Finite Group Databasebeta

← #28498372

Subgroups — #28498372

Order 15104.

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

normal subgroups only

This group has 424 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#21429439C211yesmaximal, characteristic
2#21422417C211yesmaximal, characteristic
3#21422045C212yesmaximal, complemented
4#21422043C212yesmaximal, complemented
5C59×#1465C211yesmaximal, characteristic
6C22×Dic59⋊C4C411yescharacteristic
7#16837455—21no—
8#16837335—21no—
9#16837325—21no—
10#16837286—21no—
11#16837282C411yescharacteristic
12#16837282C411yescharacteristic
13#16837275—21no—
14#16837274C2211yescharacteristic
15#16837274C2211yescharacteristic
16#16837262C411yescharacteristic
17#16837126—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C23×Dic59C2×C411yescharacteristic
24C23×Dic59C2×C411yescharacteristic
25C2×C4⋊Dic59—21no—
26C2×Dic59⋊C4—21no—
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—41no—
31C2×Dic59⋊C4—41no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59C2×C411yescharacteristic
35C2×C22⋊Dic59C2×C411yescharacteristic
36C4⋊(C59⋊C8)—41no—
37C22⋊(C59⋊C8)—22no—
38C4×C59⋊C8—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C22×D4×C59C2×C411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—41no—
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59D411yescharacteristic
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—
76C22×Dic59—42no—
77C22×Dic59—41no—
78C22×Dic59—41no—
79C22×Dic59—81no—
80M16×C59—82no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—22no—
88C22⋊Dic59—22no—
89C22⋊Dic59—42no—
90C22⋊Dic59—81no—
91C4⋊Dic59—81no—
92Dic59⋊C4—42no—
93Dic59⋊C4—42no—
94Dic59⋊C4—41no—
95Dic59⋊C4—41no—
96Dic59⋊C4—81no—
97C2×C59⋊C8—42no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59C2×C22⋊C411yescharacteristic
108C23×C59#13112yes—
109D4×C59—44no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—81no—
121C2×C4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#10112yes—
126C2×Dic59—22no—
127C2×Dic59—41no—
128C2×Dic59—42no—
129C2×Dic59—41no—
130C2×Dic59—42no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—82no—
141C8×C59—82no—
142C59⋊C8—82no—
143#11577—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—22no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—81no—
156C22×C59#40811yescharacteristic
157C22×C59C2×#9911yescharacteristic
158C22×C59C2×#10111yescharacteristic
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—22no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—41no—
165Dic59—42no—
166Dic59—81no—
167Dic59—81no—
168Dic59—81no—
169#2579—591no—
170#1465D5911yescharacteristic
171#1460—591no—
172#1326—592no—
173#1326—592no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—42no—
177C2×C59—81no—
178C2×C59#147211yescharacteristic
179C2×C59#212011yescharacteristic
180C2×C59#212111yescharacteristic
181C2×C4×D4C2×D5911yescharacteristic
182C22×C4⋊C4—591no—
183#431—1181no—
184#407—21no—
185#398—1181no—
186#396—1181no—
187#387—1181no—
188#386—591no—
189#386—591no—
190#385—591no—
191#385—591no—
192#385—1181no—
193#380—591no—
194#324—21no—
195#323—592no—
196#323—1181no—
197#323C2×D5912yes—
198C59#1157711yescharacteristic, complemented, Sylow
199C22×D4C4×D5911yescharacteristic
200C23×C4—591no—
201C23×C4—591no—
202C23×C4C4×D5911yescharacteristic
203C23×C4C22⋊D5911yescharacteristic
204C2×M16—41no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4—2361no—
213C2×C4⋊C4—2361no—
214C2×C4⋊C4C22⋊D5911yescharacteristic
215C2×C22⋊C4—591no—
216C2×C22⋊C4—591no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4D23611yescharacteristic
220C2×C22⋊C4C22×D5911yescharacteristic
221C2×C42D23611yescharacteristic
222C4⋊C8—2361no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—1182no—
226C4×C8—2361no—
227#95—1181no—
228#95—1181no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232C24(C2×D59)⋊C411yescharacteristic
233C24C2×C4×D5911yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—1181no—
244C22×C4—1181no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4—4721no—
248C22×C4—2362no—
249C22×C4—2361no—
250C22×C4—2361no—
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4C2×D23611yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256M16—82no—
257C2×C8—42no—
258C2×C8—2362no—
259C4⋊C4—41no—
260C4⋊C4—4721no—
261C4⋊C4—4721no—
262C4⋊C4—2362no—
263C4⋊C4—2362no—
264C4⋊C4—2361no—
265C4⋊C4—2361no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—1182no—
269C22⋊C4—1182no—
270C22⋊C4—4721no—
271C22⋊C4—2362no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×(C2×D59)⋊C411yescharacteristic
284C23#1427373512yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—1182no—
300C2×C4—4722no—
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—2361no—
311C2×C4—2362no—
312C2×C4—2361no—
313C2×C4—2362no—
314C2×C4#1427364111yescharacteristic
315C2×C4#1427364111yescharacteristic
316C2×C4#1427364512yes—
317C8—82no—
318C8—4722no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1683737011yescharacteristic
332C22C2×#1427364111yescharacteristic
333C22C2×#1427364511yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—4721no—
341C4—4721no—
342C4—4721no—
343C4—2362no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2142246111yescharacteristic
349C2#2142573811yescharacteristic
350C2#2142574011yescharacteristic
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#11577
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. #21422045C2semidirect
4. #21422043C2semidirect
198. C59#11577semidirect

Maximal subgroups

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

subgroupindex
#214294392
#214224172
#214220452
#214220432
C59×#14652
#1157759