Finite Group Databasebeta

← #28499044

Subgroups — #28499044

Order 15104.

436 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C4×#14273642C211yesmaximal, characteristic
3#21422118C212yesmaximal, complemented
4#21422097C212yesmaximal, complemented
5C59×#1460C211yesmaximal, 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
17C4×C22⋊Dic59—21no—
18#16837169C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic
24#14273706—41no—
25#14273706—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C4×Dic59—41no—
36#14273642—24no—
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—
56C4×C8×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—81no—
67C22×C4×C59—81no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—42no—
78C2×Dic118—41no—
79C2×Dic118—81no—
80C2×C8×C59—42no—
81C4⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C42×C59—21no—
85C42×C59—21no—
86C22⋊Dic59—24no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—81no—
91C22⋊Dic59—81no—
92C4⋊Dic59—41no—
93Dic59⋊C4—41no—
94Dic59⋊C4—41no—
95Dic59⋊C4—42no—
96Dic59⋊C4—81no—
97C4×Dic59—82no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59C2×C22⋊C411yescharacteristic
108C23×C59#13512yes—
109D4×C59—44no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—81no—
121C2×C4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#10412yes—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—44no—
129C2×Dic59—41no—
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—82no—
140C2×Dic59—82no—
141Dic118—82no—
142C8×C59—82no—
143#11731—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—22no—
147C22×C59—42no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59C2×#9911yescharacteristic
157C22×C59#41611yescharacteristic
158C22×C59C2×#10411yescharacteristic
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—22no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—41no—
165Dic59—81no—
166Dic59—81no—
167Dic59—84no—
168Dic59—81no—
169Dic59—81no—
170#2638—591no—
171C4×#99—591no—
172#1460D5911yescharacteristic
173#1351—592no—
174#1344—592no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59—42no—
178C2×C59—81no—
179C2×C59#150111yescharacteristic
180C2×C59#212111yescharacteristic
181C2×C59#212911yescharacteristic
182C2×C22⋊Q8—591no—
183C2×C4×D4C2×D5911yescharacteristic
184#431—21no—
185C2×#99—1181no—
186#398—591no—
187#397—591no—
188#397—1181no—
189#395—591no—
190#394—1181no—
191#393—591no—
192#387—1181no—
193#385—1181no—
194#381—591no—
195C4×C22⋊C4—1181no—
196#342—592no—
197#323—21no—
198#323C2×D5912yes—
199C59#1173111yescharacteristic, complemented, Sylow
200C22×Q8—591no—
201C22×D4D23611yescharacteristic
202C23×C4C4×D5911yescharacteristic
203C23×C4D23611yescharacteristic
204C22⋊Q8—2361no—
205C22⋊Q8—2361no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—591no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4C4×D5911yescharacteristic
213C2×C22⋊C4—1182no—
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4C22⋊D5911yescharacteristic
219C2×C22⋊C4C22×D5911yescharacteristic
220C2×C42—2361no—
221C2×C42C22⋊D5911yescharacteristic
222C4⋊C8—41no—
223#99—1184no—
224C22⋊C8—22no—
225C4×C8—41no—
226#95—591no—
227#95—591no—
228#95—1181no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233#95—2361no—
234C24(C2×D59)⋊C411yescharacteristic
235C24C2×D23611yescharacteristic
236C2×Q8—4721no—
237C2×Q8—2361no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1181no—
250C22×C4—1181no—
251C22×C4—2362no—
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4(C2×D59)⋊C411yescharacteristic
255C22×C4C2×C4×D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C2×C8—42no—
258C4⋊C4—41no—
259C4⋊C4—4721no—
260C4⋊C4—2361no—
261C4⋊C4—2361no—
262C4⋊C4—2361no—
263C4⋊C4—2362no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—1184no—
267C22⋊C4—4721no—
268C22⋊C4—4721no—
269C22⋊C4—2362no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C42—21no—
273C42—21no—
274C42—4722no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×(C2×D59)⋊C411yescharacteristic
285C23#1427374912yes—
286Q8—4722no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—4722no—
302C2×C4—4722no—
303C2×C4—2364no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4#1427364111yescharacteristic
317C2×C4#1427364111yescharacteristic
318C2×C4#1427365312yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#1427364111yescharacteristic
333C22#1683739911yescharacteristic
334C22C2×#1427365311yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—4721no—
342C4—4721no—
343C4—4724no—
344C4—4721no—
345C4—4721no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2142258811yescharacteristic
351C2#2142574311yescharacteristic
352C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1460C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1460D59
hypocenterC59#11731
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

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
199. C59#11731semidirect

Maximal subgroups

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

subgroupindex
#214300132
C4×#142736422
#214221182
#214220972
C59×#14602
#1173159