Finite Group Databasebeta

← #21428375

Subgroups — #21428375

Order 7552.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×(C2×D59)⋊C4C211yesmaximal, characteristic
2C2×C22⋊C4×D59C211yesmaximal, characteristic
3#16837331C212yesmaximal, complemented
4#16837329C212yesmaximal, complemented
5#16837277C211yesmaximal, characteristic, complemented
6#16837276C212yesmaximal, complemented
7#16837274C211yesmaximal, characteristic, complemented
8#16837265C211yesmaximal, characteristic, complemented
9#16837264C212yesmaximal, complemented
10#16837257C211yesmaximal, characteristic
11C59×#381C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C23×Dic59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16C2×C4⋊Dic59C2212yescomplemented
17C2×Dic59⋊C4C2212yescomplemented
18C2×C22⋊Dic59C2211yescharacteristic
19C2×C22⋊Dic59C2211yescharacteristic
20C2×(C2×D59)⋊C4—24no—
21C2×(C2×D59)⋊C4C2212yes—
22C2×(C2×D59)⋊C4C2212yes—
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C22⋊C4×D59—24no—
28C59⋊#95C2212yescomplemented
29C59⋊#95C2212yescomplemented
30C59⋊#95C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C23×C4×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2211yescharacteristic
35#95×C59C2212yescomplemented
36#95×C59C2212yescomplemented
37C24×C59C2311yescharacteristic
38C23×D59—22no—
39C23×D59—22no—
40C23×D59—24no—
41C23×D59—42no—
42C23×D59D414yes—
43C23×D59C2311yescharacteristic
44C23×D59C2311yescharacteristic
45C22×C4×C59—22no—
46C22×C4×C59—22no—
47C22×C4×C59—22no—
48C22×C4×C59C2311yescharacteristic
49C22×C4×C59C2312yes—
50C22×C4×C59C2311yescharacteristic
51C22×C4×C59C2312yes—
52C22×Dic59—22no—
53C22×Dic59—42no—
54C22×Dic59C2311yescharacteristic
55C22×Dic59C2312yes—
56C22×Dic59C2312yes—
57C22×Dic59C2311yescharacteristic
58C2×C4×D59—22no—
59C2×C4×D59—22no—
60C2×C4×D59—24no—
61C2×C4×D59—42no—
62C22⋊C4×C59—22no—
63C22⋊C4×C59—22no—
64C22⋊Dic59—22no—
65C22⋊Dic59—22no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—24no—
69(C2×D59)⋊C4—24no—
70(C2×D59)⋊C4—24no—
71(C2×D59)⋊C4—42no—
72C4⋊Dic59—42no—
73Dic59⋊C4—42no—
74C23×C59—22no—
75C23×C59—22no—
76C23×C59C2×D412yes—
77C23×C59#4112yes—
78C23×C59#4112yes—
79C23×C59C2411yescharacteristic
80C22×D59—21no—
81C22×D59—21no—
82C22×D59—22no—
83C22×D59—28no—
84C22×D59—22no—
85C22×D59—21no—
86C22×D59—21no—
87C22×D59—22no—
88C22×D59—28no—
89C22×D59—44no—
90C22×D59—48no—
91C22×D59—48no—
92C22×D59C2×D412yes—
93C22×D59C2×D412yes—
94C22×D59#4114yes—
95C2×C4×C59—22no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—44no—
106C2×Dic59—22no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59—22no—
115C2×Dic59—22no—
116C2×Dic59—44no—
117C4×D59—44no—
118C4×D59—44no—
119C22×C59—22no—
120C22×C59—22no—
121C22×C59—24no—
122C22×C59—24no—
123C22×C59#12314yes—
124C22×C59C22×D411yescharacteristic
125C22×C59C2×#4112yes—
126C22×C59C2×#4111yescharacteristic
127C22×C59C2×#4111yescharacteristic
128C22×C59H(5,𝔽2)12yes—
129C2×D59—22no—
130C2×D59—22no—
131C2×D59—24no—
132C2×D59—24no—
133C2×D59—22no—
134C2×D59—22no—
135C2×D59—22no—
136C2×D59—416no—
137C2×D59—42no—
138C2×D59—42no—
139C2×D59#12314yes—
140C4×C59—41no—
141C4×C59—42no—
142C4×C59—41no—
143C4×C59—42no—
144Dic59—41no—
145Dic59—41no—
146Dic59—42no—
147Dic59—42no—
148#2483—591nomaximal, Sylow
149C2×C59—24no—
150C2×C59C2×#12311yescharacteristic
151C2×C59C2×#12311yescharacteristic
152C2×C59#53411yescharacteristic
153C2×C59#55112yes—
154C2×C59#55112yes—
155D59—24no—
156D59—42no—
