Finite Group Databasebeta

← #28497863

Subgroups — #28497863

Order 15104.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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#21426578C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422045C212yesmaximal
4#21422043C212yesmaximal
5C59×#1458C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic
7C22×C4×Dic59C411yescharacteristic
8#16837455—21no—
9#16837298—21no—
10#16837298—21no—
11#16837274C411yescharacteristic
12#16837274C411yescharacteristic
13#16837274C2211yescharacteristic
14#16837274C2211yescharacteristic
15C4×Dic59⋊C4—21no—
16C4×Dic59⋊C4—21no—
17#16837128—21no—
18#16837126C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C23×Dic59C2×C411yescharacteristic
24C23×Dic59C2×C411yescharacteristic
25#14273686—22no—
26#14273686C814yescomplemented
27C2×C4⋊Dic59—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59C2×C411yescharacteristic
33C2×C22⋊Dic59C2×C411yescharacteristic
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36C2×C4×Dic59—21no—
37C2×C4×Dic59—41no—
38C2×C4×Dic59—41no—
39C4⋊(C59⋊C8)—41no—
40C22⋊(C59⋊C8)—22no—
41C4×C59⋊C8—41no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C23×C4×C59C2×C411yescharacteristic
47#117×C59—41no—
48#117×C59—41no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C4⋊C8×C59—41no—
56C22⋊C8×C59—22no—
57C4×C8×C59—41no—
58C24×C59C22×C411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C2×C812yes—
67C22×C4×C59M1612yes—
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—42no—
75C22×Dic59—42no—
76C22×Dic59—41no—
77C22×Dic59—41no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—21no—
88C22⋊Dic59—21no—
89C22⋊Dic59C2×C812yes—
90C22⋊Dic59C2×C812yes—
91C4⋊Dic59—24no—
92Dic59⋊C4—42no—
93Dic59⋊C4—42no—
94Dic59⋊C4—42no—
95Dic59⋊C4—42no—
96C4×Dic59—24no—
97C4×Dic59—42no—
98C4×Dic59—41no—
99C4×Dic59—41no—
100C2×C59⋊C8—42no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59C2×C22⋊C411yescharacteristic
106C23×C59C22×C811yescharacteristic
107C23×C59C2×M1611yescharacteristic
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59C22⋊C812yes—
123C2×C4×C59C22⋊C812yes—
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#9911yescharacteristic
126C2×C4×C59#10112yes—
127C2×Dic59—22no—
128C2×Dic59—22no—
129C2×Dic59—22no—
130C2×Dic59—22no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—42no—
138C2×Dic59—42no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—42no—
142C2×Dic59—42no—
143C8×C59—82no—
144C59⋊C8—82no—
145#11452—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59C2×C22⋊C811yescharacteristic
156C22×C59C2×#9911yescharacteristic
157C22×C59C2×#10111yescharacteristic
158C4×C59—22no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59#32412yes—
164C4×C59#32412yes—
165Dic59—42no—
166Dic59—42no—
167Dic59—42no—
168Dic59—42no—
169#2296—591no—
170#1458—591no—
171#1458D5911yescharacteristic
172#1326—592no—
173#1326—592no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—41no—
177C2×C59—41no—
178C2×C59C2×#32411yescharacteristic
179C2×C59#212011yescharacteristic
180C2×C59#212011yescharacteristic
181C2×#117—591no—
182C2×#117C2×D5911yescharacteristic
183C22×C42—591no—
184#431—21no—
185#431—1181no—
186#389—1181no—
187#389—1181no—
188#385—591no—
189#385—591no—
190#385—591no—
191#385—591no—
192C4×C4⋊C4—1181no—
193C4×C4⋊C4—1181no—
194#324—21no—
195#324—1181no—
196#323—592no—
197#323C2×D5912yes—
198C59#1145211yescharacteristic, complemented, Sylow
199C23×C4—591no—
200C23×C4—591no—
201C23×C4C4×D5911yescharacteristic
202#117—41no—
203#117—41no—
204#117—594no—
205#117—1182no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4D23611yescharacteristic
212C2×C4⋊C4C22⋊D5911yescharacteristic
213C2×C22⋊C4—591no—
214C2×C22⋊C4—591no—
215C2×C22⋊C4C4×D5911yescharacteristic
216C2×C22⋊C4C22×D5911yescharacteristic
217C2×C42—1181no—
218C2×C42—1181no—
219C2×C42—1181no—
220C2×C42—2361no—
221C2×C42—2361no—
222C2×C42D23611yescharacteristic
223C2×C42C22⋊D5911yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—2361no—
226C22⋊C8—22no—
227C22⋊C8—1182no—
228C4×C8—41no—
229C4×C8—2361no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234C24C2×C4×D5911yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4—2362no—
243C22×C4—2362no—
244C22×C4—2361no—
245C22×C4—2361no—
246C22×C4C8×D5912yes—
247C22×C4#1174565912yes—
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4C2×D23611yescharacteristic
253C22×C4C2×C22⋊D5911yescharacteristic
254C2×C8—42no—
255C2×C8—2362no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—1184no—
259C4⋊C4—2362no—
260C4⋊C4—2362no—
261C4⋊C4—2362no—
262C4⋊C4—2362no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—592no—
266C22⋊C4—592no—
267C22⋊C4—1181no—
268C22⋊C4—1181no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—1184no—
274C42—2362no—
275C42—2361no—
276C42—2361no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23C2×(C2×D59)⋊C411yescharacteristic
282C23C2×C8×D5911yescharacteristic
283C23C2×#1174565911yescharacteristic
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—1182no—
299C2×C4—1182no—
300C2×C4—1182no—
301C2×C4—1182no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2362no—
309C2×C4—2362no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2362no—
313C2×C4—2362no—
314C2×C4(C2×D59)⋊C812yes—
315C2×C4(C2×D59)⋊C812yes—
316C2×C4#1427364111yescharacteristic
317C2×C4#1427364111yescharacteristic
318C2×C4#1427364512yes—
319C8—82no—
320C8—4722no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22C2×(C2×D59)⋊C811yescharacteristic
331C22C2×#1427364111yescharacteristic
332C22C2×#1427364511yescharacteristic
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2362no—
339C4—2362no—
340C4—2362no—
341C4—2362no—
342C4#1683712712yes—
343C4#1683712712yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1683712711yescharacteristic
349C2#2142573511yescharacteristic
350C2#2142573811yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC23×C59C22×C8
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radicalC24×C59C22×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11452
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
26. #14273686C8semidirect
198. C59#11452semidirect

Maximal subgroups

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

subgroupindex
#214265782
#214224102
#214220452
#214220432
C59×#14582
#1145259