Finite Group Databasebeta

← #38745688

Subgroups — #38745688

Order 25216.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263296C211yesmaximal, characteristic
2C2×#20263288C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D197C211yesmaximal, characteristic, complemented
4C22×#15691335C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D197C211yesmaximal, characteristic, complemented
6#27160980C211yesmaximal, characteristic, complemented
7#27160969C211yesmaximal, characteristic, complemented
8#27160968C211yesmaximal, characteristic, complemented
9#27160957C211yesmaximal, characteristic, complemented
10#27160957C211yesmaximal, characteristic, complemented
11#27160956C211yesmaximal, characteristic, complemented
12#27160950C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
14#27160926C211yesmaximal, characteristic, complemented
15#385×C197C211yesmaximal, characteristic, complemented
16C24×D197C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×C22⋊D197C2211yescharacteristic
20C23×Dic197C2211yescharacteristic
21C22×C4×D197C2211yescharacteristic
22C22×C4×D197C2211yescharacteristic
23C22×C4×D197C2211yescharacteristic, complemented
24#20263296C418yescomplemented
25#20263288—24no—
26C4×C22⋊D197—24no—
27C2×#15691337C2211yescharacteristic, complemented
28C2×C22⋊Dic197C2211yescharacteristic
29C2×#15691335—24no—
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×#15691335C2211yescharacteristic
38C2×#15691335C2211yescharacteristic, complemented
39C2×#15691335C2211yescharacteristic, complemented
40C2×#15691335C2211yescharacteristic, complemented
41C22⋊C4×D197—24no—
42C2×C4×Dic197C2211yescharacteristic, complemented
43C2×C4×Dic197C2211yescharacteristic, complemented
44C2×C4×Dic197C2211yescharacteristic, complemented
45#20263233C2211yescharacteristic, complemented
46#20263233C2211yescharacteristic, complemented
47#20263233C2211yescharacteristic, complemented
48#20263233C2211yescharacteristic, complemented
49C23×C4×C197C2211yescharacteristic
50C2×C4⋊C4×C197C2211yescharacteristic, complemented
51C2×C4⋊C4×C197C2211yescharacteristic, complemented
52C2×C22⋊C4×C197C2211yescharacteristic
53C2×C22⋊C4×C197C2211yescharacteristic
54#95×C197C2211yescharacteristic, complemented
55#95×C197C2211yescharacteristic, complemented
56C23×D197—22no—
57C23×D197—22no—
58C23×D197—22no—
59C23×D197—22no—
60C23×D197—41no—
61C23×D197—41no—
62C23×D197C2×C414yescomplemented
63C23×D197C2311yescharacteristic
64C23×D197C2311yescharacteristic
65C23×D197C2311yescharacteristic
66C2×D788—22no—
67C2×D788—22no—
68C2×D788—41no—
69C2×D788—41no—
70C2×D788C2×C414yescomplemented
71C2×D788C2×C414yescomplemented
72C2×C22⋊D197—22no—
73C2×C22⋊D197—22no—
74C2×C22⋊D197C2×C414yescomplemented
75C22×Dic197—22no—
76C22×Dic197—22no—
77C22×Dic197—41no—
78C22×Dic197C2311yescharacteristic
79C22×Dic197C2311yescharacteristic
80C22×Dic197C2311yescharacteristic
81C22×Dic197C2311yescharacteristic
82C22×Dic197C2311yescharacteristic
83C2×C4×D197—22no—
84C2×C4×D197—22no—
85C2×C4×D197—22no—
86C2×C4×D197—22no—
87C2×C4×D197—22no—
88C2×C4×D197—22no—
89C2×C4×D197—41no—
90C2×C4×D197—41no—
91C2×C4×D197—41no—
92#15691337—22no—
93C22⋊Dic197D414yescomplemented
94#15691335—22no—
95#15691335—22no—
96#15691335—22no—
97#15691335—22no—
98#15691335—22no—
