Finite Group Databasebeta

← #28480048

Subgroups — #28480048

Order 15104.

1174 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×#14273706C211yesmaximal, characteristic
2#21424006C211yesmaximal, characteristic
3C2×#16837127C214yesmaximal, complemented
4C59×#1779C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic
6C2×#14273706—28no—
7C2×#14273706—22no—
8C2×#14273706—22no—
9C22×C4⋊Dic59C411yescharacteristic
10C22×Dic59⋊C4C2212yes—
11C22×C22⋊Dic59C412yes—
12C2×C22⋊(C59⋊C8)—22no—
13C2×C22⋊(C59⋊C8)C2212yes—
14#16837127—28no—
15C24×C4×C59C2211yescharacteristic
16C2×C22⋊C8×C59—22no—
17C2×C22⋊C8×C59C2212yes—
18C22×Dic118—22no—
19C22×Dic118—24no—
20C22×Dic118—44no—
21C23×Dic59C2×C412yes—
22C23×Dic59C2×C412yes—
23C22×C59⋊C8—22no—
24#14273706—216no—
25#14273706—216no—
26C2×C4⋊Dic59—22no—
27C2×C4⋊Dic59—24no—
28C2×Dic59⋊C4—22no—
29C2×Dic59⋊C4—24no—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4D418yes—
32C2×C22⋊Dic59—24no—
33C2×C22⋊Dic59—24no—
34C2×C22⋊Dic59—24no—
35C22⋊(C59⋊C8)—24no—
36C22⋊(C59⋊C8)—44no—
37C25×C59C2×C411yescharacteristic
38C23×C4×C59—24no—
39C23×C4×C59—44no—
40C23×C4×C59C2×C411yescharacteristic
41C23×C4×C59D412yes—
42C23×C4×C59D412yes—
43C23×C4×C59C2311yescharacteristic
44C22×C8×C59—22no—
45C22⋊C8×C59—24no—
46C22⋊C8×C59—44no—
47C24×C59—24no—
48C24×C59—22no—
49C24×C59—22no—
50C24×C59—22no—
51C24×C59—42no—
52C24×C59C22⋊C412yes—
53C24×C59C22×C411yescharacteristic
54C22×C4×C59—22no—
55C22×C4×C59—24no—
56C22×C4×C59—24no—
57C22×C4×C59—24no—
58C22×C4×C59—24no—
59C22×C4×C59—24no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59—416no—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D414yes—
68C22×Dic59—22no—
69C22×Dic59—22no—
70C22×Dic59—22no—
71C22×Dic59—22no—
72C22×Dic59—24no—
73C22×Dic59—24no—
74C22×Dic59—44no—
75C22×Dic59C22⋊C414yes—
76C22×Dic59C22⋊C414yes—
77C2×Dic118—28no—
78C2×Dic118—416no—
79C2×Dic118—44no—
80C2×Dic118—44no—
81C2×C8×C59—44no—
82C2×C8×C59—42no—
83C22⋊Dic59—216no—
84C4⋊Dic59—28no—
85Dic59⋊C4—28no—
86Dic59⋊C4—28no—
87C2×C59⋊C8—44no—
88C2×C59⋊C8—42no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—28no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—22no—
96C23×C59—22no—
97C23×C59—22no—
98C23×C59—48no—
99C23×C59—48no—
100C23×C59—44no—
101C23×C59—44no—
102C23×C59#10114yes—
103C23×C59C2×C22⋊C412yes—
104C23×C59C2×C22⋊C411yescharacteristic
105C23×C59#13112yes—
106C23×C59#13112yes—
107C2×C4×C59—28no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—44no—
113C2×C4×C59—48no—
114C2×C4×C59—48no—
115C2×C4×C59#9914yes—
116C2×C4×C59#12012yes—
117C2×C4×C59#12012yes—
118C2×Dic59—24no—
119C2×Dic59—24no—
120C2×Dic59—28no—
121C2×Dic59—44no—
122C2×Dic59—44no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—44no—
126C2×Dic59—44no—
127Dic118—416no—
128C8×C59—82no—
129C59⋊C8—82no—
130#8308—591nomaximal, Sylow
131C22×C59—24no—
132C22×C59—24no—
133C22×C59—22no—
134C22×C59—22no—
135C22×C59—24no—
136C22×C59—24no—
137C22×C59—24no—
138C22×C59—416no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59#37911yescharacteristic
144C22×C59#40814yes—
145C22×C59#40811yescharacteristic
146C22×C59#40811yescharacteristic
147C22×C59C2×#9912yes—
148C22×C59C2×#10112yes—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—24no—
152Dic59—44no—
153Dic59—44no—
154C22×C22⋊Q8—591no—
155#1779—591no—
156#1779D5911yescharacteristic
157C2×#324—594no—
