Finite Group Databasebeta

← #38745365

Subgroups — #38745365

Order 25216.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263296C211yesmaximal, characteristic
2C2×#20263288C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D197C211yesmaximal, characteristic, complemented
4#27160981C211yesmaximal, characteristic, complemented
5#27160979C211yesmaximal, characteristic, complemented
6#27160972C211yesmaximal, characteristic, complemented
7#27160969C211yesmaximal, characteristic, complemented
8#27160968C211yesmaximal, characteristic, complemented
9#27160957C211yesmaximal, characteristic, complemented
10#27160957C211yesmaximal, characteristic, complemented
11#27160955C211yesmaximal, characteristic, complemented
12#27160953C211yesmaximal, characteristic, complemented
13#27160934C211yesmaximal, characteristic, complemented
14C4×#15691335C211yesmaximal, characteristic, complemented
15#388×C197C211yesmaximal, characteristic, complemented
16C24×D197C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×C22⋊D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21C22×C4×D197C2211yescharacteristic
22C22×C4×D197C2211yescharacteristic, complemented
23#20263296C418yescomplemented
24#20263288—24no—
25C2×C4⋊Dic197C2211yescharacteristic, complemented
26C2×#15691337C2211yescharacteristic, complemented
27C2×C22⋊Dic197C2211yescharacteristic
28C2×C22⋊Dic197C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×#15691335C2211yescharacteristic, complemented
38C2×#15691335C2211yescharacteristic, complemented
39C2×#15691335C2211yescharacteristic, complemented
40C22⋊C4×D197—24no—
41C2×C4×Dic197C2211yescharacteristic, complemented
42C2×C4×Dic197C2211yescharacteristic, complemented
43#20263233C2211yescharacteristic, complemented
44#20263233C2211yescharacteristic, complemented
45#20263233C2211yescharacteristic, complemented
46#20263233C2211yescharacteristic, complemented
47C2×C4⋊C4×C197C2211yescharacteristic, complemented
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C22⋊C4×C197C2211yescharacteristic
50C2×C22⋊C4×C197C2211yescharacteristic
51C2×C42×C197C2211yescharacteristic, complemented
52#95×C197C2211yescharacteristic, complemented
53#95×C197C2211yescharacteristic, complemented
54C23×D197—22no—
55C23×D197—22no—
56C23×D197—41no—
57C23×D197—41no—
58C23×D197—41no—
59C23×D197—41no—
60C23×D197C2×C414yescomplemented
61C23×D197C2311yescharacteristic
62C23×D197C2311yescharacteristic
63C23×D197C2311yescharacteristic
64C2×D788—22no—
65C2×D788—22no—
66C2×D788—41no—
67C2×D788—41no—
68C2×D788C2×C414yes—
69C2×D788C2×C414yescomplemented
70C2×C22⋊D197—41no—
71C2×C22⋊D197—41no—
72C2×C22⋊D197C2×C414yescomplemented
73C22×Dic197C2311yescharacteristic
74C22×Dic197C2311yescharacteristic
75C22×Dic197C2311yescharacteristic
76C22×Dic197C2311yescharacteristic
77C22×Dic197C2311yescharacteristic
78C2×C4×D197—22no—
79C2×C4×D197—22no—
80C2×C4×D197—41no—
81C2×C4×D197—41no—
82C2×C4×D197—41no—
83C2×C4×D197—41no—
84C2×C4×D197—41no—
85C2×C4×D197—41no—
86C2×C4×D197D414yescomplemented
87C4⋊Dic197—41no—
88#15691337—41no—
89C22⋊Dic197—22no—
90C22⋊Dic197—41no—
91#15691335—22no—
92#15691335—22no—
93#15691335—22no—
94#15691335—22no—
