Finite Group Databasebeta

← #28748708

Subgroups — #28748708

Order 15104.

5430 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11745689C211yesmaximal, characteristic
2C2×#16837876C212yesmaximal, complemented
3#21426925C2116yesmaximal, complemented
4#21426682C211yesmaximal, characteristic, complemented
5#21426673C214yesmaximal, complemented
6#21426475C212yesmaximal, complemented
7#381×D59C214yesmaximal, complemented
8C59×#2310C211yesmaximal, characteristic, complemented
9C22×#11745689—216no—
10C22×#11745689C4116yescomplemented
11C22×#11745689C2218yes—
12C23×D236C2212yes—
13C23×D236C2211yescharacteristic
14C23×C4×D59C2212yes—
15C23×C4×D59C2211yescharacteristic
16#16837876—216no—
17C2×C4×D236C2218yescomplemented
18C2×#14273689C2218yescomplemented
19C2×#14273686C2212yes—
20C2×C22⋊C4×D59C2218yes—
21#16837309C2218yescomplemented
22#16837280C22132yescomplemented
23#16837265C2214yes—
24#16837264C22116yescomplemented
25#16837263C2214yes—
26#16837259C2218yescomplemented
27#16837257C2214yes—
28C4×(C2×D59)⋊C4C2218yescomplemented
29#95×D59C22116yescomplemented
30C23×Q8×C59C2211yescharacteristic
31C2×#117×C59C2212yes—
32C59×#391C2218yescomplemented
33C59×#381C2214yes—
34C24×D59C2311yescharacteristic
35C24×D59C2312yes—
36C2×#11745689—2128no—
37C2×#11745689—232no—
38C2×#11745689—232no—
39C2×#11745689C2×C4164yescomplemented
40C22×D236—216no—
41C22×D236C2×C418yes—
42C22×D236C2×C4116yescomplemented
43C22×D236D4116yes—
44C22×D236C2318yes—
45C22×D236C2314yes—
46C23×Dic59C2311yescharacteristic
47C22×C4×D59—216no—
48C22×C4×D59—28no—
49C22×C4×D59C2×C4116yescomplemented
50C22×C4×D59C2×C418yescomplemented
51C22×C4×D59C2318yes—
52C22×C4×D59C2314yes—
53C22×C4×D59C23116yes—
54C4×D236—232no—
55#14273689—232no—
56#14273686—28no—
57C2×C4⋊Dic59C2314yes—
58C2×C22⋊Dic59C2314yes—
59C2×(C2×D59)⋊C4C23116yes—
60C2×(C2×D59)⋊C4C23116yes—
61C22⋊C4×D59—232no—
62C2×C4×Dic59C2314yes—
63C59⋊#95C23116yes—
64C59⋊#95C23116yes—
65C22×Q8×C59—28no—
66C22×Q8×C59C2×C418yescomplemented
67C22×Q8×C59C2314yes—
68C23×C4×C59C2312yes—
69C23×C4×C59C2311yescharacteristic
70#117×C59—28no—
71C2×C4⋊C4×C59C2314yes—
72C2×C22⋊C4×C59C2314yes—
73C2×C42×C59C2314yes—
74#95×C59C23116yes—
75C24×C59C2411yescharacteristic
76C23×D59—24no—
77C23×D59—48no—
78C23×D59—44no—
79C23×D59C22×C414yes—
80C23×D59C22×C418yes—
81C23×D59C2×D4116yes—
82C23×D59C2412yes—
83C23×D59C2414yes—
84C2×Q8×C59—232no—
85C2×Q8×C59—28no—
86C2×Q8×C59—28no—
87C2×Q8×C59C22×C4116yes—
88C22×C4×C59—28no—
89C22×C4×C59C22×C418yes—
90C22×C4×C59C22×C414yes—
91C22×C4×C59C2×D418yes—
92C22×C4×C59C2418yes—
93C22×C4×C59C2414yes—
94C22×C4×C59C2412yes—
95C22×Dic59—24no—
96C22×Dic59C22×C414yes—
97C22×Dic59C2418yes—
98C22×Dic59C2412yes—
99#11745689—2256no—
100C2×D236—264no—
101C2×D236—232no—
102C2×D236C22⋊C4164yescomplemented
103C2×D236C22×C4132yes—
104C2×D236C22×C4116yes—
105C2×D236C2×D4116yes—
106C2×D236C2×D4116yes—
107C2×C4×D59—264no—
108C2×C4×D59—232no—
109C2×C4×D59—232no—
110C2×C4×D59—28no—
111C2×C4×D59—28no—
112C2×C4×D59—232no—
113C2×C4×D59—432no—
114C2×C4×D59C22×C4132yes—
