Finite Group Databasebeta

← #28486379

Subgroups — #28486379

Order 15104.

498 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 111 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#21422813C212yesmaximal, complemented
3#21422437C211yesmaximal, characteristic
4#21421952C212yesmaximal, complemented
5C59×#2563C211yesmaximal, characteristic
6#16837474—24no—
7#16837459C2212yes—
8#16837459C2212yes—
9#16837366—21no—
10C2×C22⋊(C59⋊C8)—21no—
11#16837148—24no—
12#16837126—21no—
13#16837126—21no—
14C2×C4⋊Q8×C59C2211yescharacteristic
15C2×C4×D4×C59C2211yescharacteristic
16C59×#393—21no—
17C59×#393—21no—
18C59×#390C412yes—
19C59×#382C412yes—
20C59×#380—21no—
21C59×#380—21no—
22C4×C4⋊C4×C59C2211yescharacteristic
23C2×#11745664—22no—
24C22×C59⋊C8—22no—
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)D414yes—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638—22no—
33C22×Q8×C59—21no—
34C22×D4×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C23×C4×C59D411yescharacteristic
37C4⋊Q8×C59—22no—
38C4⋊Q8×C59—22no—
39C4×D4×C59—22no—
40C4×D4×C59—22no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59C2×C412yes—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C42×C59C2×C412yes—
50C2×C42×C59C2311yescharacteristic
51#95×C59—22no—
52#95×C59—22no—
53#95×C59—21no—
54C24×C59C22⋊C411yescharacteristic
55C24×C59C22⋊C411yescharacteristic
56C2×Q8×C59—42no—
57C2×Q8×C59—82no—
58C2×D4×C59—22no—
59C2×D4×C59—22no—
60C2×D4×C59—41no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C4⋊C4×C59—22no—
75C4⋊C4×C59—24no—
76C4⋊C4×C59—42no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C22⋊C4×C59—22no—
80C22⋊C4×C59—22no—
81C22⋊C4×C59—82no—
82C22⋊C4×C59—82no—
83C42×C59—42no—
84C42×C59C2×D412yes—
85C42×C59#4112yes—
86#11745664—82no—
87C2×C59⋊C8—24no—
88C2×C59⋊C8—42no—
89C2×C59⋊C8—82no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59#10112yes—
100C23×C59C2×C22⋊C411yescharacteristic
101Q8×C59—84no—
102D4×C59—44no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—81no—
120C2×C4×C59—81no—
121C2×C4×C59C4×D412yes—
122C2×C4×C59#12011yescharacteristic
123C2×C4×C59#12112yes—
124C2×C4×C59#12311yescharacteristic
125C59⋊C8—44no—
126C59⋊C8—82no—
127#9453—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—22no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—81no—
139C22×C59—81no—
140C22×C59#38611yescharacteristic
141C22×C59C2×#10111yescharacteristic
142C22×C59#41311yescharacteristic
143C4×C59—21no—
144C4×C59—22no—
145C4×C59—22no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—82no—
149C4×C59—82no—
150C4×C59#45512yes—
151#2973—591no—
152#2563D5911yescharacteristic
153#1556—592no—
154#1466—591no—
155#1285—592no—
156C2×C59—21no—
157C2×C59—21no—
158C2×C59—42no—
159C2×C59—81no—
160C2×C59#188511yescharacteristic
161C2×C59#190111yescharacteristic
162C2×C59#191011yescharacteristic
163C2×C4⋊Q8C2×D5911yescharacteristic
164C2×C4×D4C2×D5911yescharacteristic
165#435—1184no—
166#432—592no—
167#432—592no—
168#407—1181no—
169C2×C22⋊C8—1181no—
170#393—21no—
171#393—21no—
172#390Dic5912yes—
173#382Dic5912yes—
174#380—21no—
175#380—21no—
176C4×C4⋊C4C2×D5911yescharacteristic
177#332—1184no—
178#323—1181no—
179#323—1181no—
180C59#945311yescharacteristic, complemented, Sylow
181C22×Q8—21no—
182C22×D4C2×Dic5911yescharacteristic
183C23×C4C22⋊D5911yescharacteristic
184C23×C4C22⋊D5911yescharacteristic
185C2×M16—1182no—
186C22×C8—1182no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic5912yes—
194C2×C4⋊C4C2×Dic5911yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊D5911yescharacteristic
198C2×C22⋊C4C22⋊D5911yescharacteristic
199C2×C42C2×Dic5912yes—
200C2×C42C22×D5911yescharacteristic
201C4⋊C8—594no—
202C4⋊C8—1182no—
203C22⋊C8—1182no—
204C22⋊C8—1182no—
205C22⋊C8—1182no—
206C22⋊C8—1182no—
207#97—1182no—
208#97—1182no—
209#95—22no—
210#95—22no—
211#95—21no—
212C24C22⋊Dic5911yescharacteristic
213C24C22⋊Dic5911yescharacteristic
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22⋊Dic5911yescharacteristic
228C22×C4C22⋊Dic5911yescharacteristic
229C22×C4C22×Dic5911yescharacteristic
230C22×C4C2×C22⋊D5911yescharacteristic
231C22×C4C2×C22⋊D5911yescharacteristic
232M16—4722no—
233C2×C8—1184no—
234C2×C8—4722no—
235C2×C8—2362no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42D4×D5912yes—
247C42#1174568712yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1427364612yes—
258C23C2×C22⋊Dic5911yescharacteristic
259Q8—84no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4D4×Dic5912yes—
280C2×C4#1427369511yescharacteristic
281C2×C4#1427369912yes—
282C2×C4#1427370811yescharacteristic
283C8—4722no—
284C8—2364no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1683728211yescharacteristic
298C22C2×#1427364611yescharacteristic
299C22#1683738611yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4#1683757912yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#2142448511yescharacteristic
313C2#2142456511yescharacteristic
314C2#2142462411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2563C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#2563D59
hypocenterC59#9453
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. #21422813C2semidirect
4. #21421952C2semidirect
180. C59#9453semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214228132
#214224372
#214219522
C59×#25632
#945359