Finite Group Databasebeta

← #28743162

Subgroups — #28743162

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837838C211yesmaximal, characteristic, complemented
2C22×#14273667C211yesmaximal, characteristic, complemented
3#21433677C211yesmaximal, characteristic, complemented
4#21426207C212yesmaximal, complemented
5#21426197C212yesmaximal, complemented
6#21426190C212yesmaximal, complemented
7#21426180C212yesmaximal, complemented
8#21425882C211yesmaximal, characteristic, complemented
9#21425812C211yesmaximal, characteristic, complemented
10#21425806C211yesmaximal, characteristic, complemented
11C2×C59×#466C211yesmaximal, characteristic, complemented
12C2×#14273749C2211yescharacteristic
13#16837913—21no—
14#16837838—24no—
15C2×#14273671C2212yescomplemented
16C2×#14273668C2212yescomplemented
17C2×#14273667—24no—
18C2×#14273667C2211yescharacteristic
19C2×#14273667C2211yescharacteristic
20C2×#14273667C2212yescomplemented
21C22×C59⋊C16C2212yescomplemented
22#16837643C2211yescharacteristic, complemented
23#16837643C2211yescharacteristic, complemented
24#16837638C2211yescharacteristic, complemented
25#16837638C2211yescharacteristic, complemented
26#16837631—21no—
27#16837631—21no—
28#16837440C2211yescharacteristic
29#16837399—21no—
30#16837229C2212yescomplemented
31#16837227C2212yescomplemented
32#16837205C2212yescomplemented
33#16837203C2212yescomplemented
34#16837203C2212yescomplemented
35#16837202C2212yescomplemented
36C22×D8×C59C2211yescharacteristic
37C23×C8×C59C2211yescharacteristic
38C2×#121×C59—21no—
39C59×#466—21no—
40C59×#466—21no—
41C59×#466C2212yes—
42C59×#466C2212yes—
43C2×#107×C59C2211yescharacteristic
44C2×#102×C59—21no—
45C2×#11745685—21no—
46#14273749—22no—
47#14273749—22no—
48C2×Dic236C2311yescharacteristic
49C2×#11745660—21no—
50C2×D472C2311yescharacteristic
51#14273710—41no—
52#14273706—21no—
53#14273706—21no—
54#14273701—21no—
55#14273701—21no—
56#14273696—41no—
57C4×C22⋊D59—41no—
58C4×C22⋊D59—41no—
59#14273686—21no—
60#14273671—24no—
61#14273668—24no—
62#14273667—22no—
63#14273667—24no—
64#14273667D414yescomplemented
65C2×C59⋊C16—24no—
66C2×C59⋊C16C2312yes—
67C2×C59⋊C16C2312yes—
68#14273661C2311yescharacteristic
69#14273661C2311yescharacteristic
70#14273659—21no—
71#14273652—21no—
72#14273652—21no—
73#14273649—21no—
74#14273649—21no—
75C22×D4×C59—21no—
76C22×D4×C59—41no—
77C23×C4×C59D411yescharacteristic
78C2×D8×C59—22no—
79C2×D8×C59—21no—
80C2×D8×C59—22no—
81C2×D8×C59C2312yes—
82C22×C8×C59—22no—
83C22×C8×C59—22no—
84C22×C8×C59D411yescharacteristic
85C22×C8×C59D411yescharacteristic
86C22×C8×C59D411yescharacteristic
87C22×C8×C59C2311yescharacteristic
88C22×C8×C59C2311yescharacteristic
89C22×C8×C59C2311yescharacteristic
90#121×C59—22no—
91#121×C59—22no—
92#121×C59—41no—
93#121×C59—41no—
94C2×C4⋊C4×C59—21no—
95C2×C22⋊C4×C59—41no—
96#107×C59—21no—
97#107×C59C2312yes—
98#102×C59—22no—
99#102×C59—22no—
100C24×C59—41no—
101C24×C59D811yescharacteristic
102C2×D4×C59—22no—
103C2×D4×C59—42no—
104C2×D4×C59—42no—
105C2×D4×C59—42no—
106C2×D4×C59—41no—
107C2×D4×C59—42no—
108C2×D4×C59—84no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—41no—
112C22×C4×C59D811yescharacteristic
113C22×C4×C59D811yescharacteristic
114C22×C4×C59D811yescharacteristic
115C22×C4×C59C2×D411yescharacteristic
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D411yescharacteristic
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C2×C22⋊D59—41no—
121#11745685—42no—
122#11745685—42no—
123C2×D236—21no—
124C2×C4×D59—41no—
125C2×Dic118—21no—
126D8×C59—22no—
127D8×C59—22no—
128D8×C59—42no—
129C2×C8×C59—22no—
130C2×C8×C59—22no—
131C2×C8×C59—22no—
132C2×C8×C59—21no—
133C2×C8×C59—21no—
134C2×C8×C59—21no—
135C2×C8×C59—21no—
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C2×C8×C59C2×D411yescharacteristic
139C2×C8×C59C2×D412yes—
140C2×C8×C59#4112yes—
141C2×C8×C59C2411yescharacteristic
