Finite Group Databasebeta

← #28507397

Subgroups — #28507397

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273673C211yesmaximal, characteristic, complemented
2C2×#16837644C211yesmaximal, characteristic
3C2×#16837640C211yesmaximal, characteristic
4C2×#16837207C211yesmaximal, characteristic, complemented
5C2×(C2×D59)⋊C16C211yesmaximal, characteristic
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#358C211yesmaximal, characteristic, complemented
8C22×Dic236C2211yescharacteristic
9C22×C8×D59C2211yescharacteristic
10C2×#14273707—21no—
11C2×#14273703—21no—
12C2×#14273673—22no—
13C2×#14273673—22no—
14C2×#14273673—22no—
15C22×C59⋊C16C2211yescharacteristic
16#16837644—22no—
17#16837644—22no—
18#16837640—22no—
19#16837640—22no—
20C2×#14273661C2211yescharacteristic
21C2×#14273651—21no—
22C2×#14273650—21no—
23#16837207—22no—
24(C2×D59)⋊C16—22no—
25C22×Q16×C59C2211yescharacteristic
26C22×C16×C59C2211yescharacteristic
27C2×#107×C59C2211yescharacteristic
28C59×#358—22no—
29C22×Dic118—21no—
30C22×C4×D59D411yescharacteristic
31C2×Dic236—21no—
32C2×Dic236—21no—
33C2×Dic236—21no—
34C2×Dic236—21no—
35C2×Dic236—21no—
36C2×Dic236D412yes—
37C22×C59⋊C8D411yescharacteristic
38C2×C8×D59—21no—
39C2×C8×D59—21no—
40C2×C8×D59—21no—
41C2×C8×D59—21no—
42C2×C8×D59—21no—
43C2×C8×D59—21no—
44#14273707—24no—
45#14273707—42no—
46#14273703—24no—
47#14273703—42no—
48C2×C4⋊Dic59—21no—
49C2×C4⋊Dic59—41no—
50C2×Dic59⋊C4—41no—
51C2×(C2×D59)⋊C4—41no—
52C2×(C2×D59)⋊C4—41no—
53#14273673—28no—
54C2×C59⋊C16—21no—
55C2×C59⋊C16—21no—
56C2×C59⋊C16D412yes—
57#14273661—21no—
58#14273661D412yes—
59#14273651—24no—
60#14273650—24no—
61C22×Q8×C59—21no—
62C2×Q16×C59—21no—
63C2×Q16×C59—21no—
64C2×Q16×C59—21no—
65C2×Q16×C59—21no—
66C2×Q16×C59—21no—
67C2×Q16×C59D412yes—
68C22×C8×C59C2311yescharacteristic
69C2×C4⋊C4×C59—21no—
70C2×C16×C59—21no—
71C2×C16×C59—21no—
72C2×C16×C59—21no—
73#107×C59—21no—
74#107×C59D412yes—
75C23×D59D811yescharacteristic
76C2×Q8×C59—22no—
77C2×Q8×C59—41no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C2×Q8×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59C2×D411yescharacteristic
85C22×Dic59—41no—
86C22×Dic59—41no—
87C22×Dic59D811yescharacteristic
88C2×C4×D59—21no—
89C2×C4×D59—21no—
90C2×C4×D59—21no—
91C2×C4×D59—21no—
92C2×C4×D59—21no—
93C2×C4×D59—21no—
94C2×Dic118—22no—
95C2×Dic118—41no—
96C2×Dic118—41no—
97C2×Dic118—41no—
98C2×Dic118—41no—
99C2×Dic118—41no—
100Q16×C59—22no—
101Q16×C59—22no—
102Q16×C59—41no—
103Q16×C59—41no—
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59C2×D411yescharacteristic
106C2×C8×C59C2×D411yescharacteristic
107C2×C8×C59C2×D411yescharacteristic
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59#4111yescharacteristic
110C4⋊C4×C59—22no—
111C4⋊C4×C59—41no—
112(C2×D59)⋊C4—44no—
113(C2×D59)⋊C4—44no—
114C4⋊Dic59—22no—
115C4⋊Dic59—44no—
116C4⋊Dic59—41no—
117Dic59⋊C4—44no—
118C2×C59⋊C8—21no—
119C2×C59⋊C8—21no—
120C2×C59⋊C8—21no—
121Dic236—22no—
122Dic236—22no—
123Dic236—41no—
124Dic236—41no—
125C8×D59—41no—
126C8×D59—41no—
127C8×D59—41no—
128C8×D59—41no—
129C16×C59—41no—
130C59⋊C16—22no—
131C23×C59C2×D811yescharacteristic
132C22×D59—21no—
133C22×D59—21no—
134C22×D59—21no—
135C22×D59—21no—
136C22×D59—21no—
137C22×D59—21no—
