Finite Group Databasebeta

← #21432143

Subgroups — #21432143

Order 7552.

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

normal subgroups only

This group has 580 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×#14273714C211yesmaximal, characteristic, complemented
2C2×#14273713C211yesmaximal, characteristic, complemented
3#16837317C212yesmaximal, complemented
4#16837316C212yesmaximal, complemented
5#16837315C211yesmaximal, characteristic, complemented
6#16837314C211yesmaximal, characteristic, complemented
7#16837314C212yesmaximal, complemented
8#16837310C211yesmaximal, characteristic, complemented
9#16837310C212yesmaximal, complemented
10#16837266C211yesmaximal, characteristic, complemented
11C59×#392C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×Dic118C2211yescharacteristic
14C22×C22⋊D59C2211yescharacteristic
15C22×C22⋊D59C2212yes—
16#14273714—24no—
17#14273713—24no—
18C2×Dic59⋊C4C2212yes—
19C2×C22⋊Dic59C2211yescharacteristic
20C2×C22⋊Dic59C2212yes—
21C2×C22⋊Dic59C2212yes—
22C2×C22⋊Dic59C2212yes—
23C2×C22⋊Dic59C2212yes—
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×C4×Dic59C2211yescharacteristic
28C59⋊#95C2211yescharacteristic
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×D4×C59C2211yescharacteristic
33C22×D4×C59C2212yes—
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic
37C24×C59C2312yes—
38C24×C59C2311yescharacteristic
39C23×D59C2311yescharacteristic
40C2×D4×C59—22no—
41C2×D4×C59—42no—
42C2×D4×C59—42no—
43C2×D4×C59—42no—
44C2×D4×C59—42no—
45C22×C4×C59C2312yes—
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2311yescharacteristic
48C2×C22⋊D59—22no—
49C2×C22⋊D59—42no—
50C2×C22⋊D59—42no—
51C2×C22⋊D59—42no—
52C2×C22⋊D59—42no—
53C22×Dic59C2312yes—
54C22×Dic59C2311yescharacteristic
55C22×Dic59C2312yes—
56C22×Dic59C2311yescharacteristic
57C22×Dic59C2311yescharacteristic
58C2×Dic118—22no—
59C2×Dic118—22no—
60C2×Dic118—42no—
61C2×Dic118—42no—
62C22⋊C4×C59—22no—
63C22⋊C4×C59—42no—
64C22⋊Dic59—22no—
65C22⋊Dic59—24no—
66C22⋊Dic59—24no—
67C22⋊Dic59—42no—
68C22⋊Dic59—42no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—42no—
72Dic59⋊C4—42no—
73C4×Dic59D414yescomplemented
74C23×C59—22no—
75C23×C59—21no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—21no—
84C23×C59—22no—
85C23×C59—21no—
86C23×C59C2411yescharacteristic
87C22×D59—21no—
88C22×D59—22no—
89C22×D59—22no—
90C22×D59—21no—
91C22×D59—21no—
92D4×C59—44no—
93D4×C59—48no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59C2×D412yes—
101C22⋊D59—44no—
102C22⋊D59—48no—
103C2×Dic59—21no—
104C2×Dic59—22no—
105C2×Dic59—21no—
106C2×Dic59—21no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59C2×D412yes—
115C2×Dic59C2×D412yes—
116Dic118—44no—
117Dic118—44no—
118C22×C59—41no—
119C22×C59—42no—
120C22×C59—42no—
121C22×C59—41no—
122C22×C59—42no—
123C22×C59—41no—
124C22×C59—42no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59C22×D411yescharacteristic
131C22×C59H(5,𝔽2)12yes—
132C22×C59H(5,𝔽2)12yes—
133C22×C59H(5,𝔽2)11yescharacteristic
134C22×C592−1+411yescharacteristic
135C2×D59—41no—
136C2×D59—41no—
137C2×D59—41no—
138C2×D59—42no—
139C2×D59—42no—
140C4×C59—22no—
141C4×C59—41no—
142C4×C59—42no—
143Dic59—22no—
144Dic59—22no—