142C4⋊C4×C59—22no—
143C4⋊C4×C59—41no—
144C22⋊C4×C59—42no—
145C22⋊C4×C59—42no—
146C22⋊Dic59—41no—
147C22⋊Dic59—41no—
148C22⋊Dic59—41no—
149(C2×D59)⋊C4—41no—
150(C2×D59)⋊C4—41no—
151C4⋊Dic59—21no—
152C4⋊Dic59—21no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155C4×Dic59—41no—
156Dic236—21no—
157Dic236—21no—
158D472—21no—
159D472—21no—
160#11745660—42no—
161C59⋊C16—22no—
162C59⋊C16C2×D414yescomplemented
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—41no—
166C23×C59—42no—
167C23×C59—82no—
168C23×C59—82no—
169C23×C59—82no—
170C23×C59C2×D811yescharacteristic
171C23×C59C2×D811yescharacteristic
172C23×C59C2×D811yescharacteristic
173C22×D59—41no—
174D4×C59—42no—
175D4×C59—42no—
176D4×C59—84no—
177D4×C59—84no—
178D4×C59—82no—
179C2×C4×C59—22no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—22no—
185C2×C4×C59—22no—
186C2×C4×C59—42no—
187C2×C4×C59—82no—
188C2×C4×C59#12112yes—
189C2×C4×C59C2×D811yescharacteristic
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59#13512yes—
193C2×C4×C59C22×D411yescharacteristic
194C22⋊D59—82no—
195C22⋊D59—82no—
196C22⋊D59—82no—
197C2×Dic59—41no—
198C2×Dic59—41no—
199C2×Dic59—41no—
200D236—41no—
201D236—41no—
202C4×D59—82no—
203Dic118—41no—
204Dic118—41no—
205C8×C59—21no—
206C8×C59—21no—
207C8×C59#12112yes—
208C8×C59C22×D411yescharacteristic
209C8×C59C2×#4111yescharacteristic
210#33605—591nomaximal, Sylow
211C22×C59—21no—
212C22×C59—21no—
213C22×C59—21no—
214C22×C59—21no—
215C22×C59—21no—
216C22×C59—21no—
217C22×C59—21no—
218C22×C59—44no—
219C22×C59—82no—
220C22×C59—82no—
221C22×C59—82no—
222C22×C59—164no—
223C22×C59#46612yes—
224C22×C59#50912yescomplemented
225C22×C59C22×D811yescharacteristic
226C2×D59—81no—
227C2×D59—81no—
228C2×D59—81no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—82no—
232C4×C59#46612yes—
233C4×C59C2×#12111yescharacteristic
234C4×C59C2×#13511yescharacteristic
235Dic59—81no—
236Dic59—81no—
237Dic59—81no—
238C2×#509—591no—
239C22×M32—591no—
240#3050—591no—
241C2×#466D5911yescharacteristic, complemented
242#2220—592no—
243#2218—592no—
244#2217—592no—
245#2215—592no—
246#2159—591no—
247#2140—591no—
248#2139—591no—
249C2×C59—22no—
250C2×C59—22no—
251C2×C59—41no—
252C2×C59—162no—
253C2×C59C2×#46611yescharacteristic
254C2×C59C2×#50911yescharacteristic
255C2×C59#341411yescharacteristic
256D59—161no—
257C2×#135—591no—
258C22×D8C2×D5911yescharacteristic
259C23×C8C2×D5911yescharacteristic
260#525—1181no—
261C2×#121—21no—
262#509—1184no—
263C2×SD32—592no—
264C2×D16—592no—
265C2×M32—591no—
266C2×M32—591no—
267C2×M32—592no—
268C2×M32—1184no—
269C22×C16—592no—
270#467—591no—
271#467—591no—
272#466—21no—
273#466—21no—
274#466—591no—
275#466—591no—
276#466C2×D5912yes—
277#466C2×D5912yes—
278#465—1181no—
279#465—1181no—
280#427—591no—
281C2×#107C2×D5911yescharacteristic
282#416—1181no—
283C2×#102—21no—
284#367—592no—
285#366—592no—
286#358—592no—
287#357—592no—
288#357—592no—
289#357—592no—
290C59#3360511yescharacteristic, complemented, Sylow
291C2×#41—1181no—
292C22×D4—21no—
293C22×D4—41no—
294C23×C4C22⋊D5911yescharacteristic
295#135—1182no—
296#135—1182no—
297C2×Q16—591no—
298C2×SD16—1181no—
299C2×D8—22no—
300C2×D8—22no—
301C2×D8—21no—
302C2×D8—591no—
303C2×D8C22×D5912yes—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D5911yescharacteristic
307C22×C8C22⋊D5911yescharacteristic
308C22×C8C22⋊D5911yescharacteristic
309C22×C8C22×D5911yescharacteristic
310C22×C8C22×D5911yescharacteristic
311C22×C8C22×D5911yescharacteristic
312#123—2361no—
313C22⋊Q8—1181no—
314C22⋊Q8—1181no—
315#121—22no—
316#121—22no—
317#121—41no—
318#121—41no—
319#121—1181no—
320#121—1181no—
321#120—2361no—
322C4×D4—2361no—
323C4×D4—2361no—
324#117—1181no—
325C2×C4⋊C4—21no—
326C2×C22⋊C4—41no—
327SD32—1184no—
328D16—1184no—
329M32—594no—
330M32—1184no—
331M32—1182no—
332C2×C16—592no—
333C2×C16—592no—
334C2×C16—1184no—
335#107—21no—
336#107—591no—
