Finite Group Databasebeta

← #28742039

Subgroups — #28742039

Order 15104.

660 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×#16837841C211yesmaximal, characteristic
2#21432966C211yesmaximal, characteristic
3#21432959C211yesmaximal, characteristic, complemented
4#21425968C211yesmaximal, characteristic, complemented
5#21425964C211yesmaximal, characteristic, complemented
6#21425957C211yesmaximal, characteristic, complemented
7#21425954C211yesmaximal, characteristic, complemented
8C4×#14273673C211yesmaximal, characteristic, complemented
9SD32×Dic59C211yesmaximal, characteristic, complemented
10C4×#14273671C211yesmaximal, characteristic, complemented
11#21425881C211yesmaximal, characteristic, complemented
12#21425800C211yesmaximal, characteristic
13#358×D59C211yesmaximal, characteristic, complemented
14#21425694C211yesmaximal, characteristic, complemented
15C59×#2175C211yesmaximal, characteristic, complemented
16C2×#14273746C2211yescharacteristic
17C2×Q16×D59C2211yescharacteristic
18#16837881—21no—
19#16837878—21no—
20#16837841C418yescomplemented
21C2×Dic472C2211yescharacteristic
22C2×#14273673C2211yescharacteristic
23C2×#14273671C2211yescharacteristic
24C2×#14273666C2211yescharacteristic
25C2×C16×D59C2211yescharacteristic
26#16837507—21no—
27#16837506—21no—
28#16837502—21no—
29#16837499—21no—
30C4×Dic236C2211yescharacteristic
31C4×Dic236C2211yescharacteristic
32Q16×Dic59C2211yescharacteristic
33#16837496C2211yescharacteristic
34#16837496C2211yescharacteristic
35D8×Dic59C2211yescharacteristic
36#16837439C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837409—21no—
39#16837400—21no—
40#16837356C2211yescharacteristic
41#16837230C2211yescharacteristic
42#16837229C2211yescharacteristic
43#16837227C2211yescharacteristic
44#16837207C2211yescharacteristic
45#16837207C2211yescharacteristic
46#16837206C2211yescharacteristic
47#16837205C2211yescharacteristic
48#16837205C2211yescharacteristic
49#16837203C2211yescharacteristic
50#16837177C2211yescharacteristic
51C4×C59⋊C16C2211yescharacteristic
52C16×Dic59C2211yescharacteristic
53C2×SD32×C59C2211yescharacteristic
54C4×Q16×C59C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C59×#366C2211yescharacteristic
57C59×#358C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C59×#346C2211yescharacteristic
60C2×#11745687—21no—
61C2×Q8×D59—21no—
62#14273746—22no—
63#14273746C2×C414yes—
64C2×Dic236C2311yescharacteristic
65C2×Dic236C2311yescharacteristic
66C2×#11745672—21no—
67Q16×D59—22no—
68Q16×D59C2×C414yescomplemented
69C2×#11745670—21no—
70C2×#11745659D411yescharacteristic
71C2×C8×D59C2311yescharacteristic
72Q8×Dic59—21no—
73C4×Dic118—21no—
74C4×Dic118—21no—
75C4×Dic118—41no—
76#14273692—21no—
77#14273692—21no—
78#14273692—41no—
79D4×Dic59—21no—
80C4×C22⋊D59—41no—
81(C22⋊D59)⋊C4—41no—
82#14273685—21no—
83#14273685—41no—
84#14273684—41no—
85#14273684D411yescharacteristic
86#14273683—41no—
87C2×Dic59⋊C4—41no—
88C4⋊C4×D59—21no—
89Dic472C2×C414yescomplemented
