Finite Group Databasebeta

← #28746086

Subgroups — #28746086

Order 15104.

484 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436486C211yesmaximal, characteristic
2#21436485C211yesmaximal, characteristic
3#21426370C211yesmaximal, characteristic, complemented
4#21426300C211yesmaximal, characteristic, complemented
5#21426299C211yesmaximal, characteristic, complemented
6#21426297C211yesmaximal, characteristic, complemented
7#21426247C211yesmaximal, characteristic, complemented
8#21426246C211yesmaximal, characteristic, complemented
9#21426186C211yesmaximal, characteristic, complemented
10#21426185C211yesmaximal, characteristic, complemented
11#21425952C211yesmaximal, characteristic, complemented
12(C22⋊D59)⋊C16C211yesmaximal, characteristic
13C2×#16837227C211yesmaximal, characteristic, complemented
14#366×D59C211yesmaximal, characteristic, complemented
15C59×#2236C211yesmaximal, characteristic, complemented
16#16838110—21no—
17#16838107—21no—
18C2×#14273673C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20C2×C16×D59C2211yescharacteristic
21#16837806C2211yescharacteristic
22#16837806C2211yescharacteristic
23#16837739—21no—
24#16837737—21no—
25#16837736—21no—
26#16837735—21no—
27#16837696—21no—
28#16837695—21no—
29#16837644C2211yescharacteristic
30#16837643C2211yescharacteristic
31#16837640C2211yescharacteristic
32Q16×Dic59C2211yescharacteristic
33#16837496C2211yescharacteristic
34C8×C22⋊D59C2211yescharacteristic
35C2×#14273661C2211yescharacteristic
36#107×D59C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837228C2211yescharacteristic
39#16837228C2211yescharacteristic
40#16837227C2211yescharacteristic
41#16837227C2211yescharacteristic
42#16837227C2211yescharacteristic
43#16837227C2211yescharacteristic
44#16837227C2211yescharacteristic
45Dic59⋊C16C2211yescharacteristic
46#16837207C2211yescharacteristic
47#16837207C2211yescharacteristic
48#16837206C2211yescharacteristic
49#16837205C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51C16×Dic59C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#366C2211yescharacteristic
55C59×#366C2211yescharacteristic
56C59×#358C2211yescharacteristic
57C59×#358C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×C8×D59C2311yescharacteristic
61#14273726—21no—
62#14273726—21no—
63#14273717—41no—
64#14273717—41no—
65#14273707—21no—
66#14273707—41no—
67#14273706—21no—
68#14273706—41no—
69#14273705—41no—
70#14273704—41no—
71#14273703—21no—
72Q8×Dic59—21no—
73#14273692—21no—
74C4×C22⋊D59—41no—
75C4×C22⋊D59D411yescharacteristic
76(C22⋊D59)⋊C4—41no—
77C2×C4⋊Dic59—21no—
78C4⋊C4×D59—21no—
79C4⋊C4×D59—21no—
80#14273673—22no—
81C2×C59⋊C16—22no—
82C2×C59⋊C16C2311yescharacteristic
83C2×C59⋊C16C2311yescharacteristic
84C16×D59—22no—
85#14273661—21no—
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273660—21no—
90#14273660—21no—
91#14273660—21no—
92#14273660—21no—
93Dic59⋊C8D411yescharacteristic
94#14273652—21no—
95#14273651—21no—
96#14273651—21no—
97#14273650—21no—
98#14273650—21no—
99C22⋊(C59⋊C8)D411yescharacteristic
100(C2×D59)⋊C8D411yescharacteristic
101C8×Dic59C2311yescharacteristic
102C2×Q16×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104C22⋊Q8×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#103×C59—21no—
113C2×Q8×C59—21no—
114C22×C4×C59—41no—
115C22×C4×C59C2×D411yescharacteristic
116C2×C22⋊D59D811yescharacteristic
117C22×Dic59—41no—
118C2×C4×D59—41no—
119C2×C4×D59—41no—
120C2×C4×D59C2×D411yescharacteristic
121C2×Dic118—21no—
122C2×Dic118—41no—
123C2×Dic118—41no—
124Q16×C59—21no—
125Q16×C59—21no—
126C2×C8×C59—21no—
127C2×C8×C59—21no—
128C2×C8×C59C2×D411yescharacteristic
