Finite Group Databasebeta

← #21427749

Subgroups — #21427749

Order 7552.

852 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 161 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×Q8×D59C211yesmaximal, characteristic, complemented
2C2×#14273713C212yesmaximal, complemented
3C2×#14273700C211yesmaximal, characteristic
4#16837315C211yesmaximal, characteristic, complemented
5#16837314C211yesmaximal, characteristic, complemented
6#16837313C211yesmaximal, characteristic, complemented
7#16837306C212yesmaximal, complemented
8#16837280C212yesmaximal, complemented
9#16837278C212yesmaximal, complemented
10#16837249C211yesmaximal, characteristic, complemented
11C59×#393C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×Dic118C2212yes—
14C2×Q8×D59—24no—
15C2×Q8×D59—28no—
16C22×D236C2211yescharacteristic
17C22×C22⋊D59C2211yescharacteristic
18C22×C22⋊D59C2211yescharacteristic
19C22×C4×D59C2212yes—
20C22×C4×D59C2211yescharacteristic
21#14273713—28no—
22#14273700—24no—
23C2×Dic59⋊C4C2211yescharacteristic
24C2×C22⋊Dic59C2212yes—
25C2×(C2×D59)⋊C4C2211yescharacteristic
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C2×C4×Dic59C2212yes—
30C59⋊#95C2211yescharacteristic
31C59⋊#95C2212yes—
32C59⋊#95C2212yes—
33C59⋊#95C2212yes—
34C22×Q8×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2212yes—
37#95×C59C2211yescharacteristic
38#95×C59C2212yes—
39C24×C59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C23×D59C2311yescharacteristic
42C2×Q8×C59—22no—
43C2×Q8×C59—24no—
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2312yes—
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2312yes—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—24no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—24no—
52C22×Dic59C2311yescharacteristic
53C22×Dic59C2312yes—
54C22×Dic59C2312yes—
55C22×Dic59C2311yescharacteristic
56Q8×D59—232no—
57C2×D236—22no—
58C2×D236—24no—
59C2×C4×D59—24no—
60C2×C4×D59—24no—
61C2×C4×D59—24no—
62C2×C4×D59—24no—
63C2×C4×D59D414yes—
64C2×Dic118—22no—
65C2×Dic118—24no—
66C2×Dic118—24no—
67C2×Dic118—24no—
68C2×Dic118D418yescomplemented
69C22⋊C4×C59—22no—
70C22⋊C4×C59—24no—
71C22⋊Dic59—24no—
72(C2×D59)⋊C4—24no—
73(C2×D59)⋊C4—22no—
74(C2×D59)⋊C4—24no—
75(C2×D59)⋊C4—24no—
76Dic59⋊C4—22no—
77C4×Dic59—24no—
78C23×C59—22no—
79C23×C59—21no—
80C23×C59—22no—
81C23×C59—21no—
82C23×C59—21no—
83C23×C59C2411yescharacteristic
84C22×D59—22no—
85C22×D59—21no—
86C22×D59—21no—
87C22×D59—22no—
88C22×D59—21no—
89C22×D59—22no—
90C22×D59—21no—
91C22×D59—21no—
92C22×D59C2×D412yes—
93C22×D59#4114yes—
94Q8×C59—28no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59C2×D412yes—
104C2×C4×C59C2×D414yes—
105C22⋊D59—44no—
106C22⋊D59—44no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—21no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59C2×D414yes—
115C2×Dic59C2×D414yes—
116C2×Dic59C2×D412yes—
117D236—44no—
118C4×D59—216no—
119C4×D59—28no—
120Dic118—28no—
121Dic118—216no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—41no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59C22×D411yescharacteristic
128C22×C59C22×D412yes—
129C22×C59C2×#4112yes—
130C22×C592−1+411yescharacteristic
131C22×C592−1+411yescharacteristic
132C2×D59—22no—
133C2×D59—22no—
134C2×D59—24no—
135C2×D59—24no—
136C2×D59—41no—
137C2×D59—41no—
138C2×D59—41no—
139C2×D59—42no—
140C2×D59—42no—
141C2×D59#12414yescomplemented
142C4×C59—22no—
143C4×C59—24no—
144C4×C59—41no—
145C4×C59—42no—
146Dic59—24no—
147Dic59—41no—
