Finite Group Databasebeta

← #21432122

Subgroups — #21432122

Order 7552.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#14273703C211yesmaximal, characteristic, complemented
2C2×#14273700C211yesmaximal, characteristic
3#16837339C211yesmaximal, characteristic, complemented
4#16837338C211yesmaximal, characteristic, complemented
5#16837336C212yesmaximal, complemented
6#16837320C212yesmaximal, complemented
7#16837318C212yesmaximal, complemented
8#16837317C212yesmaximal, complemented
9#16837313C211yesmaximal, characteristic, complemented
10#16837264C211yesmaximal, characteristic, complemented
11C59×#393C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59C2211yescharacteristic
14C22×C22⋊D59C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16#14273703—24no—
17#14273700—24no—
18C2×C4⋊Dic59C2211yescharacteristic
19C2×C4⋊Dic59C2212yes—
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2212yes—
22C2×Dic59⋊C4C2212yes—
23C2×C22⋊Dic59C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C59⋊#95C2211yescharacteristic
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×Q8×C59C2211yescharacteristic
33C2×C22⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2212yes—
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C24×C59C2311yescharacteristic
38C23×D59C2311yescharacteristic
39C23×D59C2311yescharacteristic
40C2×Q8×C59—22no—
41C2×Q8×C59—42no—
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2312yes—
45C22×C4×C59C2311yescharacteristic
46C2×C22⋊D59—22no—
47C2×C22⋊D59—22no—
48C2×C22⋊D59—42no—
49C2×C22⋊D59—42no—
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2312yes—
53C22×Dic59C2311yescharacteristic
54C2×D236—22no—
55C2×D236—42no—
56C2×C4×D59—42no—
57C2×C4×D59D414yes—
58C22⋊C4×C59—22no—
59C22⋊C4×C59—42no—
60C22⋊Dic59—42no—
61(C2×D59)⋊C4—24no—
62(C2×D59)⋊C4—22no—
63(C2×D59)⋊C4—42no—
64(C2×D59)⋊C4—42no—
65(C2×D59)⋊C4—42no—
66C4⋊Dic59—22no—
67C4⋊Dic59—42no—
68Dic59⋊C4—22no—
69Dic59⋊C4—24no—
70Dic59⋊C4—42no—
71C23×C59—21no—
72C23×C59—22no—
73C23×C59—22no—
74C23×C59—21no—
75C23×C59—21no—
76C23×C59C2411yescharacteristic
77C22×D59—22no—
78C22×D59—22no—
79C22×D59—21no—
80C22×D59—21no—
81C22×D59—21no—
82C22×D59—22no—
83C22×D59—22no—
84C22×D59—21no—
85C22×D59—21no—
86C22×D59C2×D412yes—
87Q8×C59—44no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59C2×D412yes—
98C22⋊D59—44no—
99C22⋊D59—44no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59C2×D412yes—
110D236—44no—
111C4×D59—44no—
112C22×C59—41no—
113C22×C59—41no—
114C22×C59—41no—
115C22×C59—42no—
116C22×C59—42no—
117C22×C59C22×D411yescharacteristic
118C22×C59H(5,𝔽2)12yes—
119C22×C59H(5,𝔽2)12yes—
120C22×C592−1+411yescharacteristic
121C22×C592−1+411yescharacteristic
122C2×D59—22no—
123C2×D59—22no—
124C2×D59—22no—
125C2×D59—42no—
126C2×D59—42no—
127C2×D59—41no—
128C2×D59—41no—
129C2×D59—41no—
130C2×D59—42no—
131C2×D59—42no—
132C4×C59—22no—
133C4×C59—42no—
134C4×C59—41no—
135C4×C59—42no—
136Dic59—22no—
137Dic59—42no—
138Dic59—41no—
139Dic59—42no—
140#2816—591nomaximal, Sylow
141C2×C59—81no—
142C2×C59#53411yescharacteristic
143C2×C59#53511yescharacteristic
144C2×C59#53611yescharacteristic
