Finite Group Databasebeta

← #28562809

Subgroups — #28562809

Order 15104.

1172 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434898C212yesmaximal, complemented
2#21429027C211yesmaximal, characteristic
3#21423121C214yesmaximal, complemented
4#21423019C214yesmaximal, complemented
5#21422616C212yesmaximal, complemented
6#21422423C211yesmaximal, characteristic, complemented
7C2×C59×#543C211yesmaximal, characteristic, complemented
8C2×#14273710—21no—
9C2×#14273701C2211yescharacteristic
10C2×#14273696—21no—
11#16837735C2214yescomplemented
12#16837715—22no—
13#16837650—24no—
14#16837552C2214yescomplemented
15#16837541—22no—
16#16837366C2212yes—
17#16837338C2212yes—
18#16837311C2212yes—
19#16837302—22no—
20C4×C22⋊Dic59C2211yescharacteristic
21#16837139C2218yescomplemented
22#16837126C2214yes—
23C59×#543—24no—
24C59×#543—22no—
25C59×#543C2214yes—
26C2×C22⋊Q8×C59—22no—
27C2×C22⋊Q8×C59—21no—
28C2×C22⋊Q8×C59—22no—
29C2×C22⋊Q8×C59C2212yes—
30C22×C22⋊C4×C59—21no—
31C22×C22⋊C4×C59C2211yescharacteristic
32C22×D236C2311yescharacteristic
33C22×C22⋊D59—21no—
34C22×C22⋊D59—22no—
35C2×#11745671—24no—
36C2×#11745664—22no—
37#14273710—44no—
38#14273701—22no—
39#14273701—22no—
40#14273696—44no—
41C2×C4⋊Dic59C2311yescharacteristic
42C2×Dic59⋊C4—22no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—22no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—22no—
47C2×C4×Dic59—21no—
48#14273660—24no—
49#14273651—24no—
50#14273647—24no—
51C22⋊(C59⋊C8)—22no—
52C22⋊(C59⋊C8)C2314yes—
53C59⋊#95C2312yes—
54C25×C59D411yescharacteristic
55C22×Q8×C59—21no—
56C22×Q8×C59D412yescomplemented
57C23×C4×C59—21no—
58C23×C4×C59—21no—
59C23×C4×C59D411yescharacteristic
60C23×C4×C59C2311yescharacteristic
61C22⋊Q8×C59—28no—
62C22⋊Q8×C59—28no—
63C22⋊Q8×C59—24no—
64C22⋊Q8×C59—28no—
65C22⋊Q8×C59—24no—
66C22⋊Q8×C59—22no—
67C22⋊Q8×C59—22no—
68C22⋊Q8×C59—28no—
69C22⋊Q8×C59D414yescomplemented
70C22⋊Q8×C59C2314yes—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59—22no—
73C2×C4⋊C4×C59—22no—
74C2×C4⋊C4×C59D412yescomplemented
75C2×C22⋊C4×C59—22no—
76C2×C22⋊C4×C59—24no—
77C2×C22⋊C4×C59—22no—
78C2×C22⋊C4×C59—24no—
79C2×C22⋊C4×C59—22no—
80C2×C22⋊C4×C59—22no—
81C2×C22⋊C4×C59D412yes—
82C2×C22⋊C4×C59C2312yes—
83C24×C59—21no—
84C24×C59—21no—
85C24×C59—24no—
86C24×C59—42no—
87C24×C59—81no—
88C24×C59C2×D411yescharacteristic
89C24×C59C2×D412yes—
90C23×D59—21no—
91C2×Q8×C59—24no—
92C2×Q8×C59—22no—
93C2×Q8×C59—82no—
94C2×Q8×C59—82no—
95C2×Q8×C59C2×D414yes—
96C22×C4×C59—22no—
97C22×C4×C59—21no—
98C22×C4×C59—22no—
99C22×C4×C59—24no—
100C22×C4×C59—24no—
101C22×C4×C59—21no—
102C22×C4×C59—42no—
103C22×C4×C59—81no—
104C22×C4×C59—82no—
105C22×C4×C59—82no—
106C22×C4×C59C2×D412yes—
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D412yes—
109C22×C4×C59C2×D412yes—
110C22×C4×C59C2411yescharacteristic
111C2×C22⋊D59—44no—
112C2×C22⋊D59—42no—
113C2×C22⋊D59—42no—
114C2×C22⋊D59—41no—
115C2×C22⋊D59—41no—
116C2×C22⋊D59—44no—
117C2×C22⋊D59—82no—
118C22×Dic59—22no—
119C22×Dic59—21no—
120C2×D236—22no—
121C2×D236—81no—
122C4⋊C4×C59—24no—
123C4⋊C4×C59—28no—
124C4⋊C4×C59—24no—
125C4⋊C4×C59—22no—
126C4⋊C4×C59—24no—
127C4⋊C4×C59C2×D414yes—
128C22⋊C4×C59—28no—
129C22⋊C4×C59—24no—
130C22⋊C4×C59—24no—
131C22⋊C4×C59—24no—