158C2×C59—24no—
159C2×C59—44no—
160C2×C59#147212yes—
161C2×C59#147212yes—
162C2×C59#178011yescharacteristic
163C2×C59#178111yescharacteristic
164C2×C59#178411yescharacteristic
165C23×Q8—591no—
166C24×C4C2×D5911yescharacteristic
167C2×C22⋊Q8—1188no—
168C2×C22⋊Q8—1182no—
169C2×C22⋊Q8—1182no—
170C22×C4⋊C4—592no—
171C22×C4⋊C4—591no—
172C22×C22⋊C4—592no—
173C2×C22⋊C8—22no—
174C2×C22⋊C8—592no—
175C2×C22⋊C8—1182no—
176C2×C22⋊C8C2×D5912yes—
177#324—1188no—
178C59#830811yescharacteristic, complemented, Sylow
179C25C4×D5911yescharacteristic
180C22×Q8—1184no—
181C22×Q8—1182no—
182C22×Q8—2364no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—592no—
186C23×C4—592no—
187C23×C4C4×D5911yescharacteristic
188C23×C4D23612yes—
189C23×C4C22⋊D5912yes—
190C23×C4C22×D5911yescharacteristic
191C22×C8—22no—
192C22×C8—1182no—
193C22⋊Q8—11816no—
194C22⋊Q8—11816no—
195C2×C4⋊C4—598no—
196C2×C4⋊C4—1184no—
197C2×C4⋊C4—1182no—
198C2×C4⋊C4—1182no—
199C2×C4⋊C4—1184no—
200C2×C4⋊C4—1182no—
201C2×C22⋊C4—1184no—
202C2×C22⋊C4—1184no—
203C2×C22⋊C4—1184no—
204C22⋊C8—24no—
205C22⋊C8—44no—
206C22⋊C8—1184no—
207C22⋊C8—2364no—
208C24—24no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—42no—
213C24(C2×D59)⋊C412yes—
214C24C2×C4×D5911yescharacteristic
215C2×Q8—1188no—
216C2×Q8—23616no—
217C2×Q8—2364no—
218C2×Q8—2364no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—416no—
229C22×C4—594no—
230C22×C4—594no—
231C22×C4—1182no—
232C22×C4—1182no—
233C22×C4—1182no—
234C22×C4—1182no—
235C22×C4—1184no—
236C22×C4—1184no—
237C22×C4—2364no—
238C22×C4(C2×D59)⋊C412yes—
239C22×C4C2×D23611yescharacteristic
240C22×C4D4×D5914yes—
241C22×C4C2×C22⋊D5911yescharacteristic
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—2364no—
245C2×C8—2362no—
246C4⋊C4—1188no—
247C4⋊C4—1188no—
248C4⋊C4—1188no—
249C22⋊C4—11816no—
250C23—22no—
251C23—22no—
252C23—28no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—48no—
260C23—48no—
261C23—44no—
262C23—44no—
263C23#1427364514yes—
264C23C22⋊C4×D5912yes—
265C23C2×(C2×D59)⋊C411yescharacteristic
266C23#1427373312yes—
267C23#1427373512yes—
268Q8—23616no—
269C2×C4—28no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—1184no—
278C2×C4—1184no—
279C2×C4—1188no—
280C2×C4—2364no—
281C2×C4—2364no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2364no—
285C2×C4—2364no—
286C2×C4#1427364114yes—
287C2×C4#1427369512yes—
288C2×C4#1427369712yes—
289C8—82no—
290C8—4722no—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—416no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22#1683725711yescharacteristic
304C22#1683736714yes—
305C22#1683736811yescharacteristic
306C22#1683737011yescharacteristic
307C22C2×#1427364112yes—
308C22C2×#1427364512yes—
309C4—22no—
310C4—22no—
311C4—24no—
312C4—2364no—
313C4—2364no—
314C2—24no—
315C2—44no—
316C2#2142245912yes—
317C2#2142246112yes—
318C2#2142400711yescharacteristic
319C2#2142401011yescharacteristic
320C2#2142401911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1779C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59C2×C4
cosocleC23×C4×C59C23
hypercenter#1779D59
hypocenterC59#8308
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. C2×#16837127C2semidirect
178. C59#8308semidirect

Maximal subgroups

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

subgroupindex
C22×#142737062
#214240062
C2×#168371272
C59×#17792
#830859