Finite Group Databasebeta

← #28484197

Subgroups — #28484197

Order 15104.

546 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 135 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#21433177C211yesmaximal, characteristic, complemented
2#21426803C211yesmaximal, characteristic
3#21422486C212yesmaximal
4#21421924C212yesmaximal, complemented
5C59×#1461C211yesmaximal, characteristic, complemented
6C2×#14273716C411yescharacteristic
7C2×C4×D236C2211yescharacteristic
8Q8×C59⋊C8—24no—
9#16837455—22no—
10#16837455C2212yes—
11#16837309C411yescharacteristic
12#16837303C411yescharacteristic
13#16837293—22no—
14#16837273—22no—
15#16837271C411yescharacteristic
16#16837248C2211yescharacteristic
17#16837129—24no—
18C2×C4×Q8×C59C2211yescharacteristic
19C59×#431C2212yes—
20C59×#324—22no—
21C22×D236C2×C411yescharacteristic
22C22×C4×D59—21no—
23#14273716—22no—
24#14273716C814yescomplemented
25C4×D236—22no—
26C4×D236—22no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—22no—
32C2×C4×Dic59C2×C411yescharacteristic
33C2×C4×Dic59C2×C411yescharacteristic
34C4⋊(C59⋊C8)—24no—
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C4×C59⋊C8—24no—
38C4×C59⋊C8D414yes—
39C59⋊#95—22no—
40C59⋊#95—21no—
41C59⋊#95—21no—
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—22no—
55C2×Q8×C59C2×C812yes—
56C2×Q8×C59M1612yes—
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×D236C2×C812yes—
69C2×C4×D59—42no—
70C2×C4×D59—82no—
71C2×C8×C59—42no—
72C4⋊C4×C59—22no—
73C4⋊C4×C59—24no—
74C42×C59—22no—
75C42×C59—22no—
76C42×C59C2×D412yes—
77C42×C59#4112yes—
78(C2×D59)⋊C4—28no—
79(C2×D59)⋊C4—41no—
80(C2×D59)⋊C4—41no—
81(C2×D59)⋊C4—82no—
82C4⋊Dic59—22no—
83Dic59⋊C4—82no—
84C4×Dic59—21no—
85C4×Dic59—22no—
86C4×Dic59C2×C812yes—
87C2×C59⋊C8—24no—
88C23×C59C2×C22⋊C411yescharacteristic
89C22×D59—22no—
90C22×D59—42no—
91C22×D59—42no—
92C22×D59—41no—
93C22×D59—41no—
94Q8×C59—24no—
95Q8×C59—24no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59C22⋊C814yes—
103C2×C4×C59C4×D412yes—
104C2×C4×C59#12011yescharacteristic
105C2×C4×C59#12112yes—
106C2×C4×C59#12311yescharacteristic
107C2×C4×C59C22×C811yescharacteristic
108C2×C4×C59C2×M1611yescharacteristic
109C2×C4×C59#13112yes—
110C2×Dic59—22no—
111C2×Dic59—41no—
112C2×Dic59—41no—
113C2×Dic59—42no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—41no—
117C2×Dic59—41no—
118D236—24no—
119D236—42no—
120C4×D59—84no—
121C8×C59—82no—
122C59⋊C8—44no—
123#9102—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59#38611yescharacteristic
127C22×C59C2×C22⋊C811yescharacteristic
128C22×C59#40811yescharacteristic
129C2×D59—44no—
130C2×D59—42no—
131C2×D59—82no—
132C2×D59—81no—
133C2×D59—81no—
134C4×C59—24no—
135C4×C59—22no—
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—42no—
139C4×C59C8×D412yes—
140C4×C59#43512yes—
141Dic59—42no—
142Dic59—81no—
143Dic59—81no—
144#2982—591no—
145#2326—591no—
146#1478—592no—
147#1461D5911yescharacteristic, complemented
148#1274—592no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59#186911yescharacteristic
152C2×C59#191011yescharacteristic
153C2×C59#191211yescharacteristic
154D59—82no—
155C2×#124—591no—
156C2×C4×Q8C2×D5911yescharacteristic
157C2×C4×D4—591no—
158C8×Q8—1184no—
