Finite Group Databasebeta

← #28499481

Subgroups — #28499481

Order 15104.

414 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 79 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#21430013C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422118C212yesmaximal, complemented
4#21422097C212yesmaximal, complemented
5C59×#1466C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic, complemented
7C2×#14273642—21no—
8#16837332C411yescharacteristic, complemented
9#16837330—21no—
10#16837327C2211yescharacteristic
11#16837322C411yescharacteristic, complemented
12#16837316—21no—
13#16837314C2211yescharacteristic
14#16837286—21no—
15#16837275—21no—
16#16837265C411yescharacteristic, complemented
17#16837262—21no—
18#16837169C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#432—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic
24#14273706—41no—
25#14273706—41no—
26C2×C4⋊Dic59—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36#14273642—42no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—41no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×D4×C59D411yescharacteristic
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C22⋊C8×C59—22no—
56#97×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×D4×C59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—81no—
68C22×C4×C59—81no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—42no—
79C2×Dic118—41no—
80C2×Dic118—81no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—42no—
88C22⋊Dic59—41no—
89C22⋊Dic59—41no—
90C22⋊Dic59—42no—
91C22⋊Dic59—42no—
92C22⋊Dic59—81no—
93C22⋊Dic59—81no—
94C4⋊Dic59—41no—
95C4⋊Dic59—82no—
96Dic59⋊C4—42no—
97Dic59⋊C4—41no—
98Dic59⋊C4—41no—
99Dic59⋊C4—81no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59AGL(1,ℤ/8)11yescharacteristic
111C23×C59#13711yescharacteristic
112D4×C59—42no—
113D4×C59—42no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59#9911yescharacteristic
129C2×C4×C59#9911yescharacteristic
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—82no—
143C2×Dic59—82no—
144C2×Dic59—82no—
145Dic118—82no—
146C8×C59—82no—
147#11826—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59—81no—
162C22×C59C2×#9911yescharacteristic
163C22×C59#41711yescharacteristic
164C22×C59#42111yescharacteristic
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—41no—
171Dic59—81no—
172Dic59—81no—
173Dic59—81no—
174Dic59—81no—
175Dic59—162no—
176#2638—591no—
177#1799—591no—
178#1466D5911yescharacteristic
179#1351—592no—
180#1344—592no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59—41no—
184C2×C59—41no—
185C2×C59—81no—
186C2×C59#151311yescharacteristic
187C2×C59#212111yescharacteristic
188C2×C59#212911yescharacteristic
189C2×C22⋊Q8—591no—
190C2×C4×D4C2×D5911yescharacteristic
191#432—21no—
192C2×#99—1181no—
193#398—591no—
194#397—591no—
195#397—1181no—
196#395—591no—
197#394—1181no—
198#393—591no—
199#387—1181no—
200#385—1181no—
201#381—591no—
202#380—1181no—
203#342—592no—
204#323—21no—
205#323C2×D5912yes—
206C59#1182611yescharacteristic, complemented, Sylow
207C22×Q8—591no—
208C22×D4D23611yescharacteristic
209C23×C4C4×D5911yescharacteristic
210C23×C4D23611yescharacteristic
211C22⋊Q8—2361no—
212C22⋊Q8—2361no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—591no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4—2361no—
220C2×C4⋊C4C4×D5911yescharacteristic
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4—1182no—
226C2×C22⋊C4C22⋊D5911yescharacteristic
227C2×C22⋊C4C22×D5911yescharacteristic
228C2×C42C22⋊D5911yescharacteristic
229C4⋊C8—41no—
230#99—2362no—
231C22⋊C8—22no—
232#97—41no—
233#95—591no—
234#95—591no—
235#95—1181no—
236#95—1181no—
237#95—1181no—
238#95—1181no—
239#95—1181no—
240#95—2361no—
241C24(C2×D59)⋊C411yescharacteristic
242C24C2×D23611yescharacteristic
243C2×Q8—4721no—
244C2×Q8—2361no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—1181no—
258C22×C4—1181no—
259C22×C4—2362no—
260C22×C4(C2×D59)⋊C411yescharacteristic
261C22×C4(C2×D59)⋊C411yescharacteristic
262C22×C4(C2×D59)⋊C411yescharacteristic
263C22×C4C2×C4×D5911yescharacteristic
264C22×C4C2×C22⋊D5911yescharacteristic
265C2×C8—42no—
266C4⋊C4—41no—
267C4⋊C4—4722no—
268C4⋊C4—4721no—
269C4⋊C4—2361no—
270C4⋊C4—2362no—
271C4⋊C4—2361no—
272C4⋊C4—2361no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—4721no—
276C22⋊C4—4721no—
277C22⋊C4—2362no—
278C22⋊C4—2362no—
279C22⋊C4—2362no—
280C22⋊C4—2361no—
281C22⋊C4—2361no—
282C42—21no—
283C42—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23C2×(C2×D59)⋊C411yescharacteristic
294C23#1427375511yescharacteristic
295C23#1427376011yescharacteristic
296Q8—4722no—
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—4722no—
314C2×C4—4722no—
315C2×C4—4722no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4#1427364111yescharacteristic
329C2×C4#1427364111yescharacteristic
330C8—82no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1427364111yescharacteristic
346C22#1683740411yescharacteristic
347C22#1683741811yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—9442no—
355C4—4721no—
356C4—4721no—
357C4—4721no—
358C4—4721no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#2142264011yescharacteristic
365C2#2142574311yescharacteristic
366C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1466C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1466D59
hypocenterC59#11826
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #21422118C2semidirect
4. #21422097C2semidirect
6. C2×#14273706C4semidirect
8. #16837332C4semidirect
11. #16837322C4semidirect
16. #16837265C4semidirect
206. C59#11826semidirect

Maximal subgroups

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

subgroupindex
#214300132
#214240842
#214221182
#214220972
C59×#14662
#1182659