95#15691335—22no—
96#15691335—41no—
97#15691335—41no—
98#15691335—41no—
99#15691335—41no—
100#15691335C2×C414yescomplemented
101#15691335C2×C414yescomplemented
102C4×Dic197—22no—
103C4×Dic197—41no—
104C24×C197C2311yescharacteristic
105C22×C4×C197C2311yescharacteristic
106C22×C4×C197C2311yescharacteristic
107C22×C4×C197C2311yescharacteristic
108C22×C4×C197C2311yescharacteristic
109C22×C4×C197C2311yescharacteristic
110C22×C4×C197C2311yescharacteristic
111C4⋊C4×C197—22no—
112C22⋊C4×C197—22no—
113C22⋊C4×C197—41no—
114C22⋊C4×C197C2×C414yescomplemented
115C42×C197—41no—
116C22×D197—21no—
117C22×D197—21no—
118C22×D197—21no—
119C22×D197—21no—
120C22×D197—21no—
121C22×D197—21no—
122C22×D197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—21no—
128C22×D197—21no—
129C22×D197—21no—
130C22×D197—21no—
131C22×D197—21no—
132C22×D197—28no—
133C22×D197—44no—
134C22×D197—44no—
135C22×D197—44no—
136C22×D197—44no—
137C22×D197—44no—
138C22×D197—44no—
139C22×D197C22×C412yes—
140C22×D197C22×C412yes—
141C22×D197C22×C412yes—
142C22×D197C2×D412yes—
143C22×D197#4112yes—
144D788—28no—
145D788—44no—
146C22⋊D197—44no—
147C2×Dic197—21no—
148C2×Dic197—21no—
149C2×Dic197—21no—
150C2×Dic197—21no—
151C2×Dic197—21no—
152C2×Dic197—21no—
153C2×Dic197—21no—
154C2×Dic197—21no—
155C2×Dic197—21no—
156C2×Dic197—21no—
157C2×Dic197—21no—
158C2×Dic197—21no—
159C2×Dic197—21no—
160C2×Dic197C22×C412yes—
161C2×Dic197C2×D412yes—
162C4×D197—44no—
163C4×D197—44no—
164C4×D197—44no—
165C23×C197—21no—
166C23×C197—21no—
167C23×C197—21no—
168C23×C197—21no—
169C23×C197—21no—
170C23×C197—21no—
171C23×C197C22×C412yes—
172C23×C197C2411yescharacteristic
173C2×C4×C197—21no—
174C2×C4×C197—21no—
175C2×C4×C197—21no—
176C2×C4×C197—21no—
177C2×C4×C197—21no—
178C2×C4×C197—21no—
179C2×C4×C197—21no—
180C2×C4×C197—21no—
181C2×C4×C197—21no—
182C2×C4×C197—21no—
183C2×C4×C197—21no—
184C2×C4×C197—21no—
185C2×C4×C197—21no—
186C2×C4×C197—21no—
187C2×C4×C197—21no—
188C2×C4×C197C22×C412yes—
189C2×C4×C197C22×C412yes—
190C2×C4×C197C2×D412yes—
191C22×C197—22no—
192C22×C197—22no—
193C22×C197—22no—
194C22×C197—41no—
195C22×C197—41no—
196C22×C197—41no—
197C22×C197—41no—
198C22×C197C23×C411yescharacteristic
199C22×C197C22×D411yescharacteristic
200C22×C197C2×#4111yescharacteristic
201C22×C197H(5,𝔽2)11yescharacteristic
202C22×C197H(5,𝔽2)11yescharacteristic
203C22×C197H(5,𝔽2)11yescharacteristic
204C22×C197H(5,𝔽2)11yescharacteristic
205C2×D197—22no—
206C2×D197—22no—
207C2×D197—22no—
208C2×D197—22no—
209C2×D197—22no—
210C2×D197—22no—
211C2×D197—22no—
212C2×D197—22no—
213C2×D197—22no—
214C2×D197—22no—
215C2×D197—22no—
216C2×D197—22no—
217C2×D197—416no—
218C2×D197—41no—
219C2×D197—41no—
220C2×D197—41no—
221C2×D197—41no—
222C2×D197—41no—
223C2×D197—41no—
224C2×D197—41no—
225C2×D197—41no—
226C2×D197C4×D414yes—
227C4×C197—22no—
228C4×C197—22no—
229C4×C197—22no—
230C4×C197—41no—
231C4×C197—41no—
232C4×C197—41no—
233Dic197—22no—
234Dic197—22no—
235Dic197—41no—
236Dic197—41no—
237Dic197—41no—
238C2×C197—42no—
239C2×C197C2×C4×D411yescharacteristic
240C2×C197#51811yescharacteristic
241C2×C197#51811yescharacteristic
242C2×C197#53411yescharacteristic