157C22×C22⋊C4—591no—
158C22×C22⋊C4—591no—
159#397—592no—
160#397—592no—
161#385—591no—
162#385—591no—
163#385—592no—
164#381—591no—
165#381—592no—
166#381D5911yescharacteristic, complemented
167#379—591no—
168C59#248311yescharacteristic, complemented, Sylow
169C25—591no—
170C23×C4—591no—
171C23×C4—591no—
172C23×C4—591no—
173C23×C4C2×D5911yescharacteristic
174C2×C4⋊C4—592no—
175C2×C4⋊C4—592no—
176C2×C22⋊C4—591no—
177C2×C22⋊C4—591no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4—591no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—591no—
182C2×C22⋊C4—592no—
183C2×C22⋊C4—592no—
184C2×C22⋊C4—1184no—
185C2×C22⋊C4—1184no—
186C2×C22⋊C4C2×D5911yescharacteristic
187C2×C22⋊C4C2×D5911yescharacteristic
188#95—592no—
189#95—592no—
190#95—592no—
191#95—592no—
192#95C2×D5912yescomplemented
193#95C2×D5912yescomplemented
194C24—591no—
195C24—591no—
196C24—594no—
197C24—1184no—
198C24—1182no—
199C24—1182no—
200C24—2362no—
201C24C22×D5911yescharacteristic
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—591no—
206C22×C4—592no—
207C22×C4—591no—
208C22×C4—592no—
209C22×C4—1182no—
210C22×C4—1182no—
211C22×C4—1184no—
212C22×C4—1182no—
213C22×C4—2362no—
214C22×C4—2362no—
215C22×C4C22×D5912yes—
216C22×C4C22×D5911yescharacteristic
217C22×C4C22×D5912yes—
218C22×C4C22×D5911yescharacteristic
219C4⋊C4—2362no—
220C4⋊C4—2362no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—1182no—
224C22⋊C4—1182no—
225C22⋊C4—1184no—
226C22⋊C4—1184no—
227C22⋊C4—1184no—
228C22⋊C4—1182no—
229C22⋊C4—1182no—
230C22⋊C4—2362no—
231C23—22no—
232C23—22no—
233C23—592no—
234C23—594no—
235C23—592no—
236C23—1188no—
237C23—1181no—
238C23—1181no—
239C23—1182no—
240C23—1188no—
241C23—1182no—
242C23—1181no—
243C23—1181no—
244C23—1182no—
245C23—2364no—
246C23—2368no—
247C23—2368no—
248C23#1174568512yes—
249C23D4×D5912yes—
250C23#1174568712yes—
251C23C23×D5911yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1181no—
266C2×C4—1182no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—1182no—
271C2×C4—1182no—
272C2×C4—1182no—
273C2×C4—2364no—
274C2×C4—2364no—
275C2×C4—2364no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—594no—
281C22—1182no—
282C22—1182no—
283C22—1184no—
284C22—1184no—
285C22—1182no—
286C22—1182no—
287C22—1182no—
288C22—23616no—
289C22—2362no—
290C22—2362no—
291C22#1427370914yes—
292C22C2×D4×D5911yescharacteristic
293C22#41×D5912yes—
294C22C2×#1174568711yescharacteristic
295C22C2×#1174568511yescharacteristic
296C22#1427377312yes—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—42no—
301C4—2361no—
302C4—2361no—
303C4—2362no—
304C4—2362no—
305C2—24no—
306C2—1184no—
307C2—2362no—
308C2#123×D5911yescharacteristic
309C2C2×#1427370911yescharacteristic
310C2#1683796311yescharacteristic
311C2#1683807512yes—
312C2#1683807612yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#381C2
hypercenter#381D59
hypocenterC59#2483
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #16837331C2semidirect
4. #16837329C2semidirect
5. #16837277C2semidirect
6. #16837276C2semidirect
7. #16837274C2semidirect
8. #16837265C2semidirect
9. #16837264C2semidirect
11. C59×#381C2semidirect
16. C2×C4⋊Dic59C22semidirect
17. C2×Dic59⋊C4C22semidirect
28. C59⋊#95C22semidirect
29. C59⋊#95C22semidirect
30. C59⋊#95C22semidirect
31. C59⋊#95C22semidirect
35. #95×C59C22semidirect
36. #95×C59C22semidirect
166. #381D59semidirect
168. C59#2483semidirect
192. #95C2×D59semidirect
193. #95C2×D59semidirect