138C22×D59Q3212yescomplemented
139Q8×C59—42no—
140Q8×C59—42no—
141Q8×C59—81no—
142Q8×C59—81no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—81no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×C4×C59#12011yescharacteristic
150C2×C4×C59#12011yescharacteristic
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59#13511yescharacteristic
153C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C2×Dic59—21no—
156C2×Dic59—21no—
157C2×Dic59—21no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160C2×Dic59—81no—
161C2×Dic59—81no—
162C2×Dic59—81no—
163C2×Dic59—81no—
164C4×D59—41no—
165C4×D59—41no—
166C4×D59—41no—
167C4×D59—41no—
168Dic118—42no—
169Dic118—42no—
170Dic118—81no—
171Dic118—81no—
172C8×C59#12011yescharacteristic
173C8×C59#12111yescharacteristic
174C8×C59#12111yescharacteristic
175C8×C59#12711yescharacteristic
176C59⋊C8—41no—
177#13306—591nomaximal, Sylow
178C22×C59#44711yescharacteristic
179C22×C59#44711yescharacteristic
180C22×C59#46611yescharacteristic
181C22×C59C2×Q3211yescharacteristic
182C22×C59#50811yescharacteristic
183C22×C59#51011yescharacteristic
184C22×C59#51011yescharacteristic
185C2×D59—22no—
186C2×D59—22no—
187C2×D59—22no—
188C2×D59—41no—
189C2×D59—41no—
190C2×D59—41no—
191C2×D59—41no—
192C4×C59—82no—
193C4×C59—82no—
194C4×C59#39211yescharacteristic
195C4×C59#44911yescharacteristic
196C4×C59#44911yescharacteristic
197C4×C59#46911yescharacteristic
198Dic59—41no—
199Dic59—82no—
200Dic59—82no—
201C22×Q32—591no—
202C2×#467—591no—
203C2×#467—591no—
204C2×#358—591no—
205C2×#358D5911yescharacteristic, complemented
206C2×C22⋊C16—591no—
207#1380—591no—
208C2×C59#199911yescharacteristic
209C2×C59#218811yescharacteristic
210C2×C59#218811yescharacteristic
211C2×C59#218911yescharacteristic
212C2×C59#218911yescharacteristic
213C2×C59#221611yescharacteristic
214C2×C59#221911yescharacteristic
215D59—42no—
216C22×Q16—591no—
217C22×Q16C2×D5911yescharacteristic
218C23×C8—591no—
219C2×C22⋊Q8—1181no—
220C2×C22⋊Q8—1181no—
221C2×Q32—1182no—
222C2×Q32—1182no—
223C2×Q32—1182no—
224C22×C16—591no—
225C22×C16C2×D5911yescharacteristic
226#467—1182no—
227#467—1182no—
228#467—1182no—
229#467—1182no—
230C2×#107—591no—
231C2×#107C2×D5911yescharacteristic
232C2×#103—1181no—
233C2×#103—1181no—
234#358—22no—
235#358—1182no—
236C22⋊C16—1182no—
237C59#1330611yescharacteristic, complemented, Sylow
238C22×Q8—21no—
239C22×Q8—1181no—
240C23×C4—591no—
241C2×Q16—21no—
242C2×Q16—21no—
243C2×Q16—21no—
244C2×Q16—21no—
245C2×Q16—21no—
246C2×Q16—592no—
247C2×Q16—1181no—
248C2×Q16—1181no—
249C2×Q16—1181no—
250C2×Q16—1181no—
251C2×Q16—1181no—
252C2×Q16D23612yes—
253C22×C8—591no—
254C22×C8—1181no—
255C22×C8—1181no—
256C22×C8—1181no—
257C22×C8—1181no—
258C22×C8—1181no—
259C22×C8—1181no—
260C22×C8C22×D5911yescharacteristic
261C22⋊Q8—1184no—
262C22⋊Q8—1184no—
263C22⋊Q8—2362no—
264C22⋊Q8—2362no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4—2361no—
268C2×C4⋊C4—2361no—
269C2×C22⋊C4—2361no—
270C2×C22⋊C4—2361no—
271Q32—1188no—
272C2×C16—21no—
273C2×C16—21no—
274C2×C16—21no—
275C2×C16—592no—
276C2×C16—1181no—
277C2×C16—1181no—
278#107—21no—
279#107—592no—
280#107—1181no—
281#107C22⋊D5912yes—
282#103—1184no—
283#103—1184no—
284C24—591no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—1182no—
