Finite Group Databasebeta

← #28620431

Subgroups — #28620431

Order 15104.

2596 conjugacy classes of subgroups; only the 263 normal subgroups are shown.

normal subgroups only

This group has 2596 conjugacy classes of subgroups — too many to draw. Only its 263 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
1C22×#14273701C212yesmaximal, complemented
2C2×#16837538C214yesmaximal, complemented
3#21425094C214yesmaximal, complemented
4#21424411C214yesmaximal, complemented
5C59×#1779C211yesmaximal, characteristic, complemented
6C23×D236C2212yescomplemented
7C23×C22⋊D59—22no—
8C22×D472C2214yescomplemented
9C2×#14273701—216no—
10C2×#14273701—22no—
11C2×#14273701—22no—
12C2×#14273701C2214yes—
13C2×#14273701C2214yescomplemented
14C22×C4⋊Dic59C2212yescomplemented
15C22×(C2×D59)⋊C4—22no—
16#16837538—216no—
17C2×#14273649C2218yescomplemented
18#16837165C2214yescomplemented
19C24×C4×C59C2211yescharacteristic
20C2×C22⋊C8×C59C2214yescomplemented
21C2×C22⋊C8×C59C2212yescomplemented
22C24×D59—22no—
23C22×D236—22no—
24C22×D236—28no—
25C22×D236—28no—
26C22×D236—44no—
27C22×D236C2314yes—
28C22×C22⋊D59—216no—
29C22×C22⋊D59—24no—
30C22×C22⋊D59—24no—
31C22×C22⋊D59—48no—
32C22×C22⋊D59—48no—
33C23×Dic59—22no—
34C2×D472—216no—
35C2×D472—44no—
36#14273701—216no—
37#14273701—216no—
38#14273701—28no—
39#14273701—28no—
40#14273701—28no—
41#14273701D4116yescomplemented
42C2×C4⋊Dic59—22no—
43C2×C4⋊Dic59—28no—
44C2×C4⋊Dic59C2314yes—
45C2×(C2×D59)⋊C4—216no—
46C2×(C2×D59)⋊C4—24no—
47C2×(C2×D59)⋊C4—24no—
48#14273649—216no—
49C25×C59D411yescharacteristic
50C23×C4×C59—28no—
51C23×C4×C59—42no—
52C23×C4×C59D411yescharacteristic
53C23×C4×C59D412yes—
54C23×C4×C59C2311yescharacteristic
55C23×C4×C59C2312yes—
56C22×C8×C59C2314yes—
57C22⋊C8×C59—28no—
58C22⋊C8×C59—42no—
59C24×C59—24no—
60C24×C59—24no—
61C24×C59—41no—
62C24×C59—41no—
63C24×C59D814yescomplemented
64C24×C59C2×D411yescharacteristic
65C24×C59C2×D412yes—
66C23×D59—28no—
67C23×D59—24no—
68C23×D59—44no—
69C23×D59—44no—
70C23×D59—44no—
71C23×D59—44no—
72C23×D59—44no—
73C23×D59—44no—
74C22×C4×C59—24no—
75C22×C4×C59—24no—
76C22×C4×C59—24no—
77C22×C4×C59—24no—
78C22×C4×C59—24no—
79C22×C4×C59—24no—
80C22×C4×C59—22no—
81C22×C4×C59—41no—
82C22×C4×C59—42no—
83C22×C4×C59—416no—
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D412yes—
86C22×C4×C59C2×D418yes—
87C22×C4×C59C2411yescharacteristic
88C2×C22⋊D59—216no—
89C2×C22⋊D59—216no—
90C2×C22⋊D59—464no—
91C2×C22⋊D59—48no—
92C2×C22⋊D59—48no—
93C2×C22⋊D59—48no—
94C2×C22⋊D59—48no—
95C22×Dic59—28no—
96C22×Dic59—24no—
97C22×Dic59—44no—
98C22×Dic59—44no—
99C2×D236—28no—
100C2×D236—28no—
101C2×D236—28no—
102C2×D236—24no—
103C2×D236—432no—
104C2×D236—44no—
105C2×D236—44no—
106C2×D236C2×D418yescomplemented
107C2×C8×C59—28no—
108C2×C8×C59—24no—
109(C2×D59)⋊C4—216no—
110(C2×D59)⋊C4—216no—
111C4⋊Dic59—28no—
112C4⋊Dic59—24no—
113C4⋊Dic59C2×D418yescomplemented
114D472—416no—
115C23×C59—22no—
116C23×C59—22no—
117C23×C59—22no—
118C23×C59—22no—
119C23×C59—22no—
120C23×C59—21no—
121C23×C59—22no—
122C23×C59—21no—
123C23×C59—216no—
124C23×C59—44no—
125C23×C59—44no—