145Dic59—41no—
146Dic59—42no—
147Dic59—42no—
148#2819—591nomaximal, Sylow
149C2×C59—81no—
150C2×C59—82no—
151C2×C59#53511yescharacteristic
152C2×C59#53511yescharacteristic
153C2×C59#53711yescharacteristic
154C2×C59H(3,𝔽4)12yes—
155C2×C59#56212yes—
156D59—81no—
157C2×#124—591no—
158C2×#124—591no—
159#394—592no—
160#394—592no—
161#393—592no—
162#393—591no—
163#393—591no—
164#392—592no—
165#392—591no—
166#392D5911yescharacteristic, complemented
167#382—591no—
168C59#281911yescharacteristic, complemented, Sylow
169C22×Q8—591no—
170C22×Q8—591no—
171C22×D4—591no—
172C22×D4—592no—
173C22×D4C2×D5911yescharacteristic
174C22×D4C2×D5912yes—
175#124—1184no—
176#124—1184no—
177C2×C4⋊C4—592no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4—591no—
181C2×C22⋊C4—591no—
182C2×C22⋊C4—591no—
183C2×C22⋊C4—592no—
184C2×C22⋊C4—592no—
185C2×C22⋊C4—592no—
186C2×C22⋊C4C2×D5911yescharacteristic
187C2×C22⋊C4C2×D5912yes—
188C2×C42—591no—
189#95—592no—
190#95—591no—
191#95—592no—
192#95—592no—
193#95C2×D5911yescharacteristic
194C24—591no—
195C24C22×D5911yescharacteristic
196C24C22×D5912yes—
197C2×Q8—1182no—
198C2×Q8—1182no—
199C2×Q8—2362no—
200C2×Q8—2362no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—1182no—
207C2×D4—2362no—
208C2×D4—2362no—
209C2×D4—2362no—
210C2×D4—2362no—
211C22×C4—591no—
212C22×C4—591no—
213C22×C4—592no—
214C22×C4—592no—
215C22×C4—591no—
216C22×C4C22×D5912yes—
217C22×C4C22×D5911yescharacteristic
218C22×C4C22×D5911yescharacteristic
219C4⋊C4—2362no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—1182no—
223C22⋊C4—1182no—
224C22⋊C4—1184no—
225C22⋊C4—1184no—
226C22⋊C4—1182no—
227C22⋊C4—2362no—
228C22⋊C4—2362no—
229C22⋊C4—2362no—
230C42—594no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—21no—
243C23—1181no—
244C23—1182no—
245C23—1181no—
246C23—1182no—
247C23—1181no—
248C23C23×D5911yescharacteristic
249Q8—2364no—
250Q8—2364no—
251D4—44no—
252D4—48no—
253D4—2364no—
254D4—2368no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—592no—
262C2×C4—592no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—1182no—
271C2×C4—1182no—
272C2×C4—1182no—
273C2×C4—1182no—
274C2×C4D4×D5912yes—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—2362no—
288C22—2362no—
289C22—2361no—
290C22—2361no—
291C22—2361no—
292C22C2×D4×D5911yescharacteristic
293C22#1427377211yescharacteristic
294C22#1427377212yes—
295C22#1427377212yes—
296C22#1427377411yescharacteristic
297C4—22no—
298C4—42no—
299C4—41no—
300C4—1182no—
301C4—1182no—
302C4—2362no—
303C4—2361no—
304C4—2362no—
305C2—81no—
306C2—82no—
307C2—4721no—
308C2#1683796511yescharacteristic
309C2#1683796711yescharacteristic
310C2#1683798111yescharacteristic
311C2#1683814512yes—
312C2#1683814712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#392C2
hypercenter#392D59
hypocenterC59#2819
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14273714C2semidirect
2. C2×#14273713C2semidirect
3. #16837317C2semidirect
4. #16837316C2semidirect
5. #16837315C2semidirect
6. #16837314C2semidirect
7. #16837314C2semidirect
8. #16837310C2semidirect
9. #16837310C2semidirect
10. #16837266C2semidirect
11. C59×#392C2semidirect
73. C4×Dic59D4semidirect
166. #392D59semidirect
168. C59#2819semidirect