145C2×C59#56212yes—
146C2×C59#56212yes—
147D59—42no—
148D59—81no—
149C2×C22⋊Q8—591no—
150C2×#121—591no—
151#399—591no—
152#399—591no—
153#398—592no—
154#394—592no—
155#394—592no—
156#394—592no—
157#393—591no—
158#393D5911yescharacteristic, complemented
159#381—591no—
160C59#281611yescharacteristic, complemented, Sylow
161C22×Q8C2×D5911yescharacteristic
162C22×D4—591no—
163C22×D4—591no—
164C22×D4—591no—
165C23×C4—591no—
166C22⋊Q8—1184no—
167#121—1184no—
168C2×C4⋊C4—591no—
169C2×C4⋊C4—592no—
170C2×C4⋊C4—591no—
171C2×C4⋊C4—592no—
172C2×C4⋊C4—592no—
173C2×C22⋊C4—591no—
174C2×C22⋊C4—592no—
175C2×C22⋊C4—592no—
176C2×C22⋊C4—592no—
177C2×C22⋊C4—592no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4C2×D5911yescharacteristic
180C2×C22⋊C4C2×D5912yes—
181#95—591no—
182#95—592no—
183#95—592no—
184#95C2×D5911yescharacteristic
185#95C2×D5912yes—
186C24—591no—
187C24—591no—
188C24C22×D5911yescharacteristic
189C2×Q8—22no—
190C2×Q8—42no—
191C2×D4—1182no—
192C2×D4—1182no—
193C2×D4—1182no—
194C2×D4—2362no—
195C2×D4—2362no—
196C2×D4—2362no—
197C22×C4—591no—
198C22×C4—594no—
199C22×C4—592no—
200C22×C4—591no—
201C22×C4—592no—
202C22×C4—2362no—
203C22×C4C22×D5912yes—
204C22×C4C22×D5912yes—
205C22×C4C22×D5911yescharacteristic
206C22×C4C22×D5911yescharacteristic
207C4⋊C4—1182no—
208C4⋊C4—1184no—
209C4⋊C4—1182no—
210C4⋊C4—2362no—
211C4⋊C4—2362no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—1184no—
215C22⋊C4—1182no—
216C22⋊C4—2362no—
217C22⋊C4—2362no—
218C22⋊C4—2362no—
219C22⋊C4—2362no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—592no—
226C23—1182no—
227C23—1182no—
228C23—1181no—
229C23—1181no—
230C23—1181no—
231C23—1182no—
232C23—1182no—
233C23—1181no—
234C23—1181no—
235C23C23×D5911yescharacteristic
236Q8—44no—
237D4—2364no—
238D4—2364no—
239D4—2364no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—592no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1181no—
253C2×C4—1182no—
254C2×C4—1182no—
255C2×C4—1182no—
256C2×C4—1182no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—2364no—
260C2×C4D4×D5912yes—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—1182no—
267C22—1182no—
268C22—1182no—
269C22—2362no—
270C22—2362no—
271C22—2361no—
272C22—2361no—
273C22—2361no—
274C22—2362no—
275C22—2362no—
276C22C2×D4×D5911yescharacteristic
277C22#1427377212yes—
278C22#1427377312yes—
279C22#1427377411yescharacteristic
280C22#1427377511yescharacteristic
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—1182no—
286C4—2362no—
287C4—2361no—
288C4—2362no—
289C2—81no—
290C2—4721no—
291C2—2362no—
292C2#1683796211yescharacteristic
293C2#1683797011yescharacteristic
294C2#1683797511yescharacteristic
295C2#1683814612yes—
296C2#1683814812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2816
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14273703C2semidirect
3. #16837339C2semidirect
4. #16837338C2semidirect
5. #16837336C2semidirect
6. #16837320C2semidirect
7. #16837318C2semidirect
8. #16837317C2semidirect
9. #16837313C2semidirect
10. #16837264C2semidirect
11. C59×#393C2semidirect
158. #393D59semidirect
160. C59#2816semidirect