Finite Group Databasebeta

← #28482026

Subgroups — #28482026

Order 15104.

530 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16837266C211yesmaximal, characteristic
2#21424162C212yesmaximal
3#21421939C211yesmaximal, characteristic
4#21421938C212yesmaximal
5C59×#1280C211yesmaximal, characteristic
6C22×Dic59⋊C4C412yes—
7C22×C4×Dic59C2211yescharacteristic
8C2×C22⋊(C59⋊C8)C2212yes—
9#16837266—22no—
10#16837266—22no—
11C2×C59⋊#95C412yes—
12C2×C59⋊#95C2211yescharacteristic
13#16837156—22no—
14#16837156—22no—
15C2×C22⋊C8×C59C2212yes—
16C2×#95×C59C2211yescharacteristic
17C23×Dic59C2×C411yescharacteristic
18C23×Dic59C2×C411yescharacteristic
19C23×Dic59C2×C412yes—
20C22×C59⋊C8—22no—
21C2×Dic59⋊C4—22no—
22C2×Dic59⋊C4—22no—
23C2×Dic59⋊C4—44no—
24C2×C4×Dic59—22no—
25C2×C4×Dic59—21no—
26C2×C4×Dic59—21no—
27C2×C4×Dic59D412yes—
28C2×C4×Dic59D412yes—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C59⋊#95—22no—
32C59⋊#95—24no—
33C23×C4×C59C2×C412yes—
34C23×C4×C59C2311yescharacteristic
35C22×C8×C59—22no—
36C22⋊C8×C59—22no—
37C22⋊C8×C59—22no—
38#95×C59—22no—
39C24×C59C22×C411yescharacteristic
40C22×C4×C59—22no—
41C22×C4×C59—22no—
42C22×C4×C59—22no—
43C22×C4×C59—42no—
44C22×C4×C59—42no—
45C22×C4×C59—42no—
46C22×C4×C59C2×D412yes—
47C22×C4×C59#4112yes—
48C22×C4×C59#4111yescharacteristic
49C22×C4×C59#4111yescharacteristic
50C22×Dic59—21no—
51C22×Dic59—21no—
52C22×Dic59—22no—
53C22×Dic59—22no—
54C22×Dic59—21no—
55C22×Dic59—21no—
56C22×Dic59—22no—
57C22×Dic59—22no—
58C22×Dic59—22no—
59C22×Dic59—22no—
60C22×Dic59—22no—
61C22×Dic59—42no—
62C22×Dic59—42no—
63C22×Dic59C22⋊C412yes—
64C22×Dic59C22⋊C411yescharacteristic
65C22×Dic59C22⋊C411yescharacteristic
66C2×C8×C59—44no—
67C2×C8×C59—42no—
68Dic59⋊C4—44no—
69Dic59⋊C4—44no—
70C4×Dic59—22no—
71C4×Dic59—22no—
72C4×Dic59—22no—
73C4×Dic59—22no—
74C2×C59⋊C8—44no—
75C2×C59⋊C8—42no—
76C23×C59—22no—
77C23×C59—21no—
78C23×C59—21no—
79C23×C59C2×C22⋊C411yescharacteristic
80C23×C59#11711yescharacteristic
81C23×C59#11711yescharacteristic
82C23×C59#13112yes—
83C23×C59#13112yes—
84C2×C4×C59—24no—
85C2×C4×C59—42no—
86C2×C4×C59—41no—
87C2×C4×C59—41no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—82no—
92C2×C4×C59—82no—
93C2×C4×C59#12312yes—
94C2×C4×C59#12312yes—
95C2×Dic59—24no—
96C2×Dic59—24no—
97C2×Dic59—21no—
98C2×Dic59—21no—
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—24no—
103C2×Dic59—41no—
104C2×Dic59—42no—
105C2×Dic59—41no—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C2×Dic59—42no—
109C2×Dic59—82no—
110C2×Dic59—82no—
111C2×Dic59#10012yes—
112C2×Dic59#10112yes—
113C8×C59—82no—
114C59⋊C8—82no—
115#8703—591nomaximal, Sylow
116C22×C59—21no—
117C22×C59—21no—
118C22×C59—21no—
119C22×C59—21no—
120C22×C59—22no—
121C22×C59—22no—
122C22×C59—22no—
123C22×C59—22no—
124C22×C59#38111yescharacteristic
125C22×C59#40812yes—
126C22×C59#41012yes—
127C22×C59C2×#10011yescharacteristic
128C22×C59C2×#10111yescharacteristic
129C22×C59#43311yescharacteristic
130C22×C59#43312yes—
131C22×C59#43311yescharacteristic
132C4×C59—22no—
133C4×C59—22no—
134C4×C59—82no—
135Dic59—22no—
136Dic59—22no—
137Dic59—42no—
138Dic59—82no—
139C2×#382—591no—
140#1817—592no—
141#1280—592no—
142#1280—591no—
