Finite Group Databasebeta

← #28499587

Subgroups — #28499587

Order 15104.

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

normal subgroups only

This group has 432 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#21424179C211yesmaximal, characteristic, complemented
2#21422410C211yesmaximal, characteristic, complemented
3#21422123C212yesmaximal, complemented
4#21422099C212yesmaximal, complemented
5C59×#2598C211yesmaximal, characteristic
6#16837455—21no—
7#16837366—21no—
8#16837171—21no—
9#16837171C2212yes—
10#16837128C2212yes—
11#16837126—21no—
12C2×C4×D4×C59C411yescharacteristic
13C2×#117×C59C2211yescharacteristic
14C59×#400C2211yescharacteristic
15C59×#399C411yescharacteristic
16C59×#394C2211yescharacteristic
17C59×#392C411yescharacteristic
18C59×#388—21no—
19C59×#386—21no—
20C59×#386—21no—
21C59×#385—21no—
22C59×#383C411yescharacteristic
23C2×#11745664—22no—
24C4⋊(C59⋊C8)—41no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—41no—
27C22⋊(C59⋊C8)—41no—
28C4×C59⋊C8—41no—
29C22×D4×C59—21no—
30C22×D4×C59C2×C411yescharacteristic
31C23×C4×C59—21no—
32C23×C4×C59D411yescharacteristic
33C4×D4×C59—41no—
34C4×D4×C59—41no—
35#117×C59—41no—
36#117×C59—41no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C4⋊C4×C59C2311yescharacteristic
41C2×C22⋊C4×C59—21no—
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C42×C59C2×C411yescharacteristic
49C2×C42×C59D411yescharacteristic
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59C2×C411yescharacteristic
55#95×C59C2×C411yescharacteristic
56C24×C59—21no—
57C24×C59C22⋊C411yescharacteristic
58C2×D4×C59—41no—
59C2×D4×C59—42no—
60C2×D4×C59—42no—
61C2×D4×C59—42no—
62C2×D4×C59—81no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—41no—
70C22×C4×C59—42no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—81no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—81no—
90C22⋊C4×C59—81no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—21no—
95#11745664—42no—
96#11745664—42no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—42no—
99C23×C59—22no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59C2×C22⋊C411yescharacteristic
112D4×C59—42no—
113D4×C59—42no—
114D4×C59—84no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—84no—
139C2×C4×C59#9911yescharacteristic
140C2×C4×C59#9911yescharacteristic
141C2×C4×C59#13512yes—
142C59⋊C8—82no—
143C59⋊C8—82no—
144#11847—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—42no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59C2×#9911yescharacteristic
160C22×C59#41611yescharacteristic
161C22×C59#42111yescharacteristic
162C4×C59—22no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—81no—
170C4×C59—81no—
171C4×C59—81no—
172#2598D5911yescharacteristic
173#1822—591no—
174#1458—591no—
175#1353—592no—
176#1345—592no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59—81no—
180C2×C59—82no—
181C2×C59#151611yescharacteristic
182C2×C59#212011yescharacteristic
183C2×C59#213411yescharacteristic
184C2×C4×D4Dic5911yescharacteristic
185C2×#117C2×D5911yescharacteristic
186#431—1181no—
187#407—1181no—
188#400C2×D5911yescharacteristic
189#399Dic5911yescharacteristic
190#394C2×D5911yescharacteristic
191#392Dic5911yescharacteristic
192#388—21no—
193#386—21no—
194#386—21no—
195#385—21no—
196#383Dic5911yescharacteristic
197#343—592no—
198#343—1181no—
199#324—592no—
200#323—1181no—
201C59#1184711yescharacteristic, complemented, Sylow
202C22×D4—21no—
203C22×D4C2×Dic5911yescharacteristic
204C23×C4—21no—
205C23×C4C22⋊D5911yescharacteristic
206C2×M16—1182no—
207C4×D4—41no—
208C4×D4—41no—
209#117—41no—
210#117—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×Dic5911yescharacteristic
213C2×C4⋊C4C22⋊D5911yescharacteristic
214C2×C4⋊C4C22×D5911yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4C2×Dic5911yescharacteristic
221C2×C22⋊C4C22⋊D5911yescharacteristic
222C2×C42C2×Dic5911yescharacteristic
223C2×C42C22⋊D5911yescharacteristic
224C4⋊C8—2361no—
225C22⋊C8—1182no—
226C22⋊C8—2361no—
227C22⋊C8—2361no—
228C4×C8—2361no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95C2×Dic5911yescharacteristic
234#95C2×Dic5911yescharacteristic
235C24—21no—
236C24C22⋊Dic5911yescharacteristic
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4C22⋊Dic5911yescharacteristic
253C22×C4C22⋊Dic5911yescharacteristic
254C22×C4C22⋊Dic5911yescharacteristic
255C22×C4C22×Dic5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C22×C4C2×C22⋊D5911yescharacteristic
258M16—2362no—
259M16—2362no—
260C2×C8—2362no—
261C2×C8—2362no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C2×C22⋊Dic5911yescharacteristic
291D4—42no—
292D4—42no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—84no—
318C2×C4#1427364211yescharacteristic
319C2×C4#1427364211yescharacteristic
320C2×C4#1427374712yes—
321C8—4722no—
322C8—4722no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22C2×#1427364211yescharacteristic
338C22#1683739811yescharacteristic
339C22#1683741911yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—82no—
354C2#2142265211yescharacteristic
355C2#2142573711yescharacteristic
356C2#2142578811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2598C2
socleC22×C59C2×#99
prime radicalC59×#392C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2598D59
hypocenterC59#11847
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
1. #21424179C2semidirect
2. #21422410C2semidirect
3. #21422123C2semidirect
4. #21422099C2semidirect
201. C59#11847semidirect

Maximal subgroups

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

subgroupindex
#214241792
#214224102
#214221232
#214220992
C59×#25982
#1184759