Finite Group Databasebeta

← #21428397

Subgroups — #21428397

Order 7552.

624 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14273709C212yesmaximal
2C2×(C22⋊D59)⋊C4C211yesmaximal, characteristic
3C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
4#16837334C211yesmaximal, characteristic, complemented
5#16837333C211yesmaximal, characteristic, complemented
6#16837327C211yesmaximal, characteristic, complemented
7#16837299C212yesmaximal, complemented
8#16837264C212yesmaximal, complemented
9#16837261C211yesmaximal, characteristic, complemented
10#16837260C212yesmaximal, complemented
11C59×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C23×Dic59C2211yescharacteristic
14C22×C4×D59C2212yes—
15C22×C4×D59C2211yescharacteristic
16#14273709—28no—
17(C22⋊D59)⋊C4—24no—
18C2×C4⋊Dic59C2212yes—
19C2×C4⋊Dic59C2211yescharacteristic
20C2×Dic59⋊C4C2212yes—
21C2×Dic59⋊C4C2212yes—
22C2×Dic59⋊C4C2212yes—
23C2×Dic59⋊C4C2211yescharacteristic
24C4⋊C4×D59—24no—
25C2×C22⋊Dic59C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2211yescharacteristic
28C2×C4×Dic59C2211yescharacteristic
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2211yescharacteristic
32C59⋊#95C2212yes—
33C2×C4⋊C4×C59C2211yescharacteristic
34C2×C22⋊C4×C59C2212yes—
35C2×C22⋊C4×C59C2211yescharacteristic
36#95×C59C2211yescharacteristic
37#95×C59C2212yes—
38C24×C59C2311yescharacteristic
39C23×D59C2311yescharacteristic
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2312yes—
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C2×C22⋊D59—22no—
45C2×C22⋊D59—22no—
46C2×C22⋊D59D414yes—
47C22×Dic59—22no—
48C22×Dic59—24no—
49C22×Dic59C2311yescharacteristic
50C22×Dic59C2312yes—
51C22×Dic59C2312yes—
52C22×Dic59C2311yescharacteristic
53C22×Dic59C2311yescharacteristic
54C2×C4×D59—24no—
55C2×C4×D59—24no—
56C2×C4×D59—24no—
57C2×C4×D59—24no—
58C2×C4×D59D414yes—
59C4⋊C4×C59—22no—
60C22⋊C4×C59—24no—
61C22⋊C4×C59—22no—
62C22⋊Dic59—24no—
63(C2×D59)⋊C4—22no—
64(C2×D59)⋊C4—24no—
65C4⋊Dic59—22no—
66C4⋊Dic59—24no—
67Dic59⋊C4—22no—
68Dic59⋊C4—24no—
69Dic59⋊C4—24no—
70Dic59⋊C4—24no—
71C4×Dic59—22no—
72C23×C59—21no—
73C23×C59—22no—
74C23×C59—21no—
75C23×C59—22no—
76C23×C59C2×D412yes—
77C23×C59C2411yescharacteristic
78C22×D59—22no—
79C22×D59—22no—
80C22×D59C2×D412yes—
81C22×D59C2×D412yes—
82C22×D59#4112yes—
83C2×C4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59C2×D412yes—
93C22⋊D59—28no—
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—21no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—44no—
105C2×Dic59C2×D412yes—
106C2×Dic59C2×D412yes—
107C2×Dic59#4114yes—
108C4×D59—28no—
109C4×D59—48no—
110C22×C59—22no—
111C22×C59—24no—
112C22×C59—41no—
113C22×C59—41no—
114C22×C59—42no—
115C22×C59C22×D411yescharacteristic
116C22×C59C22×D411yescharacteristic
117C22×C59C2×#4111yescharacteristic
118C22×C59C2×#4112yes—
119C22×C592−1+412yes—
120C2×D59—22no—
121C2×D59—22no—
122C2×D59—24no—
123C2×D59—24no—
124C2×D59#12114yes—
125C4×C59—22no—
126C4×C59—42no—
127C4×C59—41no—
128C4×C59—42no—
129Dic59—22no—
130Dic59—42no—
131Dic59—41no—
132Dic59—42no—
133Dic59#12314yescomplemented
134#2486—591nomaximal, Sylow
135C2×C59—42no—
