Finite Group Databasebeta

← #28481590

Subgroups — #28481590

Order 15104.

524 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837470C211yesmaximal, characteristic
2#21424123C211yesmaximal, characteristic, complemented
3#21424119C211yesmaximal, characteristic, complemented
4#21424081C211yesmaximal, characteristic
5#21424081C211yesmaximal, characteristic
6#21421964C211yesmaximal, characteristic, complemented
7C59×#1289C211yesmaximal, characteristic, complemented
8C22×#11745659C411yescharacteristic, complemented
9C23×C59⋊C8C2211yescharacteristic
10C2×C4×C22⋊D59C2211yescharacteristic
11#16837470—21no—
12#16837470—21no—
13#16837470—21no—
14#16837470—21no—
15C2×#14273660—21no—
16C2×#14273658—21no—
17C2×Dic59⋊C8C411yescharacteristic, complemented
18C2×#14273641—21no—
19C2×#14273641—21no—
20C2×(C2×D59)⋊C8C411yescharacteristic, complemented
21C2×#14273637C411yescharacteristic, complemented
22#16837262C2211yescharacteristic
23#16837262C2211yescharacteristic
24#16837168—21no—
25#16837168—21no—
26#16837164—21no—
27#16837164—21no—
28C2×C22⋊C8×C59C2211yescharacteristic
29C59×#380C2211yescharacteristic
30C59×#380C2211yescharacteristic
31C22×C22⋊D59D411yescharacteristic
32C22×C4×D59C2×C411yescharacteristic
33C2×#11745659—21no—
34C2×#11745659—21no—
35C2×#11745659—42no—
36C22×C59⋊C8—42no—
37C22×C59⋊C8C2×C411yescharacteristic
38C22×C59⋊C8C2×C411yescharacteristic
39C22×C59⋊C8D411yescharacteristic
40C22×C59⋊C8D411yescharacteristic
41C22×C59⋊C8D411yescharacteristic
42C22×C59⋊C8D411yescharacteristic
43C4×C22⋊D59—21no—
44C4×C22⋊D59—21no—
45C4×C22⋊D59—21no—
46C4×C22⋊D59—21no—
47C2×C4⋊Dic59—21no—
48C2×C4⋊Dic59—21no—
49C2×Dic59⋊C4Q811yescharacteristic
50C2×C22⋊Dic59—21no—
51C2×C22⋊Dic59—21no—
52C2×C22⋊Dic59D411yescharacteristic
53C2×(C2×D59)⋊C4Q811yescharacteristic
54C2×C4×Dic59C2×C411yescharacteristic
55#14273660—42no—
56#14273658—42no—
57Dic59⋊C8—21no—
58Dic59⋊C8—21no—
59#14273641—42no—
60#14273641—42no—
61(C2×D59)⋊C8—21no—
62(C2×D59)⋊C8—21no—
63#14273637—21no—
64#14273637—21no—
65C59⋊#95—21no—
66C59⋊#95—21no—
67C23×C4×C59C2311yescharacteristic
68C22×C8×C59C2×C411yescharacteristic
69C22×C8×C59C2×C411yescharacteristic
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59—21no—
72C2×C22⋊C4×C59—21no—
73C2×C22⋊C4×C59—21no—
74C22⋊C8×C59—21no—
75C22⋊C8×C59—21no—
76#95×C59—21no—
77#95×C59—21no—
78C24×C59C2×D411yescharacteristic
79C23×D59C4⋊C411yescharacteristic
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×Q811yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C2×C22⋊D59—21no—
91C2×C22⋊D59—21no—
92C2×C22⋊D59—21no—
93C2×C22⋊D59—21no—
94C2×C22⋊D59—21no—
95C2×C22⋊D59—21no—
96C22×Dic59—41no—
97C22×Dic59—41no—
98C22×Dic59C4⋊C411yescharacteristic
99C22×Dic59C4⋊C411yescharacteristic
100C22×Dic59C4⋊C411yescharacteristic
101C2×C4×D59—21no—
102C2×C4×D59—21no—
103C2×C4×D59—41no—
104C2×C4×D59—41no—
105C2×C8×C59—21no—
106C2×C8×C59—21no—
107C2×C8×C59—41no—
108C2×C8×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊Dic59—21no—
118C22⋊Dic59—21no—
119C22⋊Dic59—41no—
120C22⋊Dic59—41no—
121C22⋊Dic59—41no—
122C22⋊Dic59—41no—
123(C2×D59)⋊C4—21no—
124(C2×D59)⋊C4—21no—
125C4⋊Dic59—41no—
126C4⋊Dic59—41no—
127C4⋊Dic59—41no—
128C4⋊Dic59—41no—
129Dic59⋊C4—21no—
130Dic59⋊C4—21no—
131C4×Dic59—21no—
132C4×Dic59—21no—
133C2×C59⋊C8—22no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8—22no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C2×C59⋊C8—41no—
139C2×C59⋊C8C22⋊C411yescharacteristic
140C2×C59⋊C8C22⋊C411yescharacteristic
141C2×C59⋊C8C22⋊C411yescharacteristic
142C2×C59⋊C8C22⋊C411yescharacteristic
