Finite Group Databasebeta

← #28485759

Subgroups — #28485759

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434036C211yesmaximal, characteristic
2#21424191C211yesmaximal, characteristic, complemented
3#21422665C211yesmaximal, characteristic
4#21422457C211yesmaximal, characteristic
5#21422003C211yesmaximal, characteristic, complemented
6#21421984C211yesmaximal, characteristic, complemented
7C59×#2924C211yesmaximal, characteristic, complemented
8C2×#14273701C2211yescharacteristic
9#16837726—21no—
10#16837726—21no—
11#16837720—21no—
12#16837715—21no—
13#16837445C411yescharacteristic, complemented
14#16837445C411yescharacteristic, complemented
15#16837406C411yescharacteristic, complemented
16#16837406C411yescharacteristic, complemented
17#16837366C2211yescharacteristic
18C2×C22⋊(C59⋊C8)C2211yescharacteristic
19#16837262C2211yescharacteristic
20#16837164—21no—
21#16837164—21no—
22#16837137—21no—
23#16837137—21no—
24#16837125—22no—
25C2×#117×C59C2211yescharacteristic
26C59×#433—22no—
27C59×#432—21no—
28C59×#431—21no—
29C59×#424—21no—
30C59×#407C2211yescharacteristic
31C2×C22⋊C8×C59C2211yescharacteristic
32C59×#405—21no—
33C22×D236C2×C411yescharacteristic
34C22×C22⋊D59—21no—
35C2×#11745664C2×C411yescharacteristic
36C2×#11745664C2×C411yescharacteristic
37C22×C59⋊C8—21no—
38#14273701—21no—
39#14273701—21no—
40#14273701—21no—
41#14273701—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59C2×C411yescharacteristic
44C2×C22⋊Dic59—21no—
45C2×(C2×D59)⋊C4—21no—
46#14273660—21no—
47#14273660—21no—
48#14273658—21no—
49#14273658—21no—
50#14273647—21no—
51#14273647—21no—
52#14273647—21no—
53#14273647—21no—
54C22⋊(C59⋊C8)—21no—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57C22⋊(C59⋊C8)—21no—
58C59⋊#95—21no—
59C23×C4×C59C2311yescharacteristic
60C2×M16×C59—21no—
61C22×C8×C59—21no—
62#117×C59—21no—
63#117×C59—21no—
64#117×C59C2×C411yescharacteristic
65#117×C59C2×C411yescharacteristic
66#117×C59D411yescharacteristic
67#117×C59D411yescharacteristic
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C42×C59—21no—
72C4⋊C8×C59—22no—
73C4⋊C8×C59—22no—
74C22⋊C8×C59—21no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59D412yes—
77C22⋊C8×C59D411yescharacteristic
78C22⋊C8×C59D411yescharacteristic
79#97×C59—22no—
80C4×C8×C59—22no—
81C24×C59#4111yescharacteristic
82C23×D59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C2×C22⋊D59—41no—
94C2×C22⋊D59—41no—
95C2×C22⋊D59—81no—
96C2×C22⋊D59—81no—
97C22×Dic59—21no—
98C22×Dic59—41no—
99C2×D236—21no—
100C2×D236—21no—
101C2×D236—41no—
102C2×D236—41no—
103M16×C59—41no—
104M16×C59—41no—
105C2×C8×C59—21no—
106C2×C8×C59—22no—
107C2×C8×C59—21no—
108C2×C8×C59—41no—
109C2×C8×C59—41no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59C22⋊C411yescharacteristic
113C4⋊C4×C59C22⋊C411yescharacteristic
114C4⋊C4×C59C22⋊C411yescharacteristic
115C4⋊C4×C59C22⋊C411yescharacteristic
116C22⋊C4×C59—21no—
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—21no—
120C42×C59—21no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59—21no—
124C22⋊Dic59—81no—
125(C2×D59)⋊C4—41no—
126(C2×D59)⋊C4—41no—
127C4⋊Dic59—21no—
128C4⋊Dic59—21no—
129C4⋊Dic59—41no—
130C4⋊Dic59—41no—
131#11745664—41no—
132#11745664—41no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—21no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59C4×D411yescharacteristic
143C23×C59#12111yescharacteristic
144C23×C59#12311yescharacteristic
145C22×D59—41no—
146C22×D59—42no—
147C22×D59—81no—
148C22×D59—81no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—81no—
165C2×C4×C59C2×C22⋊C411yescharacteristic
166C2×C4×C59C4×D411yescharacteristic
167C2×C4×C59#12011yescharacteristic
168C2×C4×C59#12111yescharacteristic
169C22⋊D59—82no—
170C22⋊D59—81no—
171C22⋊D59—81no—
172C2×Dic59—41no—
173C2×Dic59—81no—
174C2×Dic59—81no—
175C2×Dic59—81no—
176C2×Dic59—81no—
177D236—41no—
178D236—41no—
179D236—81no—
180C8×C59—41no—
181C8×C59—41no—
182C8×C59—42no—
183C59⋊C8—41no—
184C59⋊C8—41no—
185C59⋊C8—81no—
186#9348—591nomaximal, Sylow
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59#38611yescharacteristic
198C22×C59#47111yescharacteristic
199C22×C59#47211yescharacteristic
200C2×D59—81no—
201C2×D59—81no—
202C2×D59—81no—
203C2×D59—162no—
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—21no—
207C4×C59—41no—
208C4×C59—41no—
209C4×C59—41no—
210C4×C59—41no—
211C4×C59#41011yescharacteristic
212C4×C59#41011yescharacteristic
213Dic59—81no—
214Dic59—161no—
215#3094—591no—
216#2924D5911yescharacteristic, complemented
217#1825—591no—
218#1519—591no—
219#1471—591no—
220#1305—591no—
