Finite Group Databasebeta

← #28497886

Subgroups — #28497886

Order 15104.

386 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#16837454C211yesmaximal, characteristic
2#21425892C212yesmaximal, complemented
3#21422036C211yesmaximal, characteristic, complemented
4#21422019C212yesmaximal
5C59×#1322C211yesmaximal, characteristic, complemented
6C2×C42×D59C411yescharacteristic
7#16837454—22no—
8#16837454—22no—
9C2×C4⋊(C59⋊C8)C411yescharacteristic
10C2×(C2×D59)⋊C8C411yescharacteristic
11C2×(C2×D59)⋊C8C2211yescharacteristic
12C2×C8×Dic59C411yescharacteristic
13C2×C8×Dic59C2211yescharacteristic
14Dic59⋊C16—22no—
15C22⋊(C59⋊C16)C2212yes—
16C16×Dic59—22no—
17C2×C4⋊C8×C59C2211yescharacteristic
18C59×C22⋊C16C2212yes—
19C22×C4×D59C811yescharacteristic
20C22×C4×D59C811yescharacteristic
21C22×C4×D59C2×C411yescharacteristic
22C22×C59⋊C8C2×C411yescharacteristic
23C22×C59⋊C8C2×C411yescharacteristic
24C2×C4×Dic59C811yescharacteristic
25C2×C4×Dic59C811yescharacteristic
26C2×C4×Dic59C2×C411yescharacteristic
27C42×D59—21no—
28C42×D59—21no—
29C42×D59—21no—
30C42×D59—21no—
31C2×C59⋊C16—22no—
32C4⋊(C59⋊C8)—22no—
33(C2×D59)⋊C8—22no—
34(C2×D59)⋊C8—22no—
35C8×Dic59—21no—
36C8×Dic59—21no—
37C8×Dic59D412yes—
38C8×Dic59D412yes—
39C22×C8×C59C2×C411yescharacteristic
40C22×C8×C59C2311yescharacteristic
41C2×C42×C59C2×C411yescharacteristic
42C2×C16×C59—22no—
43C4⋊C8×C59—22no—
44C23×D59C2×C811yescharacteristic
45C22×C4×C59C2×C811yescharacteristic
46C22×C4×C59C2×C811yescharacteristic
47C22×C4×C59C22×C411yescharacteristic
48C22×Dic59C2×C811yescharacteristic
49C22×Dic59C2×C811yescharacteristic
50C22×Dic59C2×C811yescharacteristic
51C2×C4×D59—21no—
52C2×C4×D59—21no—
53C2×C4×D59—22no—
54C2×C4×D59—21no—
55C2×C4×D59—21no—
56C2×C4×D59—21no—
57C2×C4×D59—21no—
58C2×C4×D59—42no—
59C2×C4×D59—41no—
60C2×C4×D59—41no—
61C2×C4×D59C1614yescomplemented
62C2×C8×C59—21no—
63C2×C8×C59—41no—
64C2×C8×C59—41no—
65C2×C8×C59C2×D412yes—
66C42×C59—21no—
67C42×C59—21no—
68C4×Dic59—21no—
69C4×Dic59—21no—
70C4×Dic59—21no—
71C4×Dic59—21no—
72C4×Dic59C22⋊C411yescharacteristic
73C4×Dic59C22⋊C411yescharacteristic
74C4×Dic59M1611yescharacteristic
75C4×Dic59M1611yescharacteristic
76C2×C59⋊C8—21no—
77C2×C59⋊C8—41no—
78C2×C59⋊C8—41no—
79C2×C59⋊C8C22⋊C412yes—
80C16×C59—42no—
81C59⋊C16—42no—
82C23×C59C22×C811yescharacteristic
83C22×D59—21no—
84C22×D59—21no—
85C22×D59—41no—
86C22×D59—41no—
87C22×D59C2×C1612yes—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—41no—
94C2×C4×C59C2×C1612yes—
95C2×C4×C59C2×C22⋊C411yescharacteristic
96C2×C4×C59C2×M1611yescharacteristic
97C2×Dic59—22no—
98C2×Dic59—21no—
99C2×Dic59—41no—
100C2×Dic59C22⋊C811yescharacteristic
101C2×Dic59C22⋊C811yescharacteristic
102C2×Dic59C22⋊C811yescharacteristic
103C2×Dic59C22⋊C811yescharacteristic
104C2×Dic59C2×C1612yes—
105C2×Dic59M3212yes—
106C4×D59—24no—
107C4×D59—24no—
108C4×D59—42no—
109C4×D59—42no—
110C4×D59—41no—
111C4×D59—41no—
112C4×D59—42no—
113C8×C59—22no—
114C8×C59—41no—
115C59⋊C8—22no—
116C59⋊C8—41no—
117#11459—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—21no—
120C22×C59C2×C22⋊C811yescharacteristic
121C22×C59C22×C1611yescharacteristic
122C22×C59C2×M3211yescharacteristic
123C2×D59—22no—
124C2×D59—42no—
125C2×D59—41no—
126C2×D59—41no—
127C2×D59—42no—
128C4×C59—22no—
129C4×C59—21no—
130C4×C59—41no—
131C4×C59#41011yescharacteristic
132C4×C59#41311yescharacteristic
133Dic59—21no—
134Dic59—41no—
135Dic59C22⋊C1612yes—
