Finite Group Databasebeta

← #28746190

Subgroups — #28746190

Order 15104.

484 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436501C211yesmaximal, characteristic
2#21436485C211yesmaximal, characteristic, complemented
3#21426381C211yesmaximal, characteristic, complemented
4#21426303C211yesmaximal, characteristic, complemented
5#21426296C211yesmaximal, characteristic, complemented
6#21426279C211yesmaximal, characteristic, complemented
7#21426275C211yesmaximal, characteristic, complemented
8#21426262C211yesmaximal, characteristic, complemented
9#21426231C211yesmaximal, characteristic, complemented
10#21426206C211yesmaximal, characteristic, complemented
11#21425961C211yesmaximal, characteristic, complemented
12#21425921C211yesmaximal, characteristic, complemented
13#21425883C211yesmaximal, characteristic, complemented
14C2×#16837228C211yesmaximal, characteristic, complemented
15C59×#2217C211yesmaximal, characteristic, complemented
16#16838110—21no—
17#16838104—21no—
18C2×#14273666C2211yescharacteristic
19#16837806C2211yescharacteristic
20#16837803C2211yescharacteristic
21#16837736—21no—
22#16837735—21no—
23#16837724—21no—
24#16837718—21no—
25#16837700—21no—
26#16837695—21no—
27#16837640C2211yescharacteristic
28#16837635C2211yescharacteristic
29Q16×Dic59C2211yescharacteristic
30D8×Dic59C2211yescharacteristic
31C8×C22⋊D59C2211yescharacteristic
32#16837439C2211yescharacteristic
33C2×#14273661C2211yescharacteristic
34C2×#14273661C2211yescharacteristic
35#107×D59C2211yescharacteristic
36#16837229C2211yescharacteristic
37#16837229C2211yescharacteristic
38#16837228C2211yescharacteristic
39#16837228C2211yescharacteristic
40#16837228C2211yescharacteristic
41#16837228C2211yescharacteristic
42#16837227C2211yescharacteristic
43#16837227C2211yescharacteristic
44Dic59⋊C16C2211yescharacteristic
45#16837206C2211yescharacteristic
46#16837206C2211yescharacteristic
47#16837203C2211yescharacteristic
48#16837203C2211yescharacteristic
49C22⋊(C59⋊C16)C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51#16837177C2211yescharacteristic
52C2×SD32×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C59×#466C2211yescharacteristic
56C59×#367C2211yescharacteristic
57C59×#358C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C2×C8×D59C2311yescharacteristic
60#14273726—21no—
61#14273720—41no—
62#14273718—21no—
63#14273717—41no—
64#14273711—41no—
65#14273707—41no—
66#14273704—41no—
67#14273703—21no—
68#14273699—41no—
69#14273698—21no—
70Q8×Dic59—21no—
71D4×Dic59—21no—
72D4×Dic59—41no—
73C4×C22⋊D59D411yescharacteristic
74(C22⋊D59)⋊C4—41no—
75#14273685—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C4⋊C4×D59—21no—
79#14273666—22no—
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661—21no—
83#14273661—21no—
84#14273661C2311yescharacteristic
85#14273661C2311yescharacteristic
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273660—21no—
89#14273660—21no—
90#14273658—21no—
91#14273658—21no—
92Dic59⋊C8D411yescharacteristic
93#14273651—21no—
94#14273651—21no—
95#14273648—21no—
96#14273648—21no—
97C22⋊(C59⋊C8)D411yescharacteristic
98(C2×D59)⋊C8D411yescharacteristic
99C8×Dic59C2311yescharacteristic
100C2×Q16×C59C2311yescharacteristic
101C2×D8×C59C2311yescharacteristic
102C22×C8×C59C2311yescharacteristic
103C22⋊Q8×C59—21no—
104#121×C59—21no—
105SD32×C59—22no—
106C2×C16×C59—22no—
107C2×C16×C59C2311yescharacteristic
108C2×C16×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#103×C59—21no—
112#102×C59—21no—
113C2×Q8×C59—21no—
114C2×D4×C59—21no—
115C2×D4×C59—41no—
116C22×C4×C59C2×D411yescharacteristic
117C2×C22⋊D59—41no—
118C2×C22⋊D59D811yescharacteristic
119C22×Dic59—41no—
120C22×Dic59—41no—
121C22×Dic59—41no—
122C2×C4×D59—41no—
123C2×C4×D59C2×D411yescharacteristic
