Finite Group Databasebeta

← #28486366

Subgroups — #28486366

Order 15104.

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

normal subgroups only

This group has 518 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#21433125C211yesmaximal, characteristic
2#21429088C211yesmaximal, characteristic
3#21422821C212yesmaximal, complemented
4#21421951C212yesmaximal, complemented
5C59×#1466C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7C2×#14273714C412yes—
8#16837474—24no—
9#16837459C2212yes—
10#16837366—21no—
11C2×C22⋊(C59⋊C8)—21no—
12#16837327—21no—
13#16837327—21no—
14#16837307C412yes—
15#16837262—21no—
16#16837262—21no—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837150—24no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#432C2212yes—
21C59×#323—21no—
22C59×#323—21no—
23C22×Dic118C2×C412yes—
24C2×#11745664—22no—
25C22×C59⋊C8—22no—
26#14273727—22no—
27#14273727—22no—
28#14273714—44no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—21no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59—22no—
34C2×C4×Dic59C2×C412yes—
35C4⋊(C59⋊C8)D414yes—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—22no—
39#14273638—22no—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C59⋊#95—21no—
43C22×D4×C59C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C4×D4×C59—22no—
47C4×D4×C59—22no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59C2311yescharacteristic
52C4⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54#97×C59—22no—
55C24×C59C22⋊C411yescharacteristic
56C24×C59C22⋊C411yescharacteristic
57C2×D4×C59—22no—
58C2×D4×C59—22no—
59C2×D4×C59—41no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—22no—
72C22×Dic59—22no—
73C2×Dic118—24no—
74C2×Dic118—42no—
75C2×Dic118—42no—
76C2×C8×C59—42no—
77C4⋊C4×C59—22no—
78C22⋊C4×C59—22no—
79C22⋊C4×C59—22no—
80C42×C59C2×D412yes—
81C42×C59#4112yes—
82C22⋊Dic59—44no—
83C22⋊Dic59—44no—
84C4⋊Dic59—42no—
85C4⋊Dic59—41no—
86C4⋊Dic59—42no—
87C4⋊Dic59—41no—
88C4×Dic59—42no—
89#11745664—82no—
90C2×C59⋊C8—24no—
91C2×C59⋊C8—82no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59#10012yes—
102C23×C59C2×C22⋊C411yescharacteristic
103D4×C59—44no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—41no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—81no—
115C2×C4×C59—81no—
116C2×C4×C59C4×D412yes—
117C2×C4×C59#12011yescharacteristic
118C2×C4×C59#12112yes—
119C2×C4×C59#12311yescharacteristic
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126Dic118—84no—
127C8×C59—82no—
128C59⋊C8—44no—
129#9451—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—81no—
141C22×C59—81no—
142C22×C59#38611yescharacteristic
143C22×C59C2×#10011yescharacteristic
144C22×C59#41311yescharacteristic
145C4×C59—21no—
146C4×C59—22no—
147C4×C59—22no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59#45512yes—
151Dic59—82no—
152Dic59—82no—
153#2973—591no—
154#2547—591no—
155#1558—592no—
156#1466D5911yescharacteristic
157#1285—592no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59—42no—
161C2×C59—81no—
162C2×C59#188511yescharacteristic
163C2×C59#190011yescharacteristic
164C2×C59#190911yescharacteristic
165C2×C4⋊Q8—591no—
166C2×#124—592no—
167C2×C4×D4C2×D5911yescharacteristic
168#435—1184no—
169#432—592no—
170#432C2×D5912yes—
171#407—1181no—
172C2×C22⋊C8—1181no—
173#397—1181no—
174#397—1181no—
175#391—592no—
176#380—1181no—
177#380—1181no—
178C4×C4⋊C4—591no—
179#333—1184no—
180#323—21no—
181#323—21no—
182C59#945111yescharacteristic, complemented, Sylow
183C22×Q8—592no—
184C22×D4C4×D5911yescharacteristic
185C23×C4D23611yescharacteristic
186C23×C4D23611yescharacteristic
187C2×M16—1182no—
188C22×C8—1182no—
189C4⋊Q8—1182no—
190C4⋊Q8—1182no—
191#124—2364no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4C4×D5911yescharacteristic
198C2×C22⋊C4—1182no—
199C2×C22⋊C4—1182no—
200C2×C22⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4C22⋊D5911yescharacteristic
202C2×C42—592no—
203C2×C42C22×D5911yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—594no—
206C22⋊C8—22no—
207C22⋊C8—1182no—
208C22⋊C8—1182no—
209C22⋊C8—1182no—
210#97—22no—
211#97—1182no—
212#95—1182no—
213#95—1182no—
214#95—1181no—
215C24(C2×D59)⋊C411yescharacteristic
216C24(C2×D59)⋊C411yescharacteristic
217C2×Q8—1184no—
218C2×Q8—2362no—
219C2×Q8—2362no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—1182no—
230C22×C4—1182no—
231C22×C4(C2×D59)⋊C411yescharacteristic
232C22×C4(C2×D59)⋊C411yescharacteristic
233C22×C4C2×C4×D5911yescharacteristic
234C22×C4C2×D23611yescharacteristic
235C22×C4C2×C22⋊D5911yescharacteristic
236M16—4722no—
237C2×C8—42no—
238C2×C8—1184no—
239C2×C8—4722no—
240C4⋊C4—22no—
241C4⋊C4—2362no—
242C4⋊C4—2361no—
243C4⋊C4—2362no—
244C4⋊C4—2361no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—2364no—
248C22⋊C4—2364no—
249C42—2362no—
250C42D4×D5912yes—
251C42#1174568712yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1427364312yes—
262C23C2×(C2×D59)⋊C411yescharacteristic
263Q8—4724no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—2362no—
278C2×C4—2362no—
279C2×C4—2362no—
280C2×C4—2362no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4(C22⋊D59)⋊C412yes—
284C2×C4#1427369711yescharacteristic
285C2×C4#1427369912yes—
286C2×C4#1427371111yescharacteristic
287C8—82no—
288C8—2364no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1683727911yescharacteristic
302C22C2×#1427364311yescharacteristic
303C22#1683738411yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—4722no—
310C4—4722no—
311C4#1683758212yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2142448211yescharacteristic
317C2#2142455711yescharacteristic
318C2#2142461611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1466C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1466D59
hypocenterC59#9451
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #21422821C2semidirect
4. #21421951C2semidirect
182. C59#9451semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214290882
#214228212
#214219512
C59×#14662
#945159