Finite Group Databasebeta

← #28483367

Subgroups — #28483367

Order 15104.

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

normal subgroups only

This group has 542 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
1#21433181C211yesmaximal, characteristic, complemented
2C4×#14273716C211yesmaximal, characteristic
3#21422488C212yesmaximal
4#21421928C212yesmaximal, complemented
5C59×#1461C211yesmaximal, characteristic, complemented
6C2×#14273716C411yescharacteristic
7C2×C4×D236C2211yescharacteristic
8#16837533—24no—
9#16837459—22no—
10#16837455C2212yes—
11#16837309C411yescharacteristic
12#16837303C411yescharacteristic
13#16837292—22no—
14#16837267C411yescharacteristic
15C4×(C2×D59)⋊C4—22no—
16C42×Dic59C2211yescharacteristic
17#16837131—24no—
18C2×C4×Q8×C59C2211yescharacteristic
19C59×#431C2212yes—
20C59×#324—22no—
21C22×D236C2×C411yescharacteristic
22C22×C4×D59—21no—
23#14273716—22no—
24#14273716—22no—
25C4×D236—22no—
26C4×D236—22no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—21no—
31C2×C4×Dic59—21no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59C2×C411yescharacteristic
35C4⋊(C59⋊C8)—24no—
36C4⋊(C59⋊C8)D414yes—
37C22⋊(C59⋊C8)—22no—
38#14273638—24no—
39C4×C59⋊C8—22no—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C22×Q8×C59C2×C411yescharacteristic
43C4×Q8×C59—22no—
44C4×Q8×C59—22no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C4⋊C4×C59D412yes—
47C2×C42×C59D412yes—
48C2×C42×C59C2311yescharacteristic
49C4⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C4×C8×C59—22no—
52C23×D59—21no—
53C2×Q8×C59—22no—
54C2×Q8×C59—21no—
55C2×Q8×C59—21no—
56C2×Q8×C59—22no—
57C22×C4×C59C22⋊C412yes—
58C22×C4×C59C22⋊C412yes—
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×Dic59—21no—
63C22×Dic59—21no—
64C22×Dic59—21no—
65C2×D236—22no—
66C2×D236—22no—
67C2×D236—21no—
68C2×D236M1612yes—
69C2×C4×D59—44no—
70C2×C4×D59—41no—
71C2×C4×D59—41no—
72C2×C8×C59—42no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—24no—
75C42×C59—22no—
76C42×C59—22no—
77C42×C59C2×D412yes—
78C42×C59#4112yes—
79(C2×D59)⋊C4—28no—
80(C2×D59)⋊C4—44no—
81(C2×D59)⋊C4—42no—
82C4⋊Dic59—22no—
83C4×Dic59—21no—
84C4×Dic59—22no—
85C4×Dic59—42no—
86C4×Dic59—42no—
87C4×Dic59M1612yes—
88C2×C59⋊C8—24no—
89C23×C59C2×C22⋊C411yescharacteristic
90C22×D59—22no—
91C22×D59—42no—
92C22×D59—42no—
93C22×D59—41no—
94C22×D59—41no—
95Q8×C59—42no—
96Q8×C59—42no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—41no—
103C2×C4×C59—42no—
104C2×C4×C59C4×D412yes—
105C2×C4×C59#12011yescharacteristic
106C2×C4×C59#12112yes—
107C2×C4×C59#12311yescharacteristic
108C2×C4×C59C2×M1611yescharacteristic
109C2×C4×C59C2×M1611yescharacteristic
110C2×C4×C59#13112yes—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—41no—
117C2×Dic59—42no—
118C2×Dic59—41no—
119D236—24no—
120D236—42no—
121C4×D59—82no—
122C4×D59—82no—
123C8×C59—82no—
124C59⋊C8—44no—
125#8964—591nomaximal, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59#38611yescharacteristic
129C22×C59#40711yescharacteristic
130C22×C59#40811yescharacteristic
131C2×D59—44no—
132C2×D59—42no—
133C2×D59—82no—
134C2×D59—81no—
135C2×D59—81no—
136C4×C59—22no—
137C4×C59—21no—
138C4×C59—42no—
139C4×C59—42no—
140C4×C59#43512yes—
141C4×C59#43612yes—
142C4×C59#45412yes—
143Dic59—42no—
144Dic59—42no—
145Dic59—81no—
146#2983—591no—
147C4×#124—591no—
148#1478—592no—
149#1461D5911yescharacteristic, complemented
