Finite Group Databasebeta

← #28499307

Subgroups — #28499307

Order 15104.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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
2C4×#14273642C211yesmaximal, characteristic
3#21422118C214yesmaximal, complemented
4C59×#1465C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273642—21no—
7#16837314C412yescomplemented
8#16837314C2212yes—
9#16837305—21no—
10#16837267—21no—
11#16837262—21no—
12#16837262—21no—
13C4×C4⋊Dic59C411yescharacteristic, complemented
14C4×C22⋊Dic59—21no—
15#16837169C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#407—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×C22⋊Dic59—22no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C2×C4×Dic59—21no—
30C2×C4×Dic59—41no—
31#14273642—24no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C59⋊#95—41no—
35C59⋊#95C2×C412yes—
36C22×D4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C2×M16×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—22no—
51C2×D4×C59—22no—
52C2×D4×C59—41no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—42no—
58C22×C4×C59—42no—
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×Dic59—22no—
65C22×Dic59—22no—
66C22×Dic59—42no—
67C2×Dic118—42no—
68C2×Dic118—82no—
69M16×C59—82no—
70C2×C8×C59—42no—
71C4⋊C4×C59—41no—
72C22⋊C4×C59—41no—
73C22⋊C4×C59—41no—
74C42×C59—21no—
75C42×C59—21no—
76C22⋊Dic59—24no—
77C22⋊Dic59—42no—
78C22⋊Dic59—82no—
79C22⋊Dic59—82no—
80C4⋊Dic59—42no—
81C4⋊Dic59—42no—
82C4⋊Dic59—41no—
83C4⋊Dic59—41no—
84C4×Dic59—42no—
85C4×Dic59—82no—
86C23×C59—21no—
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×C59C2×C22⋊C411yescharacteristic
96C23×C59#13712yes—
97D4×C59—44no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—81no—
110C2×C4×C59—81no—
111C2×C4×C59#9911yescharacteristic
112C2×C4×C59#9911yescharacteristic
113C2×C4×C59#10412yes—
114C2×Dic59—42no—
115C2×Dic59—44no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—82no—
122C2×Dic59—82no—
123Dic118—84no—
124C8×C59—82no—
125#11787—591nomaximal, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—22no—
129C22×C59—42no—
130C22×C59—42no—
131C22×C59—41no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59—81no—
138C22×C59C2×#9911yescharacteristic
139C22×C59#41911yescharacteristic
140C22×C59C2×#10411yescharacteristic
141C4×C59—21no—
142C4×C59—22no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—41no—
146C4×C59#35412yes—
147Dic59—82no—
148Dic59—84no—
149Dic59—82no—
150#2563—591no—
151C4×#99—591no—
152#1465D5911yescharacteristic
153#1351—594no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59—42no—
157C2×C59—81no—
158C2×C59#150811yescharacteristic
159C2×C59#212211yescharacteristic
160C2×C59C2×#35411yescharacteristic
161C2×C4⋊Q8—591no—
162C2×C4×D4C2×D5911yescharacteristic
163C2×#99—1181no—
164#407—21no—
165#393—592no—
166#393—592no—
167#390—1181no—
168#382—1181no—
169#380—1181no—
170#380—1181no—
171C4×C4⋊C4—591no—
172C4×C22⋊C4—1181no—
173#342—592no—
174#324—21no—
175#323C2×D5912yes—
176C59#1178711yescharacteristic, complemented, Sylow
177C22×Q8—592no—
178C22×D4D23611yescharacteristic
179C23×C4D23611yescharacteristic
180C23×C4C22⋊D5911yescharacteristic
181C2×M16—41no—
182C4⋊Q8—2361no—
183C4⋊Q8—2361no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4C22⋊D5911yescharacteristic
190C2×C22⋊C4—1182no—
191C2×C22⋊C4—1182no—
192C2×C22⋊C4—1182no—
193C2×C22⋊C4C4×D5911yescharacteristic
194C2×C22⋊C4C22×D5911yescharacteristic
195C2×C42—1181no—
196C2×C42—2361no—
197C2×C42C4×D5911yescharacteristic
198#99—1184no—
199C22⋊C8—22no—
200C22⋊C8—41no—
201#95—592no—
202#95—1182no—
203#95—1182no—
204#95—2361no—
205C24(C2×D59)⋊C411yescharacteristic
206C24C2×D23611yescharacteristic
207C2×Q8—4722no—
208C2×Q8—2362no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—1182no—
219C22×C4—1182no—
220C22×C4—2362no—
221C22×C4(C2×D59)⋊C411yescharacteristic
222C22×C4(C2×D59)⋊C411yescharacteristic
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4C2×C4×D5911yescharacteristic
225C22×C4C2×C22⋊D5911yescharacteristic
226M16—82no—
227C2×C8—42no—
228C4⋊C4—41no—
229C4⋊C4—2362no—
230C4⋊C4—2362no—
231C4⋊C4—2361no—
232C4⋊C4—2361no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—1184no—
236C22⋊C4—4722no—
237C22⋊C4—4722no—
238C22⋊C4—2362no—
239C42—21no—
240C42—21no—
241C42—4722no—
242C42—2362no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C2×(C2×D59)⋊C411yescharacteristic
253C23#1427376012yes—
254Q8—4724no—
255D4—44no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—4722no—
270C2×C4—4722no—
271C2×C4—2364no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4—2362no—
278C2×C4#1427364111yescharacteristic
279C2×C4#1427364111yescharacteristic
280C2×C4#1427365312yes—
281C8—82no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#1427364111yescharacteristic
295C22#1683741211yescharacteristic
296C22C2×#1427365311yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—4722no—
303C4—4724no—
304C4—4722no—
305C4#1683719612yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2142261711yescharacteristic
311C2#2142574711yescharacteristic
312C2C2×#1683719611yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1465C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1465D59
hypocenterC59#11787
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #21422118C2semidirect
5. C2×#14273727C4semidirect
7. #16837314C4semidirect
13. C4×C4⋊Dic59C4semidirect
176. C59#11787semidirect

Maximal subgroups

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

subgroupindex
#214292692
C4×#142736422
#214221182
C59×#14652
#1178759