Finite Group Databasebeta

← #28488443

Subgroups — #28488443

Order 15104.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425058C211yesmaximal, characteristic, complemented
2#21424903C211yesmaximal, characteristic
3#21424903C211yesmaximal, characteristic
4#21424899C211yesmaximal, characteristic, complemented
5#21424224C211yesmaximal, characteristic, complemented
6C2×C8×C59⋊C8C211yesmaximal, characteristic, complemented
7C2×C59×#486C211yesmaximal, characteristic
8C2×#14273651—22no—
9C2×#14273648—22no—
10C2×#14273648—22no—
11C2×C4×C59⋊C8C2211yescharacteristic
12C2×C4×C59⋊C8C2211yescharacteristic
13#16837272C2211yescharacteristic
14#16837272C2211yescharacteristic
15#16837165—22no—
16#16837164—22no—
17C8×C59⋊C8—22no—
18C2×C4⋊Q8×C59C2211yescharacteristic
19C2×#127×C59C2211yescharacteristic
20C59×#486—22no—
21C59×#486C414yescomplemented
22C2×#102×C59—22no—
23C2×C4×C8×C59C2211yescharacteristic
24C22×C59⋊C8D412yes—
25C22×C59⋊C8D412yes—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—21no—
32#14273651—44no—
33#14273648—44no—
34#14273648—44no—
35C4×C59⋊C8—21no—
36C4×C59⋊C8—21no—
37C4×C59⋊C8—21no—
38C4×C59⋊C8—21no—
39C22×Q8×C59—21no—
40C22×D4×C59—21no—
41C22×D4×C59—21no—
42C22×D4×C59—21no—
43C22×C8×C59D412yes—
44C4⋊Q8×C59—21no—
45C4⋊Q8×C59—21no—
46C4⋊Q8×C59—21no—
47C4⋊Q8×C59C2×C412yes—
48#127×C59—21no—
49#127×C59—21no—
50#127×C59—21no—
51#127×C59C2×C412yes—
52C2×C4⋊C4×C59—21no—
53C2×C4⋊C4×C59—21no—
54C2×C42×C59C2311yescharacteristic
55#102×C59—28no—
56C4×C8×C59C2×C412yes—
57C4×C8×C59D412yes—
58C24×C59—41no—
59C2×Q8×C59—22no—
60C2×Q8×C59—41no—
61C2×Q8×C59—42no—
62C2×Q8×C59—42no—
63C2×D4×C59—22no—
64C2×D4×C59—22no—
65C2×D4×C59—22no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—41no—
69C2×D4×C59—42no—
70C2×D4×C59—42no—
71C2×D4×C59—42no—
72C2×D4×C59—42no—
73C2×D4×C59—42no—
74C2×D4×C59—42no—
75C22×C4×C59—41no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×Dic59—41no—
80C22×Dic59—41no—
81C2×C8×C59—22no—
82C2×C8×C59—22no—
83C2×C8×C59C22⋊C414yes—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C42×C59C22×C411yescharacteristic
89C42×C59C2×D411yescharacteristic
90C42×C59#4111yescharacteristic
91C42×C59#4111yescharacteristic
92C4⋊Dic59—42no—
93C4⋊Dic59—42no—
94C4⋊Dic59—42no—
95C4⋊Dic59—42no—
96C4⋊Dic59—42no—
97C4⋊Dic59—42no—
98C2×C59⋊C8—22no—
99C2×C59⋊C8—22no—
100C2×C59⋊C8—22no—
101C2×C59⋊C8—22no—
102C2×C59⋊C8D814yes—
103C2×C59⋊C8SD1614yes—
104C23×C59—42no—
105C23×C59—81no—
106C23×C59—81no—
107C23×C59—81no—
108C23×C59—81no—
109C23×C59—81no—
110C23×C59—81no—
111C23×C59#12711yescharacteristic
112Q8×C59—44no—
113Q8×C59—82no—
114D4×C59—44no—
115D4×C59—44no—
116D4×C59—44no—
117D4×C59—82no—
118D4×C59—82no—
119D4×C59—82no—
120C2×C4×C59—42no—
121C2×C4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59C2×C22⋊C411yescharacteristic
124C2×C4×C59C4×D411yescharacteristic
125C2×C4×C59C4×D411yescharacteristic
126C2×C4×C59#12111yescharacteristic
127C2×C4×C59#12111yescharacteristic
128C2×C4×C59#12411yescharacteristic
129C2×C4×C59C2×D812yes—
130C2×C4×C59C2×SD1612yes—
131C2×C4×C59#13512yes—
132C2×C4×C59#13512yes—
133C2×C4×C59#13512yes—
134C2×C4×C59#13512yes—
135C2×Dic59—81no—
136C2×Dic59—81no—
137C2×Dic59—81no—
138C2×Dic59—81no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141C8×C59—24no—
142C59⋊C8—24no—
143C59⋊C8—24no—
144#9822—591nomaximal, Sylow
145C22×C59—82no—
146C22×C59—82no—
147C22×C59—82no—
148C22×C59—161no—
149C22×C59—161no—
150C22×C59—161no—
151C22×C59—161no—
152C22×C59#39011yescharacteristic
153C22×C59#49211yescharacteristic
154C22×C59#49311yescharacteristic
155C22×C59#49511yescharacteristic
156C22×C59#49511yescharacteristic
157C22×C59#49511yescharacteristic
158C22×C59#49511yescharacteristic
159C4×C59—82no—
160C4×C59#41612yes—
161C4×C59C4×D812yes—
162C4×C59C4×SD1612yes—
163C4×C59#46512yes—
164C4×C59#46612yes—
165C4×C59#48812yes—
166Dic59—161no—
167Dic59—161no—
168C2×#486D5911yescharacteristic
169#1990—591no—
170#1957—591no—
171#1957—591no—
