Finite Group Databasebeta

← #21434041

Subgroups — #21434041

Order 7552.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
2C2×#14273685C211yesmaximal, characteristic
3#16837745C212yesmaximal, complemented
4#16837741C212yesmaximal, complemented
5#16837737C211yesmaximal, characteristic, complemented
6#16837733C211yesmaximal, characteristic, complemented
7#16837730C211yesmaximal, characteristic, complemented
8#16837443C212yesmaximal, complemented
9#16837409C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic
11C59×#484C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C2×#11745659C2212yes—
15#14273707C2211yescharacteristic
16#14273706C2211yescharacteristic
17#14273705—21no—
18#14273704—21no—
19#14273703C2211yescharacteristic
20#14273685—21no—
21#14273685—21no—
22#14273685C2212yes—
23#14273685C2212yes—
24C2×C4⋊Dic59C2211yescharacteristic
25C4⋊C4×D59—21no—
26C4⋊C4×D59C2211yescharacteristic
27C2×(C2×D59)⋊C4—21no—
28C22⋊C4×D59—21no—
29C2×C4×Dic59—21no—
30#14273660C2212yescomplemented
31#14273659C2212yescomplemented
32#14273658C2212yescomplemented
33#14273652C2212yescomplemented
34#14273651C2212yescomplemented
35#14273650C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yes—
38C22⋊Q8×C59C2211yescharacteristic
39C2×C4⋊C4×C59C2211yescharacteristic
40#107×C59C2212yescomplemented
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59D411yescharacteristic
44C2×Q8×C59C2311yescharacteristic
45C22×C4×C59—21no—
46C22×C4×C59C2311yescharacteristic
47C22×Dic59—21no—
48C22×Dic59D411yescharacteristic
49Q8×D59—24no—
50C2×C4×D59—24no—
51C2×C4×D59—21no—
52C2×C4×D59—21no—
53C2×C4×D59D411yescharacteristic
54C2×C4×D59D411yescharacteristic
55C2×C4×D59C2311yescharacteristic
56C2×C4×D59C2311yescharacteristic
57C2×Dic118—21no—
58C2×Dic118C2311yescharacteristic
59C2×C8×C59C2312yes—
60C4⋊C4×C59—21no—
61C4⋊C4×C59—21no—
62C4⋊C4×C59C2312yes—
63C4⋊C4×C59C2311yescharacteristic
64C22⋊C4×C59—21no—
65C22⋊Dic59—21no—
66(C2×D59)⋊C4—21no—
67(C2×D59)⋊C4—21no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70C4⋊Dic59—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59C2312yes—
73C4⋊Dic59C2311yescharacteristic
74Dic59⋊C4—21no—
75Dic59⋊C4—21no—
76Dic59⋊C4—21no—
77C4×Dic59—22no—
78C4×Dic59—22no—
79C2×C59⋊C8C2312yes—
80#11745659—24no—
81C23×C59C2×D411yescharacteristic
82C22×D59—22no—
83C22×D59—22no—
84C22×D59—41no—
85C22×D59C2×D411yescharacteristic
86C22×D59C2×D411yescharacteristic
87Q8×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—42no—
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2411yescharacteristic
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—21no—
98C2×Dic59—21no—
99C2×Dic59—42no—
100C2×Dic59C2×D411yescharacteristic
101C2×Dic59C2×D411yescharacteristic
102C4×D59—22no—
103C4×D59—22no—
104C4×D59—22no—
105C4×D59—42no—
106C4×D59—42no—
107C4×D59#4114yes—
108Dic118—22no—
109Dic118—42no—
110C8×C59—22no—
111C59⋊C8—22no—
112C22×C59—21no—
113C22×C59—21no—
114C22×C59#13712yes—
115C22×C59C22×D411yescharacteristic
116C2×D59—21no—
117C2×D59—21no—
118C2×D59—21no—
119C2×D59—21no—
120C2×D59—21no—
121C2×D59—44no—
122C2×D59#12312yes—
123C4×C59—21no—
124C4×C59—41no—
125C4×C59—41no—
126C4×C59—42no—
127C4×C59C2×#4112yes—
128Dic59—21no—
129Dic59—41no—
130Dic59—42no—
131Dic59—41no—
132Dic59#12312yes—
133#3095—591nomaximal, Sylow
134C2×C59—22no—