99#15691335—22no—
100#15691335—22no—
101#15691335—41no—
102#15691335C2×C414yescomplemented
103#15691335C2×C414yescomplemented
104#15691335D414yescomplemented
105C4×Dic197—22no—
106C4×Dic197—22no—
107C4×Dic197—41no—
108C24×C197C2311yescharacteristic
109C22×C4×C197—22no—
110C22×C4×C197—22no—
111C22×C4×C197—41no—
112C22×C4×C197C2311yescharacteristic
113C22×C4×C197C2311yescharacteristic
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117C22×C4×C197C2311yescharacteristic
118C4⋊C4×C197—22no—
119C4⋊C4×C197—41no—
120C22⋊C4×C197—22no—
121C22⋊C4×C197C2×C414yescomplemented
122C22×D197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—28no—
127C22×D197—21no—
128C22×D197—21no—
129C22×D197—21no—
130C22×D197—21no—
131C22×D197—21no—
132C22×D197—21no—
133C22×D197—21no—
134C22×D197—21no—
135C22×D197—21no—
136C22×D197—21no—
137C22×D197—21no—
138C22×D197—21no—
139C22×D197—28no—
140C22×D197—44no—
141C22×D197—44no—
142C22×D197—44no—
143C22×D197—44no—
144C22×D197—44no—
145C22×D197C22×C412yes—
146C22×D197C22×C412yes—
147C22×D197C22×C412yes—
148C22×D197C2×D412yes—
149C22×D197#4112yes—
150D788—28no—
151D788—44no—
152C22⋊D197—28no—
153C2×Dic197—21no—
154C2×Dic197—21no—
155C2×Dic197—21no—
156C2×Dic197—21no—
157C2×Dic197—21no—
158C2×Dic197—21no—
159C2×Dic197—21no—
160C2×Dic197—21no—
161C2×Dic197—21no—
162C2×Dic197—21no—
163C2×Dic197—21no—
164C2×Dic197—44no—
165C2×Dic197C22×C412yes—
166C2×Dic197C2×D412yes—
167C2×Dic197C2×D412yes—
168C2×Dic197C2×D412yes—
169C4×D197—44no—
170C4×D197—44no—
171C4×D197—44no—
172C23×C197—21no—
173C23×C197—21no—
174C23×C197—21no—
175C23×C197—21no—
176C23×C197C22×C412yes—
177C23×C197C2×D412yes—
178C23×C197#4112yes—
179C23×C197C2411yescharacteristic
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×C197—21no—
189C2×C4×C197—21no—
190C2×C4×C197—21no—
191C2×C4×C197—21no—
192C2×C4×C197—21no—
193C2×C4×C197—21no—
194C2×C4×C197—21no—
195C2×C4×C197—44no—
196C2×C4×C197C22×C412yes—
197C2×C4×C197C22×C412yes—
198C2×C4×C197C2×D412yes—
199C22×C197—22no—
200C22×C197—22no—
201C22×C197—22no—
202C22×C197—22no—
203C22×C197—22no—
204C22×C197—22no—
205C22×C197C4×D414yescomplemented
206C22×C197C23×C411yescharacteristic
207C22×C197C22×D411yescharacteristic
208C22×C197C22×D411yescharacteristic
209C22×C197C2×#4111yescharacteristic
210C22×C197C2×#4111yescharacteristic
211C22×C197H(5,𝔽2)11yescharacteristic
212C22×C197H(5,𝔽2)11yescharacteristic
213C2×D197—22no—
214C2×D197—22no—
215C2×D197—22no—
216C2×D197—22no—
217C2×D197—22no—
218C2×D197—22no—
219C2×D197—22no—
220C2×D197—22no—
221C2×D197—22no—
222C2×D197—22no—
223C2×D197—22no—
224C2×D197—22no—
225C2×D197—416no—
226C2×D197—41no—
227C2×D197—41no—
228C2×D197—41no—
229C2×D197—41no—
230C2×D197—41no—
231C2×D197—41no—
232C2×D197—41no—
233C2×D197—41no—
234C2×D197C4×D414yescomplemented
235C4×C197—22no—
236C4×C197—22no—
237C4×C197—22no—
238C4×C197—41no—
239C4×C197—41no—
240C4×C197—41no—
241Dic197—22no—
242Dic197—22no—
243Dic197—22no—
244Dic197—22no—
245Dic197—41no—
246C2×C197—24no—
247C2×C197C2×C4×D411yescharacteristic
248C2×C197C2×C4×D411yescharacteristic
249C2×C197#51811yescharacteristic
250C2×C197D4211yescharacteristic
251C2×C197#54611yescharacteristic
252C2×C197#54611yescharacteristic
253C2×C197#55111yescharacteristic
254D197—24no—
255D197—42no—
256D197—42no—
