Finite Group Databasebeta

← #28481603

Subgroups — #28481603

Order 15104.

460 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16837470C211yesmaximal, characteristic
2#21424132C212yesmaximal, complemented
3#21422432C212yesmaximal
4#21421973C211yesmaximal, characteristic, complemented
5C59×#1293C211yesmaximal, characteristic, complemented
6C22×#11745659C411yescharacteristic
7C23×C59⋊C8C2211yescharacteristic
8C2×C4×C22⋊D59C2211yescharacteristic
9#16837470—22no—
10#16837470—22no—
11C2×#14273662—22no—
12C2×Dic59⋊C8C411yescharacteristic
13#16837366C2212yes—
14C2×(C2×D59)⋊C8C411yescharacteristic
15C2×#14273637C411yescharacteristic
16#16837166—22no—
17#16837127—22no—
18#16837125—22no—
19C59×#407C2212yes—
20C2×C22⋊C8×C59C2211yescharacteristic
21C22×C22⋊D59Q811yescharacteristic
22C22×C4×D59C2×C411yescharacteristic
23C2×#11745664—22no—
24C2×#11745659—21no—
25C2×#11745659—21no—
26C2×#11745659—42no—
27C22×C59⋊C8—42no—
28C22×C59⋊C8C2×C411yescharacteristic
29C22×C59⋊C8C2×C411yescharacteristic
30C22×C59⋊C8D412yes—
31C22×C59⋊C8D412yes—
32C4×C22⋊D59—22no—
33C4×C22⋊D59—22no—
34C2×Dic59⋊C4D411yescharacteristic
35C2×C22⋊Dic59Q811yescharacteristic
36C2×(C2×D59)⋊C4D411yescharacteristic
37C2×C4×Dic59C2×C411yescharacteristic
38#14273662—44no—
39Dic59⋊C8—22no—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—22no—
42(C2×D59)⋊C8—22no—
43#14273637—21no—
44#14273637—21no—
45C23×C4×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C22×C8×C59C2×C411yescharacteristic
48C22×C8×C59C2×C411yescharacteristic
49C22⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C22⋊C8×C59—22no—
52C24×C59C2×Q811yescharacteristic
53C23×D59C4⋊C411yescharacteristic
54C22×C4×C59—41no—
55C22×C4×C59—41no—
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D412yes—
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59#4112yes—
60C2×C22⋊D59—22no—
61C2×C22⋊D59—42no—
62C22×Dic59C4⋊C411yescharacteristic
63C22×Dic59C4⋊C411yescharacteristic
64C22×Dic59C4⋊C411yescharacteristic
65C2×C4×D59—21no—
66C2×C4×D59—21no—
67C2×C4×D59—41no—
68C2×C4×D59—41no—
69M16×C59—42no—
70M16×C59—42no—
71C2×C8×C59—21no—
72C2×C8×C59—21no—
73C2×C8×C59—41no—
74C2×C8×C59—41no—
75C2×C8×C59—42no—
76C22⋊Dic59—22no—
77(C2×D59)⋊C4—22no—
78Dic59⋊C4—22no—
79C4×Dic59—21no—
80C4×Dic59—21no—
81#11745664—42no—
82#11745664—42no—
83C2×C59⋊C8—24no—
84C2×C59⋊C8—22no—
85C2×C59⋊C8—41no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8—42no—
88C2×C59⋊C8C22⋊C411yescharacteristic
89C2×C59⋊C8C22⋊C412yes—
90C2×C59⋊C8C22⋊C411yescharacteristic
91#11745659—42no—
92#11745659—42no—
93C23×C59—22no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59C2×C4⋊C411yescharacteristic
97C23×C59C22⋊Q812yes—
98C22×D59—21no—
99C22×D59—22no—
100C22×D59—41no—
101C22×D59—41no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59C2×C22⋊C411yescharacteristic
109C2×C4×C59#11711yescharacteristic
110C2×C4×C59#12112yes—
111C22⋊D59—44no—
112C2×Dic59—21no—
113C2×Dic59—21no—
114C2×Dic59—21no—
115C2×Dic59—22no—
116C2×Dic59—41no—
117C2×Dic59—41no—
118C4×D59—42no—
119C4×D59—41no—
120C4×D59—41no—
121C8×C59—41no—
122C8×C59—41no—
123C8×C59—82no—
124C59⋊C8—21no—
125C59⋊C8—22no—
126C59⋊C8—82no—
127C59⋊C8#9912yes—
128#8613—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—22no—
132C22×C59—22no—
133C22×C59—22no—
134C22×C59—44no—
135C22×C59#38011yescharacteristic
136C22×C59#43012yes—
137C2×D59—41no—
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—82no—
141C4×C59—22no—
142C4×C59—21no—
143C4×C59C2×#9911yescharacteristic
