Finite Group Databasebeta

← #28746156

Subgroups — #28746156

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#21436465C211yesmaximal, characteristic
3#21426382C211yesmaximal, characteristic, complemented
4#21426305C211yesmaximal, characteristic, complemented
5#21426299C211yesmaximal, characteristic, complemented
6#21426294C211yesmaximal, characteristic, complemented
7#21426294C211yesmaximal, characteristic, complemented
8#21426261C211yesmaximal, characteristic, complemented
9#21426239C211yesmaximal, characteristic, complemented
10#21426204C211yesmaximal, characteristic, complemented
11#21426187C211yesmaximal, characteristic, complemented
12#21425968C211yesmaximal, characteristic, complemented
13#21425884C211yesmaximal, characteristic, complemented
14C2×#16837230C211yesmaximal, characteristic, complemented
15C59×#2168C211yesmaximal, characteristic
16#16838109—21no—
17#16838092—21no—
18C2×Dic472C2211yescharacteristic
19#16837807C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837744—21no—
22#16837738—21no—
23#16837738—21no—
24#16837733—21no—
25#16837710—21no—
26#16837698—21no—
27#16837643C2211yescharacteristic
28#16837643C2211yescharacteristic
29#16837643C2211yescharacteristic
30#16837643C2211yescharacteristic
31C4×Dic236C2211yescharacteristic
32C4×Dic236C2211yescharacteristic
33#16837440C2211yescharacteristic
34C2×#14273661C2211yescharacteristic
35C2×#14273661C2211yescharacteristic
36C2×#14273661C2211yescharacteristic
37#16837230C2211yescharacteristic
38#16837230C2211yescharacteristic
39#16837230C2211yescharacteristic
40#16837230C2211yescharacteristic
41#16837230C2211yescharacteristic
42#16837230C2211yescharacteristic
43#16837228C2211yescharacteristic
44#16837228C2211yescharacteristic
45#16837228C2211yescharacteristic
46#16837207C2211yescharacteristic
47#16837207C2211yescharacteristic
48#16837207C2211yescharacteristic
49#16837207C2211yescharacteristic
50#16837207C2211yescharacteristic
51#16837207C2211yescharacteristic
52C2×M32×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic
56C59×C22⋊C16C2211yescharacteristic
57C59×C22⋊C16C2211yescharacteristic
58C59×#346C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×Dic236C2311yescharacteristic
61#14273727—21no—
62#14273720—41no—
63#14273719—21no—
64#14273718—41no—
65#14273717—41no—
66#14273708—41no—
67#14273706—21no—
68#14273706—21no—
69#14273706—21no—
70#14273706—21no—
71#14273704—41no—
72#14273704—41no—
73C4×Dic118—21no—
74C4×Dic118—21no—
75D4×Dic59—41no—
76#14273686—21no—
77#14273683—41no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81Dic472—22no—
82#14273661—21no—
83#14273661—21no—
84#14273661—21no—
85#14273661—21no—
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273661C2311yescharacteristic
90#14273661C2311yescharacteristic
91#14273661C2311yescharacteristic
92#14273659—21no—
93#14273659—21no—
94#14273652—21no—
95#14273652—21no—
96#14273652—21no—
97#14273652—21no—
98#14273652—21no—
99#14273652—21no—
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C4×D4×C59D411yescharacteristic
103M32×C59—22no—
104C2×C16×C59—22no—
105C2×C16×C59C2311yescharacteristic
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108C2×C16×C59C2311yescharacteristic
109C4⋊C8×C59D411yescharacteristic
110C22⋊C8×C59D411yescharacteristic
111C22⋊C8×C59D411yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×D4×C59D811yescharacteristic
114C22×C4×C59C2×D411yescharacteristic
