Finite Group Databasebeta

← #28746154

Subgroups — #28746154

Order 15104.

472 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436489C211yesmaximal, characteristic
2#21426383C211yesmaximal, characteristic, complemented
3#21426304C211yesmaximal, characteristic, complemented
4#21426296C211yesmaximal, characteristic, complemented
5#21426293C211yesmaximal, characteristic, complemented
6#21426291C211yesmaximal, characteristic, complemented
7#21426257C211yesmaximal, characteristic, complemented
8#21426238C211yesmaximal, characteristic, complemented
9#21426202C211yesmaximal, characteristic, complemented
10#21426185C211yesmaximal, characteristic, complemented
11#21425965C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic
13#21425885C211yesmaximal, characteristic, complemented
14C2×#16837229C211yesmaximal, characteristic, complemented
15C59×#3336C211yesmaximal, characteristic
16#16838109—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273667C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20#16837807C2211yescharacteristic
21#16837738—21no—
22#16837738—21no—
23#16837698—21no—
24#16837643C2211yescharacteristic
25#16837643C2211yescharacteristic
26C4×Dic236C2211yescharacteristic
27C2×#14273661C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34#16837229C2211yescharacteristic
35#16837227C2211yescharacteristic
36#16837227C2211yescharacteristic
37#16837227C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837205C2211yescharacteristic
43#16837205C2211yescharacteristic
44#16837205C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#552—21no—
49C59×#499C2211yescharacteristic
50C59×#485—21no—
51C59×#483—21no—
52C59×#479—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273727—21no—
61#14273718—41no—
62#14273706—21no—
63#14273706—21no—
64#14273704—41no—
65#14273704—41no—
66C4×Dic118—21no—
67D4×Dic59—41no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70#14273673—22no—
71#14273667—22no—
72C2×C59⋊C16—22no—
73C2×C59⋊C16C2311yescharacteristic
74C2×C59⋊C16C2311yescharacteristic
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77#14273661—21no—
78#14273661—21no—
79#14273661—21no—
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273652—21no—
84#14273652—21no—
85#14273652—21no—
86C2×Q16×C59C2311yescharacteristic
87C22×C8×C59C2311yescharacteristic
88C22×C8×C59C2311yescharacteristic
89#126×C59—41no—
90#125×C59—21no—
91#125×C59—41no—
92#123×C59—41no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—21no—
95C4×Q8×C59—21no—
96C4×D4×C59D411yescharacteristic
97#117×C59—21no—
98#117×C59—41no—
99C2×C4⋊C4×C59—21no—
100#107×C59—21no—
101#107×C59C2311yescharacteristic
102#107×C59C2311yescharacteristic
103#107×C59C2311yescharacteristic
104#106×C59—21no—
105#106×C59—21no—
106C4⋊C8×C59D411yescharacteristic
107#103×C59—21no—
108#103×C59—21no—
109#103×C59—21no—
110C22⋊C8×C59D411yescharacteristic
111C22⋊C8×C59D411yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×Q8×C59—21no—
114C2×D4×C59D811yescharacteristic
115C22×C4×C59—41no—
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D411yescharacteristic
118C22×Dic59—41no—
119C22×Dic59—41no—
120C2×Dic118—21no—
121C2×Dic118—41no—
122Q16×C59—21no—
123Q16×C59—21no—
124C2×C8×C59—21no—
125C2×C8×C59—21no—
126C2×C8×C59—21no—
127C2×C8×C59—21no—
128C2×C8×C59C2×D411yescharacteristic
129C2×C8×C59C2×D411yescharacteristic
130C2×C8×C59C2×D411yescharacteristic
