Finite Group Databasebeta

← #28745684

Subgroups — #28745684

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435839C211yesmaximal, characteristic, complemented
2#21426335C211yesmaximal, characteristic, complemented
3#21426298C211yesmaximal, characteristic, complemented
4#21426276C211yesmaximal, characteristic, complemented
5#21426245C211yesmaximal, characteristic, complemented
6#21426207C211yesmaximal, characteristic, complemented
7#21426180C211yesmaximal, characteristic, complemented
8#21426151C211yesmaximal, characteristic, complemented
9#21426017C211yesmaximal, characteristic, complemented
10#21426012C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425806C211yesmaximal, characteristic, complemented
14C2×#16837205C211yesmaximal, characteristic, complemented
15C59×#3334C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×Dic236C2211yescharacteristic
18#16838061—21no—
19C2×#14273671C2211yescharacteristic
20C2×#14273671C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22C2×#14273667C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837625—21no—
28#16837566—21no—
29#16837543—21no—
30C4×Dic236C2211yescharacteristic
31#16837227C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837205C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#547—21no—
49C59×#500C2211yescharacteristic
50C59×#480—21no—
51C59×#480—21no—
52C59×#474—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×#11745685—21no—
60C22×Dic118—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236—22no—
64C2×Dic236—21no—
65C2×Dic236—41no—
66C2×Dic236C2311yescharacteristic
67C2×Dic236C2311yescharacteristic
68C2×#11745660—21no—
69C2×#11745660—21no—
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273713—41no—
73#14273706—21no—
74#14273705—41no—
75#14273701—21no—
76#14273699—41no—
77C4×Dic118—21no—
78(C22⋊D59)⋊C4—41no—
79#14273671—22no—
80#14273671—41no—
81#14273668—41no—
82#14273667—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273652—21no—
91#14273649—21no—
92C2×#41×C59—41no—
93C2×D8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95C22×C8×C59C2311yescharacteristic
96C4⋊Q8×C59—21no—
97#123×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—41no—
100#121×C59—21no—
101#121×C59—21no—
102C4×D4×C59—21no—
103C4×D4×C59D411yescharacteristic
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59—21no—
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119#41×C59—84no—
120C2×Q8×C59—41no—
121C2×D4×C59—21no—
122C2×D4×C59—41no—
123C2×D4×C59—41no—
124C2×D4×C59D811yescharacteristic
125C22×C4×C59—41no—
126C22×C4×C59—41no—
127C22×C4×C59—41no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C2×C22⋊D59—41no—
131C2×C22⋊D59—41no—
132C22×Dic59—41no—
133#11745685—42no—
134#11745685—81no—
135C2×D236—21no—
136C2×C4×D59—41no—
137C2×Dic118—21no—
138C2×Dic118—21no—
139C2×Dic118—42no—
140C2×Dic118—41no—
141C2×Dic118—81no—
142D8×C59—21no—
143D8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2411yescharacteristic
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59D811yescharacteristic
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59—41no—
162C22⋊C4×C59D811yescharacteristic
163C22⋊C4×C59D811yescharacteristic
164C42×C59C2×D411yescharacteristic
165C22⋊Dic59—41no—
166(C2×D59)⋊C4—41no—
167(C2×D59)⋊C4—41no—
168(C2×D59)⋊C4—41no—
169C4⋊Dic59—21no—
170Dic59⋊C4—41no—
171Dic59⋊C4—41no—
172C4×Dic59—41no—
173Dic236—21no—
174Dic236—21no—
175Dic236—21no—
176Dic236—42no—
177Dic236#4112yescomplemented
178D472—21no—
