Finite Group Databasebeta

← #28499366

Subgroups — #28499366

Order 15104.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429862C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic
3#21422095C214yesmaximal
4C59×#1820C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273706—21no—
7#16837459—21no—
8#16837332—21no—
9#16837332C412yescomplemented
10#16837327—21no—
11#16837327C2212yes—
12#16837308—21no—
13#16837268C411yescharacteristic, complemented
14#16837128—21no—
15#16837126C2212yes—
16C2×#117×C59C2211yescharacteristic
17C59×#381—21no—
18C59×#342—21no—
19C59×#342C2212yes—
20C22×Dic118—21no—
21#14273727—41no—
22#14273727—41no—
23#14273706—44no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—22no—
28C2×C22⋊Dic59—22no—
29C4⋊(C59⋊C8)—41no—
30C22⋊(C59⋊C8)—22no—
31#14273638—41no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95C2×C412yes—
36C23×C4×C59D411yescharacteristic
37#117×C59—41no—
38#117×C59—41no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2311yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46#95×C59—41no—
47#95×C59—41no—
48C24×C59C2×D411yescharacteristic
49C22×C4×C59—22no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—41no—
53C22×C4×C59—42no—
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×Dic59—22no—
61C22×Dic59—22no—
62C2×Dic118—42no—
63C2×Dic118—42no—
64C2×Dic118—41no—
65C2×Dic118—41no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C22⋊C4×C59—41no—
69C22⋊C4×C59—41no—
70C22⋊C4×C59—42no—
71C22⋊C4×C59—42no—
72C42×C59—21no—
73C42×C59—21no—
74C42×C59—21no—
75C42×C59—21no—
76C22⋊Dic59—44no—
77C4⋊Dic59—42no—
78C4⋊Dic59—42no—
79C4⋊Dic59—42no—
80Dic59⋊C4—44no—
81C2×C59⋊C8—42no—
82C23×C59—21no—
83C23×C59—21no—
84C23×C59—41no—
85C23×C59—41no—
86C23×C59C2×C22⋊C411yescharacteristic
87C23×C59AGL(1,ℤ/8)12yes—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—82no—
104C2×C4×C59—82no—
105C2×C4×C59—82no—
106C2×C4×C59#9911yescharacteristic
107C2×C4×C59#9911yescharacteristic
108C2×Dic59—42no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114Dic118—82no—
115Dic118—82no—
116C59⋊C8—82no—
117#11801—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59—22no—
121C22×C59—42no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—81no—
125C22×C59C2×#9911yescharacteristic
126C22×C59#41811yescharacteristic
127C22×C59#42111yescharacteristic
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—41no—
131C4×C59—41no—
132C4×C59—41no—
133C4×C59—41no—
134C4×C59—41no—
135C4×C59—162no—
136Dic59—82no—
137Dic59—82no—
138#2621—591no—
139#1820D5911yescharacteristic
140#1464—591no—
141#1344—594no—
142C2×C59—21no—
143C2×C59—21no—
144C2×C59—42no—
145C2×C59#151011yescharacteristic
146C2×C59#212311yescharacteristic
147C2×C59#212911yescharacteristic
148C2×C4⋊Q8—591no—
149C2×C22⋊Q8—1181no—
150C2×#117C2×D5911yescharacteristic
151#432—1181no—
152#398—592no—
153#398—1181no—
154#397—592no—
155#397—1181no—
156#391—1181no—
157#382—591no—
158#381—21no—
159#342—21no—
160#342C2×D5912yes—
161#324—1181no—
162#323—592no—
163C59#1180111yescharacteristic, complemented, Sylow
164C22×Q8—1181no—
165C23×C4C22⋊D5911yescharacteristic
166C4⋊Q8—2361no—
167C4⋊Q8—2361no—
168C22⋊Q8—2364no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—592no—
172C2×C4⋊C4—1182no—
173C2×C4⋊C4—1181no—
174C2×C4⋊C4—1181no—
175C2×C4⋊C4C4×D5911yescharacteristic
176C2×C4⋊C4D23611yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1182no—
179C2×C22⋊C4C22⋊D5911yescharacteristic
180C2×C22⋊C4C22×D5911yescharacteristic
181C2×C42C4×D5911yescharacteristic
182C2×C42D23611yescharacteristic
183C4⋊C8—2361no—
184C22⋊C8—1182no—
185#97—2361no—
186#95—41no—
187#95—41no—
188#95—592no—
189#95—1181no—
190#95—1182no—
191#95—1182no—
192C24C2×C22⋊D5911yescharacteristic
193C2×Q8—2362no—
194C2×Q8—2362no—
195C2×Q8—2361no—
196C2×Q8—2361no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—1182no—
203C22×C4—1182no—
204C22×C4(C2×D59)⋊C411yescharacteristic
205C22×C4(C2×D59)⋊C411yescharacteristic
206C22×C4(C2×D59)⋊C411yescharacteristic
207C22×C4(C2×D59)⋊C411yescharacteristic
208C22×C4C2×C4×D5911yescharacteristic
209C22×C4C2×D23611yescharacteristic
210C2×C8—2362no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—2362no—
214C4⋊C4—2364no—
215C4⋊C4—2362no—
216C4⋊C4—2362no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—2364no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×(C2×D59)⋊C411yescharacteristic
231C23#1427375212yes—
232Q8—4722no—
233Q8—4722no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—2362no—
253C2×C4—2362no—
254C2×C4—2362no—
255C2×C4—2362no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4#1427364111yescharacteristic
259C2×C4#1427364111yescharacteristic
260C8—4722no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1427364111yescharacteristic
269C22#1683740611yescharacteristic
270C22#1683742011yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—162no—
279C4—4722no—
280C4—4722no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2142262411yescharacteristic
285C2#2142574911yescharacteristic
286C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1820C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1820D59
hypocenterC59#11801
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
5. C2×#14273727C4semidirect
9. #16837332C4semidirect
13. #16837268C4semidirect
163. C59#11801semidirect

Maximal subgroups

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

subgroupindex
#214298622
#214224302
#214220952
C59×#18202
#1180159