115C22×C4×C59C2×D411yescharacteristic
116C22×Dic59—41no—
117C22×Dic59—41no—
118C22×Dic59—41no—
119C2×Dic118—21no—
120C2×Dic118—21no—
121C2×Dic118—41no—
122C2×C8×C59—21no—
123C2×C8×C59—21no—
124C2×C8×C59—21no—
125C2×C8×C59—21no—
126C2×C8×C59C2×D411yescharacteristic
127C2×C8×C59C2×D411yescharacteristic
128C2×C8×C59C2×D411yescharacteristic
129C2×C8×C59C2411yescharacteristic
130C4⋊C4×C59D811yescharacteristic
131C22⋊C4×C59D811yescharacteristic
132C22⋊C4×C59D811yescharacteristic
133C42×C59C2×D411yescharacteristic
134C22⋊Dic59—41no—
135C22⋊Dic59—41no—
136C22⋊Dic59—41no—
137C22⋊Dic59—41no—
138C22⋊Dic59—41no—
139C4⋊Dic59—21no—
140C4⋊Dic59—21no—
141C4⋊Dic59—21no—
142C4⋊Dic59—21no—
143C4⋊Dic59—21no—
144C4⋊Dic59—21no—
145C4⋊Dic59—41no—
146C4⋊Dic59—41no—
147C4⋊Dic59—41no—
148C4⋊Dic59—41no—
149Dic59⋊C4—41no—
150Dic59⋊C4—41no—
151Dic59⋊C4—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4—41no—
157C4×Dic59—41no—
158C4×Dic59—41no—
159C4×Dic59—41no—
160Dic236—21no—
161Dic236—21no—
162Dic236—21no—
163Dic236—21no—
164C16×C59—21no—
165C16×C59—21no—
166C16×C59—21no—
167C16×C59#4112yes—
168C23×C59C2×D811yescharacteristic
169C23×C59C2×D811yescharacteristic
170D4×C59—22no—
171C2×C4×C59—21no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59C2×D811yescharacteristic
176C2×C4×C59C2×D811yescharacteristic
177C2×C4×C59C2×D811yescharacteristic
178C2×C4×C59C2×D811yescharacteristic
179C2×C4×C59C22×D411yescharacteristic
180C2×Dic59—41no—
181C2×Dic59—41no—
182C2×Dic59—41no—
183C2×Dic59—41no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—82no—
189C2×Dic59—161no—
190C2×Dic59—161no—
191Dic118—41no—
192Dic118—41no—
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#54711yescharacteristic
214C4×C59#57811yescharacteristic
215Dic59—81no—
216Dic59—81no—
217Dic59—81no—
218Dic59—81no—
219Dic59—81no—
220Dic59—81no—
221Dic59—81no—
222Dic59—81no—
223#3338—591no—
224#3336—591no—
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#2168D5911yescharacteristic
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—1181no—
245C2×Q32—591no—
246C2×M32C2×D5911yescharacteristic
247C22×C16C2×D5911yescharacteristic
248#500—591no—
249#499—591no—
250#485—1181no—
251#484—1181no—
252#484—1181no—
253#483—1181no—
254#479—1181no—
255#477—1181no—
256#467—591no—
257#467—591no—
258#467—591no—
259#467—591no—
260C4×Q16—591no—
261C4×Q16—591no—
262C8×D4C2×D5911yescharacteristic
263#427—591no—
264C2×#107—591no—
265C2×#107—591no—
266C2×#107—591no—
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⋊C16C2×D5911yescharacteristic
277#358—591no—
278#358—591no—
279#358—591no—
280#358—591no—
281#358—591no—
282#358—591no—
283C22⋊C16C2×D5911yescharacteristic
284C22⋊C16C2×D5911yescharacteristic
285#346C2×D5911yescharacteristic
286C59#3391111yescharacteristic, complemented, Sylow
287C2×Q16—591no—
288C2×Q16—591no—
289C22×C8C22×D5911yescharacteristic
290C22×C8C22×D5911yescharacteristic
291C4⋊Q8—1181no—
292#126—2361no—
293#125—1181no—
294#125—2361no—
295#125—2361no—
296#123—2361no—
297C22⋊Q8—1181no—
298C22⋊Q8—1181no—
299C22⋊Q8—1181no—
300C22⋊Q8—1181no—
301C22⋊Q8—2361no—
302C22⋊Q8—2361no—
303C4×Q8—1181no—
304C4×Q8—1181no—
305C4×D4—2361no—
306C4×D4D23611yescharacteristic
307#117—1181no—
308#117—2361no—
309C2×C4⋊C4—1181no—
310C2×C4⋊C4—1181no—
311C2×C4⋊C4—1181no—
312Q32—1182no—
313M32—22no—
314C2×C16—22no—
315C2×C16C22×D5911yescharacteristic
316C2×C16C22×D5911yescharacteristic
317C2×C16C22×D5911yescharacteristic
