Finite Group Databasebeta

← #28486344

Subgroups — #28486344

Order 15104.

516 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 103 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#21432734C211yesmaximal, characteristic, complemented
2#21424173C211yesmaximal, characteristic, complemented
3#21422625C212yesmaximal
4#21421949C212yesmaximal, complemented
5C59×#2500C211yesmaximal, characteristic
6#16837463—24no—
7#16837426—21no—
8#16837366C2212yes—
9C4×#11745664—21no—
10#16837265C2212yes—
11#16837169—21no—
12#16837169—21no—
13#16837136—24no—
14C2×#121×C59C2211yescharacteristic
15C2×#117×C59C2211yescharacteristic
16C59×#394—21no—
17C59×#394—21no—
18C59×#390C411yescharacteristic, complemented
19C59×#390C411yescharacteristic, complemented
20C59×#388—21no—
21C59×#388—21no—
22C59×#384C411yescharacteristic, complemented
23C59×#382C411yescharacteristic, complemented
24C4×C22⋊C4×C59C2211yescharacteristic
25C2×#11745664—22no—
26C2×C22⋊Dic59—22no—
27C4⋊(C59⋊C8)—24no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)D414yes—
30#14273638—22no—
31C4×C59⋊C8—22no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C22×D4×C59—21no—
35C22×D4×C59C2×C411yescharacteristic
36C23×C4×C59C2311yescharacteristic
37#121×C59—22no—
38#121×C59—22no—
39#117×C59—22no—
40#117×C59—22no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C42×C59C2×C411yescharacteristic
55C2×C42×C59C2×C411yescharacteristic
56C2×C42×C59D411yescharacteristic
57C2×C42×C59D411yescharacteristic
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61C24×C59—21no—
62C24×C59C2×D411yescharacteristic
63C2×D4×C59—22no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C2×D4×C59—42no—
67C2×D4×C59—81no—
68C2×D4×C59—81no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22×C411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59#4112yes—
82C22×Dic59—42no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—81no—
87C4⋊C4×C59—81no—
88C22⋊C4×C59—22no—
89C22⋊C4×C59—22no—
90C22⋊C4×C59—42no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—81no—
96C22⋊C4×C59—81no—
97C42×C59—21no—
98C42×C59—21no—
99C42×C59—21no—
100C42×C59—21no—
101C42×C59—41no—
102C42×C59—41no—
103C22⋊Dic59—42no—
104C22⋊Dic59—42no—
105#11745664—42no—
106#11745664—42no—
107C2×C59⋊C8—24no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59C2×C22⋊C411yescharacteristic
120C23×C59#12112yes—
121D4×C59—82no—
122D4×C59—84no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—22no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59C4×D412yes—
147C2×C4×C59#12011yescharacteristic
148C2×C4×C59#12311yescharacteristic
149C2×Dic59—82no—
150C2×Dic59—82no—
151C2×Dic59—82no—
152C59⋊C8—44no—
153#9446—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—22no—
157C22×C59—22no—
158C22×C59—22no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59—81no—
162C22×C59—81no—
163C22×C59—81no—
164C22×C59—81no—
165C22×C59—81no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59#38611yescharacteristic
169C22×C59#41011yescharacteristic
170C22×C59#41011yescharacteristic
171C4×C59—21no—
172C4×C59—22no—
173C4×C59—21no—
174C4×C59—42no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—81no—
180C4×C59—81no—
181Dic59—162no—
182#2921—591no—
183#2500D5911yescharacteristic
184#1820—591no—
185#1510—592no—
186#1284—592no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—42no—
190C2×C59—161no—
191C2×C59#188411yescharacteristic
192C2×C59#189911yescharacteristic
193C2×C59#191011yescharacteristic
194C2×#121C2×D5911yescharacteristic
195C2×#117C2×D5911yescharacteristic
196#433—1184no—
197#423—1181no—
198#407—592no—
199C4×M16—1181no—
200#394—21no—
201#394—21no—
202#390Dic5911yescharacteristic, complemented
203#390Dic5911yescharacteristic, complemented
204#388—21no—
205#388—21no—
206#384Dic5911yescharacteristic, complemented
207#382Dic5911yescharacteristic, complemented
208#381—592no—
209C4×C22⋊C4C2×D5911yescharacteristic
210#342—1181no—
211#342—1181no—
212#327—1184no—
213C59#944611yescharacteristic, complemented, Sylow
214C22×D4—21no—
215C22×D4C2×Dic5911yescharacteristic
216C23×C4C22×D5911yescharacteristic
217C2×M16—1182no—
218#121—22no—
219#121—22no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic5911yescharacteristic
225C2×C4⋊C4C2×Dic5911yescharacteristic
226C2×C4⋊C4C2×Dic5911yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—1182no—
234C2×C22⋊C4C22⋊D5911yescharacteristic
235C2×C22⋊C4C22⋊D5911yescharacteristic
236C2×C42C2×Dic5911yescharacteristic
237C2×C42C2×Dic5911yescharacteristic
238C2×C42C22⋊D5911yescharacteristic
239C2×C42C22⋊D5911yescharacteristic
240C4⋊C8—1184no—
241C22⋊C8—594no—
242C22⋊C8—1182no—
243#97—1182no—
244C4×C8—1182no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—1182no—
249#95—1182no—
250C24—21no—
251C24C2×C22⋊D5911yescharacteristic
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—81no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—2362no—
264C22×C4C22⋊Dic5911yescharacteristic
265C22×C4C22⋊Dic5911yescharacteristic
266C22×C4C22⋊Dic5911yescharacteristic
267C22×C4C22⋊Dic5911yescharacteristic
268C22×C4D4×D5912yes—
269C22×C4#1174568712yes—
270C22×C4C22×Dic5911yescharacteristic
271C22×C4C2×C22⋊D5911yescharacteristic
272M16—2362no—
273M16—2362no—
274C2×C8—1184no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—2362no—
290C22⋊C4—2362no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—41no—
296C42—41no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×C22⋊Dic5911yescharacteristic
309C23#1427369912yes—
310D4—82no—
311D4—84no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—4722no—
336C2×C4—4722no—
337C2×C4—4722no—
338C2×C4D4×Dic5912yes—
339C2×C4#1427369511yescharacteristic
340C2×C4#1427370811yescharacteristic
341C8—2364no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1683728211yescharacteristic
357C22#1683737611yescharacteristic
358C22#1683737611yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—42no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—9442no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2#2142448011yescharacteristic
375C2#2142455511yescharacteristic
376C2#2142462411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2500C2
socleC22×C59#386
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenter#2500D59
hypocenterC59#9446
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2500

Complements

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

NH ≅ G/Ntype
1. #21432734C2semidirect
2. #21424173C2semidirect
4. #21421949C2semidirect
18. C59×#390C4semidirect
19. C59×#390C4semidirect
22. C59×#384C4semidirect
23. C59×#382C4semidirect
202. #390Dic59semidirect
203. #390Dic59semidirect
206. #384Dic59semidirect
207. #382Dic59semidirect
213. C59#9446semidirect

Maximal subgroups

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

subgroupindex
#214327342
#214241732
#214226252
#214219492
C59×#25002
#944659