Finite Group Databasebeta

← #28498385

Subgroups — #28498385

Order 15104.

552 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 87 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#21422417C211yesmaximal, characteristic
3#21422043C214yesmaximal, complemented
4C59×#1465C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic
6C2×#14273706—21no—
7#16837455—21no—
8#16837322—21no—
9#16837314—21no—
10#16837308—21no—
11#16837282C412yes—
12#16837274C2212yes—
13#16837262C411yescharacteristic
14#16837126—21no—
15#16837126C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#407—21no—
18C59×#324—21no—
19C59×#323C2212yes—
20C22×Dic118—21no—
21C22×Dic118—21no—
22C22×Dic118—44no—
23C22×Dic118—41no—
24C22×Dic118—41no—
25C23×Dic59C2×C412yes—
26#14273706—44no—
27C2×C4⋊Dic59—21no—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59C2×C412yes—
31C4⋊(C59⋊C8)—41no—
32C22⋊(C59⋊C8)—22no—
33C4×C59⋊C8—41no—
34C59⋊#95—21no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C22×D4×C59C2×C411yescharacteristic
38C23×C4×C59C2×C411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C2×M16×C59—41no—
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59D411yescharacteristic
47C22⋊C8×C59—22no—
48C22⋊C8×C59—41no—
49C24×C59C22⋊C411yescharacteristic
50C24×C59C22×C411yescharacteristic
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—41no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—44no—
68C22×Dic59—42no—
69C2×Dic118—24no—
70C2×Dic118—44no—
71C2×Dic118—42no—
72C2×Dic118—42no—
73C2×Dic118—48no—
74C2×Dic118—44no—
75C2×Dic118—41no—
76C2×Dic118—41no—
77C2×Dic118—44no—
78M16×C59—82no—
79C2×C8×C59—42no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C42×C59—21no—
84C42×C59—21no—
85C22⋊Dic59—24no—
86C22⋊Dic59—44no—
87C4⋊Dic59—42no—
88Dic59⋊C4—44no—
89C2×C59⋊C8—42no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59C2×C22⋊C411yescharacteristic
100C23×C59#13112yes—
101D4×C59—44no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—81no—
115C2×C4×C59#9911yescharacteristic
116C2×C4×C59#9911yescharacteristic
117C2×C4×C59#10112yes—
118C2×Dic59—24no—
119C2×Dic59—42no—
120C2×Dic59—44no—
121C2×Dic59—42no—
122C2×Dic59—44no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126Dic118—44no—
127Dic118—48no—
128Dic118—44no—
129C8×C59—82no—
130C59⋊C8—82no—
131#11580—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—22no—
135C22×C59—42no—
136C22×C59—42no—
137C22×C59—42no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—81no—
144C22×C59#40811yescharacteristic
145C22×C59C2×#9911yescharacteristic
146C22×C59C2×#10111yescharacteristic
147C4×C59—21no—
148C4×C59—22no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59AΣL(1,𝔽16)12yes—
153Dic59—44no—
154Dic59—82no—
155#2409—591no—
156#1465D5911yescharacteristic
157#1460—591no—
158#1326—594no—
159C2×C59—21no—
160C2×C59—21no—
161C2×C59—42no—
162C2×C59—81no—
163C2×C59#147211yescharacteristic
164C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
165C2×C59#212311yescharacteristic
166C23×Q8—591no—
167C2×C22⋊Q8—1181no—
168C2×C4×D4C2×D5911yescharacteristic
169#431—1181no—
170#407—21no—
171#395—1181no—
172#393—1181no—
173#391—1181no—
174#386—592no—
175#385—592no—
176#380—591no—
177#324—21no—
178#323—592no—
179#323—1181no—
180#323C2×D5912yes—
181C59#1158011yescharacteristic, complemented, Sylow
182C22×Q8—1181no—
183C22×Q8—1181no—
184C22×Q8—2361no—
185C22×Q8—2361no—
186C22×Q8—2364no—
187C22×D4C4×D5911yescharacteristic
188C23×C4—592no—
189C23×C4C4×D5911yescharacteristic
190C23×C4C22⋊D5911yescharacteristic
191C2×M16—41no—
192C22⋊Q8—2364no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1182no—
197C2×C4⋊C4C22⋊D5911yescharacteristic
198C2×C22⋊C4—592no—
199C2×C22⋊C4—1182no—
200C2×C22⋊C4D23611yescharacteristic
201C2×C22⋊C4C22×D5911yescharacteristic
202C2×C42D23611yescharacteristic
203C4⋊C8—2361no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—1182no—
207C4×C8—2361no—
208#95—1181no—
209#95—1182no—
210#95—1182no—
211C24(C2×D59)⋊C411yescharacteristic
212C24C2×C4×D5911yescharacteristic
213C2×Q8—1184no—
214C2×Q8—2362no—
215C2×Q8—2362no—
216C2×Q8—2364no—
217C2×Q8—2364no—
218C2×Q8—2368no—
219C2×Q8—2364no—
220C2×Q8—2361no—
221C2×Q8—2361no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—1182no—
232C22×C4—1182no—
233C22×C4—2364no—
234C22×C4—2362no—
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4C2×D23611yescharacteristic
239C22×C4C2×C22⋊D5911yescharacteristic
240M16—82no—
241C2×C8—42no—
242C2×C8—2362no—
243C4⋊C4—41no—
244C4⋊C4—2362no—
245C4⋊C4—2364no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—1184no—
249C22⋊C4—2364no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×(C2×D59)⋊C411yescharacteristic
262C23#1427373512yes—
263Q8—2364no—
264Q8—2368no—
265Q8—2364no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—1184no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2362no—
285C2×C4—2364no—
286C2×C4—2362no—
287C2×C4—2364no—
288C2×C4#1427364111yescharacteristic
289C2×C4#1427364111yescharacteristic
290C2×C4#1427364512yes—
291C8—82no—
292C8—4722no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#1683737011yescharacteristic
306C22C2×#1427364111yescharacteristic
307C22C2×#1427364511yescharacteristic
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—4722no—
314C4—2364no—
315C4#1683718812yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—81no—
320C2#2142246111yescharacteristic
321C2C2×#1683718811yescharacteristic
322C2#2142575111yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1465C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#1465D59
hypocenterC59#11580
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #21422043C2semidirect
181. C59#11580semidirect

Maximal subgroups

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

subgroupindex
#214274612
#214224172
#214220432
C59×#14652
#1158059