Finite Group Databasebeta

← #28497909

Subgroups — #28497909

Order 15104.

310 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 103 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#21432738C211yesmaximal, characteristic
2#21425895C211yesmaximal, characteristic, complemented
3#21422037C212yesmaximal, complemented
4#21422021C212yesmaximal
5C59×#2163C211yesmaximal, characteristic, complemented
6C4×#11745685C411yescharacteristic
7#16837461—21no—
8#16837461—21no—
9#16837453—21no—
10#16837453—21no—
11#16837428C411yescharacteristic
12#16837428C2211yescharacteristic
13#16837368C411yescharacteristic
14#16837368C2211yescharacteristic
15C4×#11745664C411yescharacteristic
16C4⋊(C59⋊C16)—21no—
17C22⋊(C59⋊C16)C2212yes—
18C4×C59⋊C16—21no—
19C2×C4×C8×C59C2211yescharacteristic
20C59×C4⋊C16—21no—
21C59×C22⋊C16C2212yes—
22C59×#346—21no—
23C2×#11745685C2×C411yescharacteristic
24C2×#11745664C2×C411yescharacteristic
25C2×#11745664C2×C411yescharacteristic
26C4×Dic118C811yescharacteristic
27C4×Dic118C811yescharacteristic
28C4×D236C811yescharacteristic
29C4×D236C811yescharacteristic
30C4×C22⋊D59—21no—
31C4×C22⋊D59—21no—
32#14273686C2×C411yescharacteristic
33#14273684—21no—
34C42×D59—21no—
35C2×C59⋊C16—22no—
36Dic59⋊C8—21no—
37Dic59⋊C8—21no—
38Dic59⋊C8—21no—
39Dic59⋊C8—21no—
40(C2×D59)⋊C8—21no—
41(C2×D59)⋊C8—21no—
42(C2×D59)⋊C8—21no—
43(C2×D59)⋊C8—21no—
44#14273638—21no—
45C4×C59⋊C8—21no—
46C22×C8×C59C2×C411yescharacteristic
47C22×C8×C59C2311yescharacteristic
48C2×C42×C59C2×C411yescharacteristic
49C2×C16×C59—22no—
50C4×C8×C59D411yescharacteristic
51C4×C8×C59D411yescharacteristic
52C4×C8×C59D411yescharacteristic
53C4×C8×C59D411yescharacteristic
54C22×C4×C59M1611yescharacteristic
55C22×C4×C59M1611yescharacteristic
56C22×C4×C59C22×C411yescharacteristic
57C2×C22⋊D59—21no—
58#11745685—41no—
59#11745685—41no—
60C2×D236C2×C811yescharacteristic
61C2×C4×D59—21no—
62C2×C4×D59—21no—
63C2×C4×D59—21no—
64C2×Dic118C2×C811yescharacteristic
65C2×C8×C59—41no—
66C2×C8×C59—41no—
67C2×C8×C59C22⋊C411yescharacteristic
68C2×C8×C59C22⋊C411yescharacteristic
69C2×C8×C59C2×D411yescharacteristic
70C2×C8×C59C2×D411yescharacteristic
71C42×C59C22⋊C411yescharacteristic
72C42×C59C22⋊C411yescharacteristic
73C42×C59C2×C811yescharacteristic
74C42×C59C2×C811yescharacteristic
75C22⋊Dic59—21no—
76(C2×D59)⋊C4—21no—
77(C2×D59)⋊C4—21no—
78C4⋊Dic59C2×C811yescharacteristic
79C4⋊Dic59C2×C811yescharacteristic
80Dic59⋊C4—21no—
81Dic59⋊C4—21no—
82C4×Dic59—21no—
83C4×Dic59—21no—
84C4×Dic59—21no—
85#11745664—41no—
86#11745664—41no—
87C2×C59⋊C8—21no—
88C2×C59⋊C8—21no—
89C16×C59—42no—
90C59⋊C16—42no—
91C23×C59C2×M1611yescharacteristic
92C22×D59—21no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—41no—
97C2×C4×C59C22⋊C811yescharacteristic
98C2×C4×C59C22⋊C811yescharacteristic
99C2×C4×C59C22⋊C811yescharacteristic
100C2×C4×C59C22⋊C811yescharacteristic
101C2×C4×C59C2×C22⋊C411yescharacteristic
102C2×C4×C59C22×C811yescharacteristic
103C22⋊D59—42no—
104C2×Dic59—21no—
105C2×Dic59—21no—
106C2×Dic59—21no—
107D236—41no—
108C4×D59—42no—
109C4×D59—42no—
110C4×D59—41no—
111C4×D59—41no—
112Dic118—41no—
113C8×C59—21no—
114C8×C59—21no—
115C8×C59—21no—
116C8×C59—21no—
117C59⋊C8—41no—
118C59⋊C8—41no—
119#11464—591nomaximal, Sylow
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59C2×C22⋊C811yescharacteristic
124C2×D59—41no—
125C2×D59—41no—
126C2×D59—41no—
127C4×C59—21no—
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—21no—
131C4×C59#35012yes—
132C4×C59#41011yescharacteristic
133C4×C59#41311yescharacteristic
134Dic59—41no—
135Dic59—41no—
136Dic59—41no—
137#2922—591no—
138#2163D5911yescharacteristic, complemented
139#2162—591no—
