Finite Group Databasebeta

← #21434870

Subgroups — #21434870

Order 7552.

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

normal subgroups only

This group has 416 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
1C2×#14273707C211yesmaximal, characteristic
2C22⋊Q8×D59C211yesmaximal, characteristic
3#16837739C212yesmaximal, complemented
4#16837737C211yesmaximal, characteristic, complemented
5#16837644C212yesmaximal, complemented
6#16837642C212yesmaximal, complemented
7#16837560C211yesmaximal, characteristic, complemented
8#16837559C212yesmaximal, complemented
9#16837557C211yesmaximal, characteristic, complemented
10C22⋊C8×D59C211yesmaximal, characteristic
11C59×#484C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C2×Q8×D59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C2×Dic236C2212yescomplemented
16C2×#11745672C2212yescomplemented
17C2×C8×D59C2212yes—
18#14273707—21no—
19#14273707—21no—
20#14273707C2211yescharacteristic
21#14273707C2212yes—
22#14273707C2212yes—
23#14273706C2211yescharacteristic
24#14273705—21no—
25#14273704—21no—
26#14273703C2211yescharacteristic
27C2×C4⋊Dic59—21no—
28C4⋊C4×D59—21no—
29C4⋊C4×D59C2211yescharacteristic
30C2×(C2×D59)⋊C4—21no—
31C22⋊C4×D59—21no—
32#14273660C2212yescomplemented
33#14273652C2212yescomplemented
34#14273650C2212yescomplemented
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—42no—
50C2×C4×D59—21no—
51C2×C4×D59—21no—
52C2×C4×D59—42no—
53C2×C4×D59D411yescharacteristic
54C2×C4×D59D411yescharacteristic
55C2×C4×D59C2311yescharacteristic
56C2×C4×D59C2311yescharacteristic
57C2×Dic118—21no—
58C2×Dic118—21no—
59C2×Dic118—42no—
60C2×Dic118C2311yescharacteristic
61C2×Dic118C2312yes—
62C2×C8×C59C2312yes—
63C4⋊C4×C59—21no—
64C4⋊C4×C59—21no—
65C4⋊C4×C59C2312yes—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C22⋊Dic59—21no—
69#11745672—42no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4—21no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—22no—
74C4⋊Dic59—21no—
75C4⋊Dic59—22no—
76C4⋊Dic59—22no—
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—21no—
79Dic59⋊C4—21no—
80Dic59⋊C4—21no—
81C2×C59⋊C8C2312yes—
82Dic236—42no—
83C8×D59—42no—
84C23×C59C2×D411yescharacteristic
85C22×D59—41no—
86C22×D59—41no—
87C22×D59Q1614yescomplemented
88C22×D59C2×D411yescharacteristic
89C22×D59C2×D411yescharacteristic
90Q8×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—42no—
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2411yescharacteristic
98C2×Dic59—21no—
99C2×Dic59—21no—
100C2×Dic59—22no—
101C2×Dic59—41no—
102C2×Dic59—42no—
103C2×Dic59C2×D411yescharacteristic
104C2×Dic59C2×D411yescharacteristic
105C4×D59—22no—
106C4×D59—22no—
107C4×D59—41no—
108C4×D59—41no—
109C4×D59—42no—
110C4×D59—42no—
111Dic118—22no—
112Dic118—22no—
113Dic118—22no—
114Dic118—42no—
115Dic118—42no—
116C8×C59—22no—
117C59⋊C8—22no—
118C22×C59—21no—
119C22×C59—21no—
120C22×C59C2×Q1612yes—
121C22×C59C22×D411yescharacteristic
122C2×D59—21no—
123C2×D59—21no—
124C2×D59—21no—
125C2×D59—21no—
126C2×D59—44no—
127C2×D59C2×Q1612yes—
128C2×D59C2×Q1612yes—
129C4×C59—21no—
130C4×C59—41no—
131C4×C59—41no—
