Finite Group Databasebeta

← #28499826

Subgroups — #28499826

Order 15104.

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

normal subgroups only

This group has 508 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#21429049C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422121C214yesmaximal, complemented
4C59×#1459C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273706—21no—
7C2×#14273706—21no—
8C2×#14273642—21no—
9#16837338C412yescomplemented
10#16837338C2212yes—
11#16837305—21no—
12#16837271—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×Dic118—21no—
21#14273727—41no—
22#14273727—41no—
23#14273706—44no—
24#14273706—44no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—41no—
28C2×C4⋊Dic59C2×C412yes—
29C2×Dic59⋊C4—22no—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59—22no—
34C2×C4×Dic59—21no—
35#14273642—42no—
36C59⋊#95—41no—
37C59⋊#95C2×C412yes—
38C22×D4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C22×C8×C59—41no—
42C4×D4×C59—41no—
43C4×D4×C59—41no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59C2311yescharacteristic
47C2×C42×C59C2×C411yescharacteristic
48C22⋊C8×C59—22no—
49C22⋊C8×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59SD1612yes—
55C2×D4×C59Q1612yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—42no—
70C2×Dic118—42no—
71C2×Dic118—42no—
72C2×Dic118—41no—
73C2×Dic118—41no—
74C2×C8×C59—42no—
75C2×C8×C59—82no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C42×C59—21no—
80C42×C59—21no—
81C22⋊Dic59—44no—
82C22⋊Dic59—44no—
83C22⋊Dic59—42no—
84C22⋊Dic59—42no—
85C4⋊Dic59—24no—
86C4⋊Dic59—42no—
87C4⋊Dic59—42no—
88C4⋊Dic59—82no—
89Dic59⋊C4—44no—
90Dic59⋊C4—44no—
91C4×Dic59—42no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59#10312yes—
101C23×C59C2×C22⋊C411yescharacteristic
102C23×C59C2×SD1611yescharacteristic
103C23×C59C2×Q1611yescharacteristic
104D4×C59—42no—
105D4×C59—42no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—22no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—81no—
118C2×C4×C59—81no—
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#10312yes—
122C2×Dic59—24no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—82no—
129C2×Dic59—82no—
130C2×Dic59—82no—
131Dic118—82no—
132Dic118—82no—
133C8×C59—82no—
134#11901—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—22no—
139C22×C59—21no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—81no—
148C22×C59—81no—
149C22×C59C2×#9911yescharacteristic
150C22×C59C2×#10311yescharacteristic
151C22×C59#42111yescharacteristic
152C4×C59—21no—
153C4×C59—22no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59#35312yes—
158Dic59—44no—
159Dic59—82no—
160Dic59—162no—
161#2542—591no—
162#1799—591no—
163#1459D5911yescharacteristic
164#1352—594no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—41no—
168C2×C59—42no—
169C2×C59—41no—
170C2×C59#152011yescharacteristic
171C2×C59#212311yescharacteristic
172C2×C59C2×#35311yescharacteristic
173C2×C4⋊Q8—591no—
174C2×C22⋊Q8—1181no—
175C2×C22⋊Q8—1181no—
176C2×C4×D4C2×D5911yescharacteristic
177C2×#99—1181no—
178C2×C22⋊C8—21no—
179#399—592no—
180#399—592no—
181#390—1181no—
182#384—1181no—
183#380—1181no—
184C4×C4⋊C4—591no—
185#342—592no—
186#324—21no—
187#323C2×D5912yes—
188C59#1190111yescharacteristic, complemented, Sylow
189C22×Q8—1181no—
190C22×D4D23611yescharacteristic
191C23×C4D23611yescharacteristic
192C23×C4C22⋊D5911yescharacteristic
193C22×C8—41no—
194C4⋊Q8—2361no—
195C4⋊Q8—2361no—
196C22⋊Q8—2364no—
197C22⋊Q8—2364no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—592no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4—1182no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1182no—
205C2×C4⋊C4—2361no—
206C2×C4⋊C4C22⋊D5911yescharacteristic
207C2×C22⋊C4—1182no—
208C2×C22⋊C4—1182no—
209C2×C22⋊C4—1182no—
210C2×C22⋊C4C4×D5911yescharacteristic
211C2×C22⋊C4C22×D5911yescharacteristic
212C2×C42—1181no—
213C2×C42C4×D5911yescharacteristic
214#99—2362no—
215C22⋊C8—22no—
216C22⋊C8—41no—
217#95—592no—
218#95—2361no—
219C24(C2×D59)⋊C411yescharacteristic
220C24C2×D23611yescharacteristic
221C2×Q8—2362no—
222C2×Q8—2362no—
223C2×Q8—2361no—
224C2×Q8—2361no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1174566012yes—
228C2×D4Dic23612yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1182no—
236C22×C4—1182no—
237C22×C4—2362no—
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4C2×C4×D5911yescharacteristic
242C22×C4C2×C22⋊D5911yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—41no—
246C4⋊C4—1184no—
247C4⋊C4—4722no—
248C4⋊C4—2362no—
249C4⋊C4—2362no—
250C4⋊C4—2364no—
251C4⋊C4—2364no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—2364no—
255C22⋊C4—2364no—
256C22⋊C4—2362no—
257C22⋊C4—2362no—
258C42—21no—
259C42—21no—
260C42—2362no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1427365212yes—
270C23C2×(C2×D59)⋊C411yescharacteristic
271C23C2×#1174566011yescharacteristic
272C23C2×Dic23611yescharacteristic
273Q8—4722no—
274Q8—4722no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—1184no—
291C2×C4—4722no—
292C2×C4—4722no—
293C2×C4—4722no—
294C2×C4—2362no—
295C2×C4—2362no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4#1427364111yescharacteristic
300C2×C4#1427364111yescharacteristic
301C2×C4#1427365212yes—
302C8—82no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1427364111yescharacteristic
318C22C2×#1427365211yescharacteristic
319C22#1683741811yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—9442no—
326C4—4722no—
327C4—2364no—
328C4#1683719412yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2142266911yescharacteristic
335C2#2142575111yescharacteristic
336C2C2×#1683719411yescharacteristic
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#11901
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. #21422121C2semidirect
5. C2×#14273727C4semidirect
9. #16837338C4semidirect
14. C4×C4⋊Dic59C4semidirect
188. C59#11901semidirect

Maximal subgroups

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

subgroupindex
#214290492
#214240842
#214221212
C59×#14592
#1190159