143#11745659—42no—
144#11745659—42no—
145C23×C59—41no—
146C23×C59—41no—
147C23×C59C2×C4⋊C411yescharacteristic
148C23×C59#12111yescharacteristic
149C23×C59#12111yescharacteristic
150C23×C59AGL(1,ℤ/8)11yescharacteristic
151C23×C59AGL(1,ℤ/8)11yescharacteristic
152C23×C59#13711yescharacteristic
153C23×C59#13711yescharacteristic
154C22×D59—21no—
155C22×D59—21no—
156C22×D59—21no—
157C22×D59—41no—
158C22×D59—41no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—81no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59—81no—
173C2×C4×C59—81no—
174C2×C4×C59C2×C22⋊C411yescharacteristic
175C2×C4×C59#11711yescharacteristic
176C2×C4×C59C22⋊Q811yescharacteristic
177C2×C4×C59C22⋊Q811yescharacteristic
178C22⋊D59—41no—
179C22⋊D59—41no—
180C22⋊D59—41no—
181C22⋊D59—41no—
182C2×Dic59—21no—
183C2×Dic59—21no—
184C2×Dic59—21no—
185C2×Dic59—21no—
186C2×Dic59—21no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—81no—
190C2×Dic59—81no—
191C2×Dic59—81no—
192C2×Dic59—81no—
193C2×Dic59—81no—
194C2×Dic59—81no—
195C4×D59—42no—
196C4×D59—41no—
197C4×D59—41no—
198C8×C59—41no—
199C8×C59—41no—
200C59⋊C8—21no—
201C59⋊C8—21no—
202C59⋊C8—21no—
203C59⋊C8#9912yescomplemented
204#8610—591nomaximal, Sylow
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—41no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59—41no—
215C22×C59#38011yescharacteristic
216C22×C59#42811yescharacteristic
217C22×C59#42811yescharacteristic
218C22×C59#46811yescharacteristic
219C22×C59#46811yescharacteristic
220C22×C59#46911yescharacteristic
221C22×C59#47011yescharacteristic
222C2×D59—41no—
223C2×D59—41no—
224C2×D59—41no—
225C2×D59—81no—
226C2×D59—81no—
227C4×C59—21no—
228C4×C59—21no—
229C4×C59—21no—
230C4×C59—161no—
231C4×C59—161no—
232C4×C59C2×#9911yescharacteristic
233C4×C59#41011yescharacteristic
234Dic59—41no—
235Dic59—41no—
236Dic59—41no—
237Dic59—161no—
238Dic59—161no—
239C2×#435—591no—
240#1808—591no—
241#1807—591no—
242#1799—591no—
243#1799—591no—
244#1289—591no—
245#1289D5911yescharacteristic, complemented
246C2×C59—21no—
247C2×C59—21no—
248C2×C59—21no—
249C2×C59—21no—
250C2×C59—21no—
251C2×C59—21no—
252C2×C59#179911yescharacteristic
253C2×C59#180911yescharacteristic
254C2×C59#181211yescharacteristic
255D59—81no—
256C22×M16—591no—
257C23×C8—591no—
258C2×C4×D4—591no—
259#435—1181no—
260#435—1181no—
261#435—1181no—
262#435—1181no—
263C2×#107—1181no—
264C2×#106—1181no—
265C2×C4⋊C8—591no—
266C2×#99—1181no—
267C2×#99—1181no—
268C2×C22⋊C8—591no—
269C2×C22⋊C8C2×D5911yescharacteristic
270C2×#97—591no—
271#380—591no—
272#380—591no—
273#380C2×D5911yescharacteristic
274#380C2×D5911yescharacteristic
275#342—1181no—
276#342—1181no—
277#340—1181no—
278#340—1181no—
279C59#861011yescharacteristic, complemented, Sylow
280C22×D4—591no—
281C23×C4—591no—
282C23×C4C22×D5911yescharacteristic
283C2×M16—1181no—
284C2×M16—1181no—
285C2×M16—2362no—
286C22×C8—591no—
287C22×C8—591no—
288C22×C8—591no—
289C22×C8—591no—
290C22×C8—591no—
291C22×C8—591no—
292C22×C8—2362no—
293C22×C8C4×D5911yescharacteristic
294C22×C8C4×D5911yescharacteristic
295C4×D4—1181no—
296C4×D4—1181no—
297C4×D4—1181no—
298C4×D4—1181no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—591no—
302C2×C4⋊C4—1181no—
303C2×C4⋊C4—1181no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—591no—
307C2×C22⋊C4—591no—
308C2×C22⋊C4—1181no—
309C2×C22⋊C4—1181no—
310C2×C42—591no—
311#107—2362no—
312#106—2362no—
313C4⋊C8—1181no—
314C4⋊C8—1181no—
315#99—2362no—
316#99—2362no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—1181no—
320C22⋊C8—1181no—
321#97—1181no—
322#97—1181no—
323#95—21no—
