Finite Group Databasebeta

← #28486646

Subgroups — #28486646

Order 15104.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#21428404C211yesmaximal, characteristic
3#21424174C212yesmaximal, complemented
4#21422003C212yesmaximal, complemented
5C59×#1465C211yesmaximal, characteristic
6C2×#14273708C412yes—
7C22×C4⋊Dic59C2211yescharacteristic
8#16837474—24no—
9#16837459—21no—
10#16837366C2212yes—
11C2×C22⋊(C59⋊C8)—21no—
12#16837332—21no—
13#16837327—21no—
14#16837300—21no—
15#16837263C412yes—
16#16837262—21no—
17#16837262C2211yescharacteristic
18#16837170—24no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#407C2212yes—
21C59×#324—21no—
22C59×#323—21no—
23C23×Dic59C2×C412yes—
24C2×#11745664D414yes—
25C22×C59⋊C8—22no—
26#14273708—44no—
27C2×C4⋊Dic59—22no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—22no—
32C2×Dic59⋊C4—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59C2×C412yes—
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—22no—
39#14273638—22no—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C59⋊#95—21no—
43C22×D4×C59C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C4×D4×C59—22no—
48C4×D4×C59—22no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C22⋊C8×C59—22no—
54C22⋊C8×C59—22no—
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—42no—
63C22×C4×C59—42no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C2×D412yes—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59#4112yes—
71C22×Dic59—22no—
72C22×Dic59—22no—
73C22×Dic59—24no—
74C22×Dic59—42no—
75C22×Dic59—42no—
76M16×C59—44no—
77C2×C8×C59—42no—
78C4⋊C4×C59—22no—
79C22⋊C4×C59—22no—
80C22⋊C4×C59—22no—
81C42×C59—21no—
82C42×C59—21no—
83C22⋊Dic59—44no—
84C22⋊Dic59—42no—
85C4⋊Dic59—42no—
86C4⋊Dic59—41no—
87C4⋊Dic59—42no—
88C4⋊Dic59—41no—
89Dic59⋊C4—44no—
90#11745664—44no—
91C2×C59⋊C8—24no—
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—81no—
118C2×C4×C59#12011yescharacteristic
119C2×C4×C59#12112yes—
120C2×C4×C59#12311yescharacteristic
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—84no—
128C8×C59—82no—
129C59⋊C8—44no—
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#45812yes—
152Dic59—82no—
153Dic59—82no—
154#2973—591no—
155#2486—591no—
156#1821—592no—
157#1465D5911yescharacteristic
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×#123—592no—
167C2×C4×D4C2×D5911yescharacteristic
168C22×C4⋊C4—591no—
169#435—1184no—
170#432—1181no—
171#407—592no—
172#407C2×D5912yes—
173C2×C22⋊C8—1181no—
174#398—1181no—
175#397—1181no—
176#389—1181no—
177#381—592no—
178#380—591no—
179#380—1181no—
180#343—1184no—
181#324—21no—
182#323—21no—
183C59#950711yescharacteristic, complemented, Sylow
184C22×D4C4×D5911yescharacteristic
185C23×C4—592no—
186C23×C4D23611yescharacteristic
187C23×C4C22×D5911yescharacteristic
188C2×M16—22no—
189C2×M16—594no—
190C22×C8—1182no—
191#123—2364no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1182no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—1182no—
198C2×C4⋊C4—1182no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4—592no—
202C2×C22⋊C4—1182no—
203C2×C22⋊C4C4×D5911yescharacteristic
204C2×C22⋊C4C22⋊D5911yescharacteristic
205C2×C42D23611yescharacteristic
206C4⋊C8—1182no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1182no—
210C22⋊C8—1182no—
211C22⋊C8—1182no—
212#97—1182no—
213#95—1182no—
214#95—1182no—
215#95—1181no—
216C24(C2×D59)⋊C411yescharacteristic
217C24C2×C4×D5911yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1184no—
226C22×C4—1182no—
227C22×C4—1182no—
228C22×C4—2362no—
229C22×C4—2362no—
230C22×C4(C2×D59)⋊C411yescharacteristic
231C22×C4(C2×D59)⋊C411yescharacteristic
232C22×C4(C2×D59)⋊C411yescharacteristic
233C22×C4C2×D23611yescharacteristic
234C22×C4D4×D5912yes—
235C22×C4#1174568712yes—
236C22×C4C2×C22⋊D5911yescharacteristic
237M16—44no—
238M16—2364no—
239C2×C8—42no—
240C2×C8—1184no—
241C2×C8—4722no—
242C4⋊C4—22no—
243C4⋊C4—2364no—
244C4⋊C4—2362no—
245C4⋊C4—2361no—
246C4⋊C4—2362no—
247C4⋊C4—2361no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—2364no—
251C22⋊C4—2362no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#1427364312yes—
263C23C2×(C2×D59)⋊C411yescharacteristic
264C23(C22⋊D59)⋊C412yes—
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—81no—
279C2×C4—4724no—
280C2×C4—2362no—
281C2×C4—2362no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4—2362no—
285C2×C4—2362no—
286C2×C4#1427369711yescharacteristic
287C2×C4#1427369912yes—
288C2×C4#1427371111yescharacteristic
289C8—82no—
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#1683727911yescharacteristic
304C22C2×#1427364311yescharacteristic
305C22#1683738411yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—4722no—
312C4—4722no—
313C4#1683759312yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2142450711yescharacteristic
319C2#2142455711yescharacteristic
320C2#2142464611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1465C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC23×C4×C59C23
hypercenter#1465D59
hypocenterC59#9507
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #21424174C2semidirect
4. #21422003C2semidirect
183. C59#9507semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214284042
#214241742
#214220032
C59×#14652
#950759