90#14273673C2×C414yescomplemented
91#14273671C2×C414yescomplemented
92#14273666C2×C414yescomplemented
93C2×C59⋊C16C2311yescharacteristic
94C2×C59⋊C16C2311yescharacteristic
95C16×D59—22no—
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273660—21no—
99#14273658—21no—
100#14273652—21no—
101#14273652—21no—
102#14273651—21no—
103#14273650—21no—
104#14273650—21no—
105#14273648—21no—
106#14273638D411yescharacteristic
107#14273637D411yescharacteristic
108C8×Dic59C2311yescharacteristic
109C2×Q16×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C4×Q8×C59—21no—
112C4×D4×C59—21no—
113SD32×C59C2×C414yescomplemented
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#103×C59—21no—
119#102×C59—21no—
120C4×C8×C59C2311yescharacteristic
121C2×Q8×C59—21no—
122C2×D4×C59—21no—
123C22×C4×C59—41no—
124C2×C22⋊D59—41no—
125C22×Dic59—41no—
126C22×Dic59—41no—
127Q8×D59—24no—
128Q8×D59—42no—
129#11745687—24no—
130#11745687—42no—
131C2×C4×D59—41no—
132C2×C4×D59—41no—
133C2×C4×D59C2×D411yescharacteristic
134C2×Dic118—21no—
135C2×Dic118—21no—
136C2×Dic118—41no—
137Q16×C59—21no—
138Q16×C59C22×C412yes—
139D8×C59—21no—
140D8×C59C22×C412yes—
141C2×C8×C59C2×D411yescharacteristic
142C2×C8×C59C2411yescharacteristic
143C4⋊C4×C59—21no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—41no—
146C22⋊C4×C59—41no—
147C42×C59—41no—
148C42×C59C2×D411yescharacteristic
149C22⋊Dic59—41no—
150#11745672—24no—
151#11745670—24no—
152(C2×D59)⋊C4—41no—
153(C2×D59)⋊C4D811yescharacteristic
154(C2×D59)⋊C4D811yescharacteristic
155C4⋊Dic59—21no—
156C4⋊Dic59—21no—
157C4⋊Dic59—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4D811yescharacteristic
164Dic59⋊C4D811yescharacteristic
165C4×Dic59—41no—
166C4×Dic59—41no—
167C4×Dic59—41no—
168C4×Dic59—41no—
169C4×Dic59—41no—
170C4×Dic59—41no—
171C4×Dic59C2×D411yescharacteristic
172C2×C59⋊C8C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174Dic236—21no—
175Dic236—21no—
176Dic236C22×C412yes—
177Dic236C22×C412yes—
178#11745659—22no—
179C8×D59C22×C412yes—
180C8×D59#4112yes—
181C16×C59—21no—
182C16×C59C22×C412yes—
183C59⋊C16—21no—
184C59⋊C16C22×C412yes—
185C23×C59—41no—
186C22×D59C2×D811yescharacteristic
187Q8×C59—22no—
188Q8×C59—41no—
189D4×C59—22no—
190D4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—41no—
194C2×C4×C59—161no—
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C22×D411yescharacteristic
198C22⋊D59—44no—
199C2×Dic59—41no—
200C2×Dic59—41no—
201C2×Dic59—44no—
202C2×Dic59—41no—
203C2×Dic59—41no—
204C2×Dic59—161no—
205C2×Dic59C2×D811yescharacteristic
206C2×Dic59C2×D811yescharacteristic
207C2×Dic59C2×D811yescharacteristic
208C4×D59—44no—
209C4×D59—161no—
210C4×D59C4×D412yes—
211C4×D59#13512yes—
212Dic118—22no—
213Dic118—22no—
214Dic118—41no—
215Dic118—41no—
216Dic118—44no—
217C8×C59—21no—
218C8×C59C23×C411yescharacteristic
219C8×C59C2×#4111yescharacteristic
220C59⋊C8—21no—
221C59⋊C8C4×D412yes—
222#33455—591nomaximal, Sylow
223C22×C59—42no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59C22×D811yescharacteristic
