Finite Group Databasebeta

← #21433892

Subgroups — #21433892

Order 7552.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#16837955C211yesmaximal, characteristic, complemented
2C2×#14273689C211yesmaximal, characteristic
3#16837705C212yesmaximal, complemented
4#16837701C212yesmaximal, complemented
5#16837691C212yesmaximal, complemented
6#16837681C212yesmaximal, complemented
7#16837442C211yesmaximal, characteristic, complemented
8#16837425C211yesmaximal, characteristic, complemented
9#16837408C211yesmaximal, characteristic
10#16837406C211yesmaximal, characteristic
11C59×#428C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C4×D59—21no—
14C2×#11745664C2211yescharacteristic
15#14273726C2211yescharacteristic, complemented
16#14273726C2211yescharacteristic, complemented
17#14273718C2211yescharacteristic, complemented
18#14273718C2211yescharacteristic, complemented
19#14273704—21no—
20#14273704—21no—
21C4×Dic118—21no—
22C4×Dic118—21no—
23#14273689—21no—
24#14273689—21no—
25#14273689C2212yes—
26#14273689C2212yes—
27#14273686C2211yescharacteristic
28C2×(C2×D59)⋊C4—21no—
29C2×C4×Dic59C2211yescharacteristic
30#14273660C2212yescomplemented
31#14273658C2212yescomplemented
32Dic59⋊C8C2212yescomplemented
33Dic59⋊C8C2212yescomplemented
34#14273649C2212yes—
35#14273649C2212yes—
36#14273647C2212yes—
37#14273647C2212yes—
38C2×M16×C59C2211yescharacteristic
39#117×C59C2211yescharacteristic
40C2×C4⋊C4×C59C2211yescharacteristic
41#107×C59C2212yescomplemented
42#106×C59C2212yescomplemented
43C23×D59—21no—
44C22×C4×C59—21no—
45C22×C4×C59C2311yescharacteristic
46C22×Dic59D411yescharacteristic
47C22×Dic59D411yescharacteristic
48C2×D236—21no—
49C2×D236—22no—
50C2×D236Q814yescomplemented
51C2×D236C2312yes—
52C2×C4×D59—22no—
53C2×C4×D59—22no—
54C2×C4×D59—44no—
55C2×Dic118—21no—
56C2×Dic118—21no—
57M16×C59—22no—
58C2×C8×C59C2312yes—
59C4⋊C4×C59—21no—
60C4⋊C4×C59C2311yescharacteristic
61C4⋊C4×C59C2312yes—
62C4⋊C4×C59C2311yescharacteristic
63C22⋊C4×C59—21no—
64C42×C59—21no—
65C22⋊Dic59—21no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68C4⋊Dic59—21no—
69C4⋊Dic59—21no—
70C4⋊Dic59C2311yescharacteristic
71C4⋊Dic59C2311yescharacteristic
72Dic59⋊C4—21no—
73Dic59⋊C4—21no—
74Dic59⋊C4—21no—
75Dic59⋊C4—21no—
76C4×Dic59—21no—
77C4×Dic59D411yescharacteristic
78C4×Dic59D411yescharacteristic
79C4×Dic59C2311yescharacteristic
80C4×Dic59C2311yescharacteristic
81#11745664—22no—
82C2×C59⋊C8C2312yes—
83C23×C59C2×D411yescharacteristic
84C22×D59—24no—
85C22×D59—22no—
86C22×D59—42no—
87C22×D59—42no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—42no—
92C2×C4×C59C2×D411yescharacteristic
93C2×C4×C59C2×Q812yes—
94C2×C4×C59#4112yes—
95C2×C4×C59C2411yescharacteristic
96C2×Dic59—22no—
97C2×Dic59—21no—
98C2×Dic59—21no—
99C2×Dic59—41no—
100C2×Dic59C2×D411yescharacteristic
101C2×Dic59C2×D411yescharacteristic
102C2×Dic59C2×D411yescharacteristic
103C2×Dic59C2×D411yescharacteristic
104D236—24no—
105D236—22no—
106D236C2×Q814yes—
107C4×D59—44no—
108C4×D59—44no—
109Dic118—42no—
110Dic118—42no—
111C8×C59—22no—
112C59⋊C8—22no—
113C22×C59—21no—
114C22×C59—21no—
115C22×C59C22⋊Q812yes—
116C22×C59C22×D411yescharacteristic
117C2×D59—24no—
118C2×D59—48no—
119C2×D59—42no—
120C2×D59—42no—
121C4×C59—41no—
122C4×C59—42no—
123C4×C59—41no—
124C4×C59C22⋊Q812yes—
125C4×C59C22×Q811yescharacteristic
126C4×C59C2×#4111yescharacteristic
127Dic59—21no—
128Dic59—21no—
129Dic59—21no—
130Dic59—41no—
131Dic59—41no—
132Dic59AGL(1,ℤ/8)12yes—
