Finite Group Databasebeta

← #28484412

Subgroups — #28484412

Order 15104.

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

normal subgroups only

This group has 694 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#21432435C211yesmaximal, characteristic
2C2×#16837248C211yesmaximal, characteristic
3C2×#16837140C212yesmaximal
4#21421976C212yesmaximal
5C59×#1817C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C22×C4×Dic59—21no—
8#16837308C412yescomplemented
9#16837300C412yescomplemented
10#16837286—22no—
11#16837275—21no—
12#16837265—21no—
13#16837262C2211yescharacteristic
14#16837248—24no—
15C2×C59⋊#95—21no—
16C2×C59⋊#95C2212yes—
17#16837140—24no—
18C22×C42×C59C2211yescharacteristic
19C2×C22⋊C8×C59C2212yes—
20C59×#336—22no—
21C22×Dic118C2×C411yescharacteristic
22C23×Dic59—22no—
23#14273706—22no—
24#14273706—22no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—22no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C4×Dic59—24no—
32C2×C4×Dic59—28no—
33C59⋊#95—24no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95—21no—
39C59⋊#95D414yes—
40C23×C4×C59D411yescharacteristic
41C23×C4×C59D411yescharacteristic
42C23×C4×C59C2311yescharacteristic
43C22×C8×C59—22no—
44C2×C42×C59—22no—
45C2×C42×C59—22no—
46C2×C42×C59C2×C412yes—
47C2×C42×C59D412yes—
48C22⋊C8×C59—22no—
49C22⋊C8×C59—22no—
50C24×C59C2×D411yescharacteristic
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—24no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—42no—
59C22×C4×C59C22⋊C412yes—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D412yes—
64C22×C4×C59#4112yes—
65C22×Dic59—28no—
66C22×Dic59—24no—
67C22×Dic59—22no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—44no—
71C22×Dic59—42no—
72C22×Dic59—42no—
73C2×Dic118—22no—
74C2×Dic118—42no—
75C2×C8×C59—44no—
76C2×C8×C59—42no—
77C42×C59—24no—
78C42×C59—24no—
79C22⋊Dic59—41no—
80C22⋊Dic59—41no—
81C22⋊Dic59—42no—
82C4⋊Dic59—22no—
83C4⋊Dic59—42no—
84Dic59⋊C4—44no—
85Dic59⋊C4—42no—
86C4×Dic59—216no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59C2×C22⋊C411yescharacteristic
90C23×C59#12011yescharacteristic
91C23×C59#12311yescharacteristic
92C23×C59#13512yes—
93C23×C59#13712yes—
94C2×C4×C59—24no—
95C2×C4×C59—24no—
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×C59—22no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59#10414yes—
108C2×C4×C59C4×D412yes—
109C2×C4×C59#12112yes—
110C2×Dic59—28no—
111C2×Dic59—44no—
112C2×Dic59—48no—
113C2×Dic59—48no—
114C2×Dic59—44no—
115C2×Dic59—41no—
116C2×Dic59—41no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122Dic118—82no—
123C8×C59—82no—
124#9145—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—24no—
130C22×C59—24no—
131C22×C59#38611yescharacteristic
132C22×C59#41012yes—
133C22×C59#41611yescharacteristic
134C22×C59#41911yescharacteristic
135C22×C59C2×#10412yes—
136C22×C59#45212yes—
137C22×C59#48512yes—
138C4×C59—24no—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—42no—
142Dic59—48no—
143Dic59—81no—
144Dic59—81no—
145Dic59—82no—
146#2855—591no—
147C2×#376—591no—
148#1817D5911yescharacteristic
149C2×#328—592no—
150#1294—592no—
151C2×C59—22no—
152C2×C59—22no—
153C2×C59#150012yes—
154C2×C59#150612yes—
155C2×C59#187311yescharacteristic
156C2×C59#189611yescharacteristic
157C2×C59#189911yescharacteristic
158C2×C22⋊Q8—591no—
159C22×C42—1181no—
160C22×C42C2×D5911yescharacteristic
161C2×C22⋊C8C2×D5912yes—
162#391—592no—
163#389—592no—
164#387—1182no—
165#385—1181no—
166#381—1181no—
167#380—591no—
168#376—1184no—
169C2×#95—592no—
170C2×#95—1181no—
171#336—22no—
172#328—1184no—
173C59#914511yescharacteristic, complemented, Sylow
174C22×Q8—591no—
175C23×C4—1182no—
176C23×C4D23611yescharacteristic
177C23×C4D23611yescharacteristic
178C23×C4C22×D5911yescharacteristic
179C22×C8—22no—
180C22⋊Q8—1182no—
181C22⋊Q8—1182no—
182C2×C4⋊C4—591no—
183C2×C4⋊C4—592no—
184C2×C4⋊C4—1182no—
185C2×C4⋊C4—1181no—
186C2×C22⋊C4—1181no—
187C2×C22⋊C4—1181no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—1184no—
191C2×C42—1188no—
192C2×C42C4×D5912yes—
193C2×C42C22⋊D5912yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—594no—
197#95—1184no—
198#95—1182no—
199#95—1182no—
200#95—1182no—
201#95—1182no—
202#95—1181no—
203C24C2×D23611yescharacteristic
204C2×Q8—1182no—
205C2×Q8—2362no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—1188no—
215C22×C4—1184no—
216C22×C4—1182no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4—2364no—
220C22×C4—2362no—
221C22×C4—2362no—
222C22×C4(C2×D59)⋊C412yes—
223C22×C4(C2×D59)⋊C412yes—
224C22×C4C2×C4×D5911yescharacteristic
225C22×C4D4×D5912yes—
226C22×C4#1174568712yes—
227C22×C4C2×C22⋊D5911yescharacteristic
228C2×C8—44no—
229C2×C8—42no—
230C4⋊C4—1182no—
231C4⋊C4—2362no—
232C4⋊C4—2364no—
233C4⋊C4—2362no—
234C22⋊C4—2361no—
235C22⋊C4—2361no—
236C22⋊C4—2362no—
237C42—24no—
238C42—24no—
239C42—11816no—
240C23—22no—
241C23—22no—
242C23C2×(C2×D59)⋊C411yescharacteristic
243C23#1427369711yescharacteristic
244C23#1427371111yescharacteristic
245C23#1427374912yes—
246C23#1427376012yes—
247Q8—4722no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—1188no—
262C2×C4—2362no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4—2362no—
268C2×C4—2362no—
269C2×C4—2368no—
270C2×C4—2364no—
271C2×C4—2368no—
272C2×C4—2364no—
273C2×C4#1427365314yes—
274C2×C4(C22⋊D59)⋊C412yes—
275C2×C4#1427369912yes—
276C8—82no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22#1683727911yescharacteristic
284C22#1683737512yes—
285C22#1683739911yescharacteristic
286C22#1683741211yescharacteristic
287C22C2×#1427365312yes—
288C22#1683756612yes—
289C22#1683774412yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—4721no—
295C4—4721no—
296C4—4722no—
297C4—2368no—
298C2—22no—
299C2—22no—
300C2#2142258312yes—
301C2#2142260912yes—
302C2#2142440611yescharacteristic
303C2#2142453711yescharacteristic
304C2#2142455211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1817C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#1817D59
hypocenterC59#9145
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #16837308C4semidirect
9. #16837300C4semidirect
173. C59#9145semidirect

Maximal subgroups

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

subgroupindex
#214324352
C2×#168372482
C2×#168371402
#214219762
C59×#18172
#914559