227C2×D59—21no—
228C2×D59—21no—
229C2×D59C4×D812yes—
230C4×C59—21no—
231C4×C59—21no—
232C4×C59—42no—
233C4×C59—81no—
234C4×C59—81no—
235C4×C59C2×C4×D411yescharacteristic
236C4×C59C2×#13511yescharacteristic
237Dic59—21no—
238Dic59—21no—
239Dic59—42no—
240Dic59—42no—
241Dic59—81no—
242Dic59—81no—
243Dic59C4×D812yes—
244C2×#510—591no—
245#2951—591no—
246#2950—591no—
247#2176—591no—
248#2176—591no—
249#2175—591no—
250#2175D5911yescharacteristic, complemented
251C4×Q32—591no—
252C4×Q32—591no—
253C4×SD32—591no—
254C4×SD32—591no—
255#2159—591no—
256#2138—591no—
257C2×#358—591no—
258#2108—591no—
259C2×C59—82no—
260C2×C59C2×C4×D811yescharacteristic
261C2×C59#341611yescharacteristic
262C2×C59#341711yescharacteristic
263D59—22no—
264C2×#135—591no—
265C22×Q16—591no—
266#520—1181no—
267#519—1181no—
268#510—598no—
269C2×Q32—591no—
270C2×Q32—591no—
271C2×SD32—591no—
272C2×SD32C2×D5911yescharacteristic
273C2×M32—591no—
274C22×C16—591no—
275#441—1181no—
276#441—1181no—
277#440—1181no—
278#440—1181no—
279C4×Q16—591no—
280C4×Q16—591no—
281C4×Q16—591no—
282C4×Q16—591no—
283C4×Q16—591no—
284C4×Q16C2×D5911yescharacteristic
285C4×D8—591no—
286C4×D8C2×D5911yescharacteristic
287#427—591no—
288C2×#107—591no—
289#419—1181no—
290#417—1181no—
291#405—591no—
292#367—591no—
293#367—591no—
294#366—591no—
295#366C2×D5911yescharacteristic
296#358—591no—
297#358—591no—
298#358—591no—
299#358—591no—
300#358—591no—
301#358C2×D5911yescharacteristic
302#357—591no—
303#357C2×D5911yescharacteristic
304#346—591no—
305#346C2×D5911yescharacteristic
306C4×C16—591no—
307C4×C16—591no—
308C59#3345511yescharacteristic, complemented, Sylow
309C2×#41—1181no—
310C22×Q8—1181no—
311#135—594no—
312#135—1182no—
313C2×Q16—594no—
314C2×Q16—591no—
315C2×Q16—591no—
316C2×Q16—1182no—
317C2×Q16—1181no—
318C2×Q16C22×D5911yescharacteristic
319C2×SD16—1181no—
320C2×D8C22×D5911yescharacteristic
321C2×M16—591no—
322C22×C8—591no—
323C4×Q8—21no—
324C4×Q8—1181no—
325C4×Q8—1181no—
326C4×Q8—1181no—
327C4×Q8—1181no—
328C4×Q8—1181no—
329C4×Q8—2361no—
330C4×Q8—2361no—
331C4×D4—21no—
332C4×D4—1181no—
333C4×D4—2361no—
334C4×D4—2361no—
335#117—591no—
336#117—1181no—
337#117—2361no—
338#117—2361no—
339#117—2361no—
340C2×C4⋊C4—1181no—
341C2×C4⋊C4—2361no—
342Q32—594no—
343Q32—594no—
344SD32—594no—
345SD32C4×D5914yescomplemented
346M32—594no—
347C2×C16—591no—
348C2×C16—591no—
349C2×C16—1182no—
350C2×C16C22×D5911yescharacteristic
351C2×C16C22×D5911yescharacteristic
352#107—591no—
353#107—591no—
354#107—1181no—
355#107C22×D5911yescharacteristic
356#107C22×D5911yescharacteristic
357#106—1181no—
358#103—21no—
359#103—1181no—
360#103—1181no—
361#103—1181no—
362#103—1181no—
363#103—1181no—
364#102—21no—
365#102—1181no—
366#97—591no—
367#97—591no—
368C4×C8—591no—
369C4×C8C22×D5911yescharacteristic
370#41—1184no—
371#41—2362no—
372C2×Q8—21no—
373C2×Q8—1184no—
374C2×Q8—1181no—
375C2×Q8—1181no—
376C2×Q8—2361no—
377C2×Q8—2362no—
378C2×D4—21no—
379C2×D4—2361no—
380C22×C4—41no—
