Finite Group Databasebeta

← #28499814

Subgroups — #28499814

Order 15104.

580 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 99 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#21427461C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422118C214yesmaximal, complemented
4C59×#1460C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic, complemented
6C2×#14273706—21no—
7C2×#14273642—21no—
8#16837322C412yescomplemented
9#16837314C2212yes—
10#16837308—21no—
11#16837282—21no—
12#16837274—21no—
13#16837262—21no—
14#16837262C411yescharacteristic, complemented
15#16837169C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#431—21no—
18C59×#323—21no—
19C59×#323C2212yes—
20C22×Dic118—21no—
21C22×Dic118—44no—
22C22×Dic118—41no—
23C22×Dic118—41no—
24C22×Dic118C2×C412yes—
25C23×Dic59—21no—
26#14273706—44no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—41no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33#14273642—42no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—41no—
37C59⋊#95C2×C412yes—
38C22×D4×C59D411yescharacteristic
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59D411yescharacteristic
47C4⋊C8×C59—41no—
48C22⋊C8×C59—22no—
49C4×C8×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59SD1612yes—
55C2×D4×C59Q1612yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—44no—
70C22×Dic59—42no—
71C22×Dic59—42no—
72C2×Dic118—24no—
73C2×Dic118—42no—
74C2×Dic118—48no—
75C2×Dic118—44no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×Dic118—42no—
79C2×Dic118—44no—
80C2×Dic118—44no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—44no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—42no—
91C4⋊Dic59—42no—
92C4⋊Dic59—82no—
93Dic59⋊C4—44no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59#10312yes—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59C2×SD1611yescharacteristic
105C23×C59C2×Q1611yescharacteristic
106D4×C59—42no—
107D4×C59—42no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—81no—
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59#10312yes—
124C2×Dic59—24no—
125C2×Dic59—42no—
126C2×Dic59—48no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—82no—
132C2×Dic59—82no—
133C2×Dic59—82no—
134Dic118—44no—
135Dic118—44no—
136Dic118—44no—
137Dic118—44no—
138C8×C59—82no—
139#11898—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—22no—
144C22×C59—21no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—42no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59C2×#9911yescharacteristic
155C22×C59C2×#10311yescharacteristic
156C22×C59#42111yescharacteristic
157C4×C59—21no—
158C4×C59—22no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59AΣL(1,𝔽16)12yes—
163Dic59—44no—
164Dic59—82no—
165Dic59—162no—
166#2409—591no—
167#1799—591no—
168#1460D5911yescharacteristic
169#1351—594no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—41no—
173C2×C59—42no—
174C2×C59—41no—
175C2×C59#152011yescharacteristic
176C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
177C2×C59#212911yescharacteristic
178C23×Q8—591no—
179C2×C22⋊Q8—1181no—
180C2×C4×D4C2×D5911yescharacteristic
181#431—21no—
182C2×#99—1181no—
183#395—592no—
184#393—592no—
185#391—1181no—
186#386—1181no—
187#385—1181no—
188#380—591no—
189#380—1181no—
190#342—592no—
191#323—21no—
192#323C2×D5912yes—
193C59#1189811yescharacteristic, complemented, Sylow
194C22×Q8—592no—
195C22×Q8—1181no—
196C22×Q8—2361no—
197C22×Q8—2361no—
198C22×Q8—2364no—
199C22×D4D23611yescharacteristic
200C23×C4—1181no—
201C23×C4C4×D5911yescharacteristic
202C23×C4D23611yescharacteristic
203C22⋊Q8—2364no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—1182no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—2361no—
209C2×C4⋊C4C4×D5911yescharacteristic
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1182no—
212C2×C22⋊C4—1182no—
213C2×C22⋊C4C22⋊D5911yescharacteristic
214C2×C22⋊C4C22×D5911yescharacteristic
215C2×C42C22⋊D5911yescharacteristic
216C4⋊C8—41no—
217#99—2362no—
218C22⋊C8—22no—
219C4×C8—41no—
220#95—592no—
221#95—1182no—
222#95—1182no—
223#95—2361no—
224C24(C2×D59)⋊C411yescharacteristic
225C24C2×D23611yescharacteristic
226C2×Q8—1184no—
227C2×Q8—2362no—
228C2×Q8—2362no—
229C2×Q8—2364no—
230C2×Q8—2364no—
231C2×Q8—2368no—
232C2×Q8—2364no—
233C2×Q8—2361no—
234C2×Q8—2361no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4#1174566012yes—
238C2×D4Dic23612yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1182no—
246C22×C4—1182no—
247C22×C4—2364no—
248C22×C4—2362no—
249C22×C4—2362no—
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4C2×C4×D5911yescharacteristic
254C22×C4C2×C22⋊D5911yescharacteristic
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—4722no—
258C4⋊C4—2362no—
259C4⋊C4—2364no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2364no—
263C22⋊C4—2362no—
264C22⋊C4—2362no—
265C22⋊C4—2362no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23#1427365212yes—
277C23C2×(C2×D59)⋊C411yescharacteristic
278C23C2×#1174566011yescharacteristic
279C23C2×Dic23611yescharacteristic
280Q8—2364no—
281Q8—2364no—
282Q8—2364no—
283Q8—2364no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—1184no—
300C2×C4—4722no—
301C2×C4—4722no—
302C2×C4—4722no—
303C2×C4—2362no—
304C2×C4—2362no—
305C2×C4—2362no—
306C2×C4—2362no—
307C2×C4—2368no—
308C2×C4—2362no—
309C2×C4#1427364111yescharacteristic
310C2×C4#1427364111yescharacteristic
311C2×C4#1427365212yes—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#1427364111yescharacteristic
328C22C2×#1427365211yescharacteristic
329C22#1683741811yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—9442no—
336C4—4722no—
337C4—2364no—
338C4#1683719012yes—
339C2—21no—
340C2—21no—
341C2—41no—
342C2—42no—
343C2—41no—
344C2#2142266911yescharacteristic
345C2C2×#1683719011yescharacteristic
346C2#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#11898
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
5. C23×Dic118C4semidirect
8. #16837322C4semidirect
14. #16837262C4semidirect
193. C59#11898semidirect

Maximal subgroups

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

subgroupindex
#214274612
#214240842
#214221182
C59×#14602
#1189859