Finite Group Databasebeta

← #28499306

Subgroups — #28499306

Order 15104.

460 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14273642C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422117C214yesmaximal, complemented
4C59×#2563C211yesmaximal, characteristic
5C2×#14273642—21no—
6#16837366—21no—
7C4×C22⋊Dic59—21no—
8#16837169C2212yes—
9#16837128—21no—
10#16837126C2212yes—
11C2×C4⋊Q8×C59C411yescharacteristic, complemented
12C2×C4×D4×C59C2211yescharacteristic
13C59×#393C412yescomplemented
14C59×#393C2212yes—
15C59×#390—21no—
16C59×#382—21no—
17C59×#380—21no—
18C59×#380—21no—
19C4×C4⋊C4×C59C411yescharacteristic, complemented
20C2×#11745664—41no—
21C2×C22⋊Dic59—22no—
22C2×C4×Dic59—41no—
23#14273642—24no—
24C22⋊(C59⋊C8)—22no—
25C22⋊(C59⋊C8)—41no—
26C59⋊#95—41no—
27C22×Q8×C59C2×C412yes—
28C22×D4×C59D411yescharacteristic
29C23×C4×C59D411yescharacteristic
30C23×C4×C59D411yescharacteristic
31C4⋊Q8×C59—41no—
32C4⋊Q8×C59—41no—
33C4×D4×C59—41no—
34C4×D4×C59—41no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59C2×C411yescharacteristic
42C2×C22⋊C4×C59C2311yescharacteristic
43C2×C42×C59—21no—
44C2×C42×C59C2×C411yescharacteristic
45#95×C59—22no—
46#95×C59—22no—
47#95×C59C2×C412yes—
48C24×C59C22⋊C411yescharacteristic
49C24×C59C2×D411yescharacteristic
50C2×Q8×C59—42no—
51C2×Q8×C59—82no—
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—41no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—42no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—82no—
77C22⋊C4×C59—82no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—42no—
81C22⋊Dic59—24no—
82C22⋊Dic59—42no—
83C4×Dic59—82no—
84#11745664—82no—
85C2×C59⋊C8—42no—
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—
97Q8×C59—84no—
98D4×C59—44no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—81no—
117C2×C4×C59—81no—
118C2×C4×C59#9911yescharacteristic
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#10412yes—
121C2×Dic59—44no—
122C2×Dic59—82no—
123C2×Dic59—82no—
124C59⋊C8—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—82no—
147C4×C59—82no—
148C4×C59#35412yes—
149Dic59—84no—
150#2563D5911yescharacteristic
151C4×#99—591no—
152#1465—591no—
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⋊Q8Dic5911yescharacteristic, complemented
162C2×C4×D4C2×D5911yescharacteristic
163C2×#99—1181no—
164#407—1181no—
165#393Dic5912yescomplemented
166#393C2×D5912yes—
167#390—21no—
168#382—21no—
169#380—21no—
170#380—21no—
171C4×C4⋊C4Dic5911yescharacteristic, complemented
172C4×C22⋊C4—1181no—
173#342—592no—
174#324—1181no—
175#323—592no—
176C59#1178711yescharacteristic, complemented, Sylow
177C22×Q8C2×Dic5912yes—
178C22×D4C22⋊D5911yescharacteristic
179C23×C4C22⋊D5911yescharacteristic
180C23×C4C22⋊D5911yescharacteristic
181C2×M16—2361no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22⋊D5911yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1182no—
193C2×C22⋊C4C2×Dic5911yescharacteristic
194C2×C22⋊C4C22×D5911yescharacteristic
195C2×C42—21no—
196C2×C42—2361no—
197C2×C42C2×Dic5911yescharacteristic
198#99—1184no—
199C22⋊C8—1182no—
200C22⋊C8—2361no—
201#95—22no—
202#95—22no—
203#95—2361no—
204#95C2×Dic5912yes—
205C24C22⋊Dic5911yescharacteristic
206C24C2×C22⋊D5911yescharacteristic
207C2×Q8—42no—
208C2×Q8—82no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—2362no—
221C22×C4C22⋊Dic5911yescharacteristic
222C22×C4C22⋊Dic5911yescharacteristic
223C22×C4C22⋊Dic5911yescharacteristic
224C22×C4C22×Dic5911yescharacteristic
225C22×C4C2×C22⋊D5911yescharacteristic
226M16—4722no—
227C2×C8—2362no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—1184no—
238C22⋊C4—2362no—
239C42—21no—
240C42—21no—
241C42—42no—
242C42—4722no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C2×C22⋊Dic5911yescharacteristic
253C23#1427375812yes—
254Q8—84no—
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—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—4722no—
276C2×C4—4722no—
277C2×C4—2364no—
278C2×C4#1427364211yescharacteristic
279C2×C4#1427364211yescharacteristic
280C2×C4#1427365412yes—
281C8—4722no—
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×#1427364211yescharacteristic
295C22#1683741111yescharacteristic
296C22C2×#1427365411yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4—82no—
304C4—4724no—
305C4#1683719712yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2142261611yescharacteristic
311C2#2142574611yescharacteristic
312C2C2×#1683719711yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2563C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2563D59
hypocenterC59#11787
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #21422117C2semidirect
11. C2×C4⋊Q8×C59C4semidirect
13. C59×#393C4semidirect
19. C4×C4⋊C4×C59C4semidirect
161. C2×C4⋊Q8Dic59semidirect
165. #393Dic59semidirect
171. C4×C4⋊C4Dic59semidirect
176. C59#11787semidirect

Maximal subgroups

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

subgroupindex
C4×#142736422
#214224342
#214221172
C59×#25632
#1178759