126C23×C59—44no—
127C23×C59—48no—
128C23×C59#12014yes—
129C23×C59C2×D812yes—
130C23×C59C2×D814yes—
131C23×C59AGL(1,ℤ/8)14yes—
132C23×C59C22×D411yescharacteristic
133C22×D59—28no—
134C22×D59—416no—
135C22×D59—416no—
136C22×D59—416no—
137C22×D59—44no—
138C22×D59—44no—
139C22×D59—44no—
140C22×D59—44no—
141C22×D59—44no—
142C22×D59—44no—
143C22×D59—88no—
144C22×D59—88no—
145C22×D59—88no—
146C22×D59—88no—
147C2×C4×C59—22no—
148C2×C4×C59—22no—
149C2×C4×C59—216no—
150C2×C4×C59—22no—
151C2×C4×C59—21no—
152C2×C4×C59—44no—
153C2×C4×C59—48no—
154C2×C4×C59—44no—
155C2×C4×C59#12014yes—
156C2×C4×C59C22×D414yes—
157C2×C4×C59H(5,𝔽2)12yes—
158C22⋊D59—432no—
159C22⋊D59—432no—
160C2×Dic59—28no—
161C2×Dic59—416no—
162C2×Dic59—44no—
163C2×Dic59—44no—
164D236—216no—
165D236—416no—
166D236—48no—
167C8×C59—44no—
168#23918—591nomaximal, Sylow
169C22×C59—24no—
170C22×C59—24no—
171C22×C59—24no—
172C22×C59—24no—
173C22×C59—24no—
174C22×C59—24no—
175C22×C59—22no—
176C22×C59—24no—
177C22×C59—416no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—42no—
181C22×C59#44718yescomplemented
182C22×C59C2×#12012yes—
183C22×C59#53411yescharacteristic
184C22×C59C22×D811yescharacteristic
185C22×C59C2×AGL(1,ℤ/8)12yes—
186C22×C59#57511yescharacteristic
187C2×D59—48no—
188C2×D59—48no—
189C2×D59—48no—
190C2×D59—832no—
191C2×D59—84no—
192C2×D59—84no—
193C2×D59—84no—
194C2×D59—84no—
195C4×C59—24no—
196C4×C59—41no—
197C4×C59D4214yes—
198Dic59—48no—
199C22×#121—592no—
200C2×#447—594no—
201#1999—594no—
202#1874—594no—
203#1779D5911yescharacteristic, complemented
204C2×C59—28no—
205C2×C59—42no—
206C2×C59#240311yescharacteristic
207C2×C59C2×#44712yes—
208C2×C59#300612yes—
209C2×C59#318611yescharacteristic
210C2×C59#319211yescharacteristic
211D59—88no—
212C23×D4—592no—
213C23×D4—1182no—
214C24×C4C2×D5911yescharacteristic
215C22×D8—594no—
216C2×#121—594no—
217C2×#121—594no—
218C2×#121—11816no—
219C2×#121—1182no—
220C2×#121—1182no—
221C22×C4⋊C4—592no—
222C22×C22⋊C4—1182no—
223#447—11816no—
224C2×#102—598no—
225C2×C22⋊C8C2×D5914yescomplemented
226C2×C22⋊C8C2×D5912yescomplemented
227#340—594no—
228C59#2391811yescharacteristic, complemented, Sylow
229C25—1182no—
230C25D23611yescharacteristic
231C22×D4—594no—
232C22×D4—1188no—
233C22×D4—11816no—
234C22×D4—1184no—
235C22×D4—1184no—
236C22×D4—1182no—
237C22×D4—1188no—
238C22×D4—2364no—
239C22×D4—2368no—
240C22×D4—2368no—
241C23×C4—28no—
242C23×C4—42no—
243C23×C4—1182no—
244C23×C4D23612yes—
245C23×C4D23611yescharacteristic
246C23×C4C22×D5911yescharacteristic
247C23×C4C22×D5912yes—
248C2×D8—11816no—
249C2×D8—2364no—
250C22×C8C22×D5914yes—
251#121—5916no—
252#121—1188no—
253#121—1188no—
254#121—11816no—
255#121—11816no—
256#121—1188no—
257C2×C4⋊C4—594no—
258C2×C4⋊C4—1188no—
259C2×C4⋊C4—1182no—
260C2×C22⋊C4—11816no—
261C2×C22⋊C4—1184no—
262C2×C22⋊C4—1184no—
263#102—11816no—
264C22⋊C8—28no—
265C22⋊C8—42no—
266C24—24no—
267C24—24no—
268C24—41no—
269C24—41no—
270C24—1188no—
271C24—1184no—
272C24—2364no—
273C24—2364no—
274C24—2364no—
275C24—2364no—
276C24—2364no—
277C24—2364no—
278C24D47214yescomplemented
279C24C2×D23611yescharacteristic