243C2×C197#53511yescharacteristic
244C2×C197#53811yescharacteristic
245C2×C197#53811yescharacteristic
246D197—24no—
247D197—42no—
248D197—42no—
249C197#232111yescharacteristic, complemented, Sylow
250#2321—1971nomaximal, Sylow
251C2×#120—1971no—
252C2×C4×D4—1971no—
253C22×C22⋊C4—1971no—
254#390—1971no—
255#390—1971no—
256#388—1971no—
257#388—1971no—
258#388—1971no—
259#388D19711yescharacteristic, complemented
260#386—1971no—
261#386—1971no—
262#386—1971no—
263#385—1971no—
264#379—1971no—
265C4×C22⋊C4—1971no—
266C25—1971no—
267C22×D4—1971no—
268C22×D4—1971no—
269C22×D4—1971no—
270C23×C4—1971no—
271C23×C4—1971no—
272C23×C4—1971no—
273#120—1978no—
274C4×D4—3944no—
275C2×C4⋊C4—1971no—
276C2×C4⋊C4—1971no—
277C2×C4⋊C4C2×D19711yescharacteristic, complemented
278C2×C22⋊C4—3944no—
279C2×C22⋊C4—1971no—
280C2×C22⋊C4—1971no—
281C2×C22⋊C4—1971no—
282C2×C22⋊C4—1971no—
283C2×C22⋊C4—1971no—
284C2×C22⋊C4—1971no—
285C2×C22⋊C4—1971no—
286C2×C22⋊C4—1971no—
287C2×C22⋊C4—1971no—
288C2×C22⋊C4—1971no—
289C2×C22⋊C4—1971no—
290C2×C22⋊C4—1971no—
291C2×C22⋊C4—1971no—
292C2×C22⋊C4C2×D19711yescharacteristic
293C2×C22⋊C4C2×D19711yescharacteristic
294C2×C22⋊C4C2×D19711yescharacteristic
295C2×C42—1971no—
296C2×C42—1971no—
297C2×C42C2×D19711yescharacteristic, complemented
298#95—1971no—
299#95—1971no—
300#95—1971no—
301#95—1971no—
302#95C2×D19711yescharacteristic, complemented
303#95C2×D19711yescharacteristic, complemented
304C24—7881no—
305C24—7881no—
306C24—7881no—
307C24—7881no—
308C24—3942no—
309C24—3942no—
310C24—1974no—
311C24—1971no—
312C24—1971no—
313C24—1971no—
314C24C22×D19711yescharacteristic
315C2×D4—7881no—
316C2×D4—7881no—
317C2×D4—7881no—
318C2×D4—7881no—
319C2×D4—3942no—
320C2×D4—3942no—
321C2×D4—1974no—
322C2×D4—1974no—
323C2×D4—1974no—
324C22×C4—7881no—
325C22×C4—7881no—
326C22×C4—7881no—
327C22×C4—7881no—
328C22×C4—7881no—
329C22×C4—7881no—
330C22×C4—3942no—
331C22×C4—3942no—
332C22×C4—1974no—
333C22×C4—1971no—
334C22×C4—1971no—
335C22×C4—1971no—
336C22×C4—1971no—
337C22×C4—1971no—
338C22×C4C22×D19711yescharacteristic
339C22×C4C22×D19711yescharacteristic
340C22×C4C22×D19711yescharacteristic
341C22×C4C22×D19711yescharacteristic
342C22×C4C22×D19711yescharacteristic
343C22×C4C22×D19711yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—7881no—
346C4⋊C4—7881no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—7881no—
350C22⋊C4—7881no—
351C22⋊C4—7881no—
352C22⋊C4—7881no—
353C22⋊C4—7881no—
354C22⋊C4—3942no—
355C22⋊C4—3942no—
356C22⋊C4—3942no—
357C22⋊C4—3942no—
358C22⋊C4—3942no—
359C22⋊C4—3942no—
360C22⋊C4—1974no—
361C22⋊C4—1974no—
362C22⋊C4C4×D19714yescomplemented
363C42—41no—
364C42—7881no—
365C42—3942no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—7884no—
373C23—7884no—
374C23—7884no—
375C23—7884no—
376C23—7884no—
377C23—7884no—
378C23—3948no—
379C23—3941no—
380C23—3941no—
381C23—3941no—
382C23—3941no—
383C23—3941no—
384C23—3941no—
385C23—3941no—
386C23—3941no—
387C23—3941no—
388C23—3941no—
389C23—3941no—
390C23—3941no—
391C23—3941no—
392C23—3941no—
393C23—3941no—
394C23—3941no—
395C23—1972no—
396C23—1972no—
397C23—1972no—
398C23—1972no—
399C23—1972no—
400C23C2×C4×D19712yes—
