Finite Group Databasebeta

← #28497844

Subgroups — #28497844

Order 15104.

462 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 93 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#21426502C211yesmaximal, characteristic
2#21422409C212yesmaximal
3#21422041C212yesmaximal, complemented
4#21422040C211yesmaximal, characteristic, complemented
5C59×#1324C211yesmaximal, characteristic, complemented
6C2×#14273685C411yescharacteristic
7C2×#14273685C2211yescharacteristic
8C2×C42×D59C411yescharacteristic
9#16837452—22no—
10#16837271C411yescharacteristic
11#16837271C2211yescharacteristic
12#16837268C411yescharacteristic
13#16837261—22no—
14#16837253—22no—
15#16837128C2212yes—
16#16837125—22no—
17C59×#382C2211yescharacteristic
18C59×#324C2212yes—
19C22×C4×D59—21no—
20C22×C4×D59C2×C411yescharacteristic
21#14273685—41no—
22#14273685—41no—
23#14273685—41no—
24#14273685—41no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—22no—
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4D412yes—
31C2×C4×Dic59—21no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59D412yes—
35C42×D59—22no—
36C42×D59C814yescomplemented
37Dic59⋊C8—42no—
38C22⋊(C59⋊C8)—22no—
39C8×Dic59—42no—
40C59⋊#95—22no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C22⋊C8×C59—22no—
45#95×C59—22no—
46C23×D59C22⋊C411yescharacteristic
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59C22×C411yescharacteristic
50C22×C4×C59C2×D412yes—
51C22×Dic59—21no—
52C22×Dic59—21no—
53C22×Dic59C22⋊C412yes—
54C22×Dic59C22⋊C411yescharacteristic
55C2×C4×D59—21no—
56C2×C4×D59—21no—
57C2×C4×D59—22no—
58C2×C4×D59—24no—
59C2×C4×D59—42no—
60C2×C4×D59—42no—
61C2×C4×D59C2×C812yes—
62C2×C4×D59M1612yes—
63C2×C8×C59—42no—
64C4⋊C4×C59—41no—
65C4⋊C4×C59—41no—
66C42×C59—21no—
67C42×C59C2×C812yes—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—41no—
71C4⋊Dic59—41no—
72Dic59⋊C4—44no—
73Dic59⋊C4—44no—
74C4×Dic59—21no—
75C4×Dic59—24no—
76C4×Dic59—22no—
77C4×Dic59—22no—
78C4×Dic59—41no—
79C4×Dic59—41no—
80C4×Dic59C2×C812yes—
81C2×C59⋊C8—42no—
82C23×C59C2×C22⋊C411yescharacteristic
83C22×D59—21no—
84C22×D59—21no—
85C22×D59—41no—
86C22×D59—41no—
87C22×D59C22⋊C812yes—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—41no—
97C2×C4×C59C22×C811yescharacteristic
98C2×C4×C59C2×M1611yescharacteristic
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—42no—
104C2×Dic59—42no—
105C2×Dic59—41no—
106C2×Dic59C22⋊C812yes—
107C2×Dic59#9911yescharacteristic
108C2×Dic59#9911yescharacteristic
109C2×Dic59#10112yes—
110C4×D59—22no—
111C4×D59—22no—
112C4×D59—22no—
113C4×D59—22no—
114C4×D59—48no—
115C8×C59—82no—
116C59⋊C8—82no—
117#11447—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59C2×C22⋊C811yescharacteristic
121C22×C59C2×#9911yescharacteristic
122C22×C59C2×#10111yescharacteristic
123C2×D59—22no—
124C2×D59—42no—
125C2×D59—41no—
126C2×D59—41no—
127C2×D59—42no—
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—42no—
131C4×C59—42no—
132C4×C59—81no—
133Dic59—24no—
134Dic59—42no—
135Dic59—81no—
136Dic59#32412yes—
137Dic59#32412yes—
138#2285—591no—
139#1458—592no—
140#1325—592no—