148Dic59—42no—
149Dic59#12014yes—
150#2436—591nomaximal, Sylow
151C2×C59—81no—
152C2×C59C2×#12011yescharacteristic
153C2×C59C2×#12411yescharacteristic
154C2×C59#53611yescharacteristic
155C2×C59#54812yes—
156C2×C59#54812yes—
157D59—24no—
158D59—81no—
159C23×Q8—591no—
160C2×#124—592no—
161C2×#121—591no—
162#393—591no—
163#393—591no—
164#393—591no—
165#393D5911yescharacteristic, complemented
166#391—592no—
167#386—592no—
168#386—592no—
169C4×C22⋊C4—591no—
170C59#243611yescharacteristic, complemented, Sylow
171C22×Q8—592no—
172C22×Q8—591no—
173C22×Q8—1184no—
174C22×Q8—1188no—
175C22×Q8C2×D5911yescharacteristic
176C22×D4—591no—
177C22×D4—591no—
178C22×D4—591no—
179C23×C4—591no—
180C23×C4—592no—
181#124—1188no—
182#121—1184no—
183C2×C4⋊C4—591no—
184C2×C22⋊C4—592no—
185C2×C22⋊C4—591no—
186C2×C22⋊C4—592no—
187C2×C22⋊C4—592no—
188C2×C22⋊C4—592no—
189C2×C22⋊C4C2×D5911yescharacteristic
190C2×C22⋊C4C2×D5912yes—
191C2×C42—592no—
192#95—591no—
193#95—592no—
194#95—592no—
195#95—592no—
196#95C2×D5911yescharacteristic
197#95C2×D5912yes—
198C24—591no—
199C24—591no—
200C24C22×D5911yescharacteristic
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—598no—
204C2×Q8—1184no—
205C2×Q8—1184no—
206C2×Q8—1184no—
207C2×Q8—11832no—
208C2×Q8—1182no—
209C2×D4—1182no—
210C2×D4—1184no—
211C2×D4—1182no—
212C2×D4—1184no—
213C2×D4—1182no—
214C2×D4—1184no—
215C22×C4—591no—
216C22×C4—592no—
217C22×C4—594no—
218C22×C4—591no—
219C22×C4—592no—
220C22×C4—1184no—
221C22×C4—1184no—
222C22×C4—1184no—
223C22×C4—1184no—
224C22×C4C22×D5911yescharacteristic
225C22×C4C22×D5912yes—
226C22×C4C22×D5911yescharacteristic
227C22×C4C22×D5912yes—
228C4⋊C4—1182no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—1184no—
232C22⋊C4—1184no—
233C22⋊C4—1184no—
234C22⋊C4—1182no—
235C22⋊C4—1184no—
236C42—1184no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—592no—
243C23—594no—
244C23—1182no—
245C23—1181no—
246C23—1181no—
247C23—1182no—
248C23—1181no—
249C23—1182no—
250C23—1181no—
251C23—1181no—
252C23C23×D5911yescharacteristic
253Q8—28no—
254Q8—11816no—
255Q8—1188no—
256D4—2364no—
257D4—2364no—
258D4—2364no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—594no—
268C2×C4—592no—
269C2×C4—594no—
270C2×C4—1188no—
271C2×C4—11816no—
272C2×C4—1182no—
273C2×C4—1182no—
274C2×C4—1182no—
275C2×C4—1181no—
276C2×C4—1182no—
277C2×C4—1182no—
278C2×C4—1182no—
279C2×C4D4×D5912yes—
280C2×C4D4×D5914yes—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—594no—
287C22—1182no—
288C22—1184no—
289C22—1184no—
290C22—1182no—
291C22—2361no—
292C22—2361no—
293C22—2362no—
294C22—2362no—
295C22—2361no—
296C22C2×D4×D5911yescharacteristic
297C22C2×D4×D5912yes—
298C22#41×D5912yes—
299C22#1427377411yescharacteristic
300C22#1427377511yescharacteristic
301C4—22no—
302C4—24no—
303C4—41no—
304C4—42no—
305C4—594no—
306C4—1184no—
307C4—2361no—
308C4—2362no—
309C2—81no—
310C2—1184no—
311C2—4721no—
312C2#120×D5911yescharacteristic
313C2#124×D5911yescharacteristic
314C2#1683797511yescharacteristic
315C2#1683806012yes—
316C2#1683806212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2436
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D59C2semidirect
2. C2×#14273713C2semidirect
4. #16837315C2semidirect
5. #16837314C2semidirect
6. #16837313C2semidirect
7. #16837306C2semidirect
8. #16837280C2semidirect
9. #16837278C2semidirect
10. #16837249C2semidirect
11. C59×#393C2semidirect
68. C2×Dic118D4semidirect
141. C2×D59#124semidirect
165. #393D59semidirect
170. C59#2436semidirect