Finite Group Databasebeta

← #28743859

Subgroups — #28743859

Order 15104.

556 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434748C211yesmaximal, characteristic, complemented
2#21432948C211yesmaximal, characteristic
3#21426305C211yesmaximal, characteristic, complemented
4#21426301C211yesmaximal, characteristic, complemented
5#21426255C211yesmaximal, characteristic, complemented
6#21426251C211yesmaximal, characteristic, complemented
7#21426139C211yesmaximal, characteristic, complemented
8#21426139C211yesmaximal, characteristic, complemented
9#21426025C211yesmaximal, characteristic
10#21425961C211yesmaximal, characteristic, complemented
11#21425957C211yesmaximal, characteristic, complemented
12#21425951C211yesmaximal, characteristic, complemented
13SD32×Dic59C211yesmaximal, characteristic, complemented
14#21425900C211yesmaximal, characteristic, complemented
15C59×#2187C211yesmaximal, characteristic
16#16838012—21no—
17#16837877—21no—
18C2×#14273671C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20#16837803C2211yescharacteristic, complemented
21#16837780C2211yescharacteristic, complemented
22#16837769—21no—
23#16837658—21no—
24#16837643C2211yescharacteristic
25#16837514—21no—
26#16837514—21no—
27#16837506—21no—
28Q16×Dic59C2211yescharacteristic, complemented
29#16837496C2211yescharacteristic, complemented
30D8×Dic59C2211yescharacteristic
31D8×Dic59C2211yescharacteristic
32D8×Dic59C2211yescharacteristic
33D8×Dic59C2211yescharacteristic
34#16837440C2211yescharacteristic
35#16837407—21no—
36#16837357C2211yescharacteristic
37#16837230C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837228C2211yescharacteristic, complemented
40#16837227C2211yescharacteristic, complemented
41Dic59⋊C16C2211yescharacteristic, complemented
42Dic59⋊C16C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837206C2211yescharacteristic, complemented
46#16837206C2211yescharacteristic, complemented
47#16837203C2211yescharacteristic
48#16837203C2211yescharacteristic
49C22⋊(C59⋊C16)C2211yescharacteristic
50#16837177C2211yescharacteristic, complemented
51C16×Dic59C2211yescharacteristic, complemented
52C22×D8×C59C2211yescharacteristic
53C2×SD32×C59C2211yescharacteristic
54C2×SD32×C59C2211yescharacteristic
55C59×#467C2211yescharacteristic
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×#11745672—21no—
61C2×#11745664D411yescharacteristic
62#14273725—21no—
63#14273720—41no—
64#14273720—41no—
65#14273718—21no—
66#14273718—41no—
67#14273717—41no—
68#14273706—21no—
69Q8×Dic59—21no—
70#14273692—21no—
71D4×Dic59—21no—
72D4×Dic59—21no—
73D4×Dic59—21no—
74D4×Dic59—21no—
75D4×Dic59—41no—
76D4×Dic59—41no—
77#14273686—21no—
78#14273686D411yescharacteristic
79C2×C22⋊Dic59—41no—
80C2×C22⋊Dic59—41no—
81#14273671—22no—
82#14273671—22no—
83C2×C59⋊C16C2311yescharacteristic
84C2×C59⋊C16C2311yescharacteristic
85#14273661C2311yescharacteristic
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273659—21no—
89#14273658—21no—
90#14273652—21no—
