Finite Group Databasebeta

← #28499822

Subgroups — #28499822

Order 15104.

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

normal subgroups only

This group has 452 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#21429269C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422118C214yesmaximal, complemented
4C59×#1459C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273642—21no—
7#16837314C412yescomplemented
8#16837314C2212yes—
9#16837305—21no—
10#16837267—21no—
11#16837262—21no—
12#16837262—21no—
13#16837262—21no—
14C4×C4⋊Dic59C411yescharacteristic, complemented
15#16837169C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C59×#324—21no—
19C59×#323C2212yes—
20C22×Dic118C2×C412yes—
21#14273727—41no—
22#14273727—41no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—41no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—22no—
30C2×C4×Dic59—21no—
31#14273642—42no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C59⋊#95—41no—
35C59⋊#95C2×C412yes—
36C22×D4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C22×C8×C59—41no—
40C4×D4×C59—41no—
41C4×D4×C59—41no—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59C2311yescharacteristic
45C2×C42×C59C2×C411yescharacteristic
46C22⋊C8×C59—22no—
47C22⋊C8×C59—41no—
48C24×C59C22⋊C411yescharacteristic
49C24×C59C2×D411yescharacteristic
50C2×D4×C59—41no—
51C2×D4×C59—41no—
52C2×D4×C59SD1612yes—
53C2×D4×C59Q1612yes—
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×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—42no—
68C2×Dic118—42no—
69C2×Dic118—82no—
70C2×C8×C59—42no—
71C2×C8×C59—82no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C42×C59—21no—
76C42×C59—21no—
77C22⋊Dic59—42no—
78C22⋊Dic59—42no—
79C22⋊Dic59—82no—
80C22⋊Dic59—82no—
81C4⋊Dic59—42no—
82C4⋊Dic59—42no—
83C4⋊Dic59—41no—
84C4⋊Dic59—41no—
85C4⋊Dic59—82no—
86C4×Dic59—42no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59#10312yes—
96C23×C59C2×C22⋊C411yescharacteristic
97C23×C59C2×SD1611yescharacteristic
98C23×C59C2×Q1611yescharacteristic
99D4×C59—42no—
100D4×C59—42no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—81no—
113C2×C4×C59—81no—
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59#9911yescharacteristic
116C2×C4×C59#10312yes—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—82no—
124C2×Dic59—82no—
125C2×Dic59—82no—
126Dic118—84no—
127C8×C59—82no—
128#11900—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—21no—
134C22×C59—42no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—81no—
142C22×C59—81no—
143C22×C59C2×#9911yescharacteristic
144C22×C59C2×#10311yescharacteristic
145C22×C59#42111yescharacteristic
146C4×C59—21no—
147C4×C59—22no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59#35412yes—
152Dic59—82no—
153Dic59—82no—
154Dic59—162no—
155#2563—591no—
156#1799—591no—
157#1459D5911yescharacteristic
158#1351—594no—
159C2×C59—21no—
160C2×C59—21no—
161C2×C59—41no—
162C2×C59—42no—
163C2×C59—41no—
164C2×C59#152011yescharacteristic
165C2×C59#212211yescharacteristic
166C2×C59C2×#35411yescharacteristic
167C2×C4⋊Q8—591no—
168C2×C4×D4C2×D5911yescharacteristic
169C2×#99—1181no—
170C2×C22⋊C8—21no—
171#393—592no—
172#393—592no—
173#390—1181no—
174#382—1181no—
175#380—1181no—
176#380—1181no—
177#380—1181no—
178C4×C4⋊C4—591no—
179#342—592no—
180#324—21no—
181#323C2×D5912yes—
182C59#1190011yescharacteristic, complemented, Sylow
183C22×Q8—592no—
184C22×D4D23611yescharacteristic
185C23×C4D23611yescharacteristic
186C23×C4C22⋊D5911yescharacteristic
187C22×C8—41no—
188C4⋊Q8—2361no—
189C4⋊Q8—2361no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—1181no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4—2361no—
196C2×C4⋊C4C22⋊D5911yescharacteristic
197C2×C22⋊C4—1182no—
198C2×C22⋊C4—1182no—
199C2×C22⋊C4—1182no—
200C2×C22⋊C4C4×D5911yescharacteristic
201C2×C22⋊C4C22×D5911yescharacteristic
202C2×C42—1181no—
203C2×C42C4×D5911yescharacteristic
204#99—2362no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207#95—592no—
208#95—1182no—
209#95—1182no—
210#95—2361no—
211C24(C2×D59)⋊C411yescharacteristic
212C24C2×D23611yescharacteristic
213C2×Q8—4722no—
214C2×Q8—2362no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#1174566012yes—
218C2×D4Dic23612yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1182no—
226C22×C4—1182no—
227C22×C4—2362no—
228C22×C4(C2×D59)⋊C411yescharacteristic
229C22×C4(C2×D59)⋊C411yescharacteristic
230C22×C4(C2×D59)⋊C411yescharacteristic
231C22×C4C2×C4×D5911yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233C2×C8—42no—
234C2×C8—82no—
235C4⋊C4—41no—
236C4⋊C4—4722no—
237C4⋊C4—2362no—
238C4⋊C4—2362no—
239C4⋊C4—2361no—
240C4⋊C4—2361no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—4722no—
244C22⋊C4—4722no—
245C22⋊C4—2362no—
246C22⋊C4—2362no—
247C42—21no—
248C42—21no—
249C42—2362no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1427365212yes—
259C23C2×(C2×D59)⋊C411yescharacteristic
260C23C2×#1174566011yescharacteristic
261C23C2×Dic23611yescharacteristic
262Q8—4724no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—4722no—
279C2×C4—4722no—
280C2×C4—4722no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2362no—
285C2×C4—2362no—
286C2×C4—2362no—
287C2×C4#1427364111yescharacteristic
288C2×C4#1427364111yescharacteristic
289C2×C4#1427365212yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#1427364111yescharacteristic
306C22C2×#1427365211yescharacteristic
307C22#1683741811yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—9442no—
314C4—4722no—
315C4—4722no—
316C4#1683719612yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#2142266911yescharacteristic
323C2#2142574711yescharacteristic
324C2C2×#1683719611yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11900
LCSG ▷ C22×C4×C59 ▷ C23×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. #21422118C2semidirect
5. C2×#14273727C4semidirect
7. #16837314C4semidirect
14. C4×C4⋊Dic59C4semidirect
182. C59#11900semidirect

Maximal subgroups

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

subgroupindex
#214292692
#214240842
#214221182
C59×#14592
#1190059