172#1956—591no—
173#1834—591no—
174C2×C82—591no—
175C2×C59—162no—
176C2×C59#170611yescharacteristic
177C2×C59#170811yescharacteristic
178C2×C59#170911yescharacteristic
179C2×C59#171111yescharacteristic
180C2×C59#195811yescharacteristic
181C2×C59#196811yescharacteristic
182C2×C59#197011yescharacteristic
183C2×C4⋊Q8C2×D5911yescharacteristic
184C2×#127C2×D5911yescharacteristic
185#486—22no—
186#486Dic5914yescomplemented
187C2×#103—1182no—
188C2×#102—22no—
189C2×#102—1182no—
190C2×#102—1182no—
191C2×C4×C8—591no—
192C2×C4×C8—591no—
193C2×C4×C8C2×D5911yescharacteristic
194#384—591no—
195#384—591no—
196#340—1182no—
197#340—1182no—
198C82—1182no—
199C59#982211yescharacteristic, complemented, Sylow
200C22×Q8—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—21no—
204C22×C8—592no—
205C22×C8—592no—
206C22×C8C22⋊D5912yes—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8C2×Dic5912yes—
211#127—21no—
212#127—21no—
213#127—21no—
214#127C2×Dic5912yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C42C22×D5911yescharacteristic
224#103—2364no—
225#102—28no—
226#102—2364no—
227#102—2364no—
228C4×C8—1181no—
229C4×C8—1181no—
230C4×C8—1181no—
231C4×C8—1181no—
232C4×C8C2×Dic5912yes—
233C4×C8C22⋊D5912yes—
234C24—41no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C22×C4—41no—
252C22×C4—2361no—
253C22×C4—2361no—
254C22×C4D4×D5911yescharacteristic
255C22×C4D4×D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—594no—
260C2×C8—594no—
261C2×C8—1182no—
262C2×C8—1182no—
263C2×C8—1182no—
264C2×C8—1182no—
265C2×C8C22⋊Dic5914yes—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—2362no—
271C4⋊C4—2362no—
272C4⋊C4—2362no—
273C4⋊C4—2362no—
274C4⋊C4—2362no—
275C4⋊C4—2362no—
276C42#1174568711yescharacteristic
277C42#1174568711yescharacteristic
278C42C22×Dic5911yescharacteristic
279C42C2×C22⋊D5911yescharacteristic
280C23—42no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23#1427372311yescharacteristic
288Q8—44no—
289Q8—82no—
290D4—44no—
291D4—44no—
292D4—44no—
293D4—82no—
294D4—82no—
295D4—82no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—4721no—
300C2×C4—4721no—
301C2×C4—4721no—
302C2×C4—4721no—
303C2×C4—4721no—
304C2×C4—4721no—
305C2×C4C2×C22⋊Dic5911yescharacteristic
306C2×C4D4×Dic5911yescharacteristic
307C2×C4D4×Dic5911yescharacteristic
308C2×C4#1427369811yescharacteristic
309C2×C4#1427369811yescharacteristic
310C2×C4#1427371411yescharacteristic
311C2×C4D8×D5912yes—
312C2×C4SD16×D5912yes—
313C2×C4#1427374612yes—
314C2×C4#1427374712yes—
315C2×C4#1427374712yes—
316C2×C4#1427374812yes—
317C8—24no—
318C8—1184no—
319C8—1184no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—161no—
324C22—161no—
325C22—161no—
326C22—161no—
327C22#1683730511yescharacteristic
328C22#1683777311yescharacteristic
329C22#1683777711yescharacteristic
330C22#1683778311yescharacteristic
331C22#1683778411yescharacteristic
332C22#1683778411yescharacteristic
333C22#1683778511yescharacteristic
334C4—82no—
335C4—9441no—
336C4—9441no—
337C4#1683739812yes—
338C4D8×Dic5912yes—
339C4SD16×Dic5912yes—
340C4#1683762812yes—
341C4#1683763512yes—
342C4#1683775612yes—
343C2—162no—
344C2#2142368711yescharacteristic
345C2#2142369611yescharacteristic
346C2#2142370011yescharacteristic
347C2#2142370711yescharacteristic
348C2#2142490811yescharacteristic
349C2#2142495211yescharacteristic
350C2#2142496311yescharacteristic
subgroupG/H
centerC23#14273723
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#486C2
socleC23×C59#127
prime radicalC2×#127×C59C22
cosocleC2×C42×C59C23
hypercenterC2×#486D59
hypocenterC59#9822
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #21425058C2semidirect
4. #21424899C2semidirect
5. #21424224C2semidirect
6. C2×C8×C59⋊C8C2semidirect
21. C59×#486C4semidirect
186. #486Dic59semidirect
199. C59#9822semidirect

Maximal subgroups

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

subgroupindex
#214250582
#214249032
#214248992
#214242242
C2×C8×C59⋊C82
C2×C59×#4862
#982259