129C2×C8×C59C2411yescharacteristic
130C4⋊C4×C59—21no—
131C4⋊C4×C59—21no—
132C4⋊C4×C59—21no—
133C4⋊C4×C59—41no—
134C4⋊C4×C59—41no—
135C22⋊C4×C59—41no—
136C22⋊Dic59—41no—
137C22⋊Dic59D811yescharacteristic
138(C2×D59)⋊C4—41no—
139(C2×D59)⋊C4—41no—
140(C2×D59)⋊C4D811yescharacteristic
141C4⋊Dic59—21no—
142C4⋊Dic59—21no—
143C4⋊Dic59—21no—
144C4⋊Dic59—41no—
145C4⋊Dic59—41no—
146C4⋊Dic59—41no—
147Dic59⋊C4—41no—
148Dic59⋊C4—41no—
149Dic59⋊C4—41no—
150Dic59⋊C4—41no—
151Dic59⋊C4—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4D811yescharacteristic
155C4×Dic59—41no—
156C4×Dic59—41no—
157C4×Dic59C2×D411yescharacteristic
158C2×C59⋊C8C2×D411yescharacteristic
159C2×C59⋊C8C2×D411yescharacteristic
160Dic236—21no—
161Dic236—21no—
162C8×D59—21no—
163C8×D59—21no—
164C16×C59—21no—
165C16×C59—21no—
166C59⋊C16—21no—
167C59⋊C16#4112yes—
168C23×C59C2×D811yescharacteristic
169C22×D59C2×D811yescharacteristic
170Q8×C59—41no—
171Q8×C59—41no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—161no—
179C2×C4×C59C2×D811yescharacteristic
180C2×C4×C59C22×D411yescharacteristic
181C22⋊D59Q3214yescomplemented
182C2×Dic59—41no—
183C2×Dic59—41no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186C2×Dic59—161no—
187C2×Dic59C2×D811yescharacteristic
188C2×Dic59C2×D811yescharacteristic
189C2×Dic59C2×D811yescharacteristic
190C4×D59—21no—
191C4×D59—21no—
192C4×D59—161no—
193C4×D59—161no—
194Dic118—41no—
195Dic118—41no—
196Dic118—161no—
197Dic118—161no—
198C8×C59—21no—
199C8×C59C2×#4111yescharacteristic
200C8×C592−1+411yescharacteristic
201C59⋊C8—21no—
202C59⋊C8—21no—
203#33905—591nomaximal, Sylow
204C22×C59—21no—
205C22×C59—21no—
206C22×C59C2×Q3212yes—
207C22×C59C22×D811yescharacteristic
208C2×D59—21no—
209C2×D59—21no—
210C2×D59C2×Q3212yes—
211C4×C59—21no—
212C4×C59—81no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59—81no—
216C4×C59#54711yescharacteristic
217C4×C59#57811yescharacteristic
218Dic59—21no—
219Dic59—21no—
220Dic59—81no—
221Dic59—81no—
222Dic59—81no—
223Dic59—81no—
224Dic59C2×Q3212yes—
225#3338—591no—
226#3338—591no—
227#2252—591no—
228#2236—591no—
229#2236—591no—
230#2236—591no—
231#2236D5911yescharacteristic, complemented
232#2227—591no—
233#2227—591no—
234#2216—591no—
235#2216—591no—
236C4×Q32—591no—
237D4×C16—591no—
238C2×#366—591no—
239C2×#366—591no—
240C2×C59—22no—
241C2×C59#333411yescharacteristic
242C2×C59C22×Q3211yescharacteristic
243C2×C59#341511yescharacteristic
244D59—22no—
245#554—1181no—
246#554—1181no—
247C2×Q32—591no—
248C22×C16—591no—
249C22×C16—591no—
250#500—591no—
251#500—591no—
252#484—1181no—
253#484—1181no—
254#484—1181no—
255#484—1181no—
256#477—1181no—
257#477—1181no—
258#467—591no—
259#467—591no—
260#467—591no—
261#467C2×D5911yescharacteristic
262C4×Q16—591no—
263C4×Q16—591no—
264C8×D4—591no—
265C2×#107—591no—
266C2×#107—591no—
267C2×#107—591no—
268C2×#107C2×D5911yescharacteristic
269#366—591no—
270#366—591no—
271#366—591no—
272#366—591no—
273#366—591no—
274#366—591no—
275#366—591no—
276#366C2×D5911yescharacteristic
277#366C2×D5911yescharacteristic
278C4⋊C16—591no—
279#358—591no—
280#358—591no—
281#358—591no—
282#358—591no—
283#358C2×D5911yescharacteristic
284#358C2×D5911yescharacteristic
285C22⋊C16—591no—
286C22⋊C16C2×D5911yescharacteristic
287C4×C16—591no—
288C59#3390511yescharacteristic, complemented, Sylow
289C2×Q16—591no—
290C2×Q16C22×D5911yescharacteristic
291C22×C8—591no—
292C22×C8C22×D5911yescharacteristic
293C4⋊Q8—1181no—
294C4⋊Q8—1181no—
295#125—2361no—
296#125—2361no—
297C22⋊Q8—21no—
298C22⋊Q8—1181no—
299C22⋊Q8—1181no—
300C22⋊Q8—1181no—
301C22⋊Q8—2361no—
302C22⋊Q8—2361no—
303C22⋊Q8—2361no—
304C22⋊Q8—2361no—
305C4×Q8—1181no—