131C2×C8×C59C2411yescharacteristic
132C4⋊C4×C59—21no—
133C4⋊C4×C59—21no—
134C4⋊C4×C59—21no—
135C4⋊C4×C59—41no—
136C4⋊C4×C59—41no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59D811yescharacteristic
142C22⋊C4×C59—41no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊C4×C59D811yescharacteristic
146C22⋊C4×C59D811yescharacteristic
147C42×C59—41no—
148C42×C59—41no—
149C42×C59C2×D411yescharacteristic
150C22⋊Dic59—41no—
151C22⋊Dic59—41no—
152C4⋊Dic59—21no—
153C4⋊Dic59—21no—
154C4⋊Dic59—21no—
155C4⋊Dic59—41no—
156C4⋊Dic59—41no—
157C4⋊Dic59—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161C4×Dic59—41no—
162Dic236—21no—
163Dic236—21no—
164C59⋊C16—21no—
165C59⋊C16—21no—
166C59⋊C16—21no—
167C59⋊C16#4112yes—
168C23×C59C2×D811yescharacteristic
169C23×C59C2×D811yescharacteristic
170Q8×C59—41no—
171Q8×C59—41no—
172D4×C59—22no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—82no—
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59C2×D811yescharacteristic
186C2×C4×C59C22×D411yescharacteristic
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—161no—
192C2×Dic59—161no—
193Dic118—41no—
194Dic118—41no—
195Dic118—161no—
196C8×C59—21no—
197C8×C59—21no—
198C8×C59—21no—
199C8×C59C2×#4111yescharacteristic
200C8×C592−1+411yescharacteristic
201#33911—591nomaximal, Sylow
202C22×C59—21no—
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—21no—
206C22×C59—21no—
207C22×C59#51012yes—
208C22×C59C22×D811yescharacteristic
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59—81no—
216C4×C59—81no—
217C4×C59#54711yescharacteristic
218C4×C59#57811yescharacteristic
219Dic59—81no—
220Dic59—81no—
221Dic59—81no—
222Dic59—81no—
223#3338—591no—
224#3336D5911yescharacteristic
225#2255—591no—
226#2237—591no—
227#2236—591no—
228#2235—591no—
229#2235—591no—
230#2229—591no—
231#2225—591no—
232#2219—591no—
233#2216—591no—
234#2176—591no—
235#2168—591no—
236#2159—591no—
237C2×#367—591no—
238C2×C59—22no—
239C2×C59—41no—
240C2×C59#333411yescharacteristic
241C2×C59C2×#51011yescharacteristic
242C2×C59#341611yescharacteristic
243#554—1181no—
244#552—21no—
245C2×Q32—591no—
246C2×M32—591no—
247C22×C16—591no—
248#500—591no—
249#499C2×D5911yescharacteristic
250#485—21no—
251#484—1181no—
252#484—1181no—
253#483—21no—
254#479—21no—
255#477—1181no—
256#467—591no—
257#467—591no—
258#467C2×D5911yescharacteristic
259#467C2×D5911yescharacteristic
260C4×Q16—591no—
261C4×Q16C2×D5911yescharacteristic
262C8×D4C2×D5911yescharacteristic
263#427C2×D5911yescharacteristic
264C2×#107—591no—
265C2×#107—591no—
266C2×#107C2×D5911yescharacteristic
267#367—591no—
268#367—591no—
269#367—591no—
270#367—591no—
271#367—591no—
272#367—591no—
273#366—591no—
274#366—591no—
275#366—591no—
276C4⋊C16—591no—
277#358—591no—
278#358—591no—
279#358—591no—
280#358—591no—
281#358—591no—
282#358—591no—
283C22⋊C16—591no—
284C22⋊C16—591no—
285#346—591no—
286C59#3391111yescharacteristic, complemented, Sylow
287C2×Q16—591no—
288C2×Q16C22×D5911yescharacteristic
289C22×C8C22×D5911yescharacteristic
290C22×C8C22×D5911yescharacteristic
291C4⋊Q8—1181no—
292#126—41no—
293#125—21no—
294#125—41no—
295#125—2361no—
296#123—41no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—1181no—
300C22⋊Q8—1181no—
301C22⋊Q8—2361no—
302C22⋊Q8—2361no—
303C4×Q8—21no—
304C4×Q8—1181no—
305C4×D4—2361no—
306C4×D4C22⋊D5911yescharacteristic
307#117—21no—
308#117—41no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—1181no—