91#14273651—21no—
92#14273650—21no—
93#14273648—21no—
94#14273648—21no—
95#14273648—21no—
96#14273648—21no—
97#14273637D411yescharacteristic
98#14273637D411yescharacteristic
99C8×Dic59C2311yescharacteristic
100C8×Dic59C2311yescharacteristic
101C22×D4×C59—21no—
102C2×Q16×C59C2311yescharacteristic
103C2×D8×C59—22no—
104C2×D8×C59—22no—
105C2×D8×C59—21no—
106C2×D8×C59C2311yescharacteristic
107C2×D8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22⋊Q8×C59—21no—
110SD32×C59—22no—
111SD32×C59—22no—
112C2×C16×C59C2311yescharacteristic
113C2×C16×C59C2311yescharacteristic
114#107×C59C2311yescharacteristic
115#103×C59—21no—
116C24×C59—41no—
117C2×Q8×C59—21no—
118C2×D4×C59—24no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—42no—
122C2×D4×C59—41no—
123C22×C4×C59C2×D411yescharacteristic
124C22×Dic59—41no—
125C22×Dic59—41no—
126C22×Dic59—41no—
127C22×Dic59—41no—
128C2×Dic118—21no—
129Q16×C59—21no—
130Q16×C59—21no—
131D8×C59—24no—
132D8×C59—21no—
133D8×C59—21no—
134D8×C59#4112yes—
135D8×C59#4112yes—
136C2×C8×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59C2×D411yescharacteristic
139C2×C8×C59C2411yescharacteristic
140C4⋊C4×C59—21no—
141C4⋊C4×C59—41no—
142C22⋊C4×C59—41no—
143C22⋊Dic59—41no—
144C22⋊Dic59—41no—
145C22⋊Dic59—41no—
146C22⋊Dic59—41no—
147C22⋊Dic59—41no—
148C22⋊Dic59—41no—
149C22⋊Dic59D811yescharacteristic
150C22⋊Dic59D811yescharacteristic
151#11745672—42no—
152C4⋊Dic59—21no—
153C4⋊Dic59—21no—
154C4⋊Dic59—21no—
155C4⋊Dic59—41no—
156C4⋊Dic59—41no—
157C4⋊Dic59D811yescharacteristic
158C4⋊Dic59D811yescharacteristic
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164C4×Dic59—41no—
165C4×Dic59—41no—
166C4×Dic59—41no—
167C4×Dic59C2×D411yescharacteristic
168C4×Dic59C2×D411yescharacteristic
169#11745664—41no—
170C2×C59⋊C8C2×D411yescharacteristic
171C2×C59⋊C8C2×D411yescharacteristic
172Dic236—21no—
173Dic236—21no—
174C16×C59—21no—
175C16×C59—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—44no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59—161no—
182C23×C59—161no—
183C23×C59C2×D811yescharacteristic
184Q8×C59—41no—
185Q8×C59—41no—
186D4×C59—22no—
187D4×C59—22no—
188D4×C59—44no—
189D4×C59—41no—
190D4×C59—41no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—41no—
194C2×C4×C59—41no—
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C22×D411yescharacteristic
197C2×Dic59—41no—
198C2×Dic59—41no—
199C2×Dic59—41no—
200C2×Dic59—41no—
201C2×Dic59—161no—
202C2×Dic59—161no—
203C2×Dic59—161no—
204C2×Dic59—161no—
205C2×Dic59C2×D811yescharacteristic
206C2×Dic59C2×D811yescharacteristic
207C2×Dic59C2×D811yescharacteristic
208C2×Dic59C2×D811yescharacteristic
209Dic118—41no—
210Dic118—41no—
211C8×C59—21no—
212C8×C59C2×#4111yescharacteristic
213C8×C59C2×#4111yescharacteristic
214C59⋊C8—21no—
215C59⋊C8—21no—
216#33683—591nomaximal, Sylow
217C22×C59—21no—
218C22×C59—21no—
219C22×C59—21no—
220C22×C59—42no—
221C22×C59—42no—
222C22×C59—81no—
223C22×C59—81no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59—164no—
