Finite Group Databasebeta

← #28477358

Subgroups — #28477358

Order 15104.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#21432448C211yesmaximal, characteristic
2#21424164C211yesmaximal, characteristic
3#21421920C214yesmaximal, complemented
4C59×#1797C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic
6C2×#14273706—21no—
7C2×C22⋊(C59⋊C8)—21no—
8#16837308—21no—
9#16837300—21no—
10#16837300C2212yes—
11#16837272C411yescharacteristic
12#16837262—21no—
13#16837262C412yes—
14#16837157—21no—
15#16837157C2212yes—
16C22×C42×C59C2211yescharacteristic
17C2×C4⋊C8×C59—21no—
18C2×C22⋊C8×C59—21no—
19C59×#336C2212yes—
20C22×Dic118—21no—
21C22×C59⋊C8—22no—
22#14273727—42no—
23#14273706—42no—
24#14273706—42no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C412yes—
29C2×C4⋊Dic59C2×C412yes—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—22no—
32C22⋊(C59⋊C8)—42no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59D411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C22×C8×C59—22no—
39C2×C42×C59—42no—
40C2×C42×C59C2×C411yescharacteristic
41C2×C42×C59D411yescharacteristic
42C2×C42×C59D411yescharacteristic
43C2×C42×C59C2311yescharacteristic
44C4⋊C8×C59—42no—
45C22⋊C8×C59—42no—
46C24×C59C22⋊C411yescharacteristic
47C22×C4×C59—22no—
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—41no—
52C22×C4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59—41no—
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59D812yes—
59C22×C4×C59SD1612yes—
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×Dic59—22no—
64C22×Dic59—22no—
65C2×Dic118—42no—
66C2×Dic118—42no—
67C2×Dic118—42no—
68C2×C8×C59—42no—
69C2×C8×C59—42no—
70C2×C8×C59—42no—
71C42×C59—22no—
72C42×C59—22no—
73C42×C59—41no—
74C42×C59—41no—
75C22⋊Dic59—44no—
76C4⋊Dic59—24no—
77C4⋊Dic59—42no—
78C4⋊Dic59—42no—
79C4⋊Dic59—42no—
80C4⋊Dic59—42no—
81Dic59⋊C4—42no—
82Dic59⋊C4—42no—
83C2×C59⋊C8—42no—
84C2×C59⋊C8—42no—
85C2×C59⋊C8—42no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59#10212yes—
93C23×C59C2×C22⋊C411yescharacteristic
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—44no—
109C2×C4×C59—44no—
110C2×C4×C59#9912yes—
111C2×C4×C59#10112yes—
112C2×C4×C59#10212yes—
113C2×C4×C59#13112yes—
114C2×C4×C59C2×D811yescharacteristic
115C2×C4×C59C2×SD1611yescharacteristic
116C2×C4×C59#13712yes—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123Dic118—84no—
124C8×C59—82no—
125C59⋊C8—82no—
126#7728—591nomaximal, Sylow
127C22×C59—22no—
128C22×C59—22no—
129C22×C59—22no—
130C22×C59—41no—
131C22×C59—41no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59#40811yescharacteristic
135C22×C59C2×#9911yescharacteristic
136C22×C59C2×#10111yescharacteristic
137C22×C59C2×#10211yescharacteristic
138C22×C59#41911yescharacteristic
139C22×C59#42111yescharacteristic
140C22×C59#42111yescharacteristic
141C4×C59—22no—
142C4×C59—22no—
143C4×C59—22no—
144C4×C59—22no—
145C4×C59—21no—
146C4×C59—22no—
147C4×C59—21no—
148Dic59—82no—
149Dic59—82no—
150#2857—591no—
151#1817—591no—
152#1797D5911yescharacteristic
153#1271—594no—
154C2×C59—42no—
155C2×C59#147211yescharacteristic
156C2×C59#148011yescharacteristic
157C2×C59#149711yescharacteristic
158C2×C59#151111yescharacteristic
159C2×C59#151911yescharacteristic