179D472—21no—
180#11745660—42no—
181#11745660—81no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×D811yescharacteristic
189C22×D59—41no—
190Q8×C59—82no—
191D4×C59—22no—
192D4×C59—41no—
193D4×C59—41no—
194D4×C59—82no—
195D4×C59—82no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—41no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—82no—
204C2×C4×C59—82no—
205C2×C4×C59—82no—
206C2×C4×C59C2×D811yescharacteristic
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C22×D411yescharacteristic
211C22⋊D59—82no—
212C22⋊D59—82no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—82no—
217D236—41no—
218D236—41no—
219C4×D59—82no—
220Dic118—22no—
221Dic118—41no—
222Dic118—41no—
223Dic118—41no—
224Dic118—82no—
225C8×C59—21no—
226C8×C59—21no—
227C8×C59—21no—
228C8×C59C2×#4111yescharacteristic
229C8×C59H(5,𝔽2)11yescharacteristic
230#33875—591nomaximal, Sylow
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59#50912yes—
240C22×C59C22×D811yescharacteristic
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—81no—
244C4×C59—21no—
245C4×C59—21no—
246C4×C59—21no—
247C4×C59—21no—
248C4×C59—81no—
249C4×C59—81no—
250C4×C59—81no—
251C4×C59#54611yescharacteristic
252C4×C59#57711yescharacteristic
253Dic59—42no—
254Dic59—81no—
255Dic59—81no—
256#3334D5911yescharacteristic, complemented
257#3285—591no—
258#2243—591no—
259#2236—591no—
260#2232—591no—
261#2227—591no—
262#2220—591no—
263#2215—591no—
264#2210—591no—
265#2186—591no—
266#2185—591no—
267#2175—591no—
268#2168—591no—
269#2139—591no—
270C2×#358—591no—
271C2×C59—22no—
272C2×C59—41no—
273C2×C59—161no—
274C2×C59#329411yescharacteristic
275C2×C59C2×#50911yescharacteristic
276C2×C59#341711yescharacteristic
277D59—161no—
278C2×#135—591no—
279C22×Q16—591no—
280#548—1181no—
281#547—21no—
282C2×SD32—591no—
283C2×SD32—591no—
284C2×D16—591no—
285C2×M32—591no—
286C22×C16—591no—
287#500C2×D5911yescharacteristic
288#495—591no—
289#480—21no—
290#480—21no—
291#474—21no—
292#467—591no—
293#466—591no—
294#466C2×D5911yescharacteristic
295#466C2×D5911yescharacteristic
296#464—1181no—
297#452—1181no—
298#448—1181no—
299C4×Q16—591no—
300C4×D8C2×D5911yescharacteristic
301C8×D4C2×D5911yescharacteristic
302C2×#107C2×D5911yescharacteristic
303C2×#107C2×D5911yescharacteristic
304#366—591no—
305#366—591no—
306#366—591no—
307C4⋊C16—591no—
308#358—591no—
309#358—591no—
310#358—591no—
311#358—591no—
312#358—591no—
313#358—591no—
314#357—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318C22⋊C16—591no—
319C22⋊C16—591no—
320#346—591no—
321C59#3387511yescharacteristic, complemented, Sylow
322C2×#41—41no—
323C2×#41—1181no—
324C22×Q8—1181no—
325#135—1182no—
326#135—2361no—
327C2×Q16—591no—
328C2×Q16—591no—
329C2×Q16—1182no—
330C2×Q16—1181no—
331C2×Q16—2361no—
332C2×SD16—1181no—
333C2×SD16—1181no—
334C2×D8—591no—
335C2×D8C22×D5911yescharacteristic
336C22×C8C22×D5911yescharacteristic
337C22×C8C22×D5911yescharacteristic
338C4⋊Q8—21no—
339#124—1181no—
340#124—2361no—
341#123—41no—
342#123—41no—
343C22⋊Q8—41no—
344C22⋊Q8—1181no—
345C22⋊Q8—2361no—
346#121—21no—
347#121—21no—
348#121—1181no—
349#121—2361no—
350C4×Q8—1181no—
351C4×D4—21no—
352C4×D4—2361no—
353C4×D4C22⋊D5911yescharacteristic
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356SD32—1182no—
357SD32—2361no—
358D16—2361no—
359M32—2361no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—591no—
363C2×C16—591no—
364C2×C16—2361no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—591no—
369#107C22×D5911yescharacteristic