337#107—591no—
338#107C22×D5912yes—
339#106—1181no—
340#103—1181no—
341#103—1181no—
342#102—22no—
343#102—22no—
344#102—1181no—
345#102—1181no—
346C24—41no—
347C24#1174566911yescharacteristic
348#41—2362no—
349#41—2362no—
350C2×Q8—1181no—
351C2×D4—22no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—84no—
358C2×D4—1181no—
359C2×D4—2361no—
360C2×D4—2361no—
361C2×D4—2361no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—2361no—
366C22×C4#1174566911yescharacteristic
367C22×C4#1174566911yescharacteristic
368C22×C4#1174566911yescharacteristic
369C22×C4C2×C22⋊D5911yescharacteristic
370C22×C4C2×C22⋊D5911yescharacteristic
371C22×C4C2×C22⋊D5911yescharacteristic
372Q16—1181no—
373Q16—1181no—
374SD16—2362no—
375D8—22no—
376D8—22no—
377D8—42no—
378D8—1181no—
379D8—1181no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8D4×D5912yes—
388C2×C8#1174568712yes—
389C2×C8C2×C22⋊D5911yescharacteristic
390C2×C8C2×C22⋊D5911yescharacteristic
391C2×C8C2×C22⋊D5911yescharacteristic
392C2×C8C23×D5911yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—1181no—
396C4⋊C4—1181no—
397C4⋊C4—2361no—
398C4⋊C4—2361no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—2361no—
402C22⋊C4—2361no—
403C22⋊C4—2361no—
404C22⋊C4—2361no—
405C22⋊C4—2361no—
406C42—2361no—
407C16—594no—
408C16—1182no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—42no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—2361no—
417C23C2×#1174566911yescharacteristic
418C23C2×#1174566911yescharacteristic
419C23C2×#1174566911yescharacteristic
420Q8—2361no—
421Q8—2361no—
422D4—42no—
423D4—42no—
424D4—84no—
425D4—84no—
426D4—82no—
427D4—4722no—
428D4—4722no—
429D4—4722no—
430D4—2361no—
431D4—2361no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—4722no—
442C2×C4—2361no—
443C2×C4—2361no—
444C2×C4—2361no—
445C2×C4#1427369912yes—
446C2×C4C2×#1174566911yescharacteristic
447C2×C4C2×#1174566911yescharacteristic
448C2×C4C2×#1174566911yescharacteristic
449C2×C4#1427374712yes—
450C2×C4C22×C22⋊D5911yescharacteristic
451C8—21no—
452C8—21no—
453C8#1427369912yes—
454C8C2×D4×D5911yescharacteristic
455C8C2×#1174568711yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22—4721no—
469C22—4721no—
470C22—4721no—
471C22#1683763612yes—
472C22#1683783812yescomplemented
473C22C22×#1174566911yescharacteristic
474C4—21no—
475C4—21no—
476C4—82no—
477C4—4721no—
478C4—4721no—
479C4—4721no—
480C4#1683763612yes—
481C4C2×#1427369911yescharacteristic
482C4C2×#1427374711yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—162no—
487C2—9441no—
488C2C2×#1683763611yescharacteristic
489C2C2×#1683783811yescharacteristic
490C2#2143723011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC2×C59×#466C2
socleC22×C59C22×D8
hypercenterC2×#466D59
hypocenterC59#33605
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#16837838C2semidirect
2. C22×#14273667C2semidirect
3. #21433677C2semidirect
4. #21426207C2semidirect
5. #21426197C2semidirect
6. #21426190C2semidirect
7. #21426180C2semidirect
8. #21425882C2semidirect
9. #21425812C2semidirect
10. #21425806C2semidirect
11. C2×C59×#466C2semidirect
15. C2×#14273671C22semidirect
16. C2×#14273668C22semidirect
20. C2×#14273667C22semidirect
21. C22×C59⋊C16C22semidirect
22. #16837643C22semidirect
23. #16837643C22semidirect
24. #16837638C22semidirect
25. #16837638C22semidirect
30. #16837229C22semidirect
31. #16837227C22semidirect
32. #16837205C22semidirect
33. #16837203C22semidirect
34. #16837203C22semidirect
35. #16837202C22semidirect
64. #14273667D4semidirect
162. C59⋊C16C2×D4semidirect
224. C22×C59#509semidirect
241. C2×#466D59semidirect
290. C59#33605semidirect
472. C22#16837838semidirect

Maximal subgroups

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

subgroupindex
C2×#168378382
C22×#142736672
#214336772
#214262072
#214261972
#214261902
#214261802
#214258822
#214258122
#214258062
C2×C59×#4662
#3360559