Finite Group Databasebeta

← #28486649

Subgroups — #28486649

Order 15104.

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

normal subgroups only

This group has 544 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#21424176C212yesmaximal, complemented
3#21422434C211yesmaximal, characteristic
4#21422002C212yesmaximal, complemented
5C59×#2486C211yesmaximal, characteristic
6#16837474—24no—
7#16837459—21no—
8#16837366C2212yes—
9#16837366C2212yes—
10C2×C22⋊(C59⋊C8)—21no—
11#16837171—24no—
12#16837128—21no—
13#16837126—21no—
14C2×#123×C59C412yes—
15C2×C4×D4×C59C2211yescharacteristic
16C22×C4⋊C4×C59C2211yescharacteristic
17C59×#398—21no—
18C59×#397—21no—
19C59×#389—21no—
20C59×#381C412yes—
21C59×#380—21no—
22C59×#380C2211yescharacteristic
23C2×#11745664—22no—
24C2×#11745664D414yes—
25C22×C59⋊C8—22no—
26C4⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—22no—
33C22×D4×C59C2×C411yescharacteristic
34C23×C4×C59C2×C412yes—
35C23×C4×C59D411yescharacteristic
36C23×C4×C59C2311yescharacteristic
37#123×C59—44no—
38C4×D4×C59—22no—
39C4×D4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—22no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—22no—
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59C2×C412yes—
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52#95×C59—22no—
53#95×C59—22no—
54#95×C59—21no—
55C24×C59C22⋊C411yescharacteristic
56C24×C59C22×C411yescharacteristic
57C2×D4×C59—22no—
58C2×D4×C59—22no—
59C2×D4×C59—41no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—24no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59#4112yes—
76C4⋊C4×C59—22no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—44no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—41no—
82C22⋊C4×C59—22no—
83C22⋊C4×C59—22no—
84C22⋊C4×C59—44no—
85C22⋊C4×C59—42no—
86C42×C59—21no—
87C42×C59—21no—
88#11745664—44no—
89#11745664—44no—
90C2×C59⋊C8—24no—
91C2×C59⋊C8—42no—
92C2×C59⋊C8—82no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59#10012yes—
102C23×C59C2×C22⋊C411yescharacteristic
103C23×C59C4×D412yes—
104D4×C59—44no—
105C2×C4×C59—24no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—41no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—81no—
124C2×C4×C59—84no—
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12112yes—
127C2×C4×C59#12311yescharacteristic
128C59⋊C8—44no—
129C59⋊C8—82no—
130#9507—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—22no—
134C22×C59—22no—
135C22×C59—22no—
136C22×C59—22no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—81no—
143C22×C59#38611yescharacteristic
144C22×C59C2×#10011yescharacteristic
145C22×C59#41311yescharacteristic
146C4×C59—22no—
147C4×C59—21no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—82no—
152C4×C59—82no—
153C4×C59#45812yes—
154#2973—591no—
155#2486D5911yescharacteristic
156#1821—592no—
157#1465—591no—
158#1305—592no—
159C2×C59—21no—
160C2×C59—21no—
161C2×C59—42no—
162C2×C59—42no—
163C2×C59#189011yescharacteristic
164C2×C59#190011yescharacteristic
165C2×C59#191511yescharacteristic
166C2×#123Dic5912yes—
167C2×C4×D4C2×D5911yescharacteristic
168C22×C4⋊C4C2×D5911yescharacteristic
169#435—1184no—
170#432—1181no—
171#407—592no—
172#407—592no—
173C2×C22⋊C8—1181no—
174#398—21no—
175#397—21no—
176#389—21no—
177#381Dic5912yes—
178#380—21no—
179#380C2×D5911yescharacteristic
180#343—1184no—
181#324—1181no—
182#323—1181no—
183C59#950711yescharacteristic, complemented, Sylow
184C22×D4C2×Dic5911yescharacteristic
185C23×C4C2×Dic5912yes—
186C23×C4C22⋊D5911yescharacteristic
187C23×C4C22×D5911yescharacteristic
188C2×M16—594no—
189C2×M16—1182no—
190C22×C8—1182no—
191#123—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4C2×Dic5911yescharacteristic
203C2×C22⋊C4C2×Dic5912yes—
204C2×C22⋊C4C22⋊D5911yescharacteristic
205C2×C42C22⋊D5911yescharacteristic
206C4⋊C8—1182no—
207C22⋊C8—1182no—
208C22⋊C8—1182no—
209C22⋊C8—1182no—
210C22⋊C8—1182no—
211C22⋊C8—1182no—
212#97—1182no—
213#95—22no—
214#95—22no—
215#95—21no—
216C24C22⋊Dic5911yescharacteristic
217C24C22×Dic5911yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊Dic5911yescharacteristic
231C22×C4C22⋊Dic5911yescharacteristic
232C22×C4C22⋊Dic5911yescharacteristic
233C22×C4D4×D5912yes—
234C22×C4#1174568712yes—
235C22×C4C2×C22⋊D5911yescharacteristic
236C22×C4C2×C22⋊D5911yescharacteristic
237M16—2364no—
238M16—2364no—
239C2×C8—1184no—
240C2×C8—4722no—
241C2×C8—2362no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#1427364412yes—
263C23C2×C22⋊Dic5911yescharacteristic
264C23D4×Dic5912yes—
265D4—44no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—84no—
286C2×C4#1427369511yescharacteristic
287C2×C4#1427369912yes—
288C2×C4#1427370811yescharacteristic
289C8—4722no—
290C8—2364no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1683728211yescharacteristic
304C22C2×#1427364411yescharacteristic
305C22#1683738611yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1683759112yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2142451011yescharacteristic
319C2#2142456011yescharacteristic
320C2#2142464911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2486C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC23×C4×C59C23
hypercenter#2486D59
hypocenterC59#9507
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #21424176C2semidirect
4. #21422002C2semidirect
183. C59#9507semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214241762
#214224342
#214220022
C59×#24862
#950759