257C197#235811yescharacteristic, complemented, Sylow
258#2358—1971nomaximal, Sylow
259C2×#120—1971no—
260C2×C4×D4—1971no—
261C2×C4×D4—1971no—
262C22×C22⋊C4—1971no—
263C22×C22⋊C4—1971no—
264#390—1971no—
265#388—1971no—
266#388—1971no—
267#386—1971no—
268#386—1971no—
269#386—1971no—
270#385—1971no—
271#385D19711yescharacteristic, complemented
272C4×C22⋊C4—1971no—
273C4×C22⋊C4—1971no—
274C25—1971no—
275C22×D4—1971no—
276C22×D4—1971no—
277C22×D4—1971no—
278C23×C4—1971no—
279C23×C4—1971no—
280C23×C4—1971no—
281C23×C4—1971no—
282C23×C4C2×D19711yescharacteristic
283#120—1978no—
284C4×D4—3944no—
285C4×D4—3944no—
286C2×C4⋊C4—1971no—
287C2×C4⋊C4C2×D19711yescharacteristic, complemented
288C2×C4⋊C4C2×D19711yescharacteristic, complemented
289C2×C22⋊C4—3944no—
290C2×C22⋊C4—3944no—
291C2×C22⋊C4—1971no—
292C2×C22⋊C4—1971no—
293C2×C22⋊C4—1971no—
294C2×C22⋊C4—1971no—
295C2×C22⋊C4—1971no—
296C2×C22⋊C4—1971no—
297C2×C22⋊C4—1971no—
298C2×C22⋊C4—1971no—
299C2×C22⋊C4—1971no—
300C2×C22⋊C4—1971no—
301C2×C22⋊C4—1971no—
302C2×C22⋊C4—1971no—
303C2×C22⋊C4C2×D19711yescharacteristic
304C2×C22⋊C4C2×D19711yescharacteristic
305C2×C42—1971no—
306C2×C42—1971no—
307C2×C42—1971no—
308#95—1971no—
309#95—1971no—
310#95—1971no—
311#95—1971no—
312#95C2×D19711yescharacteristic, complemented
313#95C2×D19711yescharacteristic, complemented
314C24—7881no—
315C24—7881no—
316C24—3942no—
317C24—3942no—
318C24—3942no—
319C24—3942no—
320C24—1974no—
321C24—1971no—
322C24—1971no—
323C24—1971no—
324C24C22×D19711yescharacteristic
325C2×D4—7881no—
326C2×D4—7881no—
327C2×D4—3942no—
328C2×D4—3942no—
329C2×D4—3942no—
330C2×D4—3942no—
331C2×D4—1974no—
332C2×D4—1974no—
333C2×D4—1974no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—41no—
337C22×C4—7881no—
338C22×C4—7881no—
339C22×C4—7881no—
340C22×C4—7881no—
341C22×C4—3942no—
342C22×C4—3942no—
343C22×C4—3942no—
344C22×C4—3942no—
345C22×C4—3942no—
346C22×C4—3942no—
347C22×C4—3942no—
348C22×C4—3942no—
349C22×C4—1971no—
350C22×C4—1971no—
351C22×C4—1971no—
352C22×C4—1971no—
353C22×C4—1971no—
354C22×C4C22×D19711yescharacteristic
355C22×C4C22×D19711yescharacteristic
356C22×C4C22×D19711yescharacteristic
357C22×C4C22×D19711yescharacteristic
358C22×C4C22×D19711yescharacteristic
359C22×C4C22×D19711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—3942no—
363C22⋊C4—22no—
364C22⋊C4—7881no—
365C22⋊C4—3942no—
366C22⋊C4—3942no—
367C22⋊C4—3942no—
368C22⋊C4—3942no—
369C22⋊C4—3942no—
370C22⋊C4—3942no—
371C22⋊C4—3942no—
372C22⋊C4—1974no—
373C22⋊C4—1974no—
374C22⋊C4—1974no—
375C22⋊C4—1974no—
376C22⋊C4C4×D19714yescomplemented
377C42—7881no—
378C42—3942no—
379C42—3942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—7884no—
385C23—7884no—
386C23—7884no—
387C23—7884no—
388C23—7884no—
389C23—3948no—
390C23—3948no—
391C23—3941no—
392C23—3941no—
393C23—3941no—
394C23—3941no—
395C23—3941no—
396C23—3941no—
397C23—3941no—
398C23—3941no—
399C23—3941no—
400C23—3941no—
401C23—3941no—
402C23—3941no—
403C23—3941no—
404C23—3941no—
405C23—3941no—
406C23—3941no—
407C23—1972no—
408C23—1972no—
409C23—1972no—
410C23—1972no—
411C23—1972no—
412C23C2×C4×D19712yes—
413C23D4×D19712yes—
414C23#1569135912yes—
415C23C23×D19711yescharacteristic
416D4—7884no—
417D4—3948no—
418D4—3948no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—7884no—