227C22×C59C22×D811yescharacteristic
228C4×C59—21no—
229C4×C59—81no—
230C4×C59—81no—
231C4×C59#52511yescharacteristic
232C4×C59#57511yescharacteristic
233Dic59—21no—
234Dic59—21no—
235Dic59—21no—
236Dic59—21no—
237Dic59—81no—
238Dic59—81no—
239Dic59—81no—
240Dic59—81no—
241#3177—591no—
242#2948—591no—
243#2237—591no—
244#2237—591no—
245#2228—591no—
246#2228—591no—
247#2208—591no—
248#2208—591no—
249#2187—591no—
250#2187D5911yescharacteristic
251#2175—591no—
252#2175—591no—
253C4×SD32—591no—
254C4×SD32—591no—
255#2164—591no—
256C2×C59—41no—
257C2×C59—82no—
258C2×C59—82no—
259C2×C59#310011yescharacteristic
260C2×C59#341611yescharacteristic
261C2×C59#341611yescharacteristic
262C22×D8C2×D5911yescharacteristic
263#541—1181no—
264#518—1181no—
265C2×SD32—591no—
266C2×SD32—591no—
267C2×SD32C2×D5911yescharacteristic
268C2×SD32C2×D5911yescharacteristic
269#499—591no—
270#494—591no—
271#491—1181no—
272#470—1181no—
273#467—591no—
274#467C2×D5911yescharacteristic
275#442—1181no—
276#442—1181no—
277#441—1181no—
278C4×Q16—591no—
279C4×Q16—591no—
280C4×D8—591no—
281C4×D8—591no—
282C4×D8—591no—
283C4×D8—591no—
284#427—591no—
285#418—1181no—
286#405—591no—
287#367—591no—
288#367—591no—
289#366—591no—
290#366—591no—
291C4⋊C16—591no—
292C4⋊C16—591no—
293#358—591no—
294#358—591no—
295#358—591no—
296#358—591no—
297#357—591no—
298#357—591no—
299#357C2×D5911yescharacteristic
300#357C2×D5911yescharacteristic
301C22⋊C16—591no—
302C22⋊C16C2×D5911yescharacteristic
303#346—591no—
304C4×C16—591no—
305C59#3368311yescharacteristic, complemented, Sylow
306C22×D4—21no—
307C2×Q16—591no—
308C2×Q16—1181no—
309C2×Q16C22×D5911yescharacteristic
310C2×D8—22no—
311C2×D8—22no—
312C2×D8—21no—
313C2×D8C22×D5911yescharacteristic
314C2×D8C22×D5911yescharacteristic
315C2×M16—591no—
316C22×C8C22×D5911yescharacteristic
317C4⋊Q8—1181no—
318#126—2361no—
319#126—2361no—
320#125—1181no—
321#125—2361no—
322#125—2361no—
323C22⋊Q8—21no—
324C22⋊Q8—1181no—
325C4×Q8—1181no—
326C4×Q8—1181no—
327C4×D4—1181no—
328C4×D4—1181no—
329C4×D4—1181no—
330C4×D4—1181no—
331C4×D4—2361no—
332C4×D4—2361no—
333#117—591no—
334#117—1181no—
335C2×C22⋊C4—2361no—
336C2×C22⋊C4—2361no—
337SD32—22no—
338SD32—22no—
339SD32—1182no—
340SD32—1182no—
341C2×C16—591no—
342C2×C16—591no—
343C2×C16C22×D5911yescharacteristic
344C2×C16C22×D5911yescharacteristic
345#107—591no—
346#107—591no—
347#107—591no—
348#107C22×D5911yescharacteristic
349#106—1181no—
350#106—1181no—
351#103—21no—
352#103—1181no—
353#103—1181no—
354#103—1181no—
355#102—1181no—
356#102—1181no—
357#102—1181no—
358#102—1181no—
359#97—591no—
360#97—591no—
361C4×C8—591no—
362C4×C8—591no—
363C24—41no—
364C2×Q8—21no—
365C2×Q8—1181no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—42no—
370C2×D4—41no—
371C22×C4—2361no—
372C22×C4—2361no—
373C22×C4—2361no—
374C22×C4—2361no—
375C22×C4D4×D5911yescharacteristic
376Q16—21no—
377Q16—21no—
378Q16—1181no—
379Q16—1181no—
380Q16—2362no—
381D8—24no—
382D8—21no—
383D8—21no—
384D8#1174568712yes—