132C22⋊C4×C59—24no—
133C22⋊Dic59—24no—
134C22⋊Dic59—44no—
135#11745671—84no—
136(C2×D59)⋊C4—44no—
137(C2×D59)⋊C4—42no—
138C4⋊Dic59—22no—
139Dic59⋊C4—44no—
140C4×Dic59—81no—
141#11745664—82no—
142C2×C59⋊C8—24no—
143C23×C59—24no—
144C23×C59—24no—
145C23×C59—22no—
146C23×C59—22no—
147C23×C59—22no—
148C23×C59—22no—
149C23×C59—41no—
150C23×C59—42no—
151C23×C59—42no—
152C23×C59—42no—
153C23×C59—42no—
154C23×C59—81no—
155C23×C59—81no—
156C23×C59—88no—
157C23×C59#12011yescharacteristic
158C23×C59#12011yescharacteristic
159C23×C59#13714yes—
160C23×C59C22×D411yescharacteristic
161C22×D59—41no—
162C22×D59—42no—
163C22×D59—41no—
164C22×D59—41no—
165C22×D59—42no—
166Q8×C59—44no—
167Q8×C59—82no—
168Q8×C59—84no—
169C2×C4×C59—24no—
170C2×C4×C59—22no—
171C2×C4×C59—24no—
172C2×C4×C59—42no—
173C2×C4×C59—41no—
174C2×C4×C59—41no—
175C2×C4×C59—42no—
176C2×C4×C59—44no—
177C2×C4×C59—42no—
178C2×C4×C59—44no—
179C2×C4×C59—42no—
180C2×C4×C59—81no—
181C2×C4×C59—84no—
182C2×C4×C59—84no—
183C2×C4×C59#12012yes—
184C2×C4×C59C22×D412yes—
185C22⋊D59—88no—
186C22⋊D59—82no—
187C22⋊D59—82no—
188C2×Dic59—24no—
189C2×Dic59—42no—
190C2×Dic59—42no—
191C2×Dic59—41no—
192C2×Dic59—41no—
193C2×Dic59—41no—
194D236—44no—
195C59⋊C8—44no—
196#18318—591nomaximal, Sylow
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—21no—
200C22×C59—21no—
201C22×C59—21no—
202C22×C59—24no—
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—44no—
206C22×C59—44no—
207C22×C59—48no—
208C22×C59—42no—
209C22×C59—42no—
210C22×C59—82no—
211C22×C59—84no—
212C22×C59#45312yes—
213C22×C59C2×#12011yescharacteristic
214C22×C59C2×#13712yes—
215C2×D59—82no—
216C2×D59—82no—
217C2×D59—81no—
218C2×D59—81no—
219C2×D59—81no—
220C4×C59—22no—
221C4×C59—22no—
222C4×C59—42no—
223C4×C59—44no—
224C4×C59—44no—
225Dic59—44no—
226Dic59—81no—
227C2×#543D5911yescharacteristic, complemented
228#3193—592no—
229#2539—591no—
230#1614—594no—
231#1596—594no—
232#1508—592no—
233#1462—591no—
234C2×C59—22no—
235C2×C59—22no—
236C2×C59—41no—
237C2×C59—44no—
238C2×C59#300411yescharacteristic
239C2×C59C2×#45311yescharacteristic
240C2×C59#302611yescharacteristic
241D59—161no—
242#543—24no—
243#543—22no—
244#543C2×D5914yes—
245C2×#123—1181no—
246C2×C22⋊Q8—22no—
247C2×C22⋊Q8—21no—
248C2×C22⋊Q8—22no—
249C2×C22⋊Q8C2×D5912yes—
250C2×#121—591no—
251C2×#120—1181no—
252C22×C22⋊C4—21no—
253C22×C22⋊C4C2×D5911yescharacteristic
254#484—594no—
255#480—1182no—
256#468—1184no—
257#450—594no—
258#448—1182no—
259#407—592no—
260#399—592no—
261#392—592no—
262#390—1182no—
263C4×C22⋊C4—591no—
264#328—598no—
265#323—594no—
266C59#1831811yescharacteristic, complemented, Sylow
267C25C22⋊D5911yescharacteristic
268C22×Q8—21no—
269C22×Q8C22⋊D5912yescomplemented
270C22×D4—591no—
271C22×D4—1182no—
272C22×D4—1181no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4C22⋊D5911yescharacteristic
276C23×C4C22×D5911yescharacteristic
277C2×SD16—1184no—
278C2×M16—1182no—
279#123—2364no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—24no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—28no—
288C22⋊Q8C22⋊D5914yescomplemented
289C22⋊Q8C22×D5914yes—
290#121—1182no—
291#121—1182no—
292#120—2364no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—591no—
297C2×C4⋊C4—1182no—
