Finite Group Databasebeta

← #28486578

Subgroups — #28486578

Order 15104.

458 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 99 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#21422544C212yesmaximal, complemented
3#21422430C211yesmaximal, characteristic
4#21421961C212yesmaximal, complemented
5C59×#2921C211yesmaximal, characteristic
6C2×C4×Dic118C2211yescharacteristic
7#16837459C2212yes—
8#16837332C411yescharacteristic
9#16837327C411yescharacteristic
10#16837322C411yescharacteristic
11#16837314C411yescharacteristic
12#16837290—21no—
13#16837286—21no—
14#16837283—21no—
15#16837275—21no—
16#16837270C2211yescharacteristic
17#16837134—24no—
18#16837128—21no—
19#16837126—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#433—24no—
22C59×#423—21no—
23C59×#407—21no—
24C4×M16×C59C2212yes—
25C22×Dic118C2×C411yescharacteristic
26C4×Dic118—22no—
27C4×Dic118—22no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C4×Dic59—21no—
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37#14273638—22no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48#117×C59—22no—
49#117×C59—22no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59C2311yescharacteristic
56C4⋊C8×C59—24no—
57C22⋊C8×C59—24no—
58#97×C59—22no—
59C4×C8×C59D414yes—
60C24×C59C22⋊C411yescharacteristic
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C2×Dic118—22no—
76C2×Dic118—42no—
77M16×C59—42no—
78M16×C59—42no—
79C2×C8×C59—24no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—22no—
82C22⋊C4×C59—22no—
83C22⋊C4×C59—22no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59C2×D412yes—
87C42×C59#4112yes—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—22no—
91Dic59⋊C4—42no—
92Dic59⋊C4—42no—
93Dic59⋊C4—41no—
94Dic59⋊C4—41no—
95C4×Dic59—42no—
96C2×C59⋊C8—42no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59C2×C22⋊C411yescharacteristic
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59C4×D412yes—
117C2×C4×C59#12011yescharacteristic
118C2×C4×C59#12112yes—
119C2×C4×C59#12311yescharacteristic
120C2×Dic59—22no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132Dic118—24no—
133Dic118—42no—
134C8×C59—44no—
135C59⋊C8—82no—
136#9493—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—42no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—81no—
144C22×C59#38611yescharacteristic
145C22×C59#42111yescharacteristic
146C22×C59#42111yescharacteristic
147C4×C59—22no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154Dic59—42no—
155Dic59—81no—
156Dic59—81no—
157Dic59—81no—
158#2921D5911yescharacteristic
159#2599—591no—
160#1491—592no—
161#1464—591no—
162#1288—592no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59—81no—
166C2×C59#189111yescharacteristic
167C2×C59#189711yescharacteristic
168C2×C59#190911yescharacteristic
169C2×C4×Q8—591no—
170C2×#117C2×D5911yescharacteristic
171#433—24no—
172#432—592no—
173#423—21no—
174#407—21no—
175C4×M16C2×D5912yes—
176#398—591no—
177#397—591no—
178#395—591no—
179#393—591no—
180#388—1181no—
181#387—1181no—
182#387—1181no—
183#385—1181no—
184#383—591no—
185Q8⋊C8—1184no—
186#324—1181no—
187#323—1181no—
188C59#949311yescharacteristic, complemented, Sylow
189C22×Q8—591no—
190C23×C4C22⋊D5911yescharacteristic
191C2×M16—22no—
192C4×Q8—1182no—
193C4×Q8—1182no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—591no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4C4×D5911yescharacteristic
201C2×C4⋊C4D23611yescharacteristic
202C2×C22⋊C4—1181no—
203C2×C22⋊C4—1181no—
204C2×C22⋊C4C4×D5911yescharacteristic
205C2×C22⋊C4D23611yescharacteristic
206C2×C42—1181no—
207C2×C42C22⋊D5911yescharacteristic
208C2×C42C22×D5911yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—1182no—
211C22⋊C8—24no—
212C22⋊C8—1182no—
213#97—22no—
214#97—1182no—
215C4×C8C22⋊D5914yes—
216#95—591no—
217#95—591no—
218#95—1181no—
219#95—1181no—
220#95—1181no—
221#95—1181no—
222#95—1181no—
223#95—1181no—
224C24(C2×D59)⋊C411yescharacteristic
225C2×Q8—1182no—
226C2×Q8—2362no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—1181no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4C2×C4×D5911yescharacteristic
239C22×C4C2×D23611yescharacteristic
240C22×C4C2×C22⋊D5911yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—24no—
244C2×C8—2362no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—1182no—
248C4⋊C4—2362no—
249C4⋊C4—2362no—
250C4⋊C4—2361no—
251C4⋊C4—2361no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—2362no—
255C22⋊C4—2362no—
256C42—21no—
257C42—21no—
258C42—2362no—
259C42#1174568512yes—
260C42C2×C22⋊D5912yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×(C2×D59)⋊C411yescharacteristic
267Q8—1184no—
268Q8—2362no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—1182no—
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×C4C4×C22⋊D5912yes—
295C2×C4#1427369611yescharacteristic
296C2×C4#1427370112yes—
297C2×C4#1427371011yescharacteristic
298C8—44no—
299C8—4722no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1683728111yescharacteristic
308C22#1683742011yescharacteristic
309C22#1683742011yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—4721no—
318C4—4721no—
319C4—4721no—
320C4—2362no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2142451411yescharacteristic
325C2#2142454411yescharacteristic
326C2#2142461811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2921C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2921D59
hypocenterC59#9493
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #21422544C2semidirect
4. #21421961C2semidirect
188. C59#9493semidirect

Maximal subgroups

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

subgroupindex
#214296732
#214225442
#214224302
#214219612
C59×#29212
#949359