436C2×C4—7884no—
437C2×C4—7884no—
438C2×C4—7884no—
439C2×C4—3941no—
440C2×C4—3941no—
441C2×C4—3941no—
442C2×C4—3941no—
443C2×C4—3941no—
444C2×C4—3941no—
445C2×C4—3941no—
446C2×C4—3941no—
447C2×C4—3941no—
448C2×C4—3941no—
449C2×C4—3941no—
450C2×C4—1972no—
451C2×C4—1972no—
452C2×C4—1972no—
453C2×C4—1972no—
454C2×C4C2×C4×D19712yes—
455C2×C4C2×C4×D19712yes—
456C2×C4D4×D19712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—78816no—
464C22—7881no—
465C22—7881no—
466C22—7881no—
467C22—7881no—
468C22—7881no—
469C22—7881no—
470C22—7881no—
471C22—7881no—
472C22—3942no—
473C22—3942no—
474C22—3942no—
475C22—3942no—
476C22—3942no—
477C22—3942no—
478C22—3942no—
479C22—3942no—
480C22—3942no—
481C22—3942no—
482C22—3942no—
483C22—3942no—
484C22—1974no—
485C22#2026328814yescomplemented
486C22C22×C4×D19711yescharacteristic
487C22C2×D4×D19711yescharacteristic
488C22C2×D4×D19711yescharacteristic
489C22#41×D19711yescharacteristic
490C22C2×#1569135911yescharacteristic
491C22#2026337111yescharacteristic
492C22#2026337211yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—7881no—
500C4—3942no—
501C4—3942no—
502C4—3942no—
503C4—3942no—
504C2—24no—
505C2—7882no—
506C2—7882no—
507C2—3944no—
508C2C4×D4×D19711yescharacteristic
509C2C2×#2026328811yescharacteristic
510C2#2716155111yescharacteristic
511C2#2716171111yescharacteristic
512C2#2716172011yescharacteristic
513C2#2716172111yescharacteristic
514C2#2716175611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#385×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263296C2
hypercenter#385D197
hypocenterC197#2358
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#20263288C2semidirect
3. C2×C4×C22⋊D197C2semidirect
4. C22×#15691335C2semidirect
5. C2×C22⋊C4×D197C2semidirect
6. #27160980C2semidirect
7. #27160969C2semidirect
8. #27160968C2semidirect
9. #27160957C2semidirect
10. #27160957C2semidirect
11. #27160956C2semidirect
12. #27160950C2semidirect
13. C22⋊C4×Dic197C2semidirect
14. #27160926C2semidirect
15. #385×C197C2semidirect
23. C22×C4×D197C22semidirect
24. #20263296C4semidirect
27. C2×#15691337C22semidirect
38. C2×#15691335C22semidirect
39. C2×#15691335C22semidirect
40. C2×#15691335C22semidirect
42. C2×C4×Dic197C22semidirect
43. C2×C4×Dic197C22semidirect
44. C2×C4×Dic197C22semidirect
45. #20263233C22semidirect
46. #20263233C22semidirect
47. #20263233C22semidirect
48. #20263233C22semidirect
50. C2×C4⋊C4×C197C22semidirect
51. C2×C4⋊C4×C197C22semidirect
54. #95×C197C22semidirect
55. #95×C197C22semidirect
62. C23×D197C2×C4semidirect
70. C2×D788C2×C4semidirect
71. C2×D788C2×C4semidirect
74. C2×C22⋊D197C2×C4semidirect
93. C22⋊Dic197D4semidirect
102. #15691335C2×C4semidirect
103. #15691335C2×C4semidirect
104. #15691335D4semidirect
121. C22⋊C4×C197C2×C4semidirect
205. C22×C197C4×D4semidirect
234. C2×D197C4×D4semidirect
257. C197#2358semidirect
271. #385D197semidirect
287. C2×C4⋊C4C2×D197semidirect
288. C2×C4⋊C4C2×D197semidirect
312. #95C2×D197semidirect
313. #95C2×D197semidirect
376. C22⋊C4C4×D197semidirect
485. C22#20263288semidirect

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×C4×C22⋊D1972
C22×#156913352
C2×C22⋊C4×D1972
#271609802
#271609692
#271609682
#271609572
#271609562
#271609502
C22⋊C4×Dic1972
#271609262
#385×C1972
#2358197