370#107C22×D5911yescharacteristic
371#107C22×D5911yescharacteristic
372C4⋊C8C22⋊D5911yescharacteristic
373#103—1181no—
374#103—1181no—
375#102—21no—
376#102—21no—
377#102—21no—
378#102—1181no—
379C22⋊C8C22⋊D5911yescharacteristic
380C22⋊C8C22⋊D5911yescharacteristic
381C4×C8C22×D5911yescharacteristic
382#41—84no—
383#41—4721no—
384#41—2362no—
385C2×Q8—41no—
386C2×Q8—1181no—
387C2×Q8—1181no—
388C2×Q8—4721no—
389C2×Q8—2362no—
390C2×Q8—2361no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—1181no—
395C2×D4—2361no—
396C2×D4—2361no—
397C2×D4#1174566911yescharacteristic
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—2361no—
402C22×C4—2361no—
403C22×C4C2×C22⋊D5911yescharacteristic
404C22×C4C2×C22⋊D5911yescharacteristic
405Q16—592no—
406Q16—1181no—
407Q16—1181no—
408Q16—1181no—
409Q16—2362no—
410SD16—4721no—
411SD16—2362no—
412D8—21no—
413D8—21no—
414D8—1181no—
415D8—1181no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C2×C22⋊D5911yescharacteristic
421C2×C8C2×C22⋊D5911yescharacteristic
422C2×C8C2×C22⋊D5911yescharacteristic
423C2×C8C23×D5911yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1181no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C4⋊C4#1174566911yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C22⋊C4#1174566911yescharacteristic
442C22⋊C4#1174566911yescharacteristic
443C42—2361no—
444C42C2×C22⋊D5911yescharacteristic
445C16—1181no—
446C16—1181no—
447C16—1181no—
448C16—1181no—
449C23—41no—
450C23—2361no—
451C23C2×#1174566911yescharacteristic
452C23C2×#1174566911yescharacteristic
453Q8—82no—
454Q8—1182no—
455Q8—4722no—
456Q8—2361no—
457Q8—2361no—
458Q8—2361no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—4722no—
465D4—4722no—
466D4—2361no—
467D4—2361no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—4722no—
479C2×C4—4722no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4—2361no—
483C2×C4C2×#1174566911yescharacteristic
484C2×C4C2×#1174566911yescharacteristic
485C2×C4C2×#1174566911yescharacteristic
486C2×C4C2×#1174566911yescharacteristic
487C2×C4C22×C22⋊D5911yescharacteristic
488C8—21no—
489C8—21no—
490C8—21no—
491C8#41×D5911yescharacteristic
492C8#1427377311yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4721no—
502C22—4721no—
503C22—4721no—
504C22#1683783812yes—
505C22C22×#1174566911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—4721no—
514C4—4721no—
515C4—2362no—
516C4#1683804011yescharacteristic
517C4#1683821011yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—9441no—
522C2#2143590311yescharacteristic
523C2C2×#1683783811yescharacteristic
524C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3334C2
socleC22×C59C22×D8
hypercenter#3334D59
hypocenterC59#33875
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #21435839C2semidirect
2. #21426335C2semidirect
3. #21426298C2semidirect
4. #21426276C2semidirect
5. #21426245C2semidirect
6. #21426207C2semidirect
7. #21426180C2semidirect
8. #21426151C2semidirect
9. #21426017C2semidirect
10. #21426012C2semidirect
11. #21425960C2semidirect
12. #21425922C2semidirect
13. #21425806C2semidirect
14. C2×#16837205C2semidirect
15. C59×#3334C2semidirect
177. Dic236#41semidirect
256. #3334D59semidirect
321. C59#33875semidirect

Maximal subgroups

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

subgroupindex
#214358392
#214263352
#214262982
#214262762
#214262452
#214262072
#214261802
#214261512
#214260172
#214260122
#214259602
#214259222
#214258062
C2×#168372052
C59×#33342
#3387559