Finite Group Databasebeta

← #28486643

Subgroups — #28486643

Order 15104.

682 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 107 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#21427461C211yesmaximal, characteristic
2#21424019C212yesmaximal, complemented
3#21422434C211yesmaximal, characteristic
4#21422001C212yesmaximal, complemented
5C59×#2975C211yesmaximal, characteristic
6C23×Dic118C2211yescharacteristic
7C2×#14273706—21no—
8C2×#14273645—24no—
9#16837366C2212yes—
10#16837322—21no—
11#16837314—21no—
12#16837308—21no—
13#16837282C412yes—
14#16837274C412yes—
15#16837262C2211yescharacteristic
16#16837128—21no—
17#16837126—21no—
18C2×C4×D4×C59C2211yescharacteristic
19C59×#436—24no—
20C59×#431—21no—
21C59×#407—21no—
22C59×#407C2212yes—
23C22×Dic118—22no—
24C22×Dic118—21no—
25C22×Dic118—21no—
26C22×Dic118—22no—
27C22×Dic118—44no—
28C23×Dic59C2×C412yes—
29C2×#11745664—22no—
30#14273706—44no—
31C2×C4⋊Dic59—21no—
32C2×Dic59⋊C4—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59C2×C412yes—
35#14273645—48no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—21no—
41C22×D4×C59C2×C411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C23×C4×C59C2311yescharacteristic
44C2×M16×C59—22no—
45C2×M16×C59D414yes—
46C4×D4×C59—22no—
47C4×D4×C59—22no—
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C4⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54C22⋊C8×C59—22no—
55C22⋊C8×C59—22no—
56C4×C8×C59—22no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C22×C411yescharacteristic
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C2×D412yes—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59#4112yes—
73C22×Dic59—22no—
74C22×Dic59—24no—
75C22×Dic59—22no—
76C22×Dic59—44no—
77C2×Dic118—28no—
78C2×Dic118—24no—
79C2×Dic118—44no—
80C2×Dic118—48no—
81C2×Dic118—44no—
82C2×Dic118—42no—
83C2×Dic118—41no—
84C2×Dic118—42no—
85C2×Dic118—41no—
86M16×C59—44no—
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—44no—
95C22⋊Dic59—42no—
96C4⋊Dic59—42no—
97Dic59⋊C4—44no—
98#11745664—44no—
99C2×C59⋊C8—42no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59C4×D412yes—
111D4×C59—44no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—24no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—81no—
124C2×C4×C59#10112yescomplemented
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12112yes—
127C2×C4×C59#12311yescharacteristic
128C2×Dic59—24no—
129C2×Dic59—28no—
130C2×Dic59—42no—
131C2×Dic59—44no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136Dic118—48no—
137Dic118—44no—
138Dic118—44no—
139C8×C59—44no—
140C59⋊C8—82no—
141#9506—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—22no—
146C22×C59—22no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59#38611yescharacteristic
155C22×C59C2×#10111yescharacteristic
156C22×C59#41311yescharacteristic
157C4×C59—22no—
158C4×C59—21no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59B(4,𝔽2)12yes—
163Dic59—44no—
164Dic59—82no—
165#2975D5911yescharacteristic
166#2409—591no—
167#1784—592no—
168#1465—591no—
169#1305—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—42no—
173C2×C59—81no—
174C2×C59#188911yescharacteristic
175C2×C59#190111yescharacteristic
176C2×C59#191511yescharacteristic
177C23×Q8—591no—
178C2×C22⋊Q8—1181no—
179C2×C4×D4C2×D5911yescharacteristic
180#436—24no—
181#431—21no—
182C2×#101—1184no—
183#407—21no—
184#407—592no—
185#407C2×D5912yes—
186#395—1181no—
187#393—1181no—
188#391—1181no—
189#386—592no—
190#385—592no—
191#380—591no—
192#324—1181no—
193#323—1181no—
194C59#950611yescharacteristic, complemented, Sylow
195C22×Q8—1181no—
196C22×Q8—1182no—
197C22×Q8—1181no—
198C22×Q8—1182no—
199C22×Q8—2364no—
200C22×D4C4×D5911yescharacteristic
201C23×C4—592no—
202C23×C4C22⋊D5911yescharacteristic
203C23×C4C22×D5911yescharacteristic
204C2×M16—22no—
205C2×M16—1182no—
206C2×M16C22⋊D5914yes—
207C22⋊Q8—2364no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—1182no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4D23611yescharacteristic
213C2×C22⋊C4—592no—
214C2×C22⋊C4—1182no—
215C2×C22⋊C4C4×D5911yescharacteristic
216C2×C22⋊C4D23611yescharacteristic
217C2×C42C22⋊D5911yescharacteristic
218C4⋊C8—22no—
219#101—2368no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—1182no—
224C22⋊C8—1182no—
225C4×C8—22no—
226#95—1182no—
227#95—1182no—
228#95—1181no—
229C24(C2×D59)⋊C411yescharacteristic
230C24C2×C4×D5911yescharacteristic
231C2×Q8—1188no—
232C2×Q8—1184no—
233C2×Q8—2362no—
234C2×Q8—2364no—
235C2×Q8—2364no—
236C2×Q8—2368no—
237C2×Q8—2361no—
238C2×Q8—2362no—
239C2×Q8—2361no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—1182no—
248C22×C4—1184no—
249C22×C4—1182no—
250C22×C4—2364no—
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4C2×D23611yescharacteristic
255C22×C4#1174568512yes—
256C22×C4C2×C22⋊D5912yes—
257C22×C4C2×C22⋊D5911yescharacteristic
258M16—44no—
259M16—82no—
260M16—2364no—
261C2×C8—24no—
262C2×C8—2362no—
263C4⋊C4—22no—
264C4⋊C4—2362no—
265C4⋊C4—2364no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—2364no—
269C22⋊C4—2362no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×(C2×D59)⋊C411yescharacteristic
282C23C4×C22⋊D5912yes—
283Q8—2368no—
284Q8—2364no—
285Q8—2364no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—1184no—
300C2×C4—1188no—
301C2×C4—2362no—
302C2×C4—2362no—
303C2×C4—2362no—
304C2×C4—2362no—
305C2×C4—2364no—
306C2×C4—2362no—
307C2×C4#1427364512yescomplemented
308C2×C4#1427369611yescharacteristic
309C2×C4#1427370112yes—
310C2×C4#1427371011yescharacteristic
311C8—44no—
312C8—4722no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1683728111yescharacteristic
326C22C2×#1427364511yescharacteristic
327C22#1683738411yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—4722no—
334C4—2364no—
335C4#1683758912yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#2142450411yescharacteristic
341C2#2142456411yescharacteristic
342C2#2142464811yescharacteristic
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#9506
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
2. #21424019C2semidirect
4. #21422001C2semidirect
124. C2×C4×C59#101semidirect
194. C59#9506semidirect
307. C2×C4#14273645semidirect

Maximal subgroups

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

subgroupindex
#214274612
#214240192
#214224342
#214220012
C59×#29752
#950659