Finite Group Databasebeta

← #28620617

Subgroups — #28620617

Order 15104.

1446 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1446 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C2×#16837535C212yesmaximal, complemented
2#21426676C211yesmaximal, characteristic, complemented
3#21425239C212yesmaximal, complemented
4#21425167C212yesmaximal, complemented
5#21424444C214yesmaximal, complemented
6#21424419C212yesmaximal, complemented
7#21424167C211yesmaximal, characteristic, complemented
8C2×C59×#540C211yesmaximal, characteristic, complemented
9C23×D236C2211yescharacteristic
10C22×#11745669C2214yes—
11C2×#14273686C2211yescharacteristic
12#16837535—28no—
13C2×#14273648C2214yescomplemented
14C2×#14273647C2214yes—
15C2×C22⋊(C59⋊C8)C2212yes—
16#16837304C2212yes—
17#16837304C2212yes—
18#16837303—22no—
19#16837257—22no—
20#16837164C2214yescomplemented
21#16837157C2212yescomplemented
22#16837157C2212yescomplemented
23C59×#540—28no—
24C2×#127×C59C2211yescharacteristic, complemented
25C2×#125×C59C2211yescharacteristic, complemented
26C2×#123×C59—22no—
27C2×#121×C59—22no—
28C2×#121×C59C2212yes—
29C2×C4×D4×C59C2212yescomplemented
30C2×#117×C59C2211yescharacteristic
31C24×D59—21no—
32C22×D236—21no—
33C22×D236—24no—
34C22×D236—24no—
35C22×D236—42no—
36C22×D236C2312yes—
37C2×#11745669—28no—
38C2×#11745669—28no—
39C2×#11745669—44no—
40C22×C59⋊C8C2314yes—
41#14273686—42no—
42C2×C4⋊Dic59C2312yes—
43C2×C22⋊Dic59—21no—
44C2×(C2×D59)⋊C4—24no—
45C2×(C2×D59)⋊C4—24no—
46C2×C4×Dic59—21no—
47#14273648—44no—
48#14273647—28no—
49C22⋊(C59⋊C8)—24no—
50C22×D4×C59—22no—
51C22×D4×C59—22no—
52C22×D4×C59C2312yes—
53C23×C4×C59—22no—
54C23×C4×C59C2311yescharacteristic
55#127×C59—24no—
56#125×C59—24no—
57#123×C59—216no—
58#121×C59—216no—
59#121×C59—24no—
60#121×C59D418yes—
61C4×D4×C59—24no—
62C4×D4×C59—24no—
63#117×C59—24no—
64C2×C4⋊C4×C59—22no—
65C2×C4⋊C4×C59D412yes—
66C2×C4⋊C4×C59C2312yes—
67C2×C22⋊C4×C59—22no—
68C2×C22⋊C4×C59—22no—
69C2×C22⋊C4×C59D412yes—
70C2×C42×C59C2311yescharacteristic
71C2×C42×C59C2311yescharacteristic
72C24×C59—22no—
73C24×C59C2×D411yescharacteristic
74C23×D59—22no—
75C23×D59—24no—
76C23×D59—42no—
77C23×D59—42no—
78C23×D59—84no—
79C2×D4×C59—28no—
80C2×D4×C59—28no—
81C2×D4×C59—22no—
82C2×D4×C59—24no—
83C2×D4×C59—24no—
84C2×D4×C59—84no—
85C2×D4×C59—84no—
86C2×D4×C59C2×D414yes—
87C22×C4×C59—22no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—28no—
91C22×C4×C59—84no—
92C22×C4×C59C2×D412yes—
93C22×C4×C59C2×D412yes—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D414yes—
96C22×C4×C59C2411yescharacteristic
97C22×Dic59—22no—
98C2×D236—24no—
99C2×D236—24no—
100C2×D236—24no—
101C2×D236—22no—
102C2×D236—416no—
103C2×D236—42no—
104C2×D236—42no—
105C2×D236#4114yes—
106C4⋊C4×C59—28no—
107C4⋊C4×C59—24no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59C2×D414yes—
110C22⋊C4×C59—28no—
111C22⋊C4×C59—24no—
112C22⋊C4×C59—28no—
113C42×C59—22no—
114C42×C59—22no—
115C22⋊Dic59—42no—
116#11745669—416no—
117(C2×D59)⋊C4—84no—
118(C2×D59)⋊C4—84no—
119C4⋊Dic59—24no—
