Finite Group Databasebeta

← #28479237

Subgroups — #28479237

Order 15104.

582 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C59⋊C8C211yesmaximal, characteristic
2#21423874C212yesmaximal, complemented
3#21422414C211yesmaximal, characteristic
4#21421944C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic
6C23×C59⋊C8C411yescharacteristic, complemented
7C23×C59⋊C8C2211yescharacteristic
8D4×C59⋊C8—22no—
9D4×C59⋊C8—22no—
10C2×C4⋊(C59⋊C8)C411yescharacteristic, complemented
11C2×C22⋊(C59⋊C8)C411yescharacteristic, complemented
12C2×C22⋊(C59⋊C8)C2211yescharacteristic
13C2×C22⋊(C59⋊C8)C2212yes—
14C2×#14273638—22no—
15C2×C4×C59⋊C8C411yescharacteristic, complemented
16#16837157—22no—
17#16837128C412yescomplemented
18#16837126C412yescomplemented
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59C412yes—
21C4×C22⋊C4×C59C2212yes—
22C59×#342C412yes—
23C22×C59⋊C8—22no—
24C22×C59⋊C8—21no—
25C22×C59⋊C8—21no—
26C22×C59⋊C8—42no—
27C22×C59⋊C8—42no—
28C22×C59⋊C8C2×C411yescharacteristic
29C22×C59⋊C8C2×C411yescharacteristic
30C22×C59⋊C8C2×C411yescharacteristic
31C22×C59⋊C8C2×C411yescharacteristic
32C22×C59⋊C8D412yes—
33C22×C59⋊C8D412yes—
34C4⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)C2×C414yes—
39#14273638—44no—
40C4×C59⋊C8—21no—
41C4×C59⋊C8—21no—
42C22×D4×C59C2×C411yescharacteristic
43C23×C4×C59C2×C411yescharacteristic
44C23×C4×C59C2311yescharacteristic
45C4×D4×C59—22no—
46C4×D4×C59—22no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59C2×C414yes—
50C2×C22⋊C4×C59C2×C411yescharacteristic
51C2×C42×C59—22no—
52C2×C42×C59C2×C411yescharacteristic
53#99×C59—24no—
54#95×C59—22no—
55C24×C59C4211yescharacteristic
56C24×C59C22×C411yescharacteristic
57C2×D4×C59—22no—
58C2×D4×C59—22no—
59C2×D4×C59M1614yes—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—24no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59C4211yescharacteristic
68C22×C4×C59C4211yescharacteristic
69C22×C4×C59C4211yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59#4112yes—
74C4⋊C4×C59—22no—
75C22⋊C4×C59—22no—
76C22⋊C4×C59—22no—
77C22⋊C4×C59—24no—
78C22⋊C4×C59—24no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—42no—
82C42×C59—42no—
83C2×C59⋊C8—24no—
84C2×C59⋊C8—22no—
85C2×C59⋊C8—21no—
86C2×C59⋊C8—21no—
87C2×C59⋊C8—24no—
88C2×C59⋊C8—41no—
89C2×C59⋊C8—42no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—42no—
92C2×C59⋊C8—42no—
93C2×C59⋊C8—42no—
94C2×C59⋊C8—41no—
95C2×C59⋊C8—41no—
96C2×C59⋊C8C22⋊C411yescharacteristic
97C2×C59⋊C8C22⋊C412yes—
98C2×C59⋊C8C22⋊C411yescharacteristic
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59#9712yescomplemented
106C23×C59C2×C4211yescharacteristic
107C23×C59C4×D412yes—
108C23×C59C2×M1612yes—
109C23×C59#13112yes—
110D4×C59—28no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—41no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—44no—
127C2×C4×C59—44no—
128C2×C4×C59—44no—
129C2×C4×C59#9712yescomplemented
130C2×C4×C59C2×C22⋊C411yescharacteristic
131C2×C4×C59#11711yescharacteristic
132C2×C4×C59C4×D412yes—
133C59⋊C8—21no—
134C59⋊C8—22no—
135C59⋊C8—44no—
136C59⋊C8—41no—
137C59⋊C8—41no—
138C59⋊C8#9912yescomplemented
139#8160—591nomaximal, Sylow
140C22×C59—22no—
141C22×C59—21no—
142C22×C59—22no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59C4×C22⋊C411yescharacteristic
152C22×C59C2×#9711yescharacteristic
153C22×C59#40511yescharacteristic
154C22×C59#43514yes—
155C4×C59—22no—
156C4×C59—22no—
157C4×C59—21no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—84no—
161C4×C59C2×#9911yescharacteristic
162C4×C59#41011yescharacteristic
163C2×C8×D4—591no—
164C4×#99D5911yescharacteristic
165#1752—592no—
166#1459—591no—
167#1282—592no—
168C2×C59—24no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—42no—
