Finite Group Databasebeta

← #21435885

Subgroups — #21435885

Order 7552.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#16838215C211yesmaximal, characteristic, complemented
2#16838211C212yesmaximal, complemented
3#16838207C211yesmaximal, characteristic, complemented
4#16837787C211yesmaximal, characteristic, complemented
5#16837673C211yesmaximal, characteristic, complemented
6#16837668C212yesmaximal, complemented
7#16837578C212yesmaximal, complemented
8#16837577C212yesmaximal, complemented
9#16837518C211yesmaximal, characteristic, complemented
10#16837517C211yesmaximal, characteristic, complemented
11C59×#473C211yesmaximal, characteristic, complemented
12#14273775C2212yescomplemented
13#14273773C2212yes—
14#14273759—22no—
15#14273759—22no—
16#14273758C2212yescomplemented
17#14273755—22no—
18#14273755C2212yes—
19#14273755C2212yescomplemented
20#14273754—22no—
21#14273750—22no—
22#14273750—21no—
23#14273749—21no—
24#14273749C2211yescharacteristic
25#14273747C2211yescharacteristic
26Q16×D59—21no—
27C2×#11745671C2212yescomplemented
28SD16×D59—22no—
29C2×D472—21no—
30C2×D472C2211yescharacteristic
31#14273735C2212yescomplemented
32#14273733C2211yescharacteristic, complemented
33#14273716C2212yescomplemented
34#14273662C2211yescharacteristic, complemented
35#14273654C2211yescharacteristic
36#14273653C2211yescharacteristic, complemented
37#14273653C2212yescomplemented
38#14273643C2212yescomplemented
39C8×Dic59C2211yescharacteristic, complemented
40#137×C59C2212yes—
41C2×Q16×C59C2211yescharacteristic, complemented
42#131×C59C2211yescharacteristic
43#108×C59C2211yescharacteristic, complemented
44#101×C59C2212yescomplemented
45#41×C59C2311yescharacteristic
46C2×Q8×C59C2312yes—
47#11745689—22no—
48#11745689—22no—
49#11745689—22no—
50Q8×D59—22no—
51Q8×D59—22no—
52D4×D59—22no—
53D4×D59—42no—
54#11745685—22no—
55#11745685—42no—
56#11745685C2311yescharacteristic
57#11745685C2312yes—
58C2×D236—22no—
59C2×D236C2312yes—
60Q16×C59—22no—
61Q16×C59—21no—
62Q16×C59—21no—
63SD16×C59—22no—
64M16×C59—21no—
65M16×C59C2311yescharacteristic
66M16×C59C2312yes—
67C2×C8×C59—21no—
68C2×C8×C59C2311yescharacteristic
69#11745672—22no—
70#11745672D411yescharacteristic
71#11745671—22no—
72#11745671—22no—
73#11745671—22no—
74#11745671D411yescharacteristic
75#11745670D411yescharacteristic
76#11745669D411yescharacteristic
77(C2×D59)⋊C4—42no—
78C4×Dic59C2311yescharacteristic
79#11745664C2311yescharacteristic
80C2×C59⋊C8C2311yescharacteristic
81Dic236D411yescharacteristic
82D472—22no—
83D472—22no—
84D472—21no—
85D472—21no—
86D472D411yescharacteristic
87#11745660—22no—
88#11745660—22no—
89#11745660D412yes—
90#11745659—21no—
91#11745659—22no—
92C8×D59—21no—
93C8×D59—22no—
94C22×D59—22no—
95C22×D59—42no—
96Q8×C59—22no—
97Q8×C59—22no—
98Q8×C59C2×D411yescharacteristic
99D4×C59—21no—
100D4×C59C2×D411yescharacteristic
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—21no—
105C22⋊D59—22no—
106C22⋊D59—42no—
107C2×Dic59—21no—
108D236—22no—
109D236—22no—
110D236—42no—
111D236—42no—
112D236C2×D412yes—
113D236C2×D411yescharacteristic
114C4×D59—22no—
115C4×D59—21no—
116C4×D59—42no—
117C4×D59—42no—
118C4×D59—42no—
119Dic118—42no—
120Dic118C2×D411yescharacteristic
121Dic118C2×D412yes—
122C8×C59—22no—
123C8×C59—21no—
124C8×C59C2×D411yescharacteristic
125C8×C59C2×D411yescharacteristic
126C59⋊C8—21no—
127C59⋊C8C2×D411yescharacteristic
128C59⋊C8C2×D411yescharacteristic
129C22×C59—21no—
130C22×C59H(5,𝔽2)11yescharacteristic
131C2×D59—21no—
132C2×D59—22no—
133C2×D59—42no—
134C2×D59—82no—
135C2×D59—162no—
136C4×C59—21no—
137C4×C59—42no—
138C4×C59C22×D411yescharacteristic
139C4×C59C22×D411yescharacteristic
140Dic59—21no—
141Dic59—22no—
142Dic59—41no—
143#3290—591nomaximal, Sylow
144C2×C59—21no—
145C2×C59—41no—