124C2×Dic118—41no—
125Q16×C59—21no—
126Q16×C59—21no—
127D8×C59—21no—
128D8×C59—21no—
129C2×C8×C59—21no—
130C2×C8×C59—21no—
131C2×C8×C59C2×D411yescharacteristic
132C2×C8×C59C2411yescharacteristic
133C4⋊C4×C59—21no—
134C4⋊C4×C59—21no—
135C4⋊C4×C59—41no—
136C22⋊C4×C59—41no—
137C22⋊C4×C59—41no—
138C22⋊Dic59—41no—
139C22⋊Dic59—41no—
140C22⋊Dic59D811yescharacteristic
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143(C2×D59)⋊C4D811yescharacteristic
144C4⋊Dic59—21no—
145C4⋊Dic59—21no—
146C4⋊Dic59—21no—
147C4⋊Dic59—21no—
148C4⋊Dic59—41no—
149C4⋊Dic59—41no—
150C4⋊Dic59—41no—
151C4⋊Dic59—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4D811yescharacteristic
157C4×Dic59—41no—
158C4×Dic59—41no—
159C4×Dic59C2×D411yescharacteristic
160C2×C59⋊C8C2×D411yescharacteristic
161C2×C59⋊C8C2×D411yescharacteristic
162C8×D59—21no—
163C8×D59—21no—
164C16×C59—21no—
165C16×C59#4112yes—
166C59⋊C16—21no—
167C59⋊C16—21no—
168C23×C59—41no—
169C23×C59C2×D811yescharacteristic
170C22×D59C2×D811yescharacteristic
171Q8×C59—41no—
172Q8×C59—41no—
173D4×C59—41no—
174D4×C59—41no—
175D4×C59—82no—
176C2×C4×C59—21no—
177C2×C4×C59—21no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59C2×D811yescharacteristic
182C2×C4×C59C22×D411yescharacteristic
183C22⋊D59—22no—
184C22⋊D59—161no—
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—161no—
190C2×Dic59—161no—
191C2×Dic59—161no—
192C2×Dic59C2×D811yescharacteristic
193C2×Dic59C2×D811yescharacteristic
194C2×Dic59C2×D811yescharacteristic
195C4×D59—21no—
196C4×D59—21no—
197C4×D59—161no—
198Dic118—161no—
199C8×C59—21no—
200C8×C59C2×#4111yescharacteristic
201C8×C592−1+411yescharacteristic
202C59⋊C8—21no—
203C59⋊C8—21no—
204#33913—591nomaximal, Sylow
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—81no—
208C22×C59—81no—
209C22×C59—81no—
210C22×C59#51012yes—
211C22×C59C22×D811yescharacteristic
212C2×D59—21no—
213C2×D59—21no—
214C2×D59—21no—
215C4×C59—21no—
216C4×C59—81no—
217C4×C59—81no—
218C4×C59—81no—
219C4×C59#54711yescharacteristic
220C4×C59#57811yescharacteristic
221Dic59—21no—
222Dic59—21no—
223Dic59—21no—
224Dic59—81no—
225Dic59—81no—
226Dic59—81no—
227Dic59—81no—
228#3339—591no—
229#3338—591no—
230#2255—591no—
231#2237—591no—
232#2236—591no—
233#2233—591no—
234#2232—591no—
235#2229—591no—
236#2224—591no—
237#2220—591no—
238#2217D5911yescharacteristic, complemented
239#2175—591no—
240#2168—591no—
241#2159—591no—
242C2×#366—591no—
243C2×C59—22no—
244C2×C59—161no—
245C2×C59#333411yescharacteristic
246C2×C59C2×#51011yescharacteristic
247C2×C59#341511yescharacteristic
248D59—41no—
249#554—1181no—
250#553—1181no—
251C2×SD32C2×D5911yescharacteristic
252C2×M32—591no—
253C22×C16C2×D5911yescharacteristic
254#500—591no—
255#499—591no—
256#484—1181no—
257#484—1181no—
258#482—1181no—
259#481—1181no—
260#478—1181no—
261#477—1181no—
262#467—591no—
263#467C2×D5911yescharacteristic
264#466—591no—
265#466C2×D5911yescharacteristic
266C4×Q16—591no—
267C4×D8—591no—
268C8×D4—591no—
269#427—591no—
270C2×#107—591no—
271C2×#107—591no—
272C2×#107—591no—
273#367—591no—
274#367—591no—
275#367C2×D5911yescharacteristic
276#366—591no—
277#366—591no—
278#366—591no—
279#366—591no—
280#366—591no—
281#366—591no—
282C4⋊C16—591no—
283#358—591no—
284#358—591no—
285#358C2×D5911yescharacteristic
286#357—591no—
287#357—591no—
288#357C2×D5911yescharacteristic
289C22⋊C16—591no—
290C22⋊C16—591no—
291#346—591no—
292C59#3391311yescharacteristic, complemented, Sylow
293C2×Q16C22×D5911yescharacteristic
294C2×D8C22×D5911yescharacteristic
295C22×C8—591no—
296C22×C8C22×D5911yescharacteristic
297C4⋊Q8—1181no—
298#126—2361no—
299#125—1181no—
300#125—2361no—
301#123—2361no—
302C22⋊Q8—21no—
303C22⋊Q8—1181no—
304C22⋊Q8—2361no—
305C22⋊Q8—2361no—
306#121—21no—