385D8#1174568712yes—
386M16—2361no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—591no—
390C2×C8—591no—
391C2×C8D4×D5911yescharacteristic
392C2×C8C23×D5911yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—41no—
395C4⋊C4—591no—
396C4⋊C4—591no—
397C4⋊C4—1181no—
398C4⋊C4—1181no—
399C4⋊C4—1181no—
400C4⋊C4—2361no—
401C4⋊C4—2361no—
402C4⋊C4—2361no—
403C4⋊C4—2361no—
404C4⋊C4—2361no—
405C4⋊C4—2361no—
406C4⋊C4—2361no—
407C22⋊C4—41no—
408C22⋊C4—591no—
409C22⋊C4—591no—
410C22⋊C4—2361no—
411C22⋊C4—2361no—
412C22⋊C4—2361no—
413C22⋊C4—2361no—
414C22⋊C4—2361no—
415C22⋊C4—2361no—
416C42—591no—
417C42—591no—
418C42—2361no—
419C42—2361no—
420C42—2361no—
421C16—21no—
422C16—21no—
423C16—1181no—
424C16—1181no—
425C23—44no—
426C23—41no—
427C23—41no—
428C23—161no—
429C23—161no—
430C23D8×D5911yescharacteristic
431Q8—41no—
432Q8—41no—
433Q8—2361no—
434Q8—2361no—
435D4—22no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—41no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—591no—
445C2×C4—591no—
446C2×C4—591no—
447C2×C4—591no—
448C2×C4—9441no—
449C2×C4—9441no—
450C2×C4—9441no—
451C2×C4—9441no—
452C2×C4—2361no—
453C2×C4—2361no—
454C2×C4—2361no—
455C2×C4—2361no—
456C2×C4D8×D5911yescharacteristic
457C2×C4C2×D4×D5911yescharacteristic
458C8—21no—
459C8—1181no—
460C8—1181no—
461C8C2×#1174568711yescharacteristic
462C8C2×#1174568711yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—164no—
473C22C2×D8×D5911yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—1181no—
478C4—1181no—
479C4—1181no—
480C4—1181no—
481C4—4721no—
482C4—4721no—
483C4—4721no—
484C4—4721no—
485C4#1683790811yescharacteristic
486C4#1683819811yescharacteristic
487C2—41no—
488C2—82no—
489C2—82no—
490C2#2143409111yescharacteristic
491C2#2143723811yescharacteristic
492C2#2143724211yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2187C2
socleC22×C59C22×D8
prime radicalC22×D8×C59C22
hypercenter#2187D59
hypocenterC59#33683
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #21434748C2semidirect
3. #21426305C2semidirect
4. #21426301C2semidirect
5. #21426255C2semidirect
6. #21426251C2semidirect
7. #21426139C2semidirect
8. #21426139C2semidirect
10. #21425961C2semidirect
11. #21425957C2semidirect
12. #21425951C2semidirect
13. SD32×Dic59C2semidirect
14. #21425900C2semidirect
20. #16837803C22semidirect
21. #16837780C22semidirect
28. Q16×Dic59C22semidirect
29. #16837496C22semidirect
37. #16837230C22semidirect
38. #16837229C22semidirect
39. #16837228C22semidirect
40. #16837227C22semidirect
41. Dic59⋊C16C22semidirect
42. Dic59⋊C16C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
45. #16837206C22semidirect
46. #16837206C22semidirect
50. #16837177C22semidirect
51. C16×Dic59C22semidirect
305. C59#33683semidirect

Maximal subgroups

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

subgroupindex
#214347482
#214329482
#214263052
#214263012
#214262552
#214262512
#214261392
#214260252
#214259612
#214259572
#214259512
SD32×Dic592
#214259002
C59×#21872
#3368359