Finite Group Databasebeta

← #28487494

Subgroups — #28487494

Order 15104.

558 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C4×#14273715C211yesmaximal, characteristic
2#21423842C211yesmaximal, characteristic, complemented
3C2×#16837132C212yesmaximal, complemented
4C2×#16837129C212yesmaximal
5C59×#1831C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7C2×C4×Dic118C2211yescharacteristic
8C2×C4×D236C2211yescharacteristic
9C2×C4×C59⋊C8C2212yes—
10#16837308C411yescharacteristic
11#16837304C411yescharacteristic
12#16837293—22no—
13#16837268C411yescharacteristic
14C4×(C2×D59)⋊C4—22no—
15#16837157—22no—
16#16837132—24no—
17#16837129—24no—
18C2×C4⋊C8×C59C2212yes—
19C43×C59C2211yescharacteristic
20C59×#336—22no—
21C22×Dic118C2×C411yescharacteristic
22C22×D236C2×C411yescharacteristic
23C22×C4×D59—21no—
24C22×C59⋊C8—22no—
25#14273715—22no—
26#14273715—22no—
27C4×Dic118—22no—
28C4×Dic118—22no—
29C4×D236—22no—
30C4×D236—22no—
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×Dic59⋊C4—21no—
34C2×(C2×D59)⋊C4—22no—
35C2×(C2×D59)⋊C4—22no—
36C2×(C2×D59)⋊C4—21no—
37C2×C4×Dic59—21no—
38C4×C59⋊C8—22no—
39C4×C59⋊C8D414yes—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C22×C8×C59—22no—
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D412yes—
46C2×C42×C59D412yes—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49C4⋊C8×C59—22no—
50C23×D59—21no—
51C22×C4×C59C22⋊C412yes—
52C22×C4×C59C22⋊C412yes—
53C22×C4×C59C22×C411yescharacteristic
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×Dic59—21no—
57C22×Dic59—21no—
58C22×Dic59—21no—
59C2×D236—22no—
60C2×D236—22no—
61C2×D236—21no—
62C2×D236M1612yes—
63C2×C4×D59—41no—
64C2×C4×D59—41no—
65C2×C4×D59—44no—
66C2×Dic118—22no—
67C2×Dic118—22no—
68C2×Dic118—21no—
69C2×Dic118M1612yes—
70C2×C8×C59—42no—
71C2×C8×C59—44no—
72C42×C59—21no—
73C42×C59—21no—
74C42×C59—22no—
75C42×C59—24no—
76C42×C59—42no—
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—22no—
84Dic59⋊C4—42no—
85C4×Dic59—41no—
86C4×Dic59—41no—
87C2×C59⋊C8—24no—
88C2×C59⋊C8—44no—
89C23×C59C2×C22⋊C411yescharacteristic
90C22×D59—22no—
91C22×D59—41no—
92C22×D59—41no—
93C22×D59—42no—
94C22×D59—42no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59C4×D412yes—
103C2×C4×C59#12112yes—
104C2×C4×C59#12411yescharacteristic
105C2×C4×C59#12711yescharacteristic
106C2×C4×C59C2×M1611yescharacteristic
107C2×C4×C59C2×M1611yescharacteristic
108C2×C4×C59#13112yes—
109C2×C4×C59#13514yes—
110C2×Dic59—22no—
111C2×Dic59—41no—
112C2×Dic59—41no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117D236—24no—
118D236—42no—
119C4×D59—82no—
120C4×D59—82no—
121Dic118—24no—
122Dic118—42no—
123C8×C59—82no—
124C59⋊C8—44no—
125#9645—591nomaximal, Sylow
126C22×C59#39011yescharacteristic
127C22×C59#40711yescharacteristic
128C22×C59#40811yescharacteristic
129C22×C59#41612yes—
130C22×C59#42112yes—
131C2×D59—42no—
132C2×D59—44no—
133C2×D59—81no—
134C2×D59—81no—
135C2×D59—82no—
136C4×C59—22no—
137C4×C59—42no—
138C4×C59—42no—
139C4×C59#43512yes—
140C4×C59#43512yes—
141C4×C59#48812yes—
142C4×C59#49512yes—
143Dic59—42no—
144Dic59—81no—
145Dic59—81no—
146C4×#124—591no—
147#1831D5911yescharacteristic, complemented
148#1744—591no—
149C2×Q8⋊C8—592no—
150C2×D4⋊C8—592no—
151C2×C59#148212yes—
152C2×C59#148812yes—
153C2×C59#195511yescharacteristic
154C2×C59#195811yescharacteristic
155C2×C59#196311yescharacteristic
156D59—82no—
157C2×#124—591no—