292C2×Q8—2361no—
293C2×Q8—2361no—
294C2×Q8—2361no—
295C2×Q8—2361no—
296C2×Q8—2361no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—591no—
300C22×C4—1181no—
301C22×C4—1181no—
302C22×C4—1181no—
303C22×C4—1181no—
304C22×C4—1181no—
305C22×C4—1181no—
306C22×C4—2361no—
307C22×C4—2361no—
308C22×C4D4×D5911yescharacteristic
309Q16—22no—
310Q16—22no—
311Q16—41no—
312Q16—41no—
313Q16—1182no—
314Q16—1182no—
315Q16—2361no—
316Q16—2361no—
317C2×C8—1181no—
318C2×C8—1181no—
319C2×C8—1181no—
320C2×C8—2361no—
321C2×C8—2361no—
322C2×C8—2361no—
323C2×C8—2361no—
324C2×C8C2×D23611yescharacteristic
325C2×C8#1174568511yescharacteristic
326C2×C8D4×D5911yescharacteristic
327C2×C8D4×D5911yescharacteristic
328C2×C8D4×D5911yescharacteristic
329C2×C8C2×C22⋊D5911yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—1182no—
333C4⋊C4—2364no—
334C4⋊C4—2364no—
335C4⋊C4—2361no—
336C22⋊C4—2364no—
337C22⋊C4—2364no—
338C16—41no—
339C16—1182no—
340C23—592no—
341C23—1181no—
342C23—1181no—
343C23—1181no—
344C23—1181no—
345C23—1181no—
346C23—1181no—
347C23D8×D5911yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352Q8—4721no—
353Q8—4721no—
354Q8—2362no—
355Q8—2362no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—1181no—
363C2×C4—1181no—
364C2×C4—1181no—
365C2×C4—4721no—
366C2×C4—4721no—
367C2×C4—4721no—
368C2×C4—4721no—
369C2×C4—2362no—
370C2×C4—2362no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4#1427369511yescharacteristic
376C2×C4#1427369711yescharacteristic
377C2×C4#1427370011yescharacteristic
378C2×C4#1427374811yescharacteristic
379C2×C4#1427375111yescharacteristic
380C2×C4#1427375411yescharacteristic
381C8—2361no—
382C8#1427369711yescharacteristic
383C8#1427370011yescharacteristic
384C8#1427370111yescharacteristic
385C8#1427372311yescharacteristic
386C22—1182no—
387C22—1182no—
388C22—1182no—
389C22—2361no—
390C22—2361no—
391C22—2361no—
392C22—2361no—
393C22#1683753411yescharacteristic
394C22#1683753711yescharacteristic
395C22#1683763711yescharacteristic
396C22Q32×D5911yescharacteristic
397C22#1683783311yescharacteristic
398C22#1683784111yescharacteristic
399C22#1683784411yescharacteristic
400C4—82no—
401C4—82no—
402C4—4722no—
403C4—4722no—
404C4—2361no—
405C4#1683731111yescharacteristic
406C4#1683754611yescharacteristic
407C4#1683754811yescharacteristic
408C4#1683765411yescharacteristic
409C2—2362no—
410C2#2142509311yescharacteristic
411C2#2142602811yescharacteristic
412C2#2142603111yescharacteristic
413C2#2142603311yescharacteristic
414C2#2142603911yescharacteristic
415C2#2142618611yescharacteristic
416C2#2142620311yescharacteristic
subgroupG/H
centerC23D8×D59
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×#358C2
socleC23×C59C2×D8
prime radicalC23×D59D8
hypercenterC2×#358D59
hypocenterC59#13306
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. C22×#14273673C2semidirect
4. C2×#16837207C2semidirect
6. #21422187C2semidirect
7. C2×C59×#358C2semidirect
138. C22×D59Q32semidirect
205. C2×#358D59semidirect
237. C59#13306semidirect

Maximal subgroups

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

subgroupindex
C22×#142736732
C2×#168376442
C2×#168376402
C2×#168372072
C2×(C2×D59)⋊C162
#214221872
C2×C59×#3582
#1330659