160C2×C59#153711yescharacteristic
161C2×C59#154611yescharacteristic
162C2×C4⋊Q8—591no—
163C2×C22⋊Q8—1181no—
164C22×C42C2×D5911yescharacteristic
165C2×C4⋊C8—21no—
166C2×C22⋊C8—21no—
167C2×C22⋊C8—1181no—
168#391—1181no—
169#389—592no—
170#389—1181no—
171#384—591no—
172#380—592no—
173#380—1181no—
174#336—592no—
175#336—1181no—
176#336C2×D5912yes—
177C59#772811yescharacteristic, complemented, Sylow
178C22×Q8—1181no—
179C23×C4C4×D5911yescharacteristic
180C23×C4D23611yescharacteristic
181C23×C4C22⋊D5911yescharacteristic
182C22×C8—22no—
183C22×C8—1182no—
184C4⋊Q8—2362no—
185C22⋊Q8—2362no—
186C22⋊Q8—2362no—
187C2×C4⋊C4—592no—
188C2×C4⋊C4—592no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4—1182no—
193C2×C22⋊C4—1182no—
194C2×C42—42no—
195C2×C42C4×D5911yescharacteristic
196C2×C42D23611yescharacteristic
197C2×C42C22⋊D5911yescharacteristic
198C2×C42C22×D5911yescharacteristic
199C4⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—2362no—
202#95—1182no—
203#95—1182no—
204C24(C2×D59)⋊C411yescharacteristic
205C2×Q8—2362no—
206C2×Q8—2362no—
207C2×Q8—2362no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—1182no—
217C22×C4—1182no—
218C22×C4(C2×D59)⋊C411yescharacteristic
219C22×C4(C2×D59)⋊C411yescharacteristic
220C22×C4(C2×D59)⋊C411yescharacteristic
221C22×C4#1174566912yes—
222C22×C4#1174567112yes—
223C22×C4C2×C4×D5911yescharacteristic
224C22×C4C2×D23611yescharacteristic
225C22×C4C2×C22⋊D5911yescharacteristic
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—2362no—
230C2×C8—2362no—
231C2×C8—2362no—
232C4⋊C4—1184no—
233C4⋊C4—2362no—
234C4⋊C4—2362no—
235C4⋊C4—2362no—
236C4⋊C4—2362no—
237C4⋊C4—2362no—
238C4⋊C4—2362no—
239C22⋊C4—2364no—
240C42—22no—
241C42—22no—
242C42—41no—
243C42—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1427364712yes—
251C23C2×(C2×D59)⋊C411yescharacteristic
252Q8—4724no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4—2362no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4#1427364112yes—
276C2×C4#1427364512yes—
277C2×C4#1427364712yes—
278C2×C4#1427373312yes—
279C2×C4C2×#1174566911yescharacteristic
280C2×C4C2×#1174567111yescharacteristic
281C2×C4#1427376012yes—
282C8—82no—
283C8—4722no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#1683736811yescharacteristic
292C22C2×#1427364111yescharacteristic
293C22C2×#1427364511yescharacteristic
294C22C2×#1427364711yescharacteristic
295C22#1683741211yescharacteristic
296C22#1683741811yescharacteristic
297C22#1683742011yescharacteristic
298C4—22no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—22no—
304C4—22no—
305C4—4722no—
306C4—4722no—
307C2—42no—
308C2#2142245911yescharacteristic
309C2#2142249411yescharacteristic
310C2#2142257011yescharacteristic
311C2#2142263011yescharacteristic
312C2#2142266311yescharacteristic
313C2#2142273311yescharacteristic
314C2#2142276911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1797C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1797D59
hypocenterC59#7728
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #21421920C2semidirect
177. C59#7728semidirect

Maximal subgroups

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

subgroupindex
#214324482
#214241642
#214219202
C59×#17972
#772859