158C2×C4×Q8—591no—
159C2×C4×D4—591no—
160C2×C4⋊C8C2×D5912yes—
161C2×C4×C8—592no—
162#391—591no—
163#390—591no—
164#388—1182no—
165#382—591no—
166C4×C22⋊C4—1182no—
167C43C2×D5911yescharacteristic
168#336—22no—
169#336—1182no—
170Q8⋊C8—1184no—
171D4⋊C8—1184no—
172C59#964511yescharacteristic, complemented, Sylow
173C22×Q8—591no—
174C22×D4—591no—
175C23×C4—1181no—
176C22×C8—22no—
177C22×C8—1182no—
178#124—1182no—
179#124—1182no—
180C4×Q8—1182no—
181C4×Q8—1182no—
182C4×D4—1182no—
183C4×D4—1182no—
184C2×C4⋊C4—591no—
185C2×C4⋊C4—591no—
186C2×C4⋊C4—1181no—
187C2×C22⋊C4—1182no—
188C2×C22⋊C4—1182no—
189C2×C22⋊C4—1181no—
190C2×C42—1181no—
191C2×C42C4×D5911yescharacteristic
192C2×C42C4×D5911yescharacteristic
193C2×C42D23612yes—
194C2×C42C22⋊D5912yes—
195C2×C42C22×D5911yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4×C8—594no—
199C4×C8—1182no—
200#95—1182no—
201#95—1182no—
202C24—1181no—
203C2×Q8—592no—
204C2×Q8—1181no—
205C2×Q8—1182no—
206C2×Q8—1182no—
207C2×D4—592no—
208C2×D4—1182no—
209C2×D4—1181no—
210C2×D4—1182no—
211C22×C4—1181no—
212C22×C4—1181no—
213C22×C4—1181no—
214C22×C4—2361no—
215C22×C4—2361no—
216C22×C4—2364no—
217C22×C4(C2×D59)⋊C412yes—
218C22×C4(C2×D59)⋊C412yes—
219C22×C4C2×C4×D5911yescharacteristic
220C22×C4C2×D23611yescharacteristic
221C22×C4C2×C22⋊D5911yescharacteristic
222C2×C8—42no—
223C2×C8—44no—
224C2×C8—1184no—
225C2×C8—2364no—
226C4⋊C4—1182no—
227C4⋊C4—1182no—
228C4⋊C4—2362no—
229C22⋊C4—1188no—
230C22⋊C4—2364no—
231C22⋊C4—2362no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—24no—
236C42—42no—
237C42—2361no—
238C42—2361no—
239C42D4×D5912yes—
240C42#1174568912yes—
241C23—1182no—
242C23—2362no—
243C23—2362no—
244C23—2361no—
245C23—2361no—
246C23C2×(C2×D59)⋊C411yescharacteristic
247Q8—1184no—
248Q8—2362no—
249D4—1184no—
250D4—2362no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—1182no—
259C2×C4—4722no—
260C2×C4—4722no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4—2361no—
264C2×C4—2361no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4#1427368912yes—
268C2×C4#1427370212yes—
269C2×C4#1427371611yescharacteristic
270C2×C4#1427372311yescharacteristic
271C2×C4M16×D5911yescharacteristic
272C2×C4M16×D5911yescharacteristic
273C2×C4#1427373312yes—
274C2×C4#1427374714yes—
275C8—82no—
276C8—2364no—
277C22—4721no—
278C22—4721no—
279C22—4722no—
280C22—2364no—
281C22—2362no—
282C22#1683730311yescharacteristic
283C22#1683736511yescharacteristic
284C22#1683736811yescharacteristic
285C22#1683739612yes—
286C22#1683741812yes—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—4721no—
291C4—4721no—
292C4—2362no—
293C4#1683747212yes—
294C4#1683747212yes—
295C4#1683775912yes—
296C4#1683778412yes—
297C2—4722no—
298C2#2142250212yes—
299C2#2142252812yes—
300C2#2142489311yescharacteristic
301C2#2142490511yescharacteristic
302C2#2142492711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1831C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1831D59
hypocenterC59#9645
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1831

Complements

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

NH ≅ G/Ntype
2. #21423842C2semidirect
3. C2×#16837132C2semidirect
5. C59×#1831C2semidirect
147. #1831D59semidirect
172. C59#9645semidirect

Maximal subgroups

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

subgroupindex
C4×#142737152
#214238422
C2×#168371322
C2×#168371292
C59×#18312
#964559