401C23C23×D19711yescharacteristic
402D4—7884no—
403D4—7884no—
404D4—3948no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—7884no—
421C2×C4—7884no—
422C2×C4—7884no—
423C2×C4—3941no—
424C2×C4—3941no—
425C2×C4—3941no—
426C2×C4—3941no—
427C2×C4—3941no—
428C2×C4—3941no—
429C2×C4—3941no—
430C2×C4—3941no—
431C2×C4—3941no—
432C2×C4—3941no—
433C2×C4—3941no—
434C2×C4—3941no—
435C2×C4—3941no—
436C2×C4—1972no—
437C2×C4—1972no—
438C2×C4C2×C4×D19712yes—
439C2×C4C2×C4×D19712yes—
440C2×C4D4×D19712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—78816no—
449C22—7881no—
450C22—7881no—
451C22—7881no—
452C22—7881no—
453C22—7881no—
454C22—7881no—
455C22—7881no—
456C22—7881no—
457C22—3942no—
458C22—3942no—
459C22—3942no—
460C22—3942no—
461C22—3942no—
462C22—3942no—
463C22—3942no—
464C22—3942no—
465C22—3942no—
466C22—3942no—
467C22—3942no—
468C22—3942no—
469C22—1974no—
470C22C22×C4×D19711yescharacteristic
471C22C2×D4×D19711yescharacteristic
472C22#41×D19711yescharacteristic
473C22#2026337111yescharacteristic
474C22#2026337211yescharacteristic
475C22#2026337211yescharacteristic
476C22#2026337211yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—7881no—
484C4—7881no—
485C4—7881no—
486C4—3942no—
487C4—3942no—
488C2—42no—
489C2—7882no—
490C2—7882no—
491C2—3944no—
492C2C4×D4×D19711yescharacteristic
493C2#2716155111yescharacteristic
494C2#2716155311yescharacteristic
495C2#2716163711yescharacteristic
496C2#2716164711yescharacteristic
497C2#2716166511yescharacteristic
498C2#2716166811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#388×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263296C2
hypercenter#388D197
hypocenterC197#2321
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#20263288C2semidirect
3. C2×C22⋊C4×D197C2semidirect
4. #27160981C2semidirect
5. #27160979C2semidirect
6. #27160972C2semidirect
7. #27160969C2semidirect
8. #27160968C2semidirect
9. #27160957C2semidirect
10. #27160957C2semidirect
11. #27160955C2semidirect
12. #27160953C2semidirect
13. #27160934C2semidirect
14. C4×#15691335C2semidirect
15. #388×C197C2semidirect
22. C22×C4×D197C22semidirect
23. #20263296C4semidirect
25. C2×C4⋊Dic197C22semidirect
26. C2×#15691337C22semidirect
37. C2×#15691335C22semidirect
38. C2×#15691335C22semidirect
39. C2×#15691335C22semidirect
41. C2×C4×Dic197C22semidirect
42. C2×C4×Dic197C22semidirect
43. #20263233C22semidirect
44. #20263233C22semidirect
45. #20263233C22semidirect
46. #20263233C22semidirect
47. C2×C4⋊C4×C197C22semidirect
51. C2×C42×C197C22semidirect
52. #95×C197C22semidirect
53. #95×C197C22semidirect
60. C23×D197C2×C4semidirect
69. C2×D788C2×C4semidirect
72. C2×C22⋊D197C2×C4semidirect
86. C2×C4×D197D4semidirect
100. #15691335C2×C4semidirect
101. #15691335C2×C4semidirect
114. C22⋊C4×C197C2×C4semidirect
249. C197#2321semidirect
259. #388D197semidirect
277. C2×C4⋊C4C2×D197semidirect
297. C2×C42C2×D197semidirect
302. #95C2×D197semidirect
303. #95C2×D197semidirect
362. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632962
C2×#202632882
C2×C22⋊C4×D1972
#271609812
#271609792
#271609722
#271609692
#271609682
#271609572
#271609552
#271609532
#271609342
C4×#156913352
#388×C1972
#2321197