Finite Group Databasebeta

← #38752071

Subgroups — #38752071

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#27161879C211yesmaximal, characteristic, complemented
3#27161877C211yesmaximal, characteristic, complemented
4#27161648C211yesmaximal, characteristic, complemented
5#27161258C214yesmaximal, complemented
6#27161248C214yesmaximal, complemented
7#27161097C211yesmaximal, characteristic, complemented
8C2×#20263242C211yesmaximal, characteristic, complemented
9#421×C197C211yesmaximal, characteristic, complemented
10#20263372—22no—
11#20263372—22no—
12#20263372—22no—
13#20263372C2214yes—
14C2×#15691359—21no—
15C2×#15691358—21no—
16C2×#15691358C2211yescharacteristic
17C2×#15691358C2211yescharacteristic, complemented
18C22×D788—21no—
19C22×D788C2211yescharacteristic
20C2×D4×D197—22no—
21C2×D4×D197—21no—
22#20263359—21no—
23#20263359C2212yescomplemented
24#20263357—21no—
25#20263356C2212yescomplemented
26#20263354—21no—
27#20263354C2212yescomplemented
28#20263352—21no—
29#20263351C2212yescomplemented
30#20263348—22no—
31#20263346—22no—
32C2×#15691342C2211yescharacteristic, complemented
33#20263323C2212yescomplemented
34#20263314C2212yescomplemented
35#20263301—22no—
36#20263296—22no—
37#20263254C2214yescomplemented
38#20263242C2214yescomplemented
39C2×#41×C197C2211yescharacteristic
40C2×M16×C197C2211yescharacteristic, complemented
41#117×C197C2211yescharacteristic, complemented
42#104×C197C2214yescomplemented
43C23×D197—22no—
44C23×D197D412yes—
45#15691359—24no—
46#15691359—24no—
47#15691358—24no—
48#15691358—24no—
49#15691358—21no—
50#15691358—21no—
51#15691358—21no—
52#15691358—21no—
53#15691358—21no—
54#15691358—21no—
55#15691358D412yescomplemented
56#15691358C2312yes—
57C2×Dic394D411yescharacteristic, complemented
58C2×Dic394C2311yescharacteristic, complemented
59C2×D788—22no—
60C2×D788—24no—
61C2×D788—24no—
62C2×D788—24no—
63C2×D788—21no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788D412yes—
68C2×D788D411yescharacteristic, complemented
69C2×D788C2312yes—
70C2×D788C2311yescharacteristic, complemented
71C2×C22⋊D197—22no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197—21no—
74D4×D197—28no—
75D4×D197—24no—
76D4×D197—28no—
77D4×D197—24no—
78C2×C4×D197—22no—
79C2×C4×D197—21no—
80C2×C4×D197—21no—
81Dic788—22no—
82#15691348—22no—
83#15691347—22no—
84#15691347—22no—
85#15691346—22no—
86#15691345—22no—
87D1576—22no—
88#15691343—22no—
89#15691342—21no—
90#15691342C2312yescomplemented
91C2×#13096802—21no—
92#15691335—24no—
93#41×C197—21no—
94#41×C197—21no—
95#41×C197D412yescomplemented
96#41×C197C2312yes—
97C2×Q8×C197D411yescharacteristic, complemented
98C2×D4×C197—21no—
99C2×D4×C197D411yescharacteristic, complemented
100C22×C4×C197—21no—
101C22×C4×C197C2311yescharacteristic
102M16×C197—21no—
103M16×C197C2312yescomplemented
104C2×C8×C197—21no—
105C4⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197C2312yescomplemented
108C22×D197—22no—
109C22×D197—21no—
110C22×D197—28no—
111C22×D197—22no—
112C22×D197—21no—
113C22×D197—82no—
114C22×D197—84no—
115C22×D197C2×D414yes—
116Dic394—21no—
117Dic394—21no—
118Dic394—21no—
