Finite Group Databasebeta

← #21427426

Subgroups — #21427426

Order 7552.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#11745687C211yesmaximal, characteristic
2C2×#14273709C212yesmaximal, complemented
3C2×C4⋊C4×D59C211yesmaximal, characteristic, complemented
4#16837323C211yesmaximal, characteristic, complemented
5#16837322C211yesmaximal, characteristic, complemented
6#16837316C211yesmaximal, characteristic, complemented
7#16837278C212yesmaximal, complemented
8#16837277C212yesmaximal, complemented
9#16837274C212yesmaximal, complemented
10#16837264C211yesmaximal, characteristic, complemented
11C59×#394C211yesmaximal, characteristic, complemented
12C2×#11745687—28no—
13C2×#11745687—24no—
14C22×Dic118C2211yescharacteristic
15C22×C22⋊D59C2212yes—
16C23×Dic59C2212yes—
17C22×C4×D59C2211yescharacteristic
18C22×C4×D59C2211yescharacteristic
19C22×C4×D59C2211yescharacteristic
20#14273709—28no—
21C2×C4⋊Dic59C2212yes—
22C2×C4⋊Dic59C2211yescharacteristic
23C2×Dic59⋊C4C2212yes—
24C2×Dic59⋊C4C2211yescharacteristic
25C2×Dic59⋊C4C2211yescharacteristic
26C4⋊C4×D59—24no—
27C2×C22⋊Dic59C2212yes—
28C2×C22⋊Dic59C2212yes—
29C2×(C2×D59)⋊C4C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2211yescharacteristic
32C59⋊#95C2212yes—
33C59⋊#95C2212yes—
34C22×D4×C59C2211yescharacteristic
35C2×C4⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2212yes—
37C2×C22⋊C4×C59C2212yes—
38#95×C59C2211yescharacteristic
39C24×C59C2312yes—
40C23×D59C2311yescharacteristic
41C2×D4×C59—24no—
42C2×D4×C59—22no—
43C22×C4×C59C2312yes—
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C2×C22⋊D59—24no—
48C2×C22⋊D59—24no—
49C2×C22⋊D59D418yes—
50C22×Dic59—24no—
51C22×Dic59—24no—
52C22×Dic59—24no—
53C22×Dic59C2312yes—
54C22×Dic59C2311yescharacteristic
55C22×Dic59C2311yescharacteristic
56C22×Dic59C2312yes—
57C22×Dic59C2311yescharacteristic
58#11745687—232no—
59C2×C4×D59—22no—
60C2×C4×D59—24no—
61C2×C4×D59—22no—
62C2×C4×D59—22no—
63C2×C4×D59—22no—
64C2×C4×D59—24no—
65C2×C4×D59D414yes—
66C2×Dic118—22no—
67C2×Dic118—24no—
68C4⋊C4×C59—22no—
69C22⋊C4×C59—24no—
70C22⋊C4×C59—24no—
71C22⋊Dic59—24no—
72C22⋊Dic59—24no—
73(C2×D59)⋊C4—24no—
74C4⋊Dic59—22no—
75C4⋊Dic59—24no—
76Dic59⋊C4—22no—
77Dic59⋊C4—22no—
78Dic59⋊C4—24no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—22no—
84C23×C59—22no—
85C23×C59C2×D414yes—
86C23×C59C2411yescharacteristic
87C22×D59—21no—
88C22×D59—21no—
89C22×D59—22no—
90C22×D59C2×D414yes—
91C22×D59C2×D412yes—
92D4×C59—28no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59C2×D412yes—
102C2×C4×C59#4112yescomplemented
103C22⋊D59—216no—
104C2×Dic59—216no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—21no—
110C2×Dic59—21no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59—22no—
115C2×Dic59C2×D414yes—
116C2×Dic59C2×D412yes—
117C2×Dic59#4112yescomplemented
118C4×D59—28no—
119C4×D59—28no—
120C4×D59—44no—
121Dic118—28no—
122C22×C59—24no—
123C22×C59—24no—
124C22×C59—24no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59C22×D411yescharacteristic
130C22×C59C22×D412yes—
131C22×C59C2×#4111yescharacteristic
132C22×C59C2×#4111yescharacteristic
133C22×C592−1+412yes—
134C2×D59—22no—
135C2×D59—22no—
136C2×D59—24no—
137C2×D59—24no—
138C2×D59#12014yes—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—41no—
142C4×C59—42no—
143Dic59—24no—
144Dic59—22no—
145Dic59—22no—
146Dic59—41no—
147Dic59—42no—
148#2406—591nomaximal, Sylow
149C2×C59—44no—