280C24C2×D23612yes—
281C2×D4—598no—
282C2×D4—1188no—
283C2×D4—11816no—
284C2×D4—11816no—
285C2×D4—1188no—
286C2×D4—1184no—
287C2×D4—1188no—
288C2×D4—23632no—
289C2×D4—2364no—
290C2×D4—23664no—
291C2×D4—2368no—
292C2×D4—2368no—
293C2×D4—2368no—
294C2×D4—2368no—
295C2×D4—2364no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—416no—
306C22×C4—1188no—
307C22×C4—1184no—
308C22×C4—2364no—
309C22×C4—2364no—
310C22×C4C2×D23611yescharacteristic
311C22×C4C2×D23612yes—
312C22×C4D4×D5918yes—
313C22×C4C23×D5911yescharacteristic
314D8—23616no—
315C2×C8—28no—
316C2×C8—24no—
317C4⋊C4—598no—
318C4⋊C4—1188no—
319C4⋊C4—1184no—
320C22⋊C4—11816no—
321C22⋊C4—11816no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—216no—
331C23—44no—
332C23—44no—
333C23—48no—
334C23—44no—
335C23—1188no—
336C23—4728no—
337C23—4728no—
338C23—4728no—
339C23—4728no—
340C23—23616no—
341C23—23616no—
342C23—23616no—
343C23—2364no—
344C23—2364no—
345C23—2364no—
346C23—2364no—
347C23—2364no—
348C23—2364no—
349C23#1427369714yes—
350C23C2×D47212yes—
351C23C2×D47214yes—
352C23#1427375514yes—
353C23C22×D23611yescharacteristic
354D4—11816no—
355D4—23616no—
356D4—23632no—
357D4—23632no—
358D4—2368no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—216no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—1188no—
368C2×C4—23616no—
369C2×C4—2364no—
370C2×C4—2364no—
371C2×C4#1427369714yes—
372C2×C4C2×D4×D5914yes—
373C2×C4#1427377212yes—
374C8—44no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—22no—
383C22—416no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—47232no—
388C22—4724no—
389C22—4724no—
390C22—4724no—
391C22—4724no—
392C22—2368no—
393C22—2368no—
394C22—2368no—
395C22#1683753818yescomplemented
396C22C2×#1427369712yes—
397C22#1683796411yescharacteristic
398C22C22×D47211yescharacteristic
399C22C2×#1427375512yes—
400C22#1683820211yescharacteristic
401C4—24no—
402C4—41no—
403C4—2368no—
404C4#1683803414yes—
405C2—28no—
406C2—42no—
407C2—4728no—
408C2#2142740911yescharacteristic
409C2C2×#1683753812yes—
410C2#2143335012yes—
411C2#2143484711yescharacteristic
412C2#2143489611yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1779C2
socleC23×C59C22×D4
hypercenter#1779D59
hypocenterC59#23918
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. C22×#14273701C2semidirect
2. C2×#16837538C2semidirect
3. #21425094C2semidirect
4. #21424411C2semidirect
5. C59×#1779C2semidirect
6. C23×D236C22semidirect
8. C22×D472C22semidirect
13. C2×#14273701C22semidirect
14. C22×C4⋊Dic59C22semidirect
17. C2×#14273649C22semidirect
18. #16837165C22semidirect
20. C2×C22⋊C8×C59C22semidirect
21. C2×C22⋊C8×C59C22semidirect
41. #14273701D4semidirect
63. C24×C59D8semidirect
106. C2×D236C2×D4semidirect
113. C4⋊Dic59C2×D4semidirect
181. C22×C59#447semidirect
203. #1779D59semidirect
225. C2×C22⋊C8C2×D59semidirect
226. C2×C22⋊C8C2×D59semidirect
228. C59#23918semidirect
278. C24D472semidirect
395. C22#16837538semidirect

Maximal subgroups

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

subgroupindex
C22×#142737012
C2×#168375382
#214250942
#214244112
C59×#17792
#2391859