133#3077—591nomaximal, Sylow
134C2×C59—22no—
135C2×C59C2×C22⋊Q811yescharacteristic
136C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C59#57511yescharacteristic
138D59—44no—
139#533—591no—
140C2×C4×D4—591no—
141#478—592no—
142#478—592no—
143#476—592no—
144#474—592no—
145#428—591no—
146#428D5911yescharacteristic, complemented
147#423—591no—
148#418—591no—
149#418—591no—
150C59#307711yescharacteristic, complemented, Sylow
151C22×D4—591no—
152C23×C4—1181no—
153C2×M16—591no—
154C2×M16C2×D5911yescharacteristic
155C4⋊Q8—591no—
156C4⋊Q8—591no—
157#125—591no—
158#125—591no—
159C22⋊Q8—1181no—
160C22⋊Q8—1181no—
161C4×Q8—1181no—
162C4×Q8—1181no—
163C4×D4—592no—
164C4×D4—592no—
165C4×D4—1181no—
166C4×D4—1181no—
167#117—591no—
168#117C2×D5911yescharacteristic
169C2×C4⋊C4C2×D5911yescharacteristic
170C2×C22⋊C4—1181no—
171C2×C42—591no—
172#107—592no—
173#107C2×D5912yescomplemented
174#106—592no—
175#106C2×D5912yescomplemented
176C4⋊C8—592no—
177C4⋊C8—592no—
178#102—592no—
179#102—592no—
180#102—592no—
181#102—592no—
182C24—1181no—
183C2×Q8—1181no—
184C2×Q8—1181no—
185C2×D4—594no—
186C2×D4—592no—
187C2×D4—1182no—
188C2×D4—1181no—
189C22×C4—21no—
190C22×C4—591no—
191C22×C4—591no—
192C22×C4—1182no—
193C22×C4—1182no—
194C22×C4—2364no—
195C22×C4C22×D5911yescharacteristic
196M16—22no—
197M16—1182no—
198C2×C8—592no—
199C2×C8C22×D5912yes—
200C4⋊C4—21no—
201C4⋊C4—591no—
202C4⋊C4—591no—
203C4⋊C4—1181no—
204C4⋊C4—1181no—
205C4⋊C4—1181no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4C22×D5911yescharacteristic
210C4⋊C4C22×D5911yescharacteristic
211C4⋊C4C22×D5912yes—
212C22⋊C4—21no—
213C22⋊C4—1182no—
214C22⋊C4—1182no—
215C22⋊C4—1181no—
216C42—21no—
217C42—591no—
218C42—591no—
219C42—591no—
220C42—591no—
221C42—1181no—
222C23—1184no—
223C23—1182no—
224C23—2362no—
225C23—2362no—
226C23D4×D5911yescharacteristic
227Q8—2362no—
228Q8—2362no—
229D4—594no—
230D4—1184no—
231D4—1182no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—591no—
237C2×C4—591no—
238C2×C4—591no—
239C2×C4—591no—
240C2×C4—1182no—
241C2×C4—1181no—
242C2×C4—1181no—
243C2×C4—2364no—
244C2×C4—2364no—
245C2×C4—2361no—
246C2×C4#1174568512yes—
247C2×C4D4×D5911yescharacteristic
248C2×C4Q8×D5912yes—
249C2×C4C23×D5911yescharacteristic
250C8—22no—
251C8—1182no—
252C22—21no—
253C22—21no—
254C22—1184no—
255C22—2368no—
256C22—2362no—
257C22—2362no—
258C22#1427370512yes—
259C22C2×D4×D5911yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—592no—
264C4—1181no—
265C4—1181no—
266C4—1181no—
267C4—2361no—
268C4—2361no—
269C4#1427370512yes—
270C4C2×Q8×D5911yescharacteristic
271C4C2×#1174568511yescharacteristic
272C2—22no—
273C2—2364no—
274C2C2×#1427370511yescharacteristic
275C2AGL(1,ℤ/8)×D5911yescharacteristic
276C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#428C2
socleC22×C59C22×D4
hypercenter#428D59
hypocenterC59#3077
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #16837955C2semidirect
3. #16837705C2semidirect
4. #16837701C2semidirect
5. #16837691C2semidirect
6. #16837681C2semidirect
7. #16837442C2semidirect
8. #16837425C2semidirect
11. C59×#428C2semidirect
15. #14273726C22semidirect
16. #14273726C22semidirect
17. #14273718C22semidirect
18. #14273718C22semidirect
30. #14273660C22semidirect
31. #14273658C22semidirect
32. Dic59⋊C8C22semidirect
33. Dic59⋊C8C22semidirect
41. #107×C59C22semidirect
42. #106×C59C22semidirect
50. C2×D236Q8semidirect
146. #428D59semidirect
150. C59#3077semidirect
173. #107C2×D59semidirect
175. #106C2×D59semidirect