135C2×C59C2×#12311yescharacteristic
136C2×C59C2×#13711yescharacteristic
137C2×C59#57511yescharacteristic
138D59—22no—
139D59—41no—
140C2×C22⋊Q8—591no—
141C2×#117—591no—
142#485—592no—
143#485—592no—
144#484—591no—
145#484D5911yescharacteristic, complemented
146#483—591no—
147#483—591no—
148#428—592no—
149#419—592no—
150#407—591no—
151C59#309511yescharacteristic, complemented, Sylow
152C22×Q8—591no—
153C23×C4—591no—
154C2×M16—592no—
155C22⋊Q8—591no—
156C22⋊Q8—591no—
157C22⋊Q8—591no—
158C22⋊Q8—1181no—
159C22⋊Q8—1181no—
160C22⋊Q8C2×D5911yescharacteristic
161#117—592no—
162#117—592no—
163#117—1181no—
164#117—1181no—
165C2×C4⋊C4—591no—
166C2×C4⋊C4—591no—
167C2×C4⋊C4—1181no—
168C2×C4⋊C4C2×D5911yescharacteristic
169C2×C22⋊C4—1181no—
170C2×C22⋊C4—1181no—
171C2×C42—1181no—
172#107—592no—
173#107C2×D5912yescomplemented
174#106—592no—
175#106—592no—
176#103—592no—
177#103—592no—
178#103—592no—
179#103C2×D5912yescomplemented
180C22⋊C8—591no—
181C22⋊C8—592no—
182C22⋊C8C2×D5911yescharacteristic
183C24—591no—
184C2×Q8—591no—
185C2×Q8—1181no—
186C2×Q8—1184no—
187C2×Q8C22×D5911yescharacteristic
188C22×C4—21no—
189C22×C4—591no—
190C22×C4—591no—
191C22×C4—591no—
192C22×C4—591no—
193C22×C4—591no—
194C22×C4—1184no—
195C22×C4—1181no—
196C22×C4—1181no—
197C22×C4—1181no—
198C22×C4C22×D5911yescharacteristic
199M16—1184no—
200C2×C8—592no—
201C2×C8C22×D5912yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—592no—
205C4⋊C4—591no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C4⋊C4—1181no—
211C4⋊C4C22×D5912yes—
212C4⋊C4C22×D5911yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—1182no—
215C22⋊C4—1182no—
216C22⋊C4—1181no—
217C22⋊C4—1181no—
218C22⋊C4—1181no—
219C42—1182no—
220C42—1182no—
221C23—591no—
222C23—591no—
223C23—1182no—
224C23—1182no—
225C23—2361no—
226C23D4×D5911yescharacteristic
227Q8—22no—
228Q8—1182no—
229Q8—2362no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—591no—
236C2×C4—594no—
237C2×C4—591no—
238C2×C4—1182no—
239C2×C4—1182no—
240C2×C4—1182no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—1181no—
244C2×C4—1181no—
245C2×C4—2362no—
246C2×C4—2362no—
247C2×C4—2362no—
248C2×C4D4×D5911yescharacteristic
249C2×C4C23×D5911yescharacteristic
250C8—22no—
251C8—1182no—
252C22—21no—
253C22—21no—
254C22—592no—
255C22—1181no—
256C22—1181no—
257C22—1181no—
258C22—1181no—
259C22—1181no—
260C22—2364no—
261C22#1427375912yes—
262C22C2×D4×D5911yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—592no—
268C4—1181no—
269C4—2361no—
270C4—2362no—
271C4—2361no—
272C4#41×D5912yes—
273C2—22no—
274C2—1182no—
275C2—2361no—
276C2#123×D5911yescharacteristic
277C2C2×#1427375911yescharacteristic
278C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#484C2
socleC22×C59C22×D4
hypercenter#484D59
hypocenterC59#3095
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #16837745C2semidirect
4. #16837741C2semidirect
5. #16837737C2semidirect
6. #16837733C2semidirect
7. #16837730C2semidirect
8. #16837443C2semidirect
9. #16837409C2semidirect
11. C59×#484C2semidirect
30. #14273660C22semidirect
31. #14273659C22semidirect
32. #14273658C22semidirect
33. #14273652C22semidirect
34. #14273651C22semidirect
35. #14273650C22semidirect
40. #107×C59C22semidirect
41. #103×C59C22semidirect
145. #484D59semidirect
151. C59#3095semidirect
173. #107C2×D59semidirect
179. #103C2×D59semidirect