136Dic59C22⋊C1612yes—
137Dic59#34912yes—
138C2×#431—591no—
139#2162—592no—
140#1322—591no—
141#1322D5911yescharacteristic, complemented
142#1314—592no—
143C2×C59—21no—
144C2×C59—21no—
145C2×C59#145811yescharacteristic
146C2×C59C2×C22⋊C1611yescharacteristic
147C2×C59C2×#34911yescharacteristic
148D59—42no—
149C22×C42—591no—
150#431—1182no—
151#431—1182no—
152C2×C4⋊C8—591no—
153C2×C4⋊C8C2×D5911yescharacteristic
154C2×C22⋊C8—591no—
155C2×C22⋊C8—591no—
156C2×C4×C8—591no—
157C2×C4×C8—591no—
158C4⋊C16—1182no—
159C22⋊C16—592no—
160C22⋊C16C2×D5912yes—
161C4×C16—1182no—
162C59#1145911yescharacteristic, complemented, Sylow
163C23×C4—591no—
164C23×C4—591no—
165C23×C4—591no—
166C22×C8—591no—
167C22×C8—591no—
168C22×C8C4×D5911yescharacteristic
169C22×C8C22×D5911yescharacteristic
170C2×C42—591no—
171C2×C42—591no—
172C2×C42—591no—
173C2×C42—1181no—
174C2×C42—1181no—
175C2×C42—1181no—
176C2×C42—1181no—
177C2×C42C4×D5911yescharacteristic
178C2×C16—22no—
179C2×C16—1182no—
180C4⋊C8—22no—
181C4⋊C8—1182no—
182C22⋊C8—1182no—
183C22⋊C8—1182no—
184C4×C8—592no—
185C4×C8—592no—
186C4×C8—1181no—
187C4×C8—1181no—
188C24—591no—
189C22×C4—591no—
190C22×C4—591no—
191C22×C4—594no—
192C22×C4—591no—
193C22×C4—1181no—
194C22×C4—1181no—
195C22×C4—1182no—
196C22×C4—1181no—
197C22×C4—1181no—
198C22×C4—1181no—
199C22×C4—1181no—
200C22×C4—2361no—
201C22×C4—2361no—
202C22×C4—2362no—
203C22×C4C8×D5911yescharacteristic
204C22×C4C8×D5911yescharacteristic
205C22×C4C2×C4×D5911yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—592no—
210C2×C8—1181no—
211C2×C8—2361no—
212C2×C8—2361no—
213C2×C8D4×D5912yes—
214C42—21no—
215C42—21no—
216C42—591no—
217C42—591no—
218C42—591no—
219C42—591no—
220C42—1181no—
221C42—1181no—
222C42—1181no—
223C42—1181no—
224C16—42no—
225C16—2362no—
226C23—592no—
227C23—1181no—
228C23—1181no—
229C23—2361no—
230C23—2361no—
231C23C2×C8×D5911yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—592no—
239C2×C4—592no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—591no—
243C2×C4—591no—
244C2×C4—1184no—
245C2×C4—1184no—
246C2×C4—1182no—
247C2×C4—1181no—
248C2×C4—2362no—
249C2×C4—2362no—
250C2×C4—2362no—
251C2×C4—2361no—
252C2×C4—2361no—
253C2×C4—2361no—
254C2×C4C16×D5912yes—
255C2×C4C22⋊C4×D5911yescharacteristic
256C2×C4M16×D5911yescharacteristic
257C8—22no—
258C8—41no—
259C8—1182no—
260C8—2361no—
261C22—21no—
262C22—21no—
263C22—1182no—
264C22—2362no—
265C22—2362no—
266C22—2361no—
267C22—2361no—
268C22C22⋊C8×D5911yescharacteristic
269C22C2×C16×D5911yescharacteristic
270C22M32×D5911yescharacteristic
271C4—22no—
272C4—21no—
273C4—41no—
274C4—592no—
275C4—592no—
276C4—592no—
277C4—1181no—
278C4—2361no—
279C4#1683737411yescharacteristic
280C4#1683738511yescharacteristic
281C2—21no—
282C2—21no—
283C2—2362no—
284C2#2142240911yescharacteristic
285C2C22⋊C16×D5911yescharacteristic
286C2#349×D5911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC22×C59C22×C16
FrattiniC22×C8C22×D59
FittingC59×#1322C2
socleC22×C59C2×C22⋊C8
prime radicalC23×D59C2×C8
cosocleC22×C8×C59C23
hypercenter#1322D59
hypocenterC59#11459
LCSG ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #21425892C2semidirect
3. #21422036C2semidirect
5. C59×#1322C2semidirect
61. C2×C4×D59C16semidirect
141. #1322D59semidirect
162. C59#11459semidirect

Maximal subgroups

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

subgroupindex
C2×#168374542
#214258922
#214220362
#214220192
C59×#13222
#1145959