Finite Group Databasebeta

← #28486514

Subgroups — #28486514

Order 15104.

726 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 107 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#21426699C211yesmaximal, characteristic, complemented
2#21424040C212yesmaximal
3#21422426C211yesmaximal, characteristic, complemented
4#21421948C212yesmaximal, complemented
5C59×#2406C211yesmaximal, characteristic
6C2×#14273654—24no—
7#16837366C2212yes—
8#16837288—24no—
9#16837265—21no—
10#16837262—21no—
11C4×C22⋊Dic59C2212yes—
12#16837128—22no—
13C22×#41×C59C2211yescharacteristic
14C2×#123×C59—21no—
15C22×C4⋊C4×C59C2211yescharacteristic
16C59×#395C412yescomplemented
17C59×#394C412yescomplemented
18C59×#386—21no—
19C59×#385—22no—
20C59×#381C2211yescharacteristic
21C2×#11745664—22no—
22C2×C4⋊Dic59—22no—
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59D414yes—
25#14273654—48no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—22no—
28C59⋊#95—22no—
29C59⋊#95—24no—
30C59⋊#95—22no—
31C2×#41×C59—22no—
32C2×#41×C59—22no—
33C2×#41×C59—44no—
34C22×Q8×C59C2×C411yescharacteristic
35C22×D4×C59—21no—
36C22×D4×C59C2×C411yescharacteristic
37C23×C4×C59—21no—
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41#123×C59—44no—
42C2×C4⋊C4×C59—22no—
43C2×C4⋊C4×C59—22no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59C2×C412yes—
46C2×C4⋊C4×C59D412yes—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—22no—
49C2×C22⋊C4×C59—22no—
50#95×C59—22no—
51#95×C59—21no—
52#95×C59C2×C412yes—
53C24×C59—21no—
54C24×C59C2×D411yescharacteristic
55#41×C59—28no—
56#41×C59—44no—
57#41×C59—44no—
58#41×C59—44no—
59C2×Q8×C59—22no—
60C2×Q8×C59—42no—
61C2×D4×C59—24no—
62C2×D4×C59—22no—
63C2×D4×C59—42no—
64C2×D4×C59—42no—
65C2×D4×C59—41no—
66C2×D4×C59—41no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—22no—
73C22×C4×C59—42no—
74C22×C4×C59—42no—
75C22×C4×C59—42no—
76C22×C4×C59—42no—
77C22×C4×C59—42no—
78C22×C4×C59—42no—
79C22×C4×C59C22⋊C412yes—
80C22×C4×C59C22⋊C412yes—
81C22×C4×C59C22×C411yescharacteristic
82C22×C4×C59C2×D412yes—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59#4112yes—
85C22×Dic59—24no—
86C4⋊C4×C59—24no—
87C4⋊C4×C59—24no—
88C4⋊C4×C59—44no—
89C22⋊C4×C59—44no—
90C22⋊C4×C59—44no—
91C22⋊C4×C59—42no—
92C22⋊Dic59—44no—
93C4⋊Dic59—82no—
94C4×Dic59—44no—
95#11745664—44no—
96C2×C59⋊C8—42no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—42no—
103C23×C59—41no—
104C23×C59—42no—
105C23×C59C2×C22⋊C411yescharacteristic
106C23×C59#12011yescharacteristic
107C23×C59#12311yescharacteristic
108Q8×C59—24no—
109Q8×C59—42no—
110D4×C59—24no—
111D4×C59—42no—
112D4×C59—44no—
113D4×C59—44no—
114C2×C4×C59—24no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—48no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—42no—
129C2×C4×C59—42no—
130C2×C4×C59—42no—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—81no—
134C2×C4×C59C4×D412yes—
135C2×C4×C59#12112yes—
136C2×Dic59—44no—
137C2×Dic59—44no—
138C2×Dic59—44no—
139C59⋊C8—82no—
140#9480—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—44no—
149C22×C59—44no—
150C22×C59—42no—
151C22×C59—82no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59#38611yescharacteristic
155C22×C59#42112yes—