324#95—21no—
325#95—1181no—
326#95—1181no—
327C24—591no—
328C24D4×D5911yescharacteristic
329C2×D4—1181no—
330C2×D4—1181no—
331C2×D4—1181no—
332C2×D4—1181no—
333C2×D4—1181no—
334C2×D4—1181no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—591no—
340C22×C4—591no—
341C22×C4—591no—
342C22×C4—1181no—
343C22×C4—1181no—
344C22×C4—2361no—
345C22×C4—2361no—
346C22×C4—2361no—
347C22×C4—2361no—
348C22×C4C2×C4×D5911yescharacteristic
349C22×C4#1174568511yescharacteristic
350C22×C4#1174568511yescharacteristic
351C22×C4D4×D5911yescharacteristic
352C22×C4D4×D5911yescharacteristic
353C22×C4Q8×D5911yescharacteristic
354M16—2362no—
355M16—2362no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—591no—
361C2×C8—591no—
362C2×C8—591no—
363C2×C8—591no—
364C2×C8—1182no—
365C2×C8—1182no—
366C2×C8—1182no—
367C2×C8—2361no—
368C2×C8—2361no—
369C2×C8—2361no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1181no—
375C4⋊C4—1181no—
376C4⋊C4—2361no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—1181no—
385C22⋊C4—1181no—
386C22⋊C4—1181no—
387C22⋊C4—1181no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C22⋊C4—2361no—
391C22⋊C4—2361no—
392C42—1181no—
393C42—1181no—
394C23—41no—
395C23—41no—
396C23—1181no—
397C23—1181no—
398C23—1181no—
399C23—2361no—
400C23—2361no—
401C23C4⋊C4×D5911yescharacteristic
402C23#1427370011yescharacteristic
403C23#1427370011yescharacteristic
404C23#1427375111yescharacteristic
405C23#1427375411yescharacteristic
406C23#1427375611yescharacteristic
407C23#1427375911yescharacteristic
408D4—2361no—
409D4—2361no—
410D4—2361no—
411D4—2361no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—1181no—
428C2×C4—1181no—
429C2×C4—1181no—
430C2×C4—1181no—
431C2×C4—1181no—
432C2×C4—4721no—
433C2×C4—4721no—
434C2×C4—4721no—
435C2×C4—4721no—
436C2×C4—4721no—
437C2×C4—4721no—
438C2×C4—2362no—
439C2×C4—2361no—
440C2×C4—2361no—
441C2×C4—2361no—
442C2×C4—2361no—
443C2×C4C22⋊C4×D5911yescharacteristic
444C2×C4#1427368411yescharacteristic
445C2×C4#1427370511yescharacteristic
446C2×C4#1427370511yescharacteristic
447C8—41no—
448C8—41no—
449C8—592no—
450C8—1181no—
451C8—1181no—
452C8—1181no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—4721no—
464C22—4721no—
465C22—2361no—
466C22—2361no—
467C22—2361no—
468C22#1683726111yescharacteristic
469C22#1683744111yescharacteristic
470C22#1683744311yescharacteristic
471C22#1683764711yescharacteristic
472C22#1683765111yescharacteristic
473C22#1683765411yescharacteristic
474C22#1683765911yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—161no—
479C4—161no—
480C4—9441no—
481C4—9441no—
482C4—2361no—
483C4—2361no—
484C4—2361no—
485C4#99×D5911yescharacteristic
486C4#1683737411yescharacteristic
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—4721no—
494C2#2142408311yescharacteristic
495C2#2142412811yescharacteristic
496C2#2142414111yescharacteristic
subgroupG/H
centerC22#16837261
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1289C2
socleC22×C59#380
prime radicalC22×C22⋊D59D4
cosocleC23×C4×C59C23
hypercenter#1289D59
hypocenterC59#8610
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21424123C2semidirect
3. #21424119C2semidirect
6. #21421964C2semidirect
7. C59×#1289C2semidirect
8. C22×#11745659C4semidirect
17. C2×Dic59⋊C8C4semidirect
20. C2×(C2×D59)⋊C8C4semidirect
21. C2×#14273637C4semidirect
203. C59⋊C8#99semidirect
245. #1289D59semidirect
279. C59#8610semidirect

Maximal subgroups

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

subgroupindex
C2×#168374702
#214241232
#214241192
#214240812
#214219642
C59×#12892
#861059