141#1324—591no—
142#1324D5911yescharacteristic, complemented
143C2×C59—21no—
144C2×C59—21no—
145C2×C59C2×#32411yescharacteristic
146C2×C59#212611yescharacteristic
147C2×C59#213211yescharacteristic
148D59—42no—
149C2×#117—591no—
150C2×#117—591no—
151C22×C42—591no—
152#431—1182no—
153#384—591no—
154#384—591no—
155#382—591no—
156#382C2×D5911yescharacteristic
157#380—1182no—
158C4×C4⋊C4—1182no—
159#324—592no—
160#324C2×D5912yes—
161#323—1182no—
162C59#1144711yescharacteristic, complemented, Sylow
163C23×C4—591no—
164C23×C4—1181no—
165#117—2361no—
166#117—2361no—
167#117—2361no—
168#117—2361no—
169C2×C4⋊C4—591no—
170C2×C4⋊C4—591no—
171C2×C4⋊C4—1182no—
172C2×C4⋊C4—1182no—
173C2×C4⋊C4C4×D5911yescharacteristic
174C2×C4⋊C4C22×D5911yescharacteristic
175C2×C22⋊C4—592no—
176C2×C22⋊C4—1181no—
177C2×C42—591no—
178C2×C42—594no—
179C2×C42—592no—
180C2×C42—1182no—
181C2×C42—1181no—
182C2×C42—1181no—
183C2×C42C4×D5911yescharacteristic
184C4⋊C8—2362no—
185C22⋊C8—22no—
186C22⋊C8—1182no—
187C4×C8—2362no—
188#95—22no—
189#95—1182no—
190C24—591no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—591no—
194C22×C4—592no—
195C22×C4—592no—
196C22×C4—592no—
197C22×C4—1181no—
198C22×C4—1181no—
199C22×C4—1182no—
200C22×C4—1184no—
201C22×C4—1181no—
202C22×C4—1181no—
203C22×C4—2362no—
204C22×C4—2362no—
205C22×C4C2×C4×D5911yescharacteristic
206C22×C4D4×D5912yes—
207C2×C8—42no—
208C2×C8—2362no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—2361no—
212C4⋊C4—2361no—
213C4⋊C4—2364no—
214C4⋊C4—2364no—
215C22⋊C4—2362no—
216C22⋊C4—2362no—
217C42—21no—
218C42—592no—
219C42—1182no—
220C42—1182no—
221C42—1184no—
222C42—1181no—
223C42—2361no—
224C42—2361no—
225C42C8×D5912yes—
226C23—592no—
227C23—1181no—
228C23—1181no—
229C23—2361no—
230C23—2361no—
231C23C22⋊C4×D5911yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—592no—
242C2×C4—592no—
243C2×C4—591no—
244C2×C4—591no—
245C2×C4—1182no—
246C2×C4—1182no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4—2368no—
254C2×C4—2362no—
255C2×C4—2361no—
256C2×C4—2362no—
257C2×C4C2×C8×D5911yescharacteristic
258C2×C4M16×D5911yescharacteristic
259C8—82no—
260C8—4722no—
261C22—21no—
262C22—21no—
263C22—1182no—
264C22—2362no—
265C22—2362no—
266C22—2361no—
267C22—2361no—
268C22C22⋊C8×D5911yescharacteristic
269C22#99×D5911yescharacteristic
270C22#101×D5911yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—81no—
276C4—592no—
277C4—592no—
278C4—1184no—
279C4—4721no—
280C4—2362no—
281C2—21no—
282C2—21no—
283C2—2362no—
284C2#324×D5911yescharacteristic
285C2#2142576011yescharacteristic
286C2#2142577811yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1324C2
socleC22×C59C2×#99
prime radicalC23×D59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1324D59
hypocenterC59#11447
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
4. #21422040C2semidirect
5. C59×#1324C2semidirect
36. C42×D59C8semidirect
142. #1324D59semidirect
162. C59#11447semidirect

Maximal subgroups

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

subgroupindex
#214265022
#214224092
#214220412
#214220402
C59×#13242
#1144759