318C2×C16C22×D5911yescharacteristic
319#107—591no—
320#107—591no—
321#107—591no—
322#107—591no—
323#107—591no—
324#107—591no—
325#107—1181no—
326#107—1181no—
327#107—1181no—
328#107—1181no—
329#106—1181no—
330#106—1181no—
331C4⋊C8D23611yescharacteristic
332#103—1181no—
333#103—1181no—
334#103—1181no—
335#103—1181no—
336#103—1181no—
337#103—1181no—
338C22⋊C8D23611yescharacteristic
339C22⋊C8D23611yescharacteristic
340C4×C8C22×D5911yescharacteristic
341C2×Q8—1181no—
342C2×Q8—1181no—
343C2×Q8—2361no—
344C2×D4D47211yescharacteristic
345C22×C4—2361no—
346C22×C4—2361no—
347C22×C4—2361no—
348C22×C4C2×D23611yescharacteristic
349C22×C4C2×D23611yescharacteristic
350Q16—1181no—
351Q16—1181no—
352Q16—1181no—
353Q16—1181no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×D23611yescharacteristic
359C2×C8C2×D23611yescharacteristic
360C2×C8C2×D23611yescharacteristic
361C2×C8C23×D5911yescharacteristic
362C4⋊C4—1181no—
363C4⋊C4—1181no—
364C4⋊C4—1181no—
365C4⋊C4—1181no—
366C4⋊C4—1181no—
367C4⋊C4—1181no—
368C4⋊C4—2361no—
369C4⋊C4—2361no—
370C4⋊C4—2361no—
371C4⋊C4—2361no—
372C4⋊C4—2361no—
373C4⋊C4—2361no—
374C4⋊C4—2361no—
375C4⋊C4—2361no—
376C4⋊C4—2361no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4D47211yescharacteristic
381C22⋊C4—2361no—
382C22⋊C4—2361no—
383C22⋊C4—2361no—
384C22⋊C4—2361no—
385C22⋊C4—2361no—
386C22⋊C4D47211yescharacteristic
387C22⋊C4D47211yescharacteristic
388C42—2361no—
389C42—2361no—
390C42—2361no—
391C42C2×D23611yescharacteristic
392C16—21no—
393C16—21no—
394C16—21no—
395C16#1174568712yes—
396C23C2×D47211yescharacteristic
397C23C2×D47211yescharacteristic
398Q8—9441no—
399Q8—2361no—
400Q8—2361no—
401Q8—2361no—
402Q8—2361no—
403D4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—9441no—
409C2×C4—9441no—
410C2×C4—4722no—
411C2×C4—2361no—
412C2×C4—2361no—
413C2×C4—2361no—
414C2×C4—2361no—
415C2×C4—2361no—
416C2×C4—2361no—
417C2×C4—2361no—
418C2×C4—2361no—
419C2×C4C2×D47211yescharacteristic
420C2×C4C2×D47211yescharacteristic
421C2×C4C2×D47211yescharacteristic
422C2×C4C2×D47211yescharacteristic
423C2×C4C22×D23611yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8C2×#1174568711yescharacteristic
428C8#1427377411yescharacteristic
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22#1683784512yes—
435C22C22×D47211yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—4721no—
441C4—4721no—
442C4—4721no—
443C4—4721no—
444C4—4721no—
445C4—4721no—
446C4—4721no—
447C4—4721no—
448C4#1683805611yescharacteristic
449C4#1683821911yescharacteristic
450C2—22no—
451C2—41no—
452C2#2143644411yescharacteristic
453C2C2×#1683784511yescharacteristic
454C2#2143724411yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2168C2
socleC22×C59C22×D8
prime radicalC2×D4×C59D8
hypercenter#2168D59
hypocenterC59#33911
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
3. #21426382C2semidirect
4. #21426305C2semidirect
5. #21426299C2semidirect
6. #21426294C2semidirect
7. #21426294C2semidirect
8. #21426261C2semidirect
9. #21426239C2semidirect
10. #21426204C2semidirect
11. #21426187C2semidirect
12. #21425968C2semidirect
13. #21425884C2semidirect
14. C2×#16837230C2semidirect
286. C59#33911semidirect

Maximal subgroups

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

subgroupindex
#214364892
#214364652
#214263822
#214263052
#214262992
#214262942
#214262612
#214262392
#214262042
#214261872
#214259682
#214258842
C2×#168372302
C59×#21682
#3391159