Finite Group Databasebeta

← #21434918

Subgroups — #21434918

Order 7552.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14273703C211yesmaximal, characteristic
2C22⋊Q8×D59C211yesmaximal, characteristic
3#16837733C211yesmaximal, characteristic, complemented
4#16837731C212yesmaximal, complemented
5#16837730C211yesmaximal, characteristic, complemented
6#16837659C212yesmaximal, complemented
7#16837657C212yesmaximal, complemented
8#16837558C211yesmaximal, characteristic, complemented
9#16837556C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic
11C59×#451C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C2×#11745672C2212yescomplemented
15C2×#11745659C2212yes—
16#14273707C2211yescharacteristic
17#14273706C2211yescharacteristic
18#14273705—21no—
19#14273704—21no—
20#14273703—21no—
21#14273703—21no—
22#14273703C2211yescharacteristic
23#14273703C2212yes—
24#14273703C2212yes—
25C2×C4⋊Dic59C2211yescharacteristic
26C2×Dic59⋊C4—21no—
27C4⋊C4×D59—21no—
28C4⋊C4×D59C2211yescharacteristic
29C2×(C2×D59)⋊C4—21no—
30C22⋊C4×D59—21no—
31#14273659C2212yescomplemented
32#14273658C2212yescomplemented
33#14273652C2212yescomplemented
34#14273651C2212yescomplemented
35#14273651C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yes—
38C22×Q8×C59C2211yescharacteristic
39C2×Q16×C59C2212yescomplemented
40C22⋊Q8×C59C2211yescharacteristic
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59D411yescharacteristic
44C2×Q8×C59—21no—
45C2×Q8×C59—42no—
46C2×Q8×C59C2312yes—
47C2×Q8×C59C2311yescharacteristic
48C22×C4×C59—21no—
49C22×C4×C59C2311yescharacteristic
50C22×Dic59—21no—
51C22×Dic59D411yescharacteristic
52Q8×D59—42no—
53C2×C4×D59—21no—
54C2×C4×D59—21no—
55C2×C4×D59—42no—
56C2×C4×D59D411yescharacteristic
57C2×C4×D59D411yescharacteristic
58C2×C4×D59C2311yescharacteristic
59C2×C4×D59C2311yescharacteristic
60C2×Dic118—21no—
61C2×Dic118C2311yescharacteristic
62Q16×C59—42no—
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59C2311yescharacteristic
66C22⋊C4×C59—21no—
67C22⋊Dic59—21no—
68#11745672—42no—
69(C2×D59)⋊C4—21no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—22no—
73C4⋊Dic59—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59C2311yescharacteristic
76C4⋊Dic59C2312yes—
77Dic59⋊C4—21no—
78Dic59⋊C4—21no—
79Dic59⋊C4—21no—
80Dic59⋊C4—22no—
81Dic59⋊C4—22no—
82C2×C59⋊C8C2312yes—
83#11745659—42no—
84C23×C59C2×D411yescharacteristic
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—41no—
88C22×D59C2×D411yescharacteristic
89C22×D59C2×D411yescharacteristic
90Q8×C59—22no—
91Q8×C59—22no—
92Q8×C59—22no—
93Q8×C59—42no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—42no—
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C2×Dic59—21no—
102C2×Dic59—22no—
103C2×Dic59—21no—
104C2×Dic59—41no—
105C2×Dic59—42no—
106C2×Dic59C2×D411yescharacteristic
107C2×Dic59C2×D411yescharacteristic
108C4×D59—22no—
109C4×D59—22no—
110C4×D59—41no—
111C4×D59—41no—
112C4×D59—42no—
113C4×D59—42no—
114Dic118—22no—
115Dic118—42no—
116C8×C59—22no—
117C59⋊C8—22no—
118C22×C59—21no—
119C22×C59—21no—
120C22×C59#13712yes—
121C22×C59C22×D411yescharacteristic
122C2×D59—21no—
123C2×D59—21no—
124C2×D59—21no—
125C2×D59—21no—
126C2×D59—21no—
127C2×D59—44no—
128C2×D59#13712yes—
129C4×C59—21no—
130C4×C59—41no—
131C4×C59—42no—
132C4×C59—41no—
133C4×C59H(5,𝔽2)12yes—
134Dic59—21no—
135Dic59—21no—
136Dic59—41no—
137Dic59—41no—
138Dic59—42no—
139#3195—591nomaximal, Sylow
140C2×C59—22no—
141C2×C59#53411yescharacteristic