150#1276—592no—
151C2×C59—21no—
152C2×C59—21no—
153C2×C59#187011yescharacteristic
154C2×C59#188411yescharacteristic
155C2×C59#188811yescharacteristic
156D59—82no—
157C2×#124—591no—
158C2×C4×Q8C2×D5911yescharacteristic
159C2×C4×D4—591no—
160#446—1184no—
161#432—1182no—
162#431—592no—
163#431C2×D5912yes—
164#391—591no—
165#390—591no—
166#388—1182no—
167#382—591no—
168C4×C22⋊C4—1182no—
169C43—591no—
170D4⋊C8—1184no—
171#324—22no—
172C59#896411yescharacteristic, complemented, Sylow
173C22×Q8C4×D5911yescharacteristic
174C22×D4—591no—
175C23×C4—1181no—
176#124—1182no—
177#124—1182no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1182no—
181C4×D4—1182no—
182C2×C4⋊C4—591no—
183C2×C4⋊C4C4×D5911yescharacteristic
184C2×C4⋊C4C22⋊D5912yes—
185C2×C22⋊C4—1182no—
186C2×C22⋊C4—1182no—
187C2×C22⋊C4—1181no—
188C2×C42—591no—
189C2×C42—591no—
190C2×C42—1181no—
191C2×C42—1181no—
192C2×C42D23612yes—
193C2×C42C22×D5911yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—594no—
196C4⋊C8—1184no—
197C22⋊C8—22no—
198C22⋊C8—1182no—
199#97—1184no—
200C4×C8—22no—
201C4×C8—1182no—
202#95—1182no—
203#95—1182no—
204C24—1181no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—592no—
210C2×D4—1182no—
211C2×D4—1181no—
212C2×D4—1182no—
213C22×C4—1181no—
214C22×C4—1181no—
215C22×C4—1181no—
216C22×C4—2364no—
217C22×C4—2361no—
218C22×C4—2361no—
219C22×C4(C2×D59)⋊C412yes—
220C22×C4(C2×D59)⋊C412yes—
221C22×C4C2×C4×D5911yescharacteristic
222C22×C4C2×D23611yescharacteristic
223C22×C4C2×C22⋊D5911yescharacteristic
224C2×C8—42no—
225C2×C8—1184no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—1182no—
229C22⋊C4—1188no—
230C22⋊C4—2364no—
231C22⋊C4—2362no—
232C42—22no—
233C42—22no—
234C42—592no—
235C42—1181no—
236C42—1182no—
237C42—2362no—
238C42—2362no—
239C42D4×D5912yes—
240C42#1174568712yes—
241C23—1182no—
242C23—2362no—
243C23—2362no—
244C23—2361no—
245C23—2361no—
246C23C2×(C2×D59)⋊C411yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—1184no—
250D4—2362no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—1182no—
259C2×C4—1182no—
260C2×C4—4722no—
261C2×C4—4722no—
262C2×C4—2362no—
263C2×C4—2361no—
264C2×C4—2361no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4—2361no—
268C2×C4(C22⋊D59)⋊C412yes—
269C2×C4#1427369711yescharacteristic
270C2×C4#1427369912yes—
271C2×C4#1427371111yescharacteristic
272C2×C4M16×D5911yescharacteristic
273C2×C4M16×D5911yescharacteristic
274C2×C4#1427373312yes—
275C8—82no—
276C8—2364no—
277C22—21no—
278C22—21no—
279C22—4722no—
280C22—4721no—
281C22—4721no—
282C22—2364no—
283C22—2362no—
284C22#1683727911yescharacteristic
285C22#1683736511yescharacteristic
286C22#1683736811yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—4721no—
292C4—2362no—
293C4—2362no—
294C4#1683747612yes—
295C4#1683747712yes—
296C4#1683757712yes—
297C2—21no—
298C2—21no—
299C2—4722no—
300C2#2142439111yescharacteristic
301C2#2142447711yescharacteristic
302C2#2142449711yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1461C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1461D59
hypocenterC59#8964
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
4. #21421928C2semidirect
5. C59×#1461C2semidirect
149. #1461D59semidirect
172. C59#8964semidirect

Maximal subgroups

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

subgroupindex
#214331812
C4×#142737162
#214224882
#214219282
C59×#14612
#896459