120C4×Dic59—42no—
121C2×C59⋊C8—24no—
122C2×C59⋊C8—24no—
123C2×C59⋊C8—24no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—24no—
127C23×C59—24no—
128C23×C59—42no—
129C23×C59—42no—
130C23×C59—44no—
131C23×C59—44no—
132C23×C59#12012yes—
133C23×C59C22×D411yescharacteristic
134C22×D59—24no—
135C22×D59—48no—
136C22×D59—42no—
137C22×D59—42no—
138C22×D59—44no—
139C22×D59—44no—
140C22×D59—88no—
141C22×D59—88no—
142C22×D59—88no—
143D4×C59—28no—
144D4×C59—44no—
145D4×C59—88no—
146D4×C59—88no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—28no—
150C2×C4×C59—22no—
151C2×C4×C59—22no—
152C2×C4×C59—24no—
153C2×C4×C59—24no—
154C2×C4×C59—24no—
155C2×C4×C59—44no—
156C2×C4×C59—88no—
157C2×C4×C59#12012yes—
158C2×C4×C59C22×D412yes—
159C2×C4×C59C2×#4112yes—
160C2×C4×C59H(5,𝔽2)12yes—
161C2×Dic59—42no—
162C2×Dic59—44no—
163D236—28no—
164D236—48no—
165D236—44no—
166C59⋊C8—44no—
167#23939—591nomaximal, Sylow
168C22×C59—22no—
169C22×C59—24no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59—44no—
173C22×C59—44no—
174C22×C59—44no—
175C22×C59—84no—
176C22×C59—84no—
177C22×C59C2×#12011yescharacteristic
178C22×C59#52511yescharacteristic
179C22×C59#53511yescharacteristic
180C22×C59#57614yes—
181C2×D59—48no—
182C2×D59—44no—
183C2×D59—816no—
184C2×D59—82no—
185C2×D59—82no—
186C2×D59—84no—
187C4×C59—22no—
188C4×C59—42no—
189C4×C59—42no—
190C4×C59—44no—
191C4×C59#54614yes—
192Dic59—82no—
193C2×#540D5911yescharacteristic, complemented
194C2×#447—592no—
195#2309—591no—
196#2025—592no—
197#2011—592no—
198#1879—594no—
199#1875—592no—
200#1818—591no—
201C2×C59—42no—
202C2×C59—84no—
203C2×C59#240411yescharacteristic
204C2×C59#300812yes—
205C2×C59#312212yes—
206C2×C59#322012yes—
207D59—84no—
208C23×D4—591no—
209C22×D8—594no—
210#540—28no—
211C2×#127C2×D5911yescharacteristic, complemented
212C2×#125C2×D5911yescharacteristic, complemented
213C2×#123—22no—
214C2×#121—22no—
215C2×#121C2×D5912yes—
216C2×C4×D4C2×D5912yescomplemented
217C2×#117—591no—
218C2×#117C2×D5911yescharacteristic
219#447—1188no—
220C2×#102—594no—
221C2×#102—594no—
222C2×C22⋊C8—592no—
223#390—592no—
224#390—592no—
225#390—1182no—
226#379—1182no—
227#340—594no—
228#336—592no—
229#336—592no—
230C59#2393911yescharacteristic, complemented, Sylow
231C25—1181no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—592no—
235C22×D4—1184no—
236C22×D4—1181no—
237C22×D4—1184no—
238C22×D4—2362no—
239C22×D4C22×D5912yes—
240C23×C4—22no—
241C23×C4C22×D5911yescharacteristic
242C2×D8—1188no—
243C2×D8—1188no—
244C2×D8—2364no—
245C22×C8—594no—
246#127—24no—
247#125—24no—
248#123—216no—
249#121—216no—
250#121—24no—
251#121C22⋊D5918yes—
252C4×D4—24no—
253C4×D4—24no—
254#117—24no—
255#117—2362no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—592no—
258C2×C4⋊C4C22⋊D5912yes—
259C2×C4⋊C4C22×D5912yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—1184no—
263C2×C22⋊C4—1184no—
264C2×C22⋊C4—1181no—
265C2×C22⋊C4C22⋊D5912yes—
266C2×C42—1181no—
267C2×C42C22×D5911yescharacteristic
268C2×C42C22×D5911yescharacteristic