156C22×C59B(4,𝔽2)12yes—
157C4×C59—22no—
158C4×C59—22no—
159C4×C59—42no—
160C4×C59—42no—
161C4×C59—42no—
162C4×C59—82no—
163Dic59—84no—
164#2406D5911yescharacteristic
165#2313—591no—
166#1789—592no—
167#1463—591no—
168#1284—592no—
169C2×C59—22no—
170C2×C59—22no—
171C2×C59—41no—
172C2×C59—82no—
173C2×C59#188911yescharacteristic
174C2×C59#189711yescharacteristic
175C2×C59#191211yescharacteristic
176C22×#41C2×D5911yescharacteristic
177C2×#123—21no—
178C22×C4⋊C4C2×D5911yescharacteristic
179C2×#104—1184no—
180#407—592no—
181#395Dic5912yescomplemented
182#394Dic5912yescomplemented
183#387—1184no—
184#386—21no—
185#385—22no—
186#381—1181no—
187#381C2×D5911yescharacteristic
188#380—1181no—
189C4×C22⋊C4—592no—
190#324—1182no—
191C59#948011yescharacteristic, complemented, Sylow
192C2×#41—22no—
193C2×#41—22no—
194C2×#41—44no—
195C22×Q8C2×Dic5911yescharacteristic
196C22×D4—21no—
197C22×D4C2×Dic5911yescharacteristic
198C23×C4—21no—
199C23×C4C22⋊D5911yescharacteristic
200C23×C4C22⋊D5911yescharacteristic
201C23×C4C22×D5911yescharacteristic
202C2×M16—1182no—
203#123—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—1182no—
208C2×C4⋊C4C2×Dic5912yes—
209C2×C4⋊C4C22⋊D5912yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1182no—
214C2×C42—594no—
215#104—2368no—
216C22⋊C8—1182no—
217C22⋊C8—1182no—
218#95—22no—
219#95—21no—
220#95—1182no—
221#95—1184no—
222#95—1182no—
223#95C2×Dic5912yes—
224C24—21no—
225C24C2×C22⋊D5911yescharacteristic
226#41—28no—
227#41—44no—
228#41—44no—
229#41—44no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1184no—
251C22×C4C22⋊Dic5912yes—
252C22×C4C22⋊Dic5912yes—
253C22×C4D4×D5912yes—
254C22×C4#1174568712yes—
255C22×C4C22×Dic5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257M16—2364no—
258C2×C8—2362no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—4722no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—2364no—
267C42—2364no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—42no—
275C23—41no—
276C23C2×C22⋊Dic5911yescharacteristic
277C23#1427369511yescharacteristic
278C23#1427370811yescharacteristic
279Q8—24no—
280Q8—42no—
281D4—24no—
282D4—42no—
283D4—44no—
284D4—44no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—2364no—
306C2×C4—2364no—
307C2×C4—2364no—
308C2×C4D4×Dic5912yes—
309C2×C4#1427369912yes—
310C8—4722no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#1683728211yescharacteristic
325C22#1683741912yes—
326C22#1683758812yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—82no—
333C4—4724no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2#2142450511yescharacteristic
339C2#2142454511yescharacteristic
340C2#2142463411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2406C2
socleC22×C59#386
prime radicalC22×#41×C59C22
cosocleC23×C4×C59C23
hypercenter#2406D59
hypocenterC59#9480
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. #21426699C2semidirect
3. #21422426C2semidirect
4. #21421948C2semidirect
16. C59×#395C4semidirect
17. C59×#394C4semidirect
181. #395Dic59semidirect
182. #394Dic59semidirect
191. C59#9480semidirect

Maximal subgroups

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

subgroupindex
#214266992
#214240402
#214224262
#214219482
C59×#24062
#948059