221#1298—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—41no—
225C2×C59—41no—
226C2×C59#188111yescharacteristic
227C2×C59#189911yescharacteristic
228C2×C59#190711yescharacteristic
229D59—161no—
230C2×#121—591no—
231C2×#117C2×D5911yescharacteristic
232#482—1181no—
233#482—1181no—
234#481—1181no—
235#480—1181no—
236#433—22no—
237#432—21no—
238#431—21no—
239#428—591no—
240#428—591no—
241#424—21no—
242#418—591no—
243#418—591no—
244#407—591no—
245#407C2×D5911yescharacteristic
246C2×C22⋊C8—591no—
247C2×C22⋊C8C2×D5911yescharacteristic
248#405—21no—
249#380—591no—
250#340—1181no—
251#340—1181no—
252#327—1181no—
253#327—1181no—
254#323—1182no—
255C59#934811yescharacteristic, complemented, Sylow
256C22×D4—591no—
257C22×D4—1181no—
258C23×C4C22×D5911yescharacteristic
259C2×M16—21no—
260C2×M16—591no—
261C2×M16—591no—
262C22×C8—21no—
263C22×C8—1181no—
264#121—1181no—
265#121—1181no—
266#121—1181no—
267#121—1181no—
268#117—21no—
269#117—21no—
270#117C4×D5911yescharacteristic
271#117C4×D5911yescharacteristic
272#117C22⋊D5911yescharacteristic
273#117C22⋊D5911yescharacteristic
274C2×C4⋊C4—591no—
275C2×C4⋊C4—1181no—
276C2×C4⋊C4D23611yescharacteristic
277C2×C4⋊C4D23611yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1181no—
280C2×C22⋊C4—1181no—
281C2×C42—21no—
282#107—1181no—
283#107—1181no—
284#106—1181no—
285#106—1181no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288#102—1181no—
289#102—1181no—
290#102—1181no—
291#102—1181no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—1181no—
295C22⋊C8—1181no—
296C22⋊C8—1181no—
297C22⋊C8—1181no—
298C22⋊C8C22⋊D5912yes—
299C22⋊C8C22⋊D5911yescharacteristic
300C22⋊C8C22⋊D5911yescharacteristic
301#97—22no—
302C4×C8—22no—
303#95—1181no—
304C24—1181no—
305C24#1174568511yescharacteristic
306C2×D4—1181no—
307C2×D4—1181no—
308C2×D4—4721no—
309C2×D4—4721no—
310C2×D4—2361no—
311C2×D4—2361no—
312C2×D4—2361no—
313C2×D4—2361no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—1181no—
319C22×C4—2361no—
320C22×C4C2×C4×D5911yescharacteristic
321C22×C4C2×D23611yescharacteristic
322C22×C4#1174568511yescharacteristic
323C22×C4C2×C22⋊D5911yescharacteristic
324C22×C4C2×C22⋊D5911yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326M16—41no—
327M16—41no—
328M16—2361no—
329M16—2361no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—1181no—
336C2×C8—1181no—
337C2×C8—2361no—
338C2×C8—2361no—
339C2×C8—2361no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1181no—
343C4⋊C4—1181no—
344C4⋊C4—2361no—
345C4⋊C4—2361no—
346C4⋊C4(C2×D59)⋊C411yescharacteristic
347C4⋊C4(C2×D59)⋊C411yescharacteristic
348C4⋊C4(C2×D59)⋊C411yescharacteristic
349C4⋊C4(C2×D59)⋊C411yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—4721no—
355C22⋊C4—2361no—
356C22⋊C4—2361no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—4721no—
366C23—4721no—
367C23—2362no—
368C23—2361no—
369C23C4×C22⋊D5911yescharacteristic
370C23#1427370111yescharacteristic
371C23#1427371011yescharacteristic
372D4—4721no—
373D4—4722no—
374D4—4721no—
375D4—4721no—
376D4—2361no—
377D4—2361no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—4721no—
395C2×C4—4721no—
396C2×C4—4721no—
397C2×C4—4721no—
398C2×C4—2361no—
399C2×C4C2×(C2×D59)⋊C411yescharacteristic
400C2×C4C4×C22⋊D5911yescharacteristic
401C2×C4#1427369611yescharacteristic
402C2×C4#1427370111yescharacteristic
403C8—41no—
404C8—41no—
405C8—42no—
406C8—4721no—
407C8—2361no—
408C8—2361no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—9442no—
420C22—4721no—
421C22—4721no—
422C22—4721no—
423C22#1683728111yescharacteristic
424C22#1683766511yescharacteristic
425C22#1683767211yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—9441no—
434C4—4721no—
435C4#1683737511yescharacteristic
436C4#1683737511yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—9441no—
442C2#2142445811yescharacteristic
443C2#2142455411yescharacteristic
444C2#2142460411yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2924C2
socleC22×C59#386
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#2924D59
hypocenterC59#9348
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2924

Complements

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

NH ≅ G/Ntype
2. #21424191C2semidirect
5. #21422003C2semidirect
6. #21421984C2semidirect
7. C59×#2924C2semidirect
13. #16837445C4semidirect
14. #16837445C4semidirect
15. #16837406C4semidirect
16. #16837406C4semidirect
216. #2924D59semidirect
255. C59#9348semidirect

Maximal subgroups

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

subgroupindex
#214340362
#214241912
#214226652
#214224572
#214220032
#214219842
C59×#29242
#934859