144C4×C59#41011yescharacteristic
145Dic59—41no—
146Dic59—41no—
147Dic59—41no—
148C2×#435—591no—
149#1810—592no—
150#1465—592no—
151#1293—591no—
152#1293D5911yescharacteristic, complemented
153C2×C59—22no—
154C2×C59—42no—
155C2×C59#179911yescharacteristic
156C2×C59#181111yescharacteristic
157C2×C59#181311yescharacteristic
158D59—81no—
159C22×M16—591no—
160C23×C8—591no—
161C2×C4×D4—591no—
162#435—1182no—
163#435—1182no—
164C2×#108—1182no—
165C2×C4⋊C8—591no—
166#407—592no—
167#407C2×D5912yes—
168C2×C22⋊C8—591no—
169C2×C22⋊C8C2×D5911yescharacteristic
170C2×#97—591no—
171#341—1182no—
172#324—1182no—
173#323—1182no—
174C59#861311yescharacteristic, complemented, Sylow
175C22×D4—591no—
176C23×C4—591no—
177C23×C4C22×D5911yescharacteristic
178C2×M16—22no—
179C2×M16—1181no—
180C2×M16—1181no—
181C2×M16—1182no—
182C2×M16—2362no—
183C22×C8—591no—
184C22×C8—592no—
185C22×C8—591no—
186C22×C8—592no—
187C22×C8—2362no—
188C22×C8C4×D5911yescharacteristic
189C22×C8C4×D5911yescharacteristic
190C4×D4—1182no—
191C4×D4—1182no—
192C2×C4⋊C4—591no—
193C2×C22⋊C4—591no—
194C2×C22⋊C4—591no—
195C2×C42—591no—
196#108—2364no—
197C4⋊C8—1182no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—1182no—
202C22⋊C8—1182no—
203C22⋊C8—1182no—
204#97—1181no—
205#97—1181no—
206C24—591no—
207C24Q8×D5911yescharacteristic
208C2×D4—1182no—
209C2×D4—2362no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—591no—
213C22×C4—591no—
214C22×C4—591no—
215C22×C4—1181no—
216C22×C4—1181no—
217C22×C4—2361no—
218C22×C4—2361no—
219C22×C4C2×C4×D5911yescharacteristic
220C22×C4#1174568512yes—
221C22×C4D4×D5912yes—
222C22×C4D4×D5911yescharacteristic
223M16—42no—
224M16—42no—
225M16—2362no—
226M16—2362no—
227M16—2362no—
228M16—2362no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—591no—
235C2×C8—591no—
236C2×C8—592no—
237C2×C8—1184no—
238C2×C8—1182no—
239C2×C8—2361no—
240C2×C8—2362no—
241C2×C8—2362no—
242C4⋊C4—1182no—
243C22⋊C4—1182no—
244C22⋊C4—1182no—
245C42—1181no—
246C42—1181no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—1182no—
251C23—1181no—
252C23—2361no—
253C23—2361no—
254C23C4⋊C4×D5911yescharacteristic
255C23#1427370512yes—
256D4—2364no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—1182no—
264C2×C4—1181no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—2362no—
268C2×C4—2361no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4—2361no—
272C2×C4C22⋊C4×D5911yescharacteristic
273C2×C4#1427368411yescharacteristic
274C2×C4#1427370012yes—
275C8—41no—
276C8—41no—
277C8—82no—
278C8—592no—
279C8—1182no—
280C8—1181no—
281C8—4722no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—44no—
288C22—4722no—
289C22—2361no—
290C22—2361no—
291C22—2361no—
292C22#1683726111yescharacteristic
293C22#1683744912yes—
294C4—22no—
295C4—21no—
296C4—2361no—
297C4—2361no—
298C4—2361no—
299C4#99×D5911yescharacteristic
300C4#1683737411yescharacteristic
301C2—22no—
302C2—42no—
303C2—4721no—
304C2#2142408311yescharacteristic
305C2#2142413711yescharacteristic
306C2#2142414511yescharacteristic
subgroupG/H
centerC22#16837261
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1293C2
socleC22×C59#380
prime radicalC22×C22⋊D59Q8
cosocleC23×C4×C59C23
hypercenter#1293D59
hypocenterC59#8613
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #21424132C2semidirect
4. #21421973C2semidirect
5. C59×#1293C2semidirect
152. #1293D59semidirect
174. C59#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#168374702
#214241322
#214224322
#214219732
C59×#12932
#861359