Finite Group Databasebeta

← #28499392

Subgroups — #28499392

Order 15104.

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

normal subgroups only

This group has 388 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#21422101C214yesmaximal, complemented
4C59×#1822C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic
6C2×#14273706—21no—
7#16837459—21no—
8#16837332—21no—
9#16837332C2212yes—
10#16837327—21no—
11#16837327C412yes—
12#16837308—21no—
13#16837268C411yescharacteristic
14#16837128C2212yes—
15#16837126—21no—
16C2×#117×C59C2211yescharacteristic
17C59×#407—21no—
18C59×#343—21no—
19C59×#343C2212yes—
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
37C2×M16×C59—22no—
38#117×C59—41no—
39#117×C59—41no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C22⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C22⋊C8×C59—41no—
47C22⋊C8×C59—41no—
48C24×C59C22⋊C411yescharacteristic
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59—22no—
52C22×C4×C59—41no—
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—22no—
60C22×Dic59—22no—
61C2×Dic118—42no—
62C2×Dic118—42no—
63C2×Dic118—41no—
64C2×Dic118—41no—
65M16×C59—42no—
66M16×C59—42no—
67C2×C8×C59—42no—
68C4⋊C4×C59—41no—
69C4⋊C4×C59—41no—
70C22⋊C4×C59—41no—
71C22⋊C4×C59—41no—
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—21no—
85C23×C59—41no—
86C23×C59—41no—
87C23×C59C2×C22⋊C411yescharacteristic
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—42no—
97C2×C4×C59—42no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59#9911yescharacteristic
103C2×C4×C59#9911yescharacteristic
104C2×C4×C59AGL(1,ℤ/8)12yes—
105C2×Dic59—42no—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C2×Dic59—42no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111Dic118—82no—
112Dic118—82no—
113C8×C59—82no—
114C59⋊C8—82no—
115#11807—591nomaximal, Sylow
116C22×C59—21no—
117C22×C59—21no—
118C22×C59—42no—
119C22×C59—41no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59—81no—
123C22×C59C2×#9911yescharacteristic
124C22×C59#41811yescharacteristic
125C22×C59#42111yescharacteristic
126C4×C59—22no—
127C4×C59—21no—
128C4×C59—21no—
129C4×C59—41no—
130C4×C59—41no—
131C4×C59—41no—
132C4×C59—41no—
133Dic59—82no—
134Dic59—82no—
135#2621—591no—
136#1822D5911yescharacteristic
137#1464—591no—
138#1346—594no—
139C2×C59—21no—
140C2×C59—21no—
141C2×C59—81no—
142C2×C59#151011yescharacteristic
143C2×C59#212311yescharacteristic
144C2×C59#213511yescharacteristic
145C2×C4⋊Q8—591no—
146C2×C22⋊Q8—1181no—
147C2×#117C2×D5911yescharacteristic
148#432—1181no—
149#407—21no—
150#398—592no—
151#398—1181no—
152#397—592no—
153#397—1181no—
154#391—1181no—
155#382—591no—
156#343—21no—
157#343C2×D5912yes—
158#324—592no—
159#323—1181no—
160C59#1180711yescharacteristic, complemented, Sylow
161C22×Q8—1181no—
162C23×C4C22⋊D5911yescharacteristic
163C2×M16—22no—
164C4⋊Q8—2361no—
165C4⋊Q8—2361no—
166C22⋊Q8—2364no—
167#117—41no—
168#117—41no—
169C2×C4⋊C4—592no—
170C2×C4⋊C4—1181no—
171C2×C4⋊C4—1181no—
172C2×C4⋊C4—1182no—
173C2×C4⋊C4C22⋊D5911yescharacteristic
174C2×C4⋊C4C22×D5911yescharacteristic
175C2×C22⋊C4—1182no—
176C2×C22⋊C4C4×D5911yescharacteristic
177C2×C22⋊C4D23611yescharacteristic
178C2×C42C4×D5911yescharacteristic
179C2×C42D23611yescharacteristic
180C4⋊C8—2361no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—1182no—
184#97—2361no—
185#95—592no—
186#95—1181no—
187#95—1182no—
188#95—1182no—
189C24(C2×D59)⋊C411yescharacteristic
190C2×Q8—2362no—
191C2×Q8—2362no—
192C2×Q8—2361no—
193C2×Q8—2361no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—1182no—
199C22×C4—1182no—
200C22×C4(C2×D59)⋊C411yescharacteristic
201C22×C4(C2×D59)⋊C411yescharacteristic
202C22×C4(C2×D59)⋊C411yescharacteristic
203C22×C4C2×C4×D5911yescharacteristic
204C22×C4C2×D23611yescharacteristic
205C22×C4C2×C22⋊D5911yescharacteristic
206M16—42no—
207M16—42no—
208C2×C8—42no—
209C2×C8—2362no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—2362no—
213C4⋊C4—2362no—
214C4⋊C4—2362no—
215C4⋊C4—2364no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—2364no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×(C2×D59)⋊C411yescharacteristic
229Q8—4722no—
230Q8—4722no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—2362no—
246C2×C4—2362no—
247C2×C4—2362no—
248C2×C4—2362no—
249C2×C4—2362no—
250C2×C4—2362no—
251C2×C4#1427364111yescharacteristic
252C2×C4#1427364111yescharacteristic
253C2×C4#1427375212yes—
254C8—82no—
255C8—4722no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22C2×#1427364111yescharacteristic
264C22#1683740611yescharacteristic
265C22#1683742011yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—4722no—
274C4—4722no—
275C2—21no—
276C2—21no—
277C2—81no—
278C2#2142262411yescharacteristic
279C2#2142575111yescharacteristic
280C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1822C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1822D59
hypocenterC59#11807
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #21422101C2semidirect
160. C59#11807semidirect

Maximal subgroups

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

subgroupindex
#214298622
#214224302
#214221012
C59×#18222
#1180759