Finite Group Databasebeta

← #28498162

Subgroups — #28498162

Order 15104.

460 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 77 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#21426639C211yesmaximal, characteristic
2#21422433C212yesmaximal
3#21422039C212yesmaximal, complemented
4#21422030C211yesmaximal, characteristic, complemented
5C59×#1319C211yesmaximal, characteristic, complemented
6C2×#14273716C411yescharacteristic
7C2×Q8×Dic59C411yescharacteristic
8C2×#14273689C2211yescharacteristic
9#16837364—22no—
10#16837304C411yescharacteristic
11#16837294—22no—
12#16837267C2211yescharacteristic
13#16837249—22no—
14C42×Dic59C411yescharacteristic
15#16837128C2212yes—
16#16837125—22no—
17C59×#391C2211yescharacteristic
18C59×#324C2212yes—
19C22×D236C2×C411yescharacteristic
20C22×C4×D59D412yes—
21C2×#11745659—42no—
22#14273716—22no—
23#14273716—22no—
24Q8×Dic59—41no—
25Q8×Dic59—41no—
26#14273689—41no—
27#14273689—41no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4D412yes—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36C2×C4×Dic59C2×C411yescharacteristic
37C2×C4×Dic59C2×C411yescharacteristic
38C22⋊(C59⋊C8)—22no—
39(C2×D59)⋊C8—42no—
40C59⋊#95—22no—
41C22×Q8×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C22⋊C8×C59—22no—
45#95×C59—22no—
46C23×D59C22⋊C412yes—
47C2×Q8×C59—21no—
48C2×Q8×C59—41no—
49C2×Q8×C59—41no—
50C2×Q8×C59M1612yescomplemented
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59C22×C411yescharacteristic
54C22×C4×C59C2×D412yes—
55C22×Dic59—21no—
56C22×Dic59—21no—
57C22×Dic59C22⋊C411yescharacteristic
58C22×Dic59C22⋊C411yescharacteristic
59C2×D236—21no—
60C2×D236—21no—
61C2×D236—41no—
62C2×D236—41no—
63C2×C4×D59—22no—
64C2×C4×D59—22no—
65C2×C4×D59—44no—
66C2×C8×C59—42no—
67C4⋊C4×C59—41no—
68C42×C59—41no—
69(C2×D59)⋊C4—28no—
70(C2×D59)⋊C4—42no—
71(C2×D59)⋊C4—44no—
72C4⋊Dic59—42no—
73C4⋊Dic59—41no—
74C4×Dic59—21no—
75C4×Dic59—21no—
76C4×Dic59—21no—
77C4×Dic59—42no—
78C4×Dic59—42no—
79C4×Dic59—41no—
80C4×Dic59—41no—
81C4×Dic59M1612yescomplemented
82C2×C59⋊C8—42no—
83#11745659—84no—
84C23×C59C2×C22⋊C411yescharacteristic
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—22no—
88C22×D59—42no—
89C22×D59—42no—
90Q8×C59—42no—
91Q8×C59—42no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59C2×M1611yescharacteristic
103C2×C4×C59C2×M1611yescharacteristic
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—21no—
107C2×Dic59—41no—
108C2×Dic59—41no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59—41no—
112C2×Dic59#9911yescharacteristic
113C2×Dic59#9911yescharacteristic
114D236—42no—
115D236—42no—
116C4×D59—44no—
117C4×D59—82no—
118C8×C59—82no—
119C59⋊C8—82no—
120#11531—591nomaximal, Sylow
121C22×C59—21no—
122C22×C59—21no—
123C22×C59#40711yescharacteristic
124C22×C59C2×#9911yescharacteristic
125C22×C59#41311yescharacteristic
126C2×D59—44no—
127C2×D59—42no—
128C2×D59—42no—
129C2×D59—42no—
130C2×D59—82no—
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—42no—
135C4×C59—42no—
136Dic59—21no—
137Dic59—42no—
138Dic59—41no—
139Dic59—81no—
140Dic59#35412yes—
141C4×#124—591no—
142#1465—592no—
143#1324—592no—
144#1319—591no—
145#1319D5911yescharacteristic, complemented
146C2×C59—21no—
