Finite Group Databasebeta

← #28498348

Subgroups — #28498348

Order 15104.

416 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429292C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422041C212yesmaximal, complemented
4#21422039C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic
6C2×#14273719C411yescharacteristic
7#16837366—21no—
8#16837325—21no—
9#16837325—21no—
10#16837305C411yescharacteristic
11#16837305C411yescharacteristic
12#16837275—21no—
13#16837275—21no—
14#16837271C2211yescharacteristic
15#16837267C2211yescharacteristic
16C4×C4⋊Dic59C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C2×#11745664—41no—
24#14273719—41no—
25#14273719—41no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C4×Dic59C2×C411yescharacteristic
38C2×C4×Dic59C2×C411yescharacteristic
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C22×D4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C22×C8×C59—41no—
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59C2×C411yescharacteristic
54C22⋊C8×C59—22no—
55C22⋊C8×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C24×C59C22⋊C411yescharacteristic
58C2×D4×C59—22no—
59C2×D4×C59—22no—
60C2×D4×C59—41no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—81no—
66C22×C4×C59—81no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C2×C8×C59—42no—
77C2×C8×C59—82no—
78C4⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C42×C59—21no—
82C42×C59—21no—
83C22⋊Dic59—42no—
84C22⋊Dic59—42no—
85C22⋊Dic59—81no—
86C22⋊Dic59—81no—
87C4⋊Dic59—42no—
88C4⋊Dic59—41no—
89C4⋊Dic59—41no—
90Dic59⋊C4—42no—
91Dic59⋊C4—41no—
92Dic59⋊C4—41no—
93Dic59⋊C4—81no—
94Dic59⋊C4—81no—
95C4×Dic59—22no—
96C4×Dic59—22no—
97#11745664—82no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59#10112yes—
109C23×C59C2×C22⋊C411yescharacteristic
110D4×C59—44no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—41no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—81no—
123C2×C4×C59—81no—
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#9911yescharacteristic
126C2×C4×C59#13112yes—
127C2×Dic59—22no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C8×C59—82no—
140C59⋊C8—82no—
141#11572—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59#40811yescharacteristic
155C22×C59C2×#9911yescharacteristic
156C22×C59C2×#10111yescharacteristic
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163Dic59—42no—
164Dic59—81no—
165Dic59—81no—
166Dic59—81no—
167#2566—591no—
168#1465—591no—
169#1459D5911yescharacteristic
170#1325—592no—
171#1324—592no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59—42no—
175C2×C59—81no—
176C2×C59#147211yescharacteristic
177C2×C59#212711yescharacteristic
178C2×C59#213211yescharacteristic
179C2×#125—591no—
180C2×C4×D4C2×D5911yescharacteristic
181#407—1181no—
182C2×C22⋊C8—21no—
183#396—1181no—
184#396—1181no—
185#390—591no—
186#390—591no—
187#385—1181no—
188#385—1181no—
189#384—591no—
190#382—591no—
191C4×C4⋊C4—591no—
192#324—592no—
193#324C2×D5912yes—
194#323—21no—
195#323—1181no—
196C59#1157211yescharacteristic, complemented, Sylow
197C22×D4C4×D5911yescharacteristic
198C23×C4D23611yescharacteristic
199C23×C4C22⋊D5911yescharacteristic
200C2×M16—2361no—
201C22×C8—41no—
202#125—2361no—
203#125—2361no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—591no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4C22×D5911yescharacteristic
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4D23611yescharacteristic
219C2×C22⋊C4C22⋊D5911yescharacteristic
220C2×C42—591no—
221C2×C42—591no—
222C2×C42C4×D5911yescharacteristic
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—1182no—
226C22⋊C8—2361no—
227#95—1181no—
228#95—1181no—
229#95—1181no—
230C24(C2×D59)⋊C411yescharacteristic
231C24(C2×D59)⋊C411yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—1181no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4—1181no—
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4(C2×D59)⋊C411yescharacteristic
247C22×C4C2×C4×D5911yescharacteristic
248C22×C4C2×D23611yescharacteristic
249C22×C4C2×C22⋊D5911yescharacteristic
250M16—4722no—
251C2×C8—42no—
252C2×C8—82no—
253C2×C8—2362no—
254C4⋊C4—41no—
255C4⋊C4—4721no—
256C4⋊C4—4721no—
257C4⋊C4—2361no—
258C4⋊C4—2361no—
259C4⋊C4—2362no—
260C4⋊C4—2362no—
261C4⋊C4—2361no—
262C4⋊C4—2361no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—4721no—
266C22⋊C4—4721no—
267C22⋊C4—2362no—
268C22⋊C4—2362no—
269C42—21no—
270C42—21no—
271C42—1182no—
272C42—1182no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1427364512yes—
283C23C2×(C2×D59)⋊C411yescharacteristic
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—1182no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4#1427364111yescharacteristic
311C2×C4#1427364111yescharacteristic
312C2×C4#1427373312yes—
313C8—82no—
314C8—4722no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22#1683736811yescharacteristic
328C22C2×#1427364111yescharacteristic
329C22C2×#1427364511yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—4721no—
337C4—4721no—
338C4—4721no—
339C4—2362no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2142245911yescharacteristic
345C2#2142576311yescharacteristic
346C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11572
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
4. #21422039C2semidirect
196. C59#11572semidirect

Maximal subgroups

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

subgroupindex
#214292922
#214224342
#214220412
#214220392
C59×#14592
#1157259