140#1323—592no—
141#1315—592no—
142C2×C59—41no—
143C2×C59#145811yescharacteristic
144C2×C59C2×#35011yescharacteristic
145C2×C59#211711yescharacteristic
146D59—81no—
147C4×#41—591no—
148#433—1181no—
149#433—1181no—
150#431—1181no—
151#431—1181no—
152#424—591no—
153#424—591no—
154#408—591no—
155#408—591no—
156C4×M16—591no—
157C2×C4×C8C2×D5911yescharacteristic
158C4⋊C16—21no—
159C4⋊C16—1181no—
160C22⋊C16—592no—
161C22⋊C16C2×D5912yes—
162#346—21no—
163C4×C16—1181no—
164C59#1146411yescharacteristic, complemented, Sylow
165C2×#41—591no—
166C2×M16—591no—
167C2×M16—591no—
168C22×C8C4×D5911yescharacteristic
169C22×C8C22×D5911yescharacteristic
170C4×Q8—591no—
171C4×Q8—591no—
172C4×D4—591no—
173C4×D4—591no—
174C4×D4—1181no—
175C4×D4—1181no—
176#117—591no—
177#117—1181no—
178C2×C42—1181no—
179C2×C42C4×D5911yescharacteristic
180C2×C16—22no—
181C2×C16—1182no—
182C4⋊C8—1181no—
183C4⋊C8—1181no—
184C4⋊C8—1181no—
185C4⋊C8—1181no—
186C22⋊C8—1181no—
187C22⋊C8—1181no—
188C22⋊C8—1181no—
189C22⋊C8—1181no—
190#97—1181no—
191C4×C8—1181no—
192C4×C8D23611yescharacteristic
193C4×C8D23611yescharacteristic
194C4×C8C22⋊D5911yescharacteristic
195C4×C8C22⋊D5911yescharacteristic
196#41—2361no—
197#41—2361no—
198C2×Q8—591no—
199C2×D4—591no—
200C2×D4—1181no—
201C22×C4—1181no—
202C22×C4—1181no—
203C22×C4—1181no—
204C22×C4#1174565911yescharacteristic
205C22×C4#1174565911yescharacteristic
206C22×C4C2×C4×D5911yescharacteristic
207M16—2361no—
208M16—2361no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—1181no—
212C2×C8—1181no—
213C2×C8(C2×D59)⋊C411yescharacteristic
214C2×C8(C2×D59)⋊C411yescharacteristic
215C2×C8C2×D23611yescharacteristic
216C2×C8C2×C22⋊D5911yescharacteristic
217C4⋊C4—591no—
218C4⋊C4—591no—
219C4⋊C4—1181no—
220C4⋊C4—1181no—
221C22⋊C4—1181no—
222C22⋊C4—1181no—
223C22⋊C4—1181no—
224C42—1181no—
225C42—1181no—
226C42—1181no—
227C42C8×D5911yescharacteristic
228C42C8×D5911yescharacteristic
229C42(C2×D59)⋊C411yescharacteristic
230C42(C2×D59)⋊C411yescharacteristic
231C16—42no—
232C16—2362no—
233C23—1181no—
234C23C2×#1174565911yescharacteristic
235Q8—2361no—
236D4—2361no—
237D4—2362no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—1181no—
243C2×C4—1181no—
244C2×C4—1181no—
245C2×C4—2362no—
246C2×C4—2361no—
247C2×C4—2362no—
248C2×C4—2361no—
249C2×C4(C2×D59)⋊C811yescharacteristic
250C2×C4(C2×D59)⋊C811yescharacteristic
251C2×C4(C2×D59)⋊C811yescharacteristic
252C2×C4(C2×D59)⋊C811yescharacteristic
253C2×C4C2×(C2×D59)⋊C411yescharacteristic
254C2×C4C2×C8×D5911yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—2361no—
260C8—2361no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—2361no—
265C22—2361no—
266C22—2361no—
267C22C2×(C2×D59)⋊C811yescharacteristic
268C4—21no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—2361no—
273C4—2361no—
274C4—2361no—
275C4#1683718712yes—
276C4#1683737511yescharacteristic
277C4#1683738411yescharacteristic
278C2—41no—
279C2—4721no—
280C2#2142240811yescharacteristic
281C2C2×#1683718711yescharacteristic
282C2#2142572311yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D59)⋊C4
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#2163C2
socleC22×C59C2×C22⋊C8
prime radicalC2×#11745685C2×C4
cosocleC22×C8×C59C23
hypercenter#2163D59
hypocenterC59#11464
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2163

Complements

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

NH ≅ G/Ntype
2. #21425895C2semidirect
3. #21422037C2semidirect
5. C59×#2163C2semidirect
138. #2163D59semidirect
164. C59#11464semidirect

Maximal subgroups

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

subgroupindex
#214327382
#214258952
#214220372
#214220212
C59×#21632
#1146459