115C2×C4×D59C22×C4116yes—
116C4⋊C4×C59—28no—
117C22⋊C4×C59—28no—
118C42×C59—28no—
119C22⋊Dic59—28no—
120(C2×D59)⋊C4—232no—
121(C2×D59)⋊C4—416no—
122C4⋊Dic59—28no—
123C4×Dic59—28no—
124C23×C59—24no—
125C23×C59C23×C412yes—
126C23×C59C22×D414yes—
127C23×C59C2511yescharacteristic
128C22×D59—216no—
129C22×D59—28no—
130C22×D59—24no—
131C22×D59—28no—
132C22×D59—416no—
133C22×D59—48no—
134C22×D59—432no—
135C22×D59—416no—
136C22×D59C2×C22⋊C4132yes—
137C22×D59C23×C414yes—
138C22×D59C23×C418yes—
139C22×D59C22×D4116yes—
140Q8×C59—232no—
141C2×C4×C59—216no—
142C2×C4×C59—28no—
143C2×C4×C59—28no—
144C2×C4×C59—216no—
145C2×C4×C59C2×C22⋊C4116yes—
146C2×C4×C59C23×C418yes—
147C2×C4×C59C23×C414yes—
148C2×C4×C59C22×D414yes—
149C2×C4×C59C22×D414yes—
150C2×Dic59—28no—
151C2×Dic59—24no—
152C2×Dic59—28no—
153C2×Dic59—216no—
154C2×Dic59C23×C414yes—
155D236—264no—
156D236C2×C22⋊C4164yes—
157C4×D59—264no—
158C4×D59—232no—
159C4×D59—832no—
160#34172—591nomaximal, Sylow
161C22×C59—24no—
162C22×C59—28no—
163C22×C59C22×C22⋊C414yes—
164C22×C59C24×C411yescharacteristic
165C22×C59C23×D412yes—
166C22×C59#58514yes—
167C2×D59—216no—
168C2×D59—232no—
169C2×D59—24no—
170C2×D59—28no—
171C2×D59—432no—
172C2×D59—416no—
173C2×D59—48no—
174C2×D59C22×C22⋊C4116yes—
175C4×C59—28no—
176C4×C59—48no—
177C4×C59C22×C22⋊C418yes—
178Dic59—24no—
179Dic59—48no—
180C23×#41—591no—
181C2×#518—592no—
182#2339—5916no—
183#2310—591no—
184#2310D5911yescharacteristic, complemented
185#2308—594no—
186#2280—592no—
187C2×#381—594no—
188C2×C59—24no—
189C2×C59C23×C22⋊C411yescharacteristic
190C2×C59#342812yes—
191C2×C59#348514yes—
192D59—216no—
193D59—44no—
194C22×#41—5916no—
195C22×#41—598no—
196C22×#41—11816no—
197C23×Q8C2×D5911yescharacteristic
198C23×D4—591no—
199C23×D4—592no—
200C24×C4—592no—
201C24×C4—591no—
202#518—11816no—
203C2×C4×D4—598no—
204C2×C4×D4—598no—
205C2×#117—592no—
206C2×#117C2×D5912yes—
207C22×C22⋊C4—598no—
208#391—598no—
209#391C2×D5918yescomplemented
210#386—5932no—
211#381—594no—
212#381—594no—
213#381—5916no—
214#381C2×D5914yes—
215#380—598no—
216#379—594no—
217C4×C22⋊C4—598no—
218C2×#95—5916no—
219C59#3417211yescharacteristic, complemented, Sylow
220C25—592no—
221C25—591no—
222C2×#41—5964no—
223C2×#41—118128no—
224C2×#41—11832no—
225C2×#41—11832no—
226C22×Q8—28no—
227C22×Q8C4×D5918yescomplemented
228C22×Q8C22×D5914yes—
229C22×D4—598no—
230C22×D4—5916no—
231C22×D4—5916no—
232C22×D4—594no—
233C22×D4—598no—
234C22×D4—11816no—
235C23×C4—591no—
236C23×C4—5916no—
237C23×C4—598no—
238C23×C4—598no—
239C23×C4—594no—
240C23×C4—5916no—
241C23×C4—11816no—
242C23×C4—1188no—
243C23×C4C22×D5912yes—
244C23×C4C22×D5911yescharacteristic
245C4×D4—11832no—
246C4×D4—11832no—
247#117—28no—
248#117—1188no—
249C2×C4⋊C4—594no—
250C2×C4⋊C4C22×D5914yes—
251C2×C22⋊C4—594no—
252C2×C22⋊C4—5916no—
253C2×C22⋊C4—5916no—
254C2×C22⋊C4—11832no—
255C2×C22⋊C4C22×D5914yes—
256C2×C42—594no—
257C2×C42C22×D5914yes—
258#95—5916no—
259#95—5916no—
260#95C22×D59116yes—
261C24—594no—
262C24—598no—
263C24—5916no—
264C24—592no—
265C24—594no—