381C22×C4—591no—
382C22×C4—2361no—
383C22×C4—2361no—
384C22×C4—2361no—
385C22×C4—2361no—
386Q16—21no—
387Q16—592no—
388Q16—592no—
389Q16—1184no—
390Q16—1181no—
391Q16—1181no—
392Q16C2×C4×D5912yes—
393SD16—1184no—
394D8—21no—
395D8C2×C4×D5912yes—
396M16—1182no—
397C2×C8—591no—
398C2×C8—592no—
399C2×C8—592no—
400C2×C8—591no—
401C2×C8D4×D5911yescharacteristic
402C2×C8C23×D5911yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—591no—
407C4⋊C4—591no—
408C4⋊C4—1181no—
409C4⋊C4—1181no—
410C4⋊C4—2361no—
411C4⋊C4—2361no—
412C4⋊C4—2361no—
413C4⋊C4—2361no—
414C4⋊C4—2361no—
415C4⋊C4—2361no—
416C22⋊C4—41no—
417C22⋊C4—591no—
418C22⋊C4—591no—
419C22⋊C4—2361no—
420C22⋊C4—2361no—
421C42—41no—
422C42—591no—
423C42—2361no—
424C42—2361no—
425C42—2361no—
426C42—2361no—
427C42—2361no—
428C42—2361no—
429C42D4×D5911yescharacteristic
430C16—21no—
431C16—592no—
432C16—1181no—
433C16C2×C4×D5912yes—
434C23—41no—
435C23—591no—
436Q8—22no—
437Q8—41no—
438Q8—1182no—
439Q8—1182no—
440Q8—2364no—
441Q8—2361no—
442Q8—2361no—
443D4—22no—
444D4—41no—
445D4—2364no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—161no—
450C2×C4—591no—
451C2×C4—592no—
452C2×C4—592no—
453C2×C4—591no—
454C2×C4—591no—
455C2×C4—9441no—
456C2×C4—9441no—
457C2×C4—2364no—
458C2×C4—2364no—
459C2×C4—2361no—
460C2×C4—2361no—
461C2×C4—2361no—
462C2×C4—2361no—
463C2×C4D8×D5911yescharacteristic
464C2×C4D8×D5911yescharacteristic
465C2×C4C2×D4×D5911yescharacteristic
466C8—21no—
467C8—592no—
468C8—1181no—
469C8C22×C4×D5911yescharacteristic
470C8#41×D5911yescharacteristic
471C22—42no—
472C22—81no—
473C22—81no—
474C22—592no—
475C22—1181no—
476C22—1181no—
477C22C2×D8×D5911yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—592no—
484C4—1181no—
485C4—1181no—
486C4—4721no—
487C4—4721no—
488C4—2362no—
489C4—2362no—
490C4C4×D4×D5911yescharacteristic
491C4#135×D5911yescharacteristic
492C2—82no—
493C2—1182no—
494C2C4×D8×D5911yescharacteristic
495C2#2143724211yescharacteristic
496C2#2143724811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2175C2
socleC22×C59C22×D8
prime radicalC2×#14273746C22
cosocleC2×C8×C59C24
hypercenter#2175D59
hypocenterC59#33455
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
3. #21432959C2semidirect
4. #21425968C2semidirect
5. #21425964C2semidirect
6. #21425957C2semidirect
7. #21425954C2semidirect
8. C4×#14273673C2semidirect
9. SD32×Dic59C2semidirect
10. C4×#14273671C2semidirect
11. #21425881C2semidirect
13. #358×D59C2semidirect
14. #21425694C2semidirect
15. C59×#2175C2semidirect
20. #16837841C4semidirect
68. Q16×D59C2×C4semidirect
89. Dic472C2×C4semidirect
90. #14273673C2×C4semidirect
91. #14273671C2×C4semidirect
92. #14273666C2×C4semidirect
113. SD32×C59C2×C4semidirect
250. #2175D59semidirect
308. C59#33455semidirect
345. SD32C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378412
#214329662
#214329592
#214259682
#214259642
#214259572
#214259542
C4×#142736732
SD32×Dic592
C4×#142736712
#214258812
#214258002
#358×D592
#214256942
C59×#21752
#3345559