Finite Group Databasebeta

← #28499244

Subgroups — #28499244

Order 15104.

482 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 87 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#21428404C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic
3#21422095C214yesmaximal, complemented
4C4×#99×C59C211yesmaximal, characteristic
5C2×#14273708—21no—
6C22×C4⋊Dic59C411yescharacteristic, complemented
7#16837459—21no—
8#16837332C412yescomplemented
9#16837327C2212yes—
10#16837300—21no—
11#16837263—21no—
12#16837262—21no—
13#16837262C411yescharacteristic, complemented
14#16837126—21no—
15#16837126C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59—21no—
18C4×C22⋊C4×C59—21no—
19C59×#342C2212yes—
20C23×Dic59—21no—
21#14273708—44no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—41no—
25C2×C4⋊Dic59—41no—
26C2×C4⋊Dic59C2×C412yes—
27C2×Dic59⋊C4—22no—
28C2×C22⋊Dic59—21no—
29C2×C22⋊Dic59—22no—
30C4⋊(C59⋊C8)—41no—
31C22⋊(C59⋊C8)—22no—
32#14273638—41no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95C2×C412yes—
36C22×D4×C59D411yescharacteristic
37C23×C4×C59C2×C411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C4×D4×C59—41no—
40C4×D4×C59—41no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59C2311yescharacteristic
45C2×C42×C59—41no—
46C2×C42×C59D411yescharacteristic
47#99×C59—24no—
48#95×C59—41no—
49C24×C59C22⋊C411yescharacteristic
50C24×C59C2×D411yescharacteristic
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—41no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×Dic59—22no—
67C22×Dic59—22no—
68C22×Dic59—42no—
69C22×Dic59—42no—
70C22×Dic59—42no—
71C4⋊C4×C59—41no—
72C22⋊C4×C59—24no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—42no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—82no—
79C22⋊Dic59—42no—
80C22⋊Dic59—44no—
81C4⋊Dic59—42no—
82C4⋊Dic59—41no—
83C4⋊Dic59—41no—
84C4⋊Dic59—42no—
85Dic59⋊C4—44no—
86C2×C59⋊C8—42no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59#10412yes—
96C23×C59C2×C22⋊C411yescharacteristic
97C23×C59AGL(1,ℤ/8)12yes—
98D4×C59—44no—
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—44no—
113C2×C4×C59—81no—
114C2×C4×C59—82no—
115C2×C4×C59—82no—
116C2×C4×C59#9911yescharacteristic
117C2×C4×C59#9911yescharacteristic
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—84no—
125C59⋊C8—82no—
126#11773—591nomaximal, Sylow
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—22no—
130C22×C59—22no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—81no—
139C22×C59C2×#9911yescharacteristic
140C22×C59#41811yescharacteristic
141C22×C59C2×#10411yescharacteristic
142C4×C59—21no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—41no—
146C4×C59—41no—
147C4×C59—84no—
148C4×C59#35212yes—
149Dic59—82no—
150Dic59—82no—
151#2486—591no—
152C4×#99D5911yescharacteristic
153#1466—591no—
154#1344—594no—
155C2×C59—21no—
156C2×C59—21no—
157C2×C59—42no—
158C2×C59—42no—
159C2×C59#150711yescharacteristic
160C2×C59C2×#35211yescharacteristic
161C2×C59#212911yescharacteristic
162C2×#123—1181no—
163C2×C4×D4C2×D5911yescharacteristic
164C22×C4⋊C4—591no—
165#432—1181no—
166C2×#99—21no—
167#398—592no—
168#397—592no—
169#389—1181no—
170#381—1181no—
171#380—591no—
172#380—1181no—
173C4×C22⋊C4—21no—
174#342C2×D5912yes—
175#323—592no—
176#323—1181no—
177C59#1177311yescharacteristic, complemented, Sylow
178C22×D4C22⋊D5911yescharacteristic
179C23×C4—1181no—
180C23×C4C4×D5911yescharacteristic
181C23×C4C22⋊D5911yescharacteristic
182#123—2364no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—592no—
186C2×C4⋊C4—1182no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—2361no—
190C2×C4⋊C4—2361no—
191C2×C4⋊C4C4×D5911yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—1181no—
194C2×C22⋊C4—1182no—
195C2×C22⋊C4D23611yescharacteristic
196C2×C22⋊C4C22×D5911yescharacteristic
197C2×C42—41no—
198C2×C42D23611yescharacteristic
199C4⋊C8—2361no—
200#99—24no—
201C22⋊C8—1182no—
202#97—2361no—
203#95—41no—
204#95—592no—
205#95—1182no—
206#95—1182no—
207C24(C2×D59)⋊C411yescharacteristic
208C24C2×C22⋊D5911yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1182no—
220C22×C4—1182no—
221C22×C4—2362no—
222C22×C4—2362no—
223C22×C4—2362no—
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4(C2×D59)⋊C411yescharacteristic
227C22×C4C2×C4×D5911yescharacteristic
228C22×C4C2×D23611yescharacteristic
229C2×C8—2362no—
230C4⋊C4—41no—
231C4⋊C4—2362no—
232C4⋊C4—2364no—
233C4⋊C4—2362no—
234C4⋊C4—2361no—
235C4⋊C4—2361no—
236C22⋊C4—24no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—2362no—
241C22⋊C4—2364no—
242C42—21no—
243C42—21no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1427365512yes—
254C23C2×(C2×D59)⋊C411yescharacteristic
255C23#1427375212yes—
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—4724no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4—2362no—
278C2×C4—2362no—
279C2×C4—2362no—
280C2×C4—2362no—
281C2×C4#1427364111yescharacteristic
282C2×C4#1427364111yescharacteristic
283C8—4722no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#1427364111yescharacteristic
297C22#1683740611yescharacteristic
298C22C2×#1427365511yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—84no—
305C4—4722no—
306C4—4722no—
307C4#1683719112yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2142261111yescharacteristic
313C2C2×#1683719111yescharacteristic
314C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11773
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21422095C2semidirect
6. C22×C4⋊Dic59C4semidirect
8. #16837332C4semidirect
13. #16837262C4semidirect
177. C59#11773semidirect

Maximal subgroups

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

subgroupindex
#214284042
#214224372
#214220952
C4×#99×C592
#1177359