266C24—1184no—
267C24—2368no—
268C24—2364no—
269C24C23×D5911yescharacteristic
270#41—118256no—
271C2×Q8—232no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8C2×C4×D59116yes—
275C2×D4—5964no—
276C2×D4—5916no—
277C2×D4—5932no—
278C2×D4—5916no—
279C2×D4—5916no—
280C2×D4—11832no—
281C2×D4—11864no—
282C22×C4—28no—
283C22×C4—594no—
284C22×C4—592no—
285C22×C4—5932no—
286C22×C4—5916no—
287C22×C4—598no—
288C22×C4—11832no—
289C22×C4—11864no—
290C22×C4—11832no—
291C22×C4—1184no—
292C22×C4—11832no—
293C22×C4—1188no—
294C22×C4—1188no—
295C22×C4—23632no—
296C22×C4C2×C4×D5918yes—
297C22×C4C2×C4×D5914yes—
298C22×C4D4×D5918yes—
299C22×C4C23×D5918yes—
300C22×C4C23×D5914yes—
301C22×C4C23×D5912yes—
302C4⋊C4—28no—
303C4⋊C4—1188no—
304C22⋊C4—28no—
305C22⋊C4—11832no—
306C22⋊C4—1188no—
307C22⋊C4—23616no—
308C42—28no—
309C42—1188no—
310C23—24no—
311C23—598no—
312C23—5916no—
313C23—5932no—
314C23—594no—
315C23—11816no—
316C23—1184no—
317C23—1188no—
318C23—1188no—
319C23—23632no—
320C23—23616no—
321C23—2368no—
322C23—23616no—
323C23C22×C4×D5912yes—
324C23C2×D4×D5914yes—
325C23C24×D5911yescharacteristic
326Q8—232no—
327D4—5964no—
328D4—11864no—
329C2×C4—28no—
330C2×C4—216no—
331C2×C4—28no—
332C2×C4—216no—
333C2×C4—594no—
334C2×C4—11864no—
335C2×C4—1184no—
336C2×C4—1188no—
337C2×C4—11832no—
338C2×C4—1188no—
339C2×C4—11816no—
340C2×C4—47232no—
341C2×C4C22⋊C4×D59116yes—
342C2×C4C22×C4×D5918yes—
343C2×C4C22×C4×D5914yes—
344C2×C4C2×D4×D5914yes—
345C2×C4C2×D4×D5914yes—
346C22—28no—
347C22—24no—
348C22—5916no—
349C22—11832no—
350C22—11816no—
351C22—1184no—
352C22—1188no—
353C22—23632no—
354C22—23616no—
355C22—2368no—
356C22C2×C22⋊C4×D5914yes—
357C22C23×C4×D5911yescharacteristic
358C22C22×D4×D5912yes—
359C22#1683823914yes—
360C4—28no—
361C4—48no—
362C4—1184no—
363C4—2368no—
364C4C2×C22⋊C4×D5918yes—
365C2—24no—
366C2—11816no—
367C2—2364no—
368C2C22×C22⋊C4×D5911yescharacteristic
369C2#2143729812yes—
370C2#2143773914yes—
subgroupG/H
center, FrattiniC23C24×D59
commutatorC22×C59C24×C4
FittingC59×#2310C2
socle, cosocleC23×C59C25
prime radicalC23×#11745689C2
hypercenter#2310D59
hypocenterC59#34172
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#16837876C2semidirect
3. #21426925C2semidirect
4. #21426682C2semidirect
5. #21426673C2semidirect
6. #21426475C2semidirect
7. #381×D59C2semidirect
8. C59×#2310C2semidirect
10. C22×#11745689C4semidirect
17. C2×C4×D236C22semidirect
18. C2×#14273689C22semidirect
21. #16837309C22semidirect
22. #16837280C22semidirect
24. #16837264C22semidirect
26. #16837259C22semidirect
28. C4×(C2×D59)⋊C4C22semidirect
29. #95×D59C22semidirect
32. C59×#391C22semidirect
39. C2×#11745689C2×C4semidirect
42. C22×D236C2×C4semidirect
49. C22×C4×D59C2×C4semidirect
50. C22×C4×D59C2×C4semidirect
66. C22×Q8×C59C2×C4semidirect
102. C2×D236C22⋊C4semidirect
184. #2310D59semidirect
209. #391C2×D59semidirect
219. C59#34172semidirect
227. C22×Q8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C23×#117456892
C2×#168378762
#214269252
#214266822
#214266732
#214264752
#381×D592
C59×#23102
#3417259