311C2×C4⋊C4—1181no—
312Q32—1182no—
313M32—1182no—
314C2×C16—591no—
315C2×C16—591no—
316C2×C16—591no—
317C2×C16—591no—
318C2×C16—1182no—
319#107—21no—
320#107—591no—
321#107—591no—
322#107—591no—
323#107—1181no—
324#107—1181no—
325#107—1181no—
326#107C22×D5911yescharacteristic
327#107C22×D5911yescharacteristic
328#107C22×D5911yescharacteristic
329#106—21no—
330#106—21no—
331C4⋊C8C22⋊D5911yescharacteristic
332#103—21no—
333#103—21no—
334#103—21no—
335#103—1181no—
336#103—1181no—
337#103—1181no—
338C22⋊C8C22⋊D5911yescharacteristic
339C22⋊C8C22⋊D5911yescharacteristic
340C4×C8C22×D5911yescharacteristic
341C2×Q8—21no—
342C2×Q8—1181no—
343C2×Q8—2361no—
344C2×D4#1174566911yescharacteristic
345C22×C4—41no—
346C22×C4—2361no—
347C22×C4—2361no—
348C22×C4C2×C22⋊D5911yescharacteristic
349C22×C4C2×C22⋊D5911yescharacteristic
350Q16—21no—
351Q16—21no—
352Q16—1181no—
353Q16—1181no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×C22⋊D5911yescharacteristic
359C2×C8C2×C22⋊D5911yescharacteristic
360C2×C8C2×C22⋊D5911yescharacteristic
361C2×C8C23×D5911yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1181no—
372C4⋊C4—1181no—
373C4⋊C4—1181no—
374C4⋊C4—2361no—
375C4⋊C4—2361no—
376C4⋊C4—2361no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4#1174566911yescharacteristic
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—2361no—
385C22⋊C4—2361no—
386C22⋊C4#1174566911yescharacteristic
387C22⋊C4#1174566911yescharacteristic
388C42—41no—
389C42—41no—
390C42—2361no—
391C42C2×C22⋊D5911yescharacteristic
392C16—592no—
393C16—1181no—
394C16—1181no—
395C16—1181no—
396C23C2×#1174566911yescharacteristic
397C23C2×#1174566911yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—9441no—
401Q8—2361no—
402Q8—2361no—
403D4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—82no—
413C2×C4—9441no—
414C2×C4—9441no—
415C2×C4—2361no—
416C2×C4—2361no—
417C2×C4—2361no—
418C2×C4—2361no—
419C2×C4C2×#1174566911yescharacteristic
420C2×C4C2×#1174566911yescharacteristic
421C2×C4C2×#1174566911yescharacteristic
422C2×C4C2×#1174566911yescharacteristic
423C2×C4C22×C22⋊D5911yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8#41×D5911yescharacteristic
428C8#1427377411yescharacteristic
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22#1683784212yes—
435C22C22×#1174566911yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—4721no—
445C4—4721no—
446C4—4721no—
447C4—4721no—
448C4#1683805111yescharacteristic
449C4#1683821711yescharacteristic
450C2—22no—
451C2—41no—
452C2#2143643911yescharacteristic
453C2C2×#1683784211yescharacteristic
454C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3336C2
socleC22×C59C22×D8
prime radicalC2×D4×C59D8
hypercenter#3336D59
hypocenterC59#33911
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21426383C2semidirect
3. #21426304C2semidirect
4. #21426296C2semidirect
5. #21426293C2semidirect
6. #21426291C2semidirect
7. #21426257C2semidirect
8. #21426238C2semidirect
9. #21426202C2semidirect
10. #21426185C2semidirect
11. #21425965C2semidirect
13. #21425885C2semidirect
14. C2×#16837229C2semidirect
286. C59#33911semidirect

Maximal subgroups

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

subgroupindex
#214364892
#214263832
#214263042
#214262962
#214262932
#214262912
#214262572
#214262382
#214262022
#214261852
#214259652
#214259222
#214258852
C2×#168372292
C59×#33362
#3391159