307#121—1181no—
308#121—2361no—
309C4×Q8—1181no—
310C4×D4—591no—
311C4×D4—1181no—
312C4×D4—2361no—
313C4×D4—2361no—
314#117—1181no—
315C2×C4⋊C4—1181no—
316C2×C4⋊C4—1181no—
317C2×C4⋊C4—1181no—
318SD32—22no—
319M32—1182no—
320C2×C16—22no—
321C2×C16—591no—
322C2×C16—591no—
323C2×C16C22×D5911yescharacteristic
324C2×C16C22×D5911yescharacteristic
325#107—591no—
326#107—591no—
327#107—591no—
328#107—591no—
329#107—1181no—
330#107—1181no—
331#107—1181no—
332#107—1181no—
333#107C22×D5911yescharacteristic
334#107C22×D5911yescharacteristic
335#106—1181no—
336#106—1181no—
337C4⋊C8—591no—
338#103—21no—
339#103—1181no—
340#103—1181no—
341#102—21no—
342#102—1181no—
343#102—1181no—
344C22⋊C8—591no—
345C22⋊C8—591no—
346C4×C8—591no—
347C2×Q8—21no—
348C2×Q8—2361no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4—591no—
352C2×D4—2361no—
353C22×C4—591no—
354C22×C4—2361no—
355C22×C4—2361no—
356C22×C4—2361no—
357C22×C4—2361no—
358C22×C4D4×D5911yescharacteristic
359Q16—21no—
360Q16—21no—
361D8—21no—
362D8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—591no—
366C2×C8—591no—
367C2×C8—1181no—
368C2×C8—1181no—
369C2×C8D4×D5911yescharacteristic
370C2×C8C23×D5911yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—591no—
375C4⋊C4—1181no—
376C4⋊C4—1181no—
377C4⋊C4—1181no—
378C4⋊C4—1181no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C4⋊C4—2361no—
382C4⋊C4—2361no—
383C4⋊C4—2361no—
384C4⋊C4—2361no—
385C4⋊C4—2361no—
386C4⋊C4—2361no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—591no—
390C22⋊C4—591no—
391C22⋊C4—2361no—
392C22⋊C4—2361no—
393C22⋊C4—2361no—
394C22⋊C4—2361no—
395C42—591no—
396C42—2361no—
397C42—2361no—
398C16—21no—
399C16—1181no—
400C16—1181no—
401C16#1174568712yes—
402C23—41no—
403C23—591no—
404C23D8×D5911yescharacteristic
405Q8—41no—
406Q8—41no—
407Q8—9441no—
408D4—41no—
409D4—41no—
410D4—82no—
411D4—1182no—
412D4—9441no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—591no—
419C2×C4—591no—
420C2×C4—591no—
421C2×C4—1181no—
422C2×C4—1181no—
423C2×C4—9441no—
424C2×C4—9441no—
425C2×C4—9441no—
426C2×C4—9441no—
427C2×C4—2361no—
428C2×C4—2361no—
429C2×C4—2361no—
430C2×C4—2361no—
431C2×C4D8×D5911yescharacteristic
432C2×C4C2×D4×D5911yescharacteristic
433C8—21no—
434C8—1181no—
435C8—1181no—
436C8C2×#1174568711yescharacteristic
437C8#1427377411yescharacteristic
438C22—21no—
439C22—21no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—1181no—
444C22—1181no—
445C22—1181no—
446C22#1683784112yes—
447C22C2×D8×D5911yescharacteristic
448C4—21no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—1181no—
453C4—1181no—
454C4—1181no—
455C4—4721no—
456C4—4721no—
457C4—4721no—
458C4—4721no—
459C4#1683805211yescharacteristic
460C4#1683821611yescharacteristic
461C2—22no—
462C2—161no—
463C2—2361no—
464C2#2143644011yescharacteristic
465C2C2×#1683784111yescharacteristic
466C2#2143723411yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2217C2
socleC22×C59C22×D8
prime radical#21436501C2
hypercenter#2217D59
hypocenterC59#33913
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
2. #21436485C2semidirect
3. #21426381C2semidirect
4. #21426303C2semidirect
5. #21426296C2semidirect
6. #21426279C2semidirect
7. #21426275C2semidirect
8. #21426262C2semidirect
9. #21426231C2semidirect
10. #21426206C2semidirect
11. #21425961C2semidirect
12. #21425921C2semidirect
13. #21425883C2semidirect
14. C2×#16837228C2semidirect
15. C59×#2217C2semidirect
238. #2217D59semidirect
292. C59#33913semidirect

Maximal subgroups

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

subgroupindex
#214365012
#214364852
#214263812
#214263032
#214262962
#214262792
#214262752
#214262622
#214262312
#214262062
#214259612
#214259212
#214258832
C2×#168372282
C59×#22172
#3391359