Finite Group Databasebeta

← #21435780

Subgroups — #21435780

Order 7552.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#14273750C211yesmaximal, characteristic, complemented
2C2×D8×D59C211yesmaximal, characteristic, complemented
3#16838045C212yesmaximal, complemented
4#16837640C211yesmaximal, characteristic, complemented
5#16837635C211yesmaximal, characteristic, complemented
6#16837614C212yesmaximal, complemented
7#16837551C212yesmaximal, complemented
8#16837546C212yesmaximal, complemented
9C4×D472C211yesmaximal, characteristic, complemented
10C8×D236C211yesmaximal, characteristic, complemented
11C59×#495C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yescomplemented
13C2×D4×D59C2212yes—
14#14273750—24no—
15C2×#11745671C2212yescomplemented
16C2×D472C2211yescharacteristic
17C2×#11745669C2212yes—
18D8×D59—24no—
19C2×C8×D59C2211yescharacteristic
20C2×C8×D59C2211yescharacteristic, complemented
21#14273709—22no—
22#14273703C2212yes—
23#14273700—22no—
24#14273698C2212yescomplemented
25#14273697—22no—
26C4×D236C2211yescharacteristic
27C4×D236C2212yescomplemented
28C22⋊C4×D59—22no—
29#14273661C2211yescharacteristic, complemented
30C4⋊(C59⋊C8)C2211yescharacteristic, complemented
31#14273651C2212yescomplemented
32#14273649C2212yescomplemented
33#14273648C2212yescomplemented
34(C2×D59)⋊C8C2211yescharacteristic
35(C2×D59)⋊C8C2211yescharacteristic, complemented
36C2×Q16×C59C2211yescharacteristic, complemented
37C2×SD16×C59C2212yescomplemented
38C2×D8×C59C2211yescharacteristic
39#124×C59C2212yescomplemented
40C4×C8×C59C2211yescharacteristic, complemented
41C23×D59—22no—
42C2×Q8×C59C2312yes—
43C2×D4×C59C2312yes—
44C2×C22⋊D59—22no—
45C2×C22⋊D59—22no—
46#11745689—24no—
47#11745689—42no—
48D4×D59—24no—
49D4×D59—42no—
50C2×D236—22no—
51C2×D236D411yescharacteristic
52C2×D236C2312yes—
53C2×C4×D59—22no—
54C2×C4×D59—22no—
55C2×C4×D59C2311yescharacteristic
56C2×C4×D59C2311yescharacteristic
57Q16×C59—22no—
58SD16×C59—42no—
59D8×C59—22no—
60C2×C8×C59C2311yescharacteristic
61C2×C8×C59C2311yescharacteristic
62C22⋊C4×C59—22no—
63C22⋊C4×C59—22no—
64C42×C59C2311yescharacteristic
65C22⋊Dic59—22no—
66#11745671—24no—
67#11745669—24no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4D411yescharacteristic
71(C2×D59)⋊C4D411yescharacteristic
72C4⋊Dic59D411yescharacteristic
73C4⋊Dic59C2312yes—
74Dic59⋊C4—22no—
75C2×C59⋊C8C2311yescharacteristic
76C2×C59⋊C8C2311yescharacteristic
77D472D414yescomplemented
78C8×D59—22no—
79C8×D59—22no—
80C23×C59—22no—
81C22×D59—22no—
82C22×D59—44no—
83C22×D59—44no—
84C22×D59—82no—
85C22×D59C2×D411yescharacteristic
86C22×D59C2×D411yescharacteristic
87Q8×C59—22no—
88Q8×C59—22no—
89D4×C59—22no—
90D4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59C2×D411yescharacteristic
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2411yescharacteristic
95C22⋊D59—44no—
96C22⋊D59—82no—
97C2×Dic59—22no—
98C2×Dic59C2×D411yescharacteristic
99C2×Dic59C2×D411yescharacteristic
100D236—22no—
101D236—22no—
102D236—44no—
103D236C2×D414yes—
104C4×D59—21no—
105C4×D59—21no—
106C4×D59—21no—
107C4×D59—21no—
108C4×D59—44no—
109C4×D59—82no—
110C8×C59—21no—
111C8×C59C2×D412yes—
112C59⋊C8—21no—
113C59⋊C8—21no—
114C22×C59—42no—
115C22×C59—42no—
116C22×C59—42no—
117C22×C59C22×D411yescharacteristic
118C2×D59—21no—
119C2×D59—21no—
120C2×D59—21no—
121C2×D59—24no—
122C2×D59—21no—
123C2×D59—21no—
124C2×D59—42no—
125C2×D59—42no—
126C2×D59—88no—
127C2×D59#13512yescomplemented
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—42no—
131C4×C59C22×D411yescharacteristic
132C4×C59H(5,𝔽2)11yescharacteristic
133Dic59—21no—
134Dic59—21no—
135Dic59—42no—
136#3280—591nomaximal, Sylow
137C2×C59—82no—
138C2×C59D4211yescharacteristic
139C2×C59C2×#13511yescharacteristic
140C2×C59#57611yescharacteristic
