Finite Group Databasebeta

← #28499817

Subgroups — #28499817

Order 15104.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#21424084C211yesmaximal, characteristic
2#21422417C211yesmaximal, characteristic
3#21422120C214yesmaximal, complemented
4C59×#2487C211yesmaximal, characteristic
5#16837455—21no—
6C2×#14273642—21no—
7#16837262—21no—
8#16837169C2212yes—
9#16837126—21no—
10#16837126C2212yes—
11C2×C4×D4×C59C2211yescharacteristic
12C22×C4⋊C4×C59C411yescharacteristic, complemented
13C59×#400C412yescomplemented
14C59×#399C2212yes—
15C59×#389—21no—
16C59×#386—21no—
17C59×#385—21no—
18C59×#380—21no—
19C59×#380C411yescharacteristic, complemented
20C2×C4⋊Dic59—41no—
21C2×C22⋊Dic59—22no—
22C4⋊(C59⋊C8)—41no—
23#14273642—42no—
24C22⋊(C59⋊C8)—22no—
25C4×C59⋊C8—41no—
26C59⋊#95—41no—
27C22×D4×C59D411yescharacteristic
28C23×C4×C59—21no—
29C23×C4×C59C2×C411yescharacteristic
30C23×C4×C59D411yescharacteristic
31C4×D4×C59—41no—
32C4×D4×C59—41no—
33C2×C4⋊C4×C59—21no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—41no—
37C2×C4⋊C4×C59—41no—
38C2×C4⋊C4×C59C2×C412yes—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C22⋊C4×C59—21no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2311yescharacteristic
44C2×C42×C59D411yescharacteristic
45#95×C59—22no—
46#95×C59—22no—
47#95×C59C2×C412yes—
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—22no—
59C22×C4×C59—22no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59—82no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—42no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—82no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—42no—
79C22⋊C4×C59—82no—
80C42×C59—21no—
81C42×C59—21no—
82C22⋊Dic59—42no—
83C22⋊Dic59—42no—
84C4⋊Dic59—82no—
85C2×C59⋊C8—42no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59#10312yes—
95C23×C59C2×C22⋊C411yescharacteristic
96C23×C59C2×SD1611yescharacteristic
97C23×C59C2×Q1611yescharacteristic
98D4×C59—42no—
99D4×C59—42no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—81no—
119C2×C4×C59—84no—
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#10312yes—
123C2×Dic59—82no—
124C2×Dic59—82no—
125C2×Dic59—82no—
126C59⋊C8—82no—
127#11899—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—22no—
132C22×C59—21no—
133C22×C59—42no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—42no—
137C22×C59—41no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—81no—
142C22×C59C2×#9911yescharacteristic
143C22×C59C2×#10311yescharacteristic
144C22×C59#42111yescharacteristic
145C4×C59—21no—
146C4×C59—22no—
147C4×C59—41no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—82no—
151C4×C59—82no—
152C4×C59#35212yes—
153Dic59—162no—
154#2487D5911yescharacteristic
155#1799—591no—
156#1460—591no—
157#1352—594no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59—41no—
161C2×C59—42no—
162C2×C59—41no—
163C2×C59#152011yescharacteristic
164C2×C59C2×#35211yescharacteristic
165C2×C59#212811yescharacteristic
166C2×C4×D4C2×D5911yescharacteristic
167C22×C4⋊C4Dic5911yescharacteristic, complemented
168#431—1181no—
169C2×#99—1181no—
170#400Dic5912yescomplemented
171#399C2×D5912yes—
172#389—21no—
173#386—21no—
174#385—21no—
175#380—21no—
176#380—1181no—
177#380Dic5911yescharacteristic, complemented
178#342—592no—
179#323—592no—
180#323—1181no—
181C59#1189911yescharacteristic, complemented, Sylow
182C22×D4C22⋊D5911yescharacteristic
183C23×C4—21no—
184C23×C4C2×Dic5911yescharacteristic
185C23×C4C22⋊D5911yescharacteristic
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—2361no—
194C2×C4⋊C4C2×Dic5912yes—
195C2×C4⋊C4C2×Dic5911yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1182no—
199C2×C22⋊C4C22⋊D5911yescharacteristic
200C2×C22⋊C4C22×D5911yescharacteristic
201C2×C42C22⋊D5911yescharacteristic
202C4⋊C8—2361no—
203#99—2362no—
204C22⋊C8—1182no—
205C4×C8—2361no—
206#95—22no—
207#95—22no—
208#95—2361no—
209#95C2×Dic5912yes—
210C24C22⋊Dic5911yescharacteristic
211C24C2×C22⋊D5911yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#1174567112yes—
215C2×D4#1174567212yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—2362no—
227C22×C4C22⋊Dic5911yescharacteristic
228C22×C4C22⋊Dic5911yescharacteristic
229C22×C4C22⋊Dic5911yescharacteristic
230C22×C4C22×Dic5911yescharacteristic
231C22×C4C2×C22⋊D5911yescharacteristic
232C2×C8—2362no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—82no—
239C4⋊C4—4722no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1427365112yes—
257C23C2×C22⋊Dic5911yescharacteristic
258C23C2×#1174567111yescharacteristic
259C23C2×#1174567211yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4—4722no—
283C2×C4—4722no—
284C2×C4—4722no—
285C2×C4#1427364211yescharacteristic
286C2×C4#1427364211yescharacteristic
287C2×C4#1427365112yes—
288C8—4722no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1427364211yescharacteristic
304C22C2×#1427365111yescharacteristic
305C22#1683741911yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—9442no—
314C4#1683719212yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#2142266811yescharacteristic
321C2C2×#1683719211yescharacteristic
322C2#2142576811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2487C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2487D59
hypocenterC59#11899
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #21422120C2semidirect
12. C22×C4⋊C4×C59C4semidirect
13. C59×#400C4semidirect
19. C59×#380C4semidirect
167. C22×C4⋊C4Dic59semidirect
170. #400Dic59semidirect
177. #380Dic59semidirect
181. C59#11899semidirect

Maximal subgroups

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

subgroupindex
#214240842
#214224172
#214221202
C59×#24872
#1189959