306C4×Q8—1181no—
307C4×D4—591no—
308C4×D4—2361no—
309C4×D4—2361no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—1181no—
312C2×C4⋊C4—1181no—
313C2×C4⋊C4—1181no—
314Q32—1182no—
315C2×C16—591no—
316C2×C16—591no—
317C2×C16—1182no—
318C2×C16—1182no—
319C2×C16C22×D5911yescharacteristic
320C2×C16C22×D5911yescharacteristic
321#107—21no—
322#107—591no—
323#107—591no—
324#107—591no—
325#107—1181no—
326#107—1181no—
327#107—1181no—
328#107—1181no—
329#107—1181no—
330#107C22×D5911yescharacteristic
331#107C22×D5911yescharacteristic
332#107C22×D5911yescharacteristic
333C4⋊C8—591no—
334#103—21no—
335#103—1181no—
336#103—1181no—
337#103—1181no—
338#103—1181no—
339#103—1181no—
340C22⋊C8—591no—
341C22⋊C8—591no—
342C4×C8—591no—
343C2×Q8—21no—
344C2×Q8—1181no—
345C2×Q8—2361no—
346C2×Q8—2361no—
347C2×D4—591no—
348C22×C4—41no—
349C22×C4—591no—
350C22×C4—2361no—
351C22×C4—2361no—
352C22×C4—2361no—
353C22×C4D4×D5911yescharacteristic
354Q16—21no—
355Q16—21no—
356Q16—1181no—
357Q16—1181no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—591no—
361C2×C8—591no—
362C2×C8—1181no—
363C2×C8—1181no—
364C2×C8D4×D5911yescharacteristic
365C2×C8C23×D5911yescharacteristic
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—591no—
372C4⋊C4—1181no—
373C4⋊C4—1181no—
374C4⋊C4—1181no—
375C4⋊C4—2361no—
376C4⋊C4—2361no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C4⋊C4—2361no—
382C4⋊C4—2361no—
383C4⋊C4—2361no—
384C4⋊C4—2361no—
385C22⋊C4—41no—
386C22⋊C4—591no—
387C22⋊C4—591no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C22⋊C4—2361no—
391C42—591no—
392C42—2361no—
393C42—2361no—
394C16—21no—
395C16—21no—
396C16—592no—
397C16—1181no—
398C23—591no—
399C23D8×D5911yescharacteristic
400Q8—41no—
401Q8—41no—
402Q8—9441no—
403Q8—9441no—
404Q8—2361no—
405Q8—2361no—
406D4—594no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—161no—
414C2×C4—591no—
415C2×C4—591no—
416C2×C4—591no—
417C2×C4—1181no—
418C2×C4—1181no—
419C2×C4—9441no—
420C2×C4—9441no—
421C2×C4—9441no—
422C2×C4—2361no—
423C2×C4—2361no—
424C2×C4—2361no—
425C2×C4—2361no—
426C2×C4D8×D5911yescharacteristic
427C2×C4C2×D4×D5911yescharacteristic
428C8—21no—
429C8—1181no—
430C8—1181no—
431C8#41×D5911yescharacteristic
432C8#1427377411yescharacteristic
433C22—21no—
434C22—21no—
435C22—592no—
436C22—1181no—
437C22—1181no—
438C22Q32×D5912yes—
439C22C2×D8×D5911yescharacteristic
440C4—21no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—592no—
446C4—1181no—
447C4—1181no—
448C4—4721no—
449C4—4721no—
450C4—4721no—
451C4—4721no—
452C4#1683805011yescharacteristic
453C4#1683821811yescharacteristic
454C2—22no—
455C2—1182no—
456C2#2143643811yescharacteristic
457C2C2×Q32×D5911yescharacteristic
458C2#2143723611yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2236C2
socleC22×C59C22×D8
prime radicalC2×C22⋊D59D8
hypercenter#2236D59
hypocenterC59#33905
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2236

Complements

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

NH ≅ G/Ntype
3. #21426370C2semidirect
4. #21426300C2semidirect
5. #21426299C2semidirect
6. #21426297C2semidirect
7. #21426247C2semidirect
8. #21426246C2semidirect
9. #21426186C2semidirect
10. #21426185C2semidirect
11. #21425952C2semidirect
13. C2×#16837227C2semidirect
14. #366×D59C2semidirect
15. C59×#2236C2semidirect
181. C22⋊D59Q32semidirect
231. #2236D59semidirect
288. C59#33905semidirect

Maximal subgroups

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

subgroupindex
#214364862
#214364852
#214263702
#214263002
#214262992
#214262972
#214262472
#214262462
#214261862
#214261852
#214259522
(C22⋊D59)⋊C162
C2×#168372272
#366×D592
C59×#22362
#3390559