141D59—22no—
142D59—41no—
143D59—44no—
144C2×#135—591no—
145C22×D8—591no—
146#546—592no—
147#495D5911yescharacteristic, complemented
148#467—591no—
149#466—591no—
150#463—592no—
151#450—592no—
152#449—592no—
153C4×D8—591no—
154C8×D4—591no—
155C59#328011yescharacteristic, complemented, Sylow
156C2×#41—592no—
157C22×D4—592no—
158#135—1184no—
159C2×Q16C2×D5911yescharacteristic, complemented
160C2×SD16—592no—
161C2×SD16C2×D5912yescomplemented
162C2×D8—592no—
163C2×D8—591no—
164C2×D8—1184no—
165C2×D8C2×D5911yescharacteristic
166C22×C8—591no—
167C22×C8—591no—
168#124C2×D5912yescomplemented
169#123—1182no—
170C22⋊Q8—592no—
171#121—592no—
172#121—1182no—
173#120—1182no—
174C4×D4—591no—
175C4×D4—592no—
176C2×C22⋊C4—1182no—
177#107—591no—
178C4⋊C8—591no—
179#103—592no—
180#102—592no—
181#102—592no—
182C22⋊C8—591no—
183C22⋊C8—591no—
184C4×C8C2×D5911yescharacteristic, complemented
185C24—1182no—
186#41—1184no—
187#41—2362no—
188C2×Q8C22×D5912yes—
189C2×D4—591no—
190C2×D4—592no—
191C2×D4—1182no—
192C2×D4—1182no—
193C2×D4—1184no—
194C2×D4—1182no—
195C2×D4—2362no—
196C2×D4C22×D5912yes—
197C22×C4—591no—
198C22×C4—591no—
199C22×C4—1182no—
200C22×C4—1182no—
201Q16—22no—
202SD16—42no—
203SD16—1184no—
204D8—22no—
205D8—594no—
206D8—1184no—
207C2×C8—591no—
208C2×C8—591no—
209C2×C8—1182no—
210C2×C8—1182no—
211C2×C8C22×D5911yescharacteristic
212C2×C8C22×D5911yescharacteristic
213C4⋊C4—591no—
214C4⋊C4—592no—
215C4⋊C4—1182no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—591no—
219C22⋊C4—591no—
220C22⋊C4—1182no—
221C22⋊C4—1182no—
222C22⋊C4—1182no—
223C42C22×D5911yescharacteristic
224C23—22no—
225C23—591no—
226C23—591no—
227C23—1182no—
228C23—4722no—
229C23—2364no—
230C23—2364no—
231Q8—22no—
232Q8—22no—
233D4—22no—
234D4—22no—
235D4—594no—
236D4—1182no—
237D4—1182no—
238D4—4722no—
239D4—2364no—
240D4—2364no—
241C2×C4—22no—
242C2×C4—591no—
243C2×C4—591no—
244C2×C4—1181no—
245C2×C4—1181no—
246C2×C4—1181no—
247C2×C4—1181no—
248C2×C4—1182no—
249C2×C4—4722no—
250C2×C4—2364no—
251C2×C4D4×D5911yescharacteristic
252C2×C4D4×D5911yescharacteristic
253C2×C4C23×D5911yescharacteristic
254C8—21no—
255C8—1181no—
256C8—1181no—
257C8D4×D5912yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—592no—
262C22—1181no—
263C22—1181no—
264C22—1181no—
265C22—1181no—
266C22—1181no—
267C22—1184no—
268C22—4728no—
269C22—2362no—
270C22—2362no—
271C22C2×D4×D5911yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—1181no—
276C4—1181no—
277C4—2362no—
278C4C2×D4×D5911yescharacteristic
279C4#1427377211yescharacteristic
280C2—82no—
281C2—1182no—
282C2—2361no—
283C2—2364no—
284C2#1683803711yescharacteristic
285C2#135×D5911yescharacteristic
286C2#1683820411yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#495C2
socleC22×C59C22×D4
hypercenter#495D59
hypocenterC59#3280
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#14273750C2semidirect
2. C2×D8×D59C2semidirect
3. #16838045C2semidirect
4. #16837640C2semidirect
5. #16837635C2semidirect
6. #16837614C2semidirect
7. #16837551C2semidirect
8. #16837546C2semidirect
9. C4×D472C2semidirect
10. C8×D236C2semidirect
11. C59×#495C2semidirect
12. C2×#11745689C22semidirect
15. C2×#11745671C22semidirect
20. C2×C8×D59C22semidirect
24. #14273698C22semidirect
27. C4×D236C22semidirect
29. #14273661C22semidirect
30. C4⋊(C59⋊C8)C22semidirect
31. #14273651C22semidirect
32. #14273649C22semidirect
33. #14273648C22semidirect
35. (C2×D59)⋊C8C22semidirect
36. C2×Q16×C59C22semidirect
37. C2×SD16×C59C22semidirect
39. #124×C59C22semidirect
40. C4×C8×C59C22semidirect
77. D472D4semidirect
127. C2×D59#135semidirect
147. #495D59semidirect
155. C59#3280semidirect
159. C2×Q16C2×D59semidirect
161. C2×SD16C2×D59semidirect
168. #124C2×D59semidirect
184. C4×C8C2×D59semidirect