132C4×C59—42no—
133C4×C59H(5,𝔽2)12yes—
134Dic59—21no—
135Dic59—21no—
136Dic59—41no—
137Dic59—42no—
138Dic59—41no—
139#3189—591nomaximal, Sylow
140C2×C59—22no—
141C2×C59#53411yescharacteristic
142C2×C59C22×Q1611yescharacteristic
143C2×C59#57511yescharacteristic
144D59—22no—
145D59—22no—
146C2×C22⋊Q8—591no—
147C2×C22⋊Q8—591no—
148#484—591no—
149#484—592no—
150#484D5911yescharacteristic, complemented
151#467—592no—
152#467—592no—
153#451—591no—
154#451—592no—
155#451—591no—
156C2×C22⋊C8—591no—
157C59#318911yescharacteristic, complemented, Sylow
158C22×Q8—591no—
159C22×Q8—591no—
160C23×C4—591no—
161C2×Q16—592no—
162C2×Q16—592no—
163C22×C8—592no—
164C22⋊Q8—591no—
165C22⋊Q8—591no—
166C22⋊Q8—592no—
167C22⋊Q8—591no—
168C22⋊Q8—592no—
169C22⋊Q8—1181no—
170C22⋊Q8—1181no—
171C22⋊Q8—1181no—
172C22⋊Q8—1181no—
173C22⋊Q8C2×D5911yescharacteristic
174C2×C4⋊C4—591no—
175C2×C4⋊C4—1181no—
176C2×C4⋊C4—1181no—
177C2×C4⋊C4C2×D5911yescharacteristic
178C2×C22⋊C4—1181no—
179C2×C22⋊C4—1181no—
180#107—592no—
181#107C2×D5912yescomplemented
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—592no—
191C2×Q8—591no—
192C2×Q8—1181no—
193C2×Q8—1181no—
194C2×Q8—2362no—
195C2×Q8—2362no—
196C2×Q8C22×D5911yescharacteristic
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—2362no—
209Q16—2362no—
210C2×C8—592no—
211C2×C8—2362no—
212C2×C8C22×D5912yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—591no—
216C4⋊C4—1181no—
217C4⋊C4—1181no—
218C4⋊C4—1181no—
219C4⋊C4—1182no—
220C4⋊C4—1182no—
221C4⋊C4—1181no—
222C4⋊C4C22×D5911yescharacteristic
223C4⋊C4C22×D5912yes—
224C22⋊C4—21no—
225C22⋊C4—1181no—
226C22⋊C4—1181no—
227C22⋊C4—1182no—
228C22⋊C4—1182no—
229C22⋊C4—1181no—
230C23—591no—
231C23—591no—
232C23—594no—
233C23—2361no—
234C23—2361no—
235C23D4×D5911yescharacteristic
236Q8—22no—
237Q8—1182no—
238Q8—1182no—
239Q8—1182no—
240Q8—2362no—
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—1181no—
253C2×C4—1182no—
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—592no—
268C22—1181no—
269C22—1181no—
270C22—1181no—
271C22—1181no—
272C22—2364no—
273C22Q16×D5912yes—
274C22C2×D4×D5911yescharacteristic
275C4—21no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—1181no—
280C4—1181no—
281C4—2361no—
282C4—2361no—
283C4—2362no—
284C4#1427377312yes—
285C2—22no—
286C2—1182no—
287C2—1182no—
288C2#1683796311yescharacteristic
289C2C2×Q16×D5911yescharacteristic
290C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#484C2
socleC22×C59C22×D4
hypercenter#484D59
hypocenterC59#3189
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. #16837739C2semidirect
4. #16837737C2semidirect
5. #16837644C2semidirect
6. #16837642C2semidirect
7. #16837560C2semidirect
8. #16837559C2semidirect
9. #16837557C2semidirect
11. C59×#484C2semidirect
15. C2×Dic236C22semidirect
16. C2×#11745672C22semidirect
32. #14273660C22semidirect
33. #14273652C22semidirect
34. #14273650C22semidirect
35. #14273650C22semidirect
40. #107×C59C22semidirect
41. #103×C59C22semidirect
87. C22×D59Q16semidirect
150. #484D59semidirect
157. C59#3189semidirect
181. #107C2×D59semidirect
185. #103C2×D59semidirect