411yescharacteristic
182C4×C197C22×D411yescharacteristic
183Dic197—22no—
184Dic197—42no—
185C2×C197—21no—
186C2×C197—21no—
187C2×C197—21no—
188C2×C197—42no—
189C2×C197—81no—
190C2×C197C2×#12011yescharacteristic
191D197—44no—
192D197—42no—
193C197#301911yescharacteristic, complemented, Sylow
194#3019—1971nomaximal, Sylow
195C2×H(5,𝔽2)—1971no—
196#575—1971no—
197#575D19711yescharacteristic, complemented
198#535—1971no—
199#455—1974no—
200#453—1974no—
201#421—1971no—
202#421—1971no—
203C2×#100—1971no—
204H(5,𝔽2)—3942no—
205H(5,𝔽2)—3942no—
206H(5,𝔽2)—3942no—
207H(5,𝔽2)—1974no—
208C2×#41—3941no—
209C2×#41—3941no—
210C2×#41—1971no—
211C2×#41C2×D19711yescharacteristic
212C2×#41C2×D19711yescharacteristic, complemented
213C22×D4—3941no—
214C22×D4—3942no—
215C22×D4—3941no—
216C22×D4—1971no—
217#137—21no—
218#137—3941no—
219#137—1972no—
220#137C2×D19712yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—3941no—
223AGL(1,ℤ/8)—1972no—
224AGL(1,ℤ/8)C2×D19712yescomplemented
225#135—22no—
226#135—3942no—
227C2×M16—1971no—
228C2×M16C2×D19711yescharacteristic, complemented
229#127—1972no—
230#124—1972no—
231#121—3942no—
232#120—3942no—
233#117—1971no—
234#104—1974no—
235#104—1974no—
236#100—1974no—
237C24—3942no—
238C24—1972no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—3944no—
246#41—3944no—
247#41—3944no—
248#41—3944no—
249#41—3941no—
250#41—3941no—
251#41—1972no—
252#41—1972no—
253#41C22⋊D19712yescomplemented
254#41C22×D19712yes—
255C2×Q8—1971no—
256C2×Q8C22⋊D19711yescharacteristic, complemented
257C2×Q8C22×D19711yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—3942no—
261C2×D4—3944no—
262C2×D4—3944no—
263C2×D4—3948no—
264C2×D4—3944no—
265C2×D4—3948no—
266C2×D4—3944no—
267C2×D4—3942no—
268C2×D4—3941no—
269C2×D4—3944no—
270C2×D4—3941no—
271C2×D4—3941no—
272C2×D4—3941no—
273C2×D4—3941no—
274C2×D4—1972no—
275C2×D4—1971no—
276C2×D4—1972no—
277C2×D4C22⋊D19711yescharacteristic, complemented
278C2×D4C22×D19711yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3942no—
282C22×C4—3941no—
283C22×C4C22×D19711yescharacteristic
284Q16—22no—
285Q16—3942no—
286SD16—22no—
287SD16—22no—
288SD16—3942no—
289SD16—3942no—
290D8—22no—
291D8—3942no—
292M16—21no—
293M16—3941no—
294M16—1972no—
295M16C22×D19712yescomplemented
296C2×C8—21no—
297C2×C8—3941no—
298C4⋊C4—3941no—
299C22⋊C4—3944no—
300C22⋊C4—3941no—
301C42—1972no—
302C23—21no—
303C23—21no—
304C23—15762no—
305C23—15764no—
306C23—3948no—
307C23—3942no—
308C23—3941no—
309C23—3942no—
310C23—1974no—
311C23D4×D19711yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—3941no—
316Q8—1972no—
317Q8C2×C22⋊D19712yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—15764no—
326D4—3948no—
327D4—3948no—
328D4—3942no—
329D4—3942no—
330D4—3942no—
331D4—3941no—
332D4—3942no—
333D4—3942no—
334D4—3942no—
335D4—1972no—
336D4C2×C22⋊D19712yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—3948no—
344C2×C4—3942no—
345C2×C4—3941no—
346C2×C4—3942no—
347C2×C4—3942no—
348C2×C4D4×D19711yescharacteristic
349C2×C4D4×D19711yescharacteristic
350C2×C4D4×D19711yescharacteristic
351C2×C4C2×C22⋊D19711yescharacteristic
352C2×C4C2×C22⋊D19711yescharacteristic
353C2×C4C23×D19711yescharacteristic
354C8—22no—
355C8—3942no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—7884no—
362C22—7884no—
363C22—7882no—
364C22—15762no—
365C22—15768no—
366C22—3944no—
367C22—3942no—
368C22#2026329411yescharacteristic
369C22#2026329411yescharacteristic
370C22C2×D4×D19711yescharacteristic
371C4—22no—
372C4—41no—
373C4—7882no—
374C4—3942no—
375C4#2026329411yescharacteristic
376C4#2026329411yescharacteristic
377C4C2×D4×D19711yescharacteristic
378C4C22×C22⋊D19711yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—7884no—
385C2—7882no—
386C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#575×C197C2
socleC2×C197C2×#120
hypercenter#575D197
hypocenterC197#3019
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. #27161877C2semidirect
3. #27161643C2semidirect
4. #27161255C2semidirect
5. #27161245C2semidirect
6. #27161096C2semidirect
7. #27161095C2semidirect
8. C2×#20263242C2semidirect
9. #575×C197C2semidirect
21. #20263357C22semidirect
23. #20263352C22semidirect
26. C2×#15691342C22semidirect
27. #20263322C22semidirect
28. #20263315C22semidirect
31. #20263285C22semidirect
32. #20263253C22semidirect
33. #20263252C22semidirect
34. #20263242C22semidirect
36. C2×#41×C197C22semidirect
38. #137×C197C22semidirect
40. AGL(1,ℤ/8)×C197C22semidirect
42. C2×M16×C197C22semidirect
51. #15691358D4semidirect
53. C2×Dic394D4semidirect
62. C2×D788D4semidirect
78. #15691342C23semidirect
83. C4×Dic197C23semidirect
90. #41×C197D4semidirect
92. C2×Q8×C197D4semidirect
93. C2×Q8×C197C23semidirect
96. C2×D4×C197D4semidirect
97. C2×D4×C197C23semidirect
106. M16×C197C23semidirect
193. C197#3019semidirect
197. #575D197semidirect
212. C2×#41C2×D197semidirect
220. #137C2×D197semidirect
224. AGL(1,ℤ/8)C2×D197semidirect
228. C2×M16C2×D197semidirect
253. #41C22⋊D197semidirect
256. C2×Q8C22⋊D197semidirect
257. C2×Q8C22×D197semidirect
277. C2×D4C22⋊D197semidirect
278. C2×D4C22×D197semidirect
295. M16C22×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
#271618772
#271616432
#271612552
#271612452
#271610962
#271610952
C2×#202632422
#575×C1972
#3019197