119Dic394C2×D412yes—
120D788—28no—
121D788—22no—
122D788—22no—
123D788—21no—
124D788—22no—
125D788—22no—
126D788—21no—
127D788—21no—
128D788—84no—
129D788C2×D412yes—
130C22⋊D197—28no—
131C22⋊D197—22no—
132C22⋊D197—22no—
133C22⋊D197—41no—
134C22⋊D197—41no—
135C2×Dic197—21no—
136C2×Dic197—21no—
137C4×D197—28no—
138C4×D197—22no—
139C4×D197—22no—
140C4×D197—41no—
141C4×D197—41no—
142#13096802—22no—
143C23×C197—21no—
144C23×C197C2×D411yescharacteristic
145Q8×C197—21no—
146Q8×C197C2×D412yes—
147D4×C197—22no—
148D4×C197—22no—
149D4×C197—21no—
150D4×C197C2×D412yes—
151C2×C4×C197—21no—
152C2×C4×C197—22no—
153C2×C4×C197—22no—
154C2×C4×C197—22no—
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#12011yescharacteristic
166C22×C197#12011yescharacteristic
167C22×C197C22×D411yescharacteristic
168C2×D197—24no—
169C2×D197—22no—
170C2×D197—44no—
171C2×D197—42no—
172C2×D197—44no—
173C2×D197—42no—
174C2×D197—41no—
175C2×D197—82no—
176C2×D197—88no—
177C4×C197—22no—
178C4×C197—42no—
179C4×C197#12011yescharacteristic
180C4×C197#12011yescharacteristic
181C4×C197C22×D411yescharacteristic
182C4×C197C22×D411yescharacteristic
183Dic197—22no—
184Dic197—41no—
185C2×C197—21no—
186C2×C197—21no—
187C2×C197—21no—
188C2×C197—42no—
189C2×C197C2×#12011yescharacteristic
190D197—44no—
191D197—42no—
192D197—81no—
193C197#301911yescharacteristic, complemented, Sylow
194#3019—1971nomaximal, Sylow
195C2×H(5,𝔽2)—1971no—
196#575—1971no—
197#575—1971no—
198#535—1971no—
199#455—1974no—
200#453—1974no—
201#421—1971no—
202#421D19711yescharacteristic, complemented
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×#41—1971no—
212C2×#41C2×D19711yescharacteristic
213C22×D4—3941no—
214C22×D4—3942no—
215C22×D4—3941no—
216C22×D4—1971no—
217#137—3941no—
218#137—3941no—
219#137—1972no—
220#137—1972no—
221AGL(1,ℤ/8)—3941no—
222AGL(1,ℤ/8)—3941no—
223AGL(1,ℤ/8)—1972no—
224AGL(1,ℤ/8)—1972no—
225#135—3942no—
226#135—3942no—
227C2×M16—1971no—
228C2×M16C2×D19711yescharacteristic, complemented
229#127—1972no—
230#124—1972no—
231#121—3942no—
232#120—3942no—
233#117C2×D19711yescharacteristic, complemented
234#104—1974no—
235#104C2×D19714yescomplemented
236#100—1974no—
237C24—3942no—
238C24—1972no—
239#41—21no—
240#41—21no—
241#41—3944no—
242#41—3944no—
243#41—3941no—
244#41—3941no—
245#41—3941no—
246#41—3941no—
247#41—3944no—
248#41—3944no—
249#41—3941no—
250#41—3941no—
251#41—1972no—
252#41—1972no—
253#41D78812yescomplemented
254#41C22×D19712yes—
255C2×Q8—1971no—
256C2×Q8—1971no—
257C2×Q8D78811yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—3944no—
260C2×D4—3942no—
261C2×D4—3944no—
262C2×D4—3944no—
263C2×D4—3948no—
264C2×D4—3944no—
265C2×D4—3948no—
266C2×D4—3941no—
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—1971no—
277C2×D4—1972no—
278C2×D4D78811yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—3942no—
281C22×C4—3941no—
282C22×C4—3941no—
283C22×C4C22×D19711yescharacteristic
284Q16—3942no—
285Q16—3942no—
286SD16—3942no—
287SD16—3942no—
288SD16—3942no—
289SD16—3942no—
290D8—3942no—
291D8—3942no—
292M16—21no—
293M16—3941no—
294M16—1972no—
295M16C22×D19712yescomplemented
296C2×C8—21no—