159#431—592no—
160#431—1182no—
161#431C2×D5912yes—
162#391—591no—
163#390—591no—
164#388—1182no—
165#385—1182no—
166#384—591no—
167#376—591no—
168D4⋊C8—1184no—
169#324—22no—
170C59#910211yescharacteristic, complemented, Sylow
171C22×Q8C4×D5911yescharacteristic
172C22×D4—591no—
173C23×C4—1181no—
174#124—594no—
175#124—1182no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1182no—
179C4×D4—1182no—
180C2×C4⋊C4—591no—
181C2×C4⋊C4—1182no—
182C2×C4⋊C4C4×D5911yescharacteristic
183C2×C4⋊C4C22⋊D5912yes—
184C2×C22⋊C4—1182no—
185C2×C22⋊C4—1182no—
186C2×C22⋊C4—1181no—
187C2×C42—591no—
188C2×C42—591no—
189C2×C42D23612yes—
190C2×C42C22×D5911yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—1184no—
193C4⋊C8—1182no—
194C22⋊C8—22no—
195C22⋊C8—1182no—
196C4×C8—22no—
197C4×C8—594no—
198C4×C8—1184no—
199#95—1182no—
200#95—1181no—
201#95—1181no—
202C24—1181no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C8×D5912yes—
206C2×Q8#1174565912yes—
207C2×D4—592no—
208C2×D4—1182no—
209C2×D4—1181no—
210C2×D4—1182no—
211C22×C4—1181no—
212C22×C4—1181no—
213C22×C4—1181no—
214C22×C4—4722no—
215C22×C4—2362no—
216C22×C4(C2×D59)⋊C412yes—
217C22×C4(C2×D59)⋊C412yes—
218C22×C4C2×C4×D5911yescharacteristic
219C22×C4C2×D23611yescharacteristic
220C22×C4C2×C22⋊D5911yescharacteristic
221C2×C8—42no—
222C2×C8—1184no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—1182no—
226C4⋊C4—4722no—
227C22⋊C4—1188no—
228C22⋊C4—4722no—
229C22⋊C4—2361no—
230C22⋊C4—2361no—
231C42—22no—
232C42—22no—
233C42—592no—
234C42—1181no—
235C42—1182no—
236C42D4×D5912yes—
237C42#1174568712yes—
238C23—1182no—
239C23—2362no—
240C23—2362no—
241C23—2361no—
242C23—2361no—
243C23C2×(C2×D59)⋊C411yescharacteristic
244Q8—24no—
245Q8—24no—
246D4—1184no—
247D4—2362no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—1182no—
255C2×C4—4724no—
256C2×C4—2361no—
257C2×C4—2362no—
258C2×C4—2361no—
259C2×C4—2361no—
260C2×C4—2361no—
261C2×C4—2361no—
262C2×C4—2361no—
263C2×C4(C2×D59)⋊C814yes—
264C2×C4(C22⋊D59)⋊C412yes—
265C2×C4#1427369711yescharacteristic
266C2×C4#1427369912yes—
267C2×C4#1427371111yescharacteristic
268C2×C4C2×C8×D5911yescharacteristic
269C2×C4C2×#1174565911yescharacteristic
270C2×C4#1427373512yes—
271C8—82no—
272C8—2364no—
273C22—21no—
274C22—21no—
275C22—4722no—
276C22—4721no—
277C22—4721no—
278C22—2364no—
279C22—2362no—
280C22#1683727911yescharacteristic
281C22C2×(C2×D59)⋊C811yescharacteristic
282C22#1683737011yescharacteristic
283C4—24no—
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—4721no—
289C4—4721no—
290C4—2362no—
291C4(C22⋊D59)⋊C812yes—
292C4#1683747012yes—
293C2—21no—
294C2—21no—
295C2—4722no—
296C2#2142438611yescharacteristic
297C2#2142462111yescharacteristic
298C2#2142463111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1461C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1461D59
hypocenterC59#9102
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #21433177C2semidirect
4. #21421924C2semidirect
5. C59×#1461C2semidirect
24. #14273716C8semidirect
147. #1461D59semidirect
170. C59#9102semidirect

Maximal subgroups

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

subgroupindex
#214331772
#214268032
#214224862
#214219242
C59×#14612
#910259