150C2×C59C2×#12011yescharacteristic
151C2×C59#52511yescharacteristic
152C2×C59#53611yescharacteristic
153C2×C59#54712yes—
154C2×C59#54712yes—
155D59—24no—
156C22×#41—591no—
157C2×#123—592no—
158C22×C4⋊C4—591no—
159#395—591no—
160#395—591no—
161#394—591no—
162#394D5911yescharacteristic, complemented
163#386—592no—
164#385—592no—
165#385—592no—
166#381—591no—
167C59#240611yescharacteristic, complemented, Sylow
168C2×#41—1184no—
169C2×#41—1188no—
170C22×Q8—591no—
171C22×D4—592no—
172C22×D4C2×D5911yescharacteristic
173C23×C4—591no—
174C23×C4—592no—
175C23×C4—591no—
176C23×C4—591no—
177#123—1188no—
178C2×C4⋊C4—591no—
179C2×C4⋊C4—591no—
180C2×C4⋊C4—591no—
181C2×C4⋊C4—592no—
182C2×C4⋊C4—592no—
183C2×C4⋊C4—1184no—
184C2×C4⋊C4C2×D5911yescharacteristic
185C2×C22⋊C4—592no—
186C2×C22⋊C4—592no—
187C2×C22⋊C4—592no—
188C2×C22⋊C4C2×D5912yes—
189C2×C22⋊C4C2×D5912yes—
190#95—592no—
191#95—591no—
192#95—592no—
193#95—592no—
194#95C2×D5911yescharacteristic
195C24—591no—
196C24C22×D5912yes—
197#41—11832no—
198C2×Q8—1184no—
199C2×Q8—1182no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—598no—
203C2×D4—1184no—
204C2×D4—1184no—
205C22×C4—591no—
206C22×C4—591no—
207C22×C4—592no—
208C22×C4—594no—
209C22×C4—592no—
210C22×C4—591no—
211C22×C4—1182no—
212C22×C4—1182no—
213C22×C4—1184no—
214C22×C4—1182no—
215C22×C4—1184no—
216C22×C4—1184no—
217C22×C4—1184no—
218C22×C4—1184no—
219C22×C4—1182no—
220C22×C4C22×D5912yes—
221C22×C4C22×D5911yescharacteristic
222C22×C4C22×D5911yescharacteristic
223C22×C4C22×D5911yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—1182no—
226C4⋊C4—1182no—
227C4⋊C4—1184no—
228C4⋊C4—1184no—
229C4⋊C4—1182no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—1184no—
233C22⋊C4—1184no—
234C22⋊C4—1184no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—594no—
242C23—592no—
243C23—1181no—
244C23—1181no—
245C23—1182no—
246C23D4×D5914yes—
247C23C23×D5911yescharacteristic
248Q8—1188no—
249D4—28no—
250D4—11816no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—592no—
260C2×C4—594no—
261C2×C4—592no—
262C2×C4—1188no—
263C2×C4—1188no—
264C2×C4—11816no—
265C2×C4—1181no—
266C2×C4—1182no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1181no—
270C2×C4—1182no—
271C2×C4—1182no—
272C2×C4—1181no—
273C2×C4—1182no—
274C2×C4—1182no—
275C2×C4—2364no—
276C2×C4D4×D5912yes—
277C2×C4#1174568712yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—594no—
286C22—1182no—
287C22—1182no—
288C22—1184no—
289C22—1184no—
290C22C2×D4×D5911yescharacteristic
291C22C2×D4×D5912yes—
292C22#41×D5911yescharacteristic
293C22C2×#1174568711yescharacteristic
294C22#1427377412yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—1182no—
300C4—1182no—
301C4—1184no—
302C4—2362no—
303C4—2361no—
304C2—44no—
305C2—1184no—
306C2#120×D5911yescharacteristic
307C2#1683791211yescharacteristic
308C2#1683797211yescharacteristic
309C2#1683805012yes—
310C2#1683805212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#394C2
hypercenter#394D59
hypocenterC59#2406
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14273709C2semidirect
3. C2×C4⋊C4×D59C2semidirect
4. #16837323C2semidirect
5. #16837322C2semidirect
6. #16837316C2semidirect
7. #16837278C2semidirect
8. #16837277C2semidirect
9. #16837274C2semidirect
10. #16837264C2semidirect
11. C59×#394C2semidirect
102. C2×C4×C59#41semidirect
117. C2×Dic59#41semidirect
162. #394D59semidirect
167. C59#2406semidirect
277. C2×C4#11745687semidirect