297C2×C8—3941no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3944no—
301C42C22×D19712yescomplemented
302C23—21no—
303C23—15762no—
304C23—15764no—
305C23—3948no—
306C23—3941no—
307C23—3942no—
308C23—3941no—
309C23—3942no—
310C23—1974no—
311C23D4×D19711yescharacteristic
312Q8—21no—
313Q8—3941no—
314Q8—3941no—
315Q8—3941no—
316Q8—1972no—
317Q8C2×D78812yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—7881no—
322D4—7881no—
323D4—15764no—
324D4—3948no—
325D4—3948no—
326D4—3942no—
327D4—3942no—
328D4—3942no—
329D4—3941no—
330D4—3942no—
331D4—3941no—
332D4—3942no—
333D4—3941no—
334D4—3942no—
335D4—1972no—
336D4C2×D78812yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—7881no—
342C2×C4—7881no—
343C2×C4—3948no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3942no—
347C2×C4—3942no—
348C2×C4D4×D19711yescharacteristic
349C2×C4D4×D19711yescharacteristic
350C2×C4D4×D19711yescharacteristic
351C2×C4C2×D78811yescharacteristic
352C2×C4C2×D78811yescharacteristic
353C2×C4C23×D19711yescharacteristic
354C8—22no—
355C8—3942no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—7884no—
360C22—7884no—
361C22—7882no—
362C22—7882no—
363C22—7881no—
364C22—15762no—
365C22—15768no—
366C22—3944no—
367C22—3942no—
368C22#2026329611yescharacteristic
369C22#2026329611yescharacteristic
370C22C2×D4×D19711yescharacteristic
371C4—22no—
372C4—42no—
373C4—7881no—
374C4—3942no—
375C4#2026329611yescharacteristic
376C4#2026329611yescharacteristic
377C4C2×D4×D19711yescharacteristic
378C4C22×D78811yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—7884no—
384C2—7882no—
385C2—15761no—
386C2C2×#2026329611yescharacteristic
subgroupG/H
centerC2C2×#20263296
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#421×C197C2
socleC2×C197C2×#120
hypercenter#421D197
hypocenterC197#3019
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. #27161879C2semidirect
3. #27161877C2semidirect
4. #27161648C2semidirect
5. #27161258C2semidirect
6. #27161248C2semidirect
7. #27161097C2semidirect
8. C2×#20263242C2semidirect
9. #421×C197C2semidirect
17. C2×#15691358C22semidirect
23. #20263359C22semidirect
25. #20263356C22semidirect
27. #20263354C22semidirect
29. #20263351C22semidirect
32. C2×#15691342C22semidirect
33. #20263323C22semidirect
34. #20263314C22semidirect
37. #20263254C22semidirect
38. #20263242C22semidirect
40. C2×M16×C197C22semidirect
41. #117×C197C22semidirect
42. #104×C197C22semidirect
55. #15691358D4semidirect
57. C2×Dic394D4semidirect
58. C2×Dic394C23semidirect
68. C2×D788D4semidirect
70. C2×D788C23semidirect
90. #15691342C23semidirect
95. #41×C197D4semidirect
97. C2×Q8×C197D4semidirect
99. C2×D4×C197D4semidirect
103. M16×C197C23semidirect
107. C42×C197C23semidirect
193. C197#3019semidirect
202. #421D197semidirect
228. C2×M16C2×D197semidirect
233. #117C2×D197semidirect
235. #104C2×D197semidirect
253. #41D788semidirect
257. C2×Q8D788semidirect
278. C2×D4D788semidirect
295. M16C22×D197semidirect
301. C42C22×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
#271618792
#271618772
#271616482
#271612582
#271612482
#271610972
C2×#202632422
#421×C1972
#3019197