Finite Group Databasebeta

← #28499563

Subgroups — #28499563

Order 15104.

390 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429673C211yesmaximal, characteristic
2#21424173C211yesmaximal, characteristic
3#21422118C212yesmaximal
4#21422097C212yesmaximal
5C59×#1458C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic, complemented
7#16837332C411yescharacteristic, complemented
8#16837327C2211yescharacteristic
9#16837322C411yescharacteristic, complemented
10#16837314C2211yescharacteristic
11#16837290—21no—
12#16837286—21no—
13#16837283—21no—
14#16837275—21no—
15#16837270C411yescharacteristic, complemented
16#16837265—21no—
17#16837169—21no—
18#16837169C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—22no—
33C2×C4×Dic59—21no—
34C59⋊#95—21no—
35C59⋊#95—21no—
36C59⋊#95—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—41no—
41C59⋊#95—41no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45#117×C59—41no—
46#117×C59—41no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—41no—
54C22⋊C8×C59—22no—
55C4×C8×C59—41no—
56C24×C59C2×D411yescharacteristic
57C22×C4×C59—22no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—41no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—42no—
72C2×Dic118—41no—
73C2×Dic118—42no—
74C2×C8×C59—42no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C22⋊Dic59—42no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C22⋊Dic59—42no—
87C4⋊Dic59—41no—
88Dic59⋊C4—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—41no—
91Dic59⋊C4—41no—
92C4×Dic59—42no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59C2×C22⋊C411yescharacteristic
98C23×C59#13512yes—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59#9911yescharacteristic
116C2×Dic59—22no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—82no—
129C2×Dic59—82no—
130C2×Dic59—82no—
131Dic118—42no—
132Dic118—42no—
133C8×C59—82no—
134#11842—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—22no—
138C22×C59—42no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—81no—
142C22×C59C2×#9911yescharacteristic
143C22×C59#41611yescharacteristic
144C22×C59#42111yescharacteristic
145C4×C59—21no—
146C4×C59—21no—
147C4×C59—41no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152Dic59—42no—
153Dic59—81no—
154Dic59—81no—
155Dic59—81no—
156Dic59—162no—
157#2599—591no—
158#1820—591no—
159#1458D5911yescharacteristic
160#1351—592no—
161#1344—592no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—42no—
165C2×C59#151611yescharacteristic
166C2×C59#212011yescharacteristic
167C2×C59#212911yescharacteristic
168C2×C4×Q8—591no—
169C2×#117C2×D5911yescharacteristic
170#431—21no—
171#398—591no—
172#397—591no—
173#395—591no—
174#393—591no—
175#388—1181no—
176#387—1181no—
177#387—1181no—
178#385—1181no—
179#383—591no—
180#381—1181no—
181#342—592no—
182#342—1181no—
183#324—21no—
184#323C2×D5912yes—
185C59#1184211yescharacteristic, complemented, Sylow
186C22×Q8—591no—
187C23×C4D23611yescharacteristic
188C4×Q8—2361no—
189C4×Q8—2361no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—591no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4C4×D5911yescharacteristic
197C2×C4⋊C4C22⋊D5911yescharacteristic
198C2×C22⋊C4—1181no—
199C2×C22⋊C4—1181no—
200C2×C22⋊C4—1182no—
201C2×C22⋊C4D23611yescharacteristic
202C2×C22⋊C4C22×D5911yescharacteristic
203C2×C42—1181no—
204C2×C42C4×D5911yescharacteristic
205C2×C42C22⋊D5911yescharacteristic
206C4⋊C8—41no—
207C22⋊C8—22no—
208C4×C8—41no—
209#95—591no—
210#95—591no—
211#95—1181no—
212#95—1181no—
213#95—1181no—
214#95—1181no—
215#95—1181no—
216#95—1181no—
217#95—2361no—
218#95—2361no—
219C24C2×D23611yescharacteristic
220C2×Q8—2361no—
221C2×Q8—2362no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—1181no—
227C22×C4—1181no—
228C22×C4—1181no—
229C22×C4—1181no—
230C22×C4—2362no—
231C22×C4(C2×D59)⋊C411yescharacteristic
232C22×C4(C2×D59)⋊C411yescharacteristic
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4C2×C4×D5911yescharacteristic
236C22×C4C2×C22⋊D5911yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—2361no—
241C4⋊C4—2362no—
242C4⋊C4—2362no—
243C4⋊C4—2361no—
244C4⋊C4—2361no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2362no—
248C22⋊C4—2362no—
249C22⋊C4—2362no—
250C22⋊C4—2362no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—2362no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×(C2×D59)⋊C411yescharacteristic
261C23#1427374912yes—
262Q8—2362no—
263Q8—2362no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—1182no—
280C2×C4—4722no—
281C2×C4—4722no—
282C2×C4—4722no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4#1427364111yescharacteristic
295C2×C4#1427364111yescharacteristic
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1427364111yescharacteristic
305C22#1683739911yescharacteristic
306C22#1683741811yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—9442no—
315C4—4721no—
316C4—4721no—
317C4—4721no—
318C4—2362no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2142265311yescharacteristic
323C2#2142573811yescharacteristic
324C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11842
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic118C4semidirect
7. #16837332C4semidirect
9. #16837322C4semidirect
15. #16837270C4semidirect
185. C59#11842semidirect

Maximal subgroups

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

subgroupindex
#214296732
#214241732
#214221182
#214220972
C59×#14582
#1184259