172C2×C59#175211yescharacteristic
173C2×C59C4×#9911yescharacteristic
174C2×C59#175711yescharacteristic
175C23×C8—591no—
176C23×C8—591no—
177C2×C4×D4C2×D5911yescharacteristic
178C8×D4—1182no—
179C8×D4—1182no—
180C2×C4⋊C8—591no—
181C2×#99Dic5912yes—
182C2×C22⋊C8—592no—
183C2×C22⋊C8—591no—
184C2×C22⋊C8—591no—
185C2×#97—1182no—
186C2×C4×C8—591no—
187C4×C22⋊C4C2×D5912yes—
188#342Dic5912yes—
189#336—1182no—
190#324—592no—
191#323—592no—
192C59#816011yescharacteristic, complemented, Sylow
193C22×D4C2×Dic5911yescharacteristic
194C23×C4C4×D5911yescharacteristic
195C23×C4C22×D5911yescharacteristic
196C22×C8—591no—
197C22×C8—592no—
198C22×C8—591no—
199C22×C8—592no—
200C22×C8—591no—
201C22×C8—591no—
202C22×C8—1181no—
203C22×C8—1181no—
204C22×C8—1182no—
205C22×C8—2362no—
206C22×C8—2362no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4C4×D5911yescharacteristic
210C2×C22⋊C4C4×D5911yescharacteristic
211C2×C22⋊C4C2×Dic5914yes—
212C2×C22⋊C4C2×Dic5911yescharacteristic
213C2×C42—22no—
214C2×C42C4×D5911yescharacteristic
215C4⋊C8—1182no—
216#99—24no—
217C22⋊C8—594no—
218C22⋊C8—1182no—
219C22⋊C8—1182no—
220C22⋊C8—1182no—
221#97—2364no—
222C4×C8—1181no—
223C4×C8—1181no—
224#95—22no—
225C24C4×Dic5911yescharacteristic
226C24C22×Dic5911yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4#1174566414yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C4×Dic5911yescharacteristic
238C22×C4C4×Dic5911yescharacteristic
239C22×C4C4×Dic5911yescharacteristic
240C22×C4C2×C4×D5911yescharacteristic
241C22×C4C2×C4×D5911yescharacteristic
242C22×C4D4×D5912yes—
243C22×C4#1174568712yes—
244C2×C8—591no—
245C2×C8—591no—
246C2×C8—592no—
247C2×C8—1184no—
248C2×C8—1182no—
249C2×C8—1181no—
250C2×C8—1181no—
251C2×C8—1184no—
252C2×C8—2362no—
253C2×C8—2361no—
254C2×C8—2362no—
255C2×C8—2362no—
256C2×C8—2362no—
257C2×C8—2362no—
258C2×C8—2361no—
259C2×C8—2361no—
260C4⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C42—21no—
266C42—21no—
267C42—42no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23#1427363812yescomplemented
276C23C2×C4×Dic5911yescharacteristic
277C23D4×Dic5912yes—
278C23C2×#1174566412yes—
279C23#1427373412yes—
280D4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4#1427363812yescomplemented
300C2×C4C22⋊C4×D5911yescharacteristic
301C2×C4#1427368311yescharacteristic
302C2×C4(C22⋊D59)⋊C412yes—
303C8—592no—
304C8—1182no—
305C8—1181no—
306C8—2364no—
307C8—2361no—
308C8—2361no—
309C22—22no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22C22⋊C4×Dic5911yescharacteristic
321C22C2×#1427363811yescharacteristic
322C22#1683735811yescharacteristic
323C22#1683747414yes—
324C4—22no—
325C4—22no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—84no—
330C4#99×D5911yescharacteristic
331C4#1683737411yescharacteristic
332C2—24no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2142387411yescharacteristic
337C2#99×Dic5911yescharacteristic
338C2#2142389811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC23×C59C2×C42
FrattiniC23×C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C4×C22⋊C4
prime radicalC22×D4×C59C2×C4
cosocleC23×C4×C59C23
hypercenterC4×#99D59
hypocenterC59#8160
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #21423874C2semidirect
4. #21421944C2semidirect
6. C23×C59⋊C8C4semidirect
10. C2×C4⋊(C59⋊C8)C4semidirect
11. C2×C22⋊(C59⋊C8)C4semidirect
15. C2×C4×C59⋊C8C4semidirect
17. #16837128C4semidirect
18. #16837126C4semidirect
105. C23×C59#97semidirect
129. C2×C4×C59#97semidirect
138. C59⋊C8#99semidirect
192. C59#8160semidirect
275. C23#14273638semidirect
299. C2×C4#14273638semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C59⋊C82
#214238742
#214224142
#214219442
C4×#99×C592
#816059