143#1280D5911yescharacteristic
144C2×C59—21no—
145C2×C59—22no—
146C2×C59—21no—
147C2×C59#146912yes—
148C2×C59#147012yes—
149C2×C59#181811yescharacteristic
150C2×C59#182011yescharacteristic
151C2×C59#182111yescharacteristic
152C22×C4⋊C4—592no—
153C22×C42—591no—
154C2×C22⋊C8—592no—
155C2×C22⋊C8C2×D5912yes—
156#382—1182no—
157#382—1182no—
158C2×#95—592no—
159C2×#95—591no—
160C2×#95C2×D5911yescharacteristic
161#336—1182no—
162#336—1182no—
163C59#870311yescharacteristic, complemented, Sylow
164C23×C4—591no—
165C23×C4—591no—
166C23×C4—592no—
167C23×C4C4×D5912yes—
168C23×C4C22×D5911yescharacteristic
169C22×C8—22no—
170C22×C8—1182no—
171C2×C4⋊C4—1182no—
172C2×C4⋊C4—1182no—
173C2×C4⋊C4—2364no—
174C2×C42—592no—
175C2×C42—592no—
176C2×C42—1182no—
177C2×C42—1181no—
178C2×C42—1181no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—1182no—
182C22⋊C8—1182no—
183#95—22no—
184#95—1184no—
185#95—1182no—
186C24C2×C4×D5911yescharacteristic
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—591no—
194C22×C4—591no—
195C22×C4—592no—
196C22×C4—1182no—
197C22×C4—1182no—
198C22×C4—1181no—
199C22×C4—1181no—
200C22×C4—1181no—
201C22×C4—1181no—
202C22×C4—1182no—
203C22×C4—1182no—
204C22×C4—1182no—
205C22×C4—1182no—
206C22×C4—1182no—
207C22×C4—2362no—
208C22×C4—2362no—
209C22×C4#1174568512yes—
210C22×C4D4×D5912yes—
211C22×C4#1174568711yescharacteristic
212C22×C4#1174568911yescharacteristic
213C2×C8—44no—
214C2×C8—42no—
215C2×C8—2364no—
216C2×C8—2362no—
217C4⋊C4—2364no—
218C4⋊C4—2364no—
219C42—1182no—
220C42—1182no—
221C42—1182no—
222C42—1182no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23C22⋊C4×D5911yescharacteristic
227C23#1427368411yescharacteristic
228C23#1427368511yescharacteristic
229C23#1427373312yes—
230C23#1427373512yes—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—592no—
241C2×C4—592no—
242C2×C4—1184no—
243C2×C4—1184no—
244C2×C4—1181no—
245C2×C4—1181no—
246C2×C4—1181no—
247C2×C4—1181no—
248C2×C4—1182no—
249C2×C4—1184no—
250C2×C4—4722no—
251C2×C4—4722no—
252C2×C4—2362no—
253C2×C4—2362no—
254C2×C4—2362no—
255C2×C4—2361no—
256C2×C4—2362no—
257C2×C4—2361no—
258C2×C4#1427370912yes—
259C2×C4#1427371212yes—
260C8—82no—
261C8—4722no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22#1683726411yescharacteristic
271C22#1683736712yes—
272C22#1683737412yes—
273C22#100×D5911yescharacteristic
274C22#101×D5911yescharacteristic
275C22#1683746111yescharacteristic
276C22#1683746212yes—
277C22#1683746411yescharacteristic
278C4—22no—
279C4—22no—
280C4—82no—
281C4—1182no—
282C4—1182no—
283C4—4722no—
284C4—2362no—
285C2—21no—
286C2—22no—
287C2—21no—
288C2#2142244612yes—
289C2#2142245012yes—
290C2#2142416611yescharacteristic
291C2#2142417211yescharacteristic
292C2#2142417511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutator, prime radicalC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1280C2
socleC23×C59C2×C22⋊C4
cosocleC23×C4×C59C23
hypercenter#1280D59
hypocenterC59#8703
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
163. C59#8703semidirect

Maximal subgroups

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

subgroupindex
C2×#168372662
#214241622
#214219392
#214219382
C59×#12802
#870359