146C2×C59D4211yescharacteristic
147D59—41no—
148D59—42no—
149D59—82no—
150#578—591no—
151#577—592no—
152#576—591no—
153#495—591no—
154#473D5911yescharacteristic, complemented
155#472—591no—
156#471—592no—
157#455—592no—
158#454—592no—
159#443—591no—
160#443—591no—
161C59#329011yescharacteristic, complemented, Sylow
1622−1+4—592no—
163H(5,𝔽2)—592no—
164#137—592no—
165#137—1182no—
166#137—1182no—
167#137C2×D5912yes—
168AGL(1,ℤ/8)—592no—
169AGL(1,ℤ/8)—592no—
170AGL(1,ℤ/8)—1182no—
171AGL(1,ℤ/8)—1182no—
172#135—591no—
173#135—591no—
174#135—1181no—
175#135—1181no—
176#135—1182no—
177C2×Q16—1181no—
178C2×Q16C2×D5911yescharacteristic, complemented
179C2×SD16—592no—
180C2×SD16—1182no—
181C2×D8—591no—
182C2×D8—1181no—
183#131—592no—
184#131—591no—
185#131C2×D5911yescharacteristic
186#124—592no—
187#108—591no—
188#108C2×D5911yescharacteristic, complemented
189#104—591no—
190#104—591no—
191#104—592no—
192#101C2×D5912yescomplemented
193#100—592no—
194C4×C8—591no—
195#41—591no—
196#41—592no—
197#41—1182no—
198#41—1182no—
199#41—1182no—
200#41—1182no—
201#41—2362no—
202#41C22×D5911yescharacteristic
203C2×Q8—1182no—
204C2×Q8—1182no—
205C2×Q8C22×D5912yes—
206C2×D4—592no—
207C2×D4—1182no—
208C2×D4—1182no—
209C2×D4—2362no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—591no—
214Q16—591no—
215Q16—1182no—
216SD16—22no—
217SD16—591no—
218SD16—591no—
219SD16—592no—
220SD16—1182no—
221SD16—1182no—
222SD16—1182no—
223SD16—1182no—
224SD16—1182no—
225D8—591no—
226D8—591no—
227D8—1182no—
228D8—1182no—
229D8—1181no—
230D8—1181no—
231M16—21no—
232M16—591no—
233M16—1182no—
234M16—1181no—
235M16C22×D5912yes—
236M16C22×D5911yescharacteristic
237C2×C8—21no—
238C2×C8—591no—
239C2×C8—1182no—
240C2×C8—1181no—
241C2×C8C22×D5911yescharacteristic
242C22⋊C4—2362no—
243C42—591no—
244C23—1182no—
245C23—2362no—
246Q8—22no—
247Q8—22no—
248Q8—591no—
249Q8—592no—
250Q8—2362no—
251Q8D4×D5911yescharacteristic
252D4—21no—
253D4—591no—
254D4—592no—
255D4—1182no—
256D4—1181no—
257D4—1182no—
258D4—1182no—
259D4—2362no—
260D4—2362no—
261D4—2362no—
262D4D4×D5911yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—1182no—
266C2×C4—1181no—
267C2×C4—1181no—
268C2×C4—2362no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4C23×D5911yescharacteristic
272C8—21no—
273C8—22no—
274C8—591no—
275C8—591no—
276C8—1181no—
277C8D4×D5911yescharacteristic
278C8D4×D5911yescharacteristic
279C22—21no—
280C22—1181no—
281C22—1182no—
282C22—9442no—
283C22—4722no—
284C22—2362no—
285C22#1427377311yescharacteristic
286C4—21no—
287C4—42no—
288C4—1182no—
289C4—1181no—
290C4—2361no—
291C4C2×D4×D5911yescharacteristic
292C4C2×D4×D5911yescharacteristic
293C2—21no—
294C2—41no—
295C2—4722no—
296C2—2362no—
297C2—2361no—
298C2#1683803611yescharacteristic
subgroupG/H
centerC2#16838036
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#473C2
socleC2×C59D42
hypercenter#473D59
hypocenterC59#3290
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #16838215C2semidirect
2. #16838211C2semidirect
3. #16838207C2semidirect
4. #16837787C2semidirect
5. #16837673C2semidirect
6. #16837668C2semidirect
7. #16837578C2semidirect
8. #16837577C2semidirect
9. #16837518C2semidirect
10. #16837517C2semidirect
11. C59×#473C2semidirect
12. #14273775C22semidirect
16. #14273758C22semidirect
19. #14273755C22semidirect
27. C2×#11745671C22semidirect
31. #14273735C22semidirect
32. #14273733C22semidirect
33. #14273716C22semidirect
34. #14273662C22semidirect
36. #14273653C22semidirect
37. #14273653C22semidirect
38. #14273643C22semidirect
39. C8×Dic59C22semidirect
41. C2×Q16×C59C22semidirect
43. #108×C59C22semidirect
44. #101×C59C22semidirect
154. #473D59semidirect
161. C59#3290semidirect
178. C2×Q16C2×D59semidirect
188. #108C2×D59semidirect
192. #101C2×D59semidirect