269#102—1188no—
270#102—2364no—
271C22⋊C8—1184no—
272C24—22no—
273C24—1184no—
274C24—1182no—
275C24—4724no—
276C24—2362no—
277C24—2362no—
278C24C2×C22⋊D5911yescharacteristic
279C2×D4—28no—
280C2×D4—28no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—84no—
285C2×D4—84no—
286C2×D4—594no—
287C2×D4—1184no—
288C2×D4—1184no—
289C2×D4—1182no—
290C2×D4—1184no—
291C2×D4—23616no—
292C2×D4—2362no—
293C2×D4—2362no—
294C2×D4C2×C22⋊D5914yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—28no—
299C22×C4—84no—
300C22×C4—1182no—
301C22×C4C2×C22⋊D5912yes—
302C22×C4C2×C22⋊D5912yes—
303C22×C4C2×C22⋊D5911yescharacteristic
304C22×C4C2×C22⋊D5914yes—
305C22×C4C23×D5911yescharacteristic
306D8—23616no—
307C2×C8—1184no—
308C2×C8—1184no—
309C2×C8—1184no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—1184no—
314C4⋊C4C2×C22⋊D5914yes—
315C22⋊C4—28no—
316C22⋊C4—24no—
317C22⋊C4—28no—
318C22⋊C4—4724no—
319C22⋊C4—4724no—
320C22⋊C4—2362no—
321C42—22no—
322C42—22no—
323C42—2362no—
324C23—21no—
325C23—24no—
326C23—21no—
327C23—24no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—1184no—
333C23—4728no—
334C23—4728no—
335C23—4728no—
336C23—2368no—
337C23—2362no—
338C23—2364no—
339C23—2364no—
340C23—2362no—
341C23#1427369612yes—
342C23C22×C22⋊D5911yescharacteristic
343D4—28no—
344D4—44no—
345D4—88no—
346D4—88no—
347D4—1188no—
348D4—2368no—
349D4—2364no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—44no—
359C2×C4—88no—
360C2×C4—2364no—
361C2×C4—2362no—
362C2×C4#1427369612yes—
363C2×C4C22×C22⋊D5912yes—
364C2×C4#41×D5912yes—
365C2×C4#1427377312yes—
366C8—2364no—
367C22—24no—
368C22—22no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—84no—
375C22—84no—
376C22—47216no—
377C22—4722no—
378C22—4722no—
379C22—4724no—
380C22—2364no—
381C22—2368no—
382C22C2×#1427369611yescharacteristic
383C22#1683790911yescharacteristic
384C22#1683796611yescharacteristic
385C22#1683820514yes—
386C4—22no—
387C4—42no—
388C4—42no—
389C4—44no—
390C4—4722no—
391C4#1683804014yes—
392C2—42no—
393C2—84no—
394C2—4724no—
395C2#2142741111yescharacteristic
396C2#2143336412yes—
397C2#2143433412yes—
398C2#2143514012yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#540C2
socleC23×C59C22×D4
hypercenterC2×#540D59
hypocenterC59#23939
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#16837535C2semidirect
2. #21426676C2semidirect
3. #21425239C2semidirect
4. #21425167C2semidirect
5. #21424444C2semidirect
6. #21424419C2semidirect
7. #21424167C2semidirect
8. C2×C59×#540C2semidirect
13. C2×#14273648C22semidirect
20. #16837164C22semidirect
21. #16837157C22semidirect
22. #16837157C22semidirect
24. C2×#127×C59C22semidirect
25. C2×#125×C59C22semidirect
29. C2×C4×D4×C59C22semidirect
193. C2×#540D59semidirect
211. C2×#127C2×D59semidirect
212. C2×#125C2×D59semidirect
216. C2×C4×D4C2×D59semidirect
230. C59#23939semidirect

Maximal subgroups

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

subgroupindex
C2×#168375352
#214266762
#214252392
#214251672
#214244442
#214244192
#214241672
C2×C59×#5402
#2393959