298C2×C4⋊C4C22⋊D5912yescomplemented
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—1181no—
306C2×C22⋊C4—1182no—
307C2×C22⋊C4—1181no—
308C2×C22⋊C4—1182no—
309C2×C22⋊C4C22⋊D5912yes—
310C2×C22⋊C4C22×D5912yes—
311C2×C42—1181no—
312#107—1184no—
313#103—1184no—
314#102—1184no—
315C22⋊C8—594no—
316C22⋊C8—1182no—
317#95—592no—
318C24—21no—
319C24—21no—
320C24—24no—
321C24—42no—
322C24—81no—
323C24—1181no—
324C24C2×C22⋊D5911yescharacteristic
325C24C2×C22⋊D5912yes—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8C2×C22⋊D5914yes—
331C2×D4—1182no—
332C2×D4—4721no—
333C2×D4—4722no—
334C2×D4—2364no—
335C2×D4—2362no—
336C2×D4—2362no—
337C2×D4—2361no—
338C2×D4—2361no—
339C2×D4—2364no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—81no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—1182no—
351C22×C4—1181no—
352C22×C4C2×C22⋊D5912yes—
353C22×C4C2×C22⋊D5911yescharacteristic
354C22×C4C2×C22⋊D5912yes—
355C22×C4C2×C22⋊D5912yes—
356C22×C4C23×D5911yescharacteristic
357SD16—4724no—
358M16—4722no—
359C2×C8—1184no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—1182no—
366C4⋊C4—2364no—
367C4⋊C4C2×C22⋊D5914yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—1184no—
374C22⋊C4—2364no—
375C22⋊C4—2364no—
376C22⋊C4—2362no—
377C42—4721no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—81no—
390C23—81no—
391C23—88no—
392C23—2361no—
393C23—2362no—
394C23—2361no—
395C23—2361no—
396C23—2362no—
397C23#1427369611yescharacteristic
398C23#1427369611yescharacteristic
399C23#1427375814yes—
400C23C22×C22⋊D5911yescharacteristic
401Q8—44no—
402Q8—82no—
403Q8—84no—
404D4—4722no—
405D4—4722no—
406D4—4728no—
407D4—2364no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—81no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—1184no—
423C2×C4—2361no—
424C2×C4—2361no—
425C2×C4—2361no—
426C2×C4—2362no—
427C2×C4—2362no—
428C2×C4#1427369612yes—
429C2×C4C22×C22⋊D5912yes—
430C8—2364no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—44no—
440C22—44no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—84no—
446C22—4722no—
447C22—4722no—
448C22—4721no—
449C22—4721no—
450C22—4721no—
451C22#1683757012yes—
452C22C2×#1427369611yescharacteristic
453C22C2×#1427375812yes—
454C4—22no—
455C4—22no—
456C4—42no—
457C4—44no—
458C4—44no—
459C4—4721no—
460C4—2364no—
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—9441no—
466C2#2143333011yescharacteristic
467C2C2×#1683757011yescharacteristic
468C2#2143349311yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#543C2
socleC22×C59C2×#120
prime radical#21429027C2
hypercenterC2×#543D59
hypocenterC59#18318
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #21434898C2semidirect
3. #21423121C2semidirect
4. #21423019C2semidirect
5. #21422616C2semidirect
6. #21422423C2semidirect
7. C2×C59×#543C2semidirect
11. #16837735C22semidirect
14. #16837552C22semidirect
21. #16837139C22semidirect
56. C22×Q8×C59D4semidirect
69. C22⋊Q8×C59D4semidirect
74. C2×C4⋊C4×C59D4semidirect
227. C2×#543D59semidirect
266. C59#18318semidirect
269. C22×Q8C22⋊D59semidirect
288. C22⋊Q8C22⋊D59semidirect
298. C2×C4⋊C4C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214348982
#214290272
#214231212
#214230192
#214226162
#214224232
C2×C59×#5432
#1831859