136C2×C59C2×#12111yescharacteristic
137C2×C59C2×#12311yescharacteristic
138C2×C59#53611yescharacteristic
139C2×C59#54712yes—
140C2×C59#55212yes—
141D59—24no—
142C2×#123—592no—
143C2×C4×D4—591no—
144C22×C4⋊C4—591no—
145#398—591no—
146#398—591no—
147#397—591no—
148#397D5911yescharacteristic, complemented
149#389—592no—
150#381—592no—
151#380—592no—
152#380—591no—
153C59#248611yescharacteristic, complemented, Sylow
154C22×D4—591no—
155C23×C4—591no—
156C23×C4—591no—
157C23×C4—592no—
158#123—1188no—
159C4×D4—1184no—
160C2×C4⋊C4—592no—
161C2×C4⋊C4—591no—
162C2×C4⋊C4—591no—
163C2×C4⋊C4—592no—
164C2×C4⋊C4—592no—
165C2×C4⋊C4—592no—
166C2×C4⋊C4—1184no—
167C2×C4⋊C4C2×D5911yescharacteristic
168C2×C22⋊C4—592no—
169C2×C22⋊C4—591no—
170C2×C22⋊C4—592no—
171C2×C22⋊C4C2×D5911yescharacteristic
172C2×C22⋊C4C2×D5912yes—
173C2×C42—591no—
174#95—592no—
175#95—592no—
176#95—591no—
177#95—592no—
178#95C2×D5911yescharacteristic
179#95C2×D5912yes—
180C24—591no—
181C24C22×D5911yescharacteristic
182C2×D4—594no—
183C2×D4—1182no—
184C2×D4—1182no—
185C22×C4—591no—
186C22×C4—591no—
187C22×C4—594no—
188C22×C4—591no—
189C22×C4—592no—
190C22×C4—592no—
191C22×C4—1184no—
192C22×C4—1184no—
193C22×C4—1184no—
194C22×C4—1184no—
195C22×C4—1182no—
196C22×C4—1184no—
197C22×C4C22×D5911yescharacteristic
198C22×C4C22×D5912yes—
199C22×C4C22×D5912yes—
200C22×C4C22×D5911yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—1184no—
203C4⋊C4—1184no—
204C4⋊C4—1184no—
205C4⋊C4—1184no—
206C4⋊C4—1182no—
207C4⋊C4—1182no—
208C22⋊C4—24no—
209C22⋊C4—22no—
210C22⋊C4—1182no—
211C22⋊C4—1184no—
212C22⋊C4—1184no—
213C42—1182no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—592no—
219C23—592no—
220C23—592no—
221C23—1182no—
222C23—1182no—
223C23D4×D5912yes—
224C23C23×D5911yescharacteristic
225D4—1188no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—592no—
236C2×C4—594no—
237C2×C4—592no—
238C2×C4—1188no—
239C2×C4—1181no—
240C2×C4—1181no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—1181no—
244C2×C4—1182no—
245C2×C4—1182no—
246C2×C4—1182no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—2368no—
250C2×C4—2364no—
251C2×C4D4×D5912yes—
252C22—22no—
253C22—24no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—594no—
258C22—1182no—
259C22—1182no—
260C22—1184no—
261C22—1184no—
262C22C2×D4×D5911yescharacteristic
263C22C2×D4×D5911yescharacteristic
264C22#41×D5911yescharacteristic
265C22#41×D5912yes—
266C22#1427377412yes—
267C4—22no—
268C4—42no—
269C4—41no—
270C4—42no—
271C4—594no—
272C4—1182no—
273C4—2362no—
274C4—2361no—
275C4—2362no—
276C2—42no—
277C2—1184no—
278C2#121×D5911yescharacteristic
279C2#123×D5911yescharacteristic
280C2#1683797211yescharacteristic
281C2#1683805012yes—
282C2#1683808812yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#397C2
hypercenter#397D59
hypocenterC59#2486
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D59C2semidirect
4. #16837334C2semidirect
5. #16837333C2semidirect
6. #16837327C2semidirect
7. #16837299C2semidirect
8. #16837264C2semidirect
9. #16837261C2semidirect
10. #16837260C2semidirect
11. C59×#397C2semidirect
133. Dic59#123semidirect
148. #397D59semidirect
153. C59#2486semidirect