142C2×C59C2×#13711yescharacteristic
143C2×C59C2×#13711yescharacteristic
144D59—22no—
145D59—41no—
146C2×C22⋊Q8—591no—
147C2×C22⋊Q8—591no—
148#483—591no—
149#483—592no—
150#483—591no—
151#470—592no—
152#470—592no—
153#451—591no—
154#451—592no—
155#451D5911yescharacteristic, complemented
156#407—591no—
157C59#319511yescharacteristic, complemented, Sylow
158C22×Q8—591no—
159C22×Q8C2×D5911yescharacteristic
160C23×C4—591no—
161C2×Q16—592no—
162C2×Q16C2×D5912yescomplemented
163C2×M16—592no—
164C22⋊Q8—592no—
165C22⋊Q8—591no—
166C22⋊Q8—592no—
167C22⋊Q8—591no—
168C22⋊Q8—591no—
169C22⋊Q8—1181no—
170C22⋊Q8—1181no—
171C22⋊Q8—1181no—
172C22⋊Q8—1181no—
173C22⋊Q8C2×D5911yescharacteristic
174C2×C4⋊C4—591no—
175C2×C4⋊C4—591no—
176C2×C4⋊C4—1181no—
177C2×C4⋊C4—1181no—
178C2×C22⋊C4—1181no—
179C2×C22⋊C4—1181no—
180#106—592no—
181#106—592no—
182#103—592no—
183#103—592no—
184#103—592no—
185#103C2×D5912yescomplemented
186C22⋊C8—591no—
187C22⋊C8—592no—
188C22⋊C8C2×D5911yescharacteristic
189C24—591no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—591no—
193C2×Q8—1181no—
194C2×Q8—2362no—
195C2×Q8C22×D5911yescharacteristic
196C2×Q8C22×D5912yes—
197C22×C4—21no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—591no—
201C22×C4—591no—
202C22×C4—591no—
203C22×C4—1181no—
204C22×C4—1181no—
205C22×C4—1181no—
206C22×C4—2362no—
207C22×C4C22×D5911yescharacteristic
208Q16—42no—
209Q16—2362no—
210M16—2362no—
211C2×C8—592no—
212C2×C8C22×D5912yes—
213C4⋊C4—21no—
214C4⋊C4—591no—
215C4⋊C4—592no—
216C4⋊C4—1181no—
217C4⋊C4—1181no—
218C4⋊C4—1182no—
219C4⋊C4—1182no—
220C4⋊C4—1181no—
221C4⋊C4—1181no—
222C4⋊C4—1181no—
223C4⋊C4C22×D5911yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1181no—
226C22⋊C4—1181no—
227C22⋊C4—1182no—
228C22⋊C4—1182no—
229C22⋊C4—1181no—
230C23—591no—
231C23—591no—
232C23—1182no—
233C23—1182no—
234C23—2361no—
235C23D4×D5911yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—1182no—
241Q8—2362no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—591no—
248C2×C4—591no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1181no—
252C2×C4—1182no—
253C2×C4—1181no—
254C2×C4—2361no—
255C2×C4—2361no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4—2361no—
259C2×C4—2362no—
260C2×C4D4×D5911yescharacteristic
261C2×C4C23×D5911yescharacteristic
262C8—22no—
263C8—1182no—
264C22—21no—
265C22—21no—
266C22—592no—
267C22—1181no—
268C22—1181no—
269C22—1181no—
270C22—1181no—
271C22—1181no—
272C22—2364no—
273C22#1427375912yes—
274C22C2×D4×D5911yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—1181no—
280C4—1181no—
281C4—2361no—
282C4—2362no—
283C4—2361no—
284C4#1427377212yes—
285C2—22no—
286C2—1182no—
287C2—2361no—
288C2#1683796211yescharacteristic
289C2#137×D5911yescharacteristic
290C2C2×#1427375911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#451C2
socleC22×C59C22×D4
hypercenter#451D59
hypocenterC59#3195
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #16837733C2semidirect
4. #16837731C2semidirect
5. #16837730C2semidirect
6. #16837659C2semidirect
7. #16837657C2semidirect
8. #16837558C2semidirect
9. #16837556C2semidirect
11. C59×#451C2semidirect
14. C2×#11745672C22semidirect
31. #14273659C22semidirect
32. #14273658C22semidirect
33. #14273652C22semidirect
34. #14273651C22semidirect
35. #14273651C22semidirect
39. C2×Q16×C59C22semidirect
41. #103×C59C22semidirect
155. #451D59semidirect
157. C59#3195semidirect
162. C2×Q16C2×D59semidirect
185. #103C2×D59semidirect