Finite Group Databasebeta

← #28486703

Subgroups — #28486703

Order 15104.

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

normal subgroups only

This group has 518 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#21429439C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422425C212yesmaximal, complemented
4#21422003C212yesmaximal, complemented
5C59×#2975C211yesmaximal, characteristic
6C22×Dic59⋊C4C2211yescharacteristic
7#16837366C2212yes—
8#16837335—21no—
9#16837325—21no—
10#16837286—21no—
11#16837282C411yescharacteristic
12#16837282C411yescharacteristic
13#16837275—21no—
14#16837274C411yescharacteristic
15#16837274C411yescharacteristic
16#16837262C2211yescharacteristic
17#16837128—21no—
18#16837127—24no—
19#16837126—21no—
20C2×C4×D4×C59C2211yescharacteristic
21C59×#436—24no—
22C59×#431—21no—
23C59×#407—21no—
24C59×#407C2212yes—
25C23×Dic59C2×C411yescharacteristic
26C23×Dic59C2×C411yescharacteristic
27C2×#11745664—22no—
28C2×C4⋊Dic59—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—22no—
34C2×Dic59⋊C4—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59C2×C411yescharacteristic
38C2×C22⋊Dic59C2×C411yescharacteristic
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—22no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×D4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59C2311yescharacteristic
49C2×M16×C59—22no—
50C2×M16×C59—22no—
51C4×D4×C59—22no—
52C4×D4×C59—22no—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C42×C59D411yescharacteristic
57C4⋊C8×C59—22no—
58C22⋊C8×C59—22no—
59C22⋊C8×C59—22no—
60C22⋊C8×C59D414yes—
61C4×C8×C59—22no—
62C24×C59C22⋊C411yescharacteristic
63C24×C59C22×C411yescharacteristic
64C2×D4×C59—22no—
65C2×D4×C59—22no—
66C2×D4×C59—41no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—81no—
70C22×C4×C59—81no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C2×D412yes—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59#4112yes—
78C22×Dic59—21no—
79C22×Dic59—22no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—22no—
84C22×Dic59—42no—
85C22×Dic59—81no—
86M16×C59—82no—
87M16×C59—82no—
88C2×C8×C59—24no—
89C4⋊C4×C59—22no—
90C22⋊C4×C59—22no—
91C22⋊C4×C59—22no—
92C42×C59—21no—
93C42×C59—21no—
94C22⋊Dic59—42no—
95C22⋊Dic59—41no—
96C22⋊Dic59—41no—
97C22⋊Dic59—81no—
98C4⋊Dic59—81no—
99Dic59⋊C4—42no—
100Dic59⋊C4—41no—
101Dic59⋊C4—42no—
102Dic59⋊C4—41no—
103Dic59⋊C4—81no—
104#11745664—82no—
105C2×C59⋊C8—42no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59C2×C22⋊C411yescharacteristic
116C23×C59C4×D412yes—
117D4×C59—44no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—81no—
130C2×C4×C59#10112yes—
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12112yes—
133C2×C4×C59#12311yescharacteristic
134C2×Dic59—22no—
135C2×Dic59—24no—
136C2×Dic59—41no—
137C2×Dic59—42no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—82no—
149C8×C59—44no—
150C59⋊C8—82no—
151#9518—591nomaximal, Sylow
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—22no—
156C22×C59—22no—
157C22×C59—42no—
158C22×C59—42no—
159C22×C59—42no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59#38611yescharacteristic
165C22×C59C2×#10111yescharacteristic
166C22×C59#41311yescharacteristic
167C4×C59—22no—
168C4×C59—22no—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—41no—
172C4×C59—41no—
173Dic59—42no—
174Dic59—81no—
175Dic59—81no—
176Dic59—81no—
177#2975D5911yescharacteristic
178#2579—591no—
179#1465—591no—
180#1463—592no—
181#1305—592no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—42no—
185C2×C59—81no—
186C2×C59#189111yescharacteristic
187C2×C59#190111yescharacteristic
188C2×C59#191311yescharacteristic
189C2×C4×D4C2×D5911yescharacteristic
190C22×C4⋊C4—591no—
191#436—24no—
192#431—21no—
193#407—21no—
194#407—592no—
195#407C2×D5912yes—
196#398—1181no—
197#396—1181no—
198#387—1181no—
199#386—591no—
200#386—591no—
201#385—591no—
202#385—591no—
203#385—1181no—
204#380—591no—
205#324—1181no—
206#324—1184no—
207#323—1181no—
208C59#951811yescharacteristic, complemented, Sylow
209C22×D4C4×D5911yescharacteristic
210C23×C4—591no—
211C23×C4—591no—
212C23×C4C22⋊D5911yescharacteristic
213C23×C4C22×D5911yescharacteristic
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—1182no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1182no—
224C2×C4⋊C4—1182no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4D23611yescharacteristic
227C2×C22⋊C4—591no—
228C2×C22⋊C4—591no—
229C2×C22⋊C4—1181no—
230C2×C22⋊C4—1181no—
231C2×C22⋊C4C4×D5911yescharacteristic
232C2×C22⋊C4D23611yescharacteristic
233C2×C42C22⋊D5911yescharacteristic
234C4⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—1182no—
238C22⋊C8—1182no—
239C22⋊C8C22⋊D5914yes—
240C4×C8—22no—
241#95—1181no—
242#95—1181no—
243#95—1181no—
244#95—1181no—
245#95—1181no—
246C24(C2×D59)⋊C411yescharacteristic
247C24C2×C4×D5911yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—1181no—
256C22×C4—1182no—
257C22×C4—1182no—
258C22×C4—1181no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—4721no—
262C22×C4—2362no—
263C22×C4(C2×D59)⋊C411yescharacteristic
264C22×C4(C2×D59)⋊C411yescharacteristic
265C22×C4(C2×D59)⋊C411yescharacteristic
266C22×C4C2×D23611yescharacteristic
267C22×C4#1174568512yes—
268C22×C4C2×C22⋊D5912yes—
269C22×C4C2×C22⋊D5911yescharacteristic
270M16—82no—
271M16—82no—
272M16—4722no—
273C2×C8—24no—
274C2×C8—2362no—
275C4⋊C4—22no—
276C4⋊C4—4721no—
277C4⋊C4—4721no—
278C4⋊C4—2362no—
279C4⋊C4—2361no—
280C4⋊C4—2362no—
281C4⋊C4—2361no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—4721no—
285C22⋊C4—2362no—
286C22⋊C4—2361no—
287C22⋊C4—2361no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×(C2×D59)⋊C411yescharacteristic
300C23C4×C22⋊D5912yes—
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—1182no—
315C2×C4—1184no—
316C2×C4—4722no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2362no—
328C2×C4—2361no—
329C2×C4#1427364512yes—
330C2×C4#1427369611yescharacteristic
331C2×C4#1427370112yes—
332C2×C4#1427371011yescharacteristic
333C8—44no—
334C8—4722no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1683728111yescharacteristic
348C22C2×#1427364511yescharacteristic
349C22#1683738411yescharacteristic
350C4—22no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—4721no—
357C4—4721no—
358C4—4721no—
359C4—2362no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—81no—
364C2#2142451411yescharacteristic
365C2#2142456411yescharacteristic
366C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2975C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC23×C4×C59C23
hypercenter#2975D59
hypocenterC59#9518
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #21422425C2semidirect
4. #21422003C2semidirect
208. C59#9518semidirect

Maximal subgroups

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

subgroupindex
#214294392
#214224342
#214224252
#214220032
C59×#29752
#951859