147C2×C59—21no—
148C2×C59#146511yescharacteristic
149C2×C59C2×#35411yescharacteristic
150C2×C59#213211yescharacteristic
151D59—82no—
152C2×#124—591no—
153C2×C4×Q8—591no—
154C2×C4×D4—591no—
155#407—1182no—
156#391C2×D5911yescharacteristic
157#390—591no—
158#388—1182no—
159#382—591no—
160C4×C22⋊C4—1182no—
161C43—591no—
162#324—592no—
163#324C2×D5912yes—
164#323—1182no—
165C59#1153111yescharacteristic, complemented, Sylow
166C22×Q8C4×D5911yescharacteristic
167C22×D4—591no—
168C23×C4—592no—
169C2×M16—2362no—
170#124—1182no—
171#124—1182no—
172C4×Q8—2361no—
173C4×Q8—2361no—
174C4×D4—2361no—
175C4×D4—2361no—
176C2×C4⋊C4—591no—
177C2×C4⋊C4—1181no—
178C2×C4⋊C4C22×D5911yescharacteristic
179C2×C22⋊C4—592no—
180C2×C22⋊C4—1182no—
181C2×C22⋊C4—1182no—
182C2×C42—591no—
183C2×C42—591no—
184C2×C42—1181no—
185C2×C42—1181no—
186C2×C42—1181no—
187C2×C42C4×D5911yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—1182no—
190C22⋊C8—2362no—
191#95—22no—
192#95—1182no—
193C24—592no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8#1174565912yescomplemented
198C2×D4—1181no—
199C2×D4—1181no—
200C2×D4—2361no—
201C2×D4—2361no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—591no—
205C22×C4—591no—
206C22×C4—1182no—
207C22×C4—1182no—
208C22×C4—1181no—
209C22×C4—1181no—
210C22×C4—2364no—
211C22×C4C2×C4×D5911yescharacteristic
212C22×C4D4×D5912yes—
213M16—4724no—
214C2×C8—42no—
215C2×C8—2362no—
216C4⋊C4—41no—
217C4⋊C4—2361no—
218C4⋊C4—2362no—
219C22⋊C4—1188no—
220C22⋊C4—2362no—
221C22⋊C4—2364no—
222C42—41no—
223C42—592no—
224C42—1181no—
225C42—1181no—
226C42—1181no—
227C42—2362no—
228C42—2361no—
229C42—2361no—
230C42—2362no—
231C23—1182no—
232C23—1182no—
233C23—1182no—
234C23—2362no—
235C23—2362no—
236C23C22⋊C4×D5911yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—2362no—
240D4—2362no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—591no—
252C2×C4—591no—
253C2×C4—1182no—
254C2×C4—1182no—
255C2×C4—1181no—
256C2×C4—4722no—
257C2×C4—2364no—
258C2×C4—2362no—
259C2×C4—2361no—
260C2×C4—2361no—
261C2×C4—2361no—
262C2×C4—2361no—
263C2×C4M16×D5911yescharacteristic
264C2×C4C2×#1174565911yescharacteristic
265C8—82no—
266C8—4722no—
267C22—21no—
268C22—21no—
269C22—4722no—
270C22—2364no—
271C22—2362no—
272C22—2362no—
273C22—2362no—
274C22#1683736411yescharacteristic
275C22#99×D5911yescharacteristic
276C22#1683738311yescharacteristic
277C4—21no—
278C4—21no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—592no—
283C4—1181no—
284C4—4721no—
285C4—2362no—
286C4—2361no—
287C2—21no—
288C2—21no—
289C2—4722no—
290C2#2142243111yescharacteristic
291C2#354×D5911yescharacteristic
292C2#2142578011yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1319C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1319D59
hypocenterC59#11531
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #21422039C2semidirect
4. #21422030C2semidirect
5. C59×#1319C2semidirect
50. C2×Q8×C59M16semidirect
81. C4×Dic59M16semidirect
145. #1319D59semidirect
165. C59#11531semidirect
197. C2×Q8#11745659semidirect

Maximal subgroups

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

subgroupindex
#214266392
#214224332
#214220392
#214220302
C59×#13192
#1153159