Finite Group Databasebeta

← #21434544

Subgroups — #21434544

Order 7552.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14273755C211yesmaximal, characteristic, complemented
2C2×#14273753C211yesmaximal, characteristic, complemented
3C2×#14273752C211yesmaximal, characteristic, complemented
4#16837948C211yesmaximal, characteristic, complemented
5C2×#14273723C211yesmaximal, characteristic, complemented
6#16837790C212yesmaximal, complemented
7#16837788C212yesmaximal, complemented
8#16837777C212yesmaximal, complemented
9#16837773C212yesmaximal, complemented
10M16×Dic59C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C59C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C22×D236C2211yescharacteristic
14C22×C22⋊D59—22no—
15#14273755—24no—
16#14273753—24no—
17#14273752—24no—
18C2×#11745660C2212yescomplemented
19C2×#11745670C2212yescomplemented
20C2×#11745671C2212yescomplemented
21C2×D472C2212yescomplemented
22C2×#11745669C2212yescomplemented
23C2×#11745669C2212yescomplemented
24C2×#11745664C2211yescharacteristic
25#14273725C2211yescharacteristic, complemented
26#14273723—22no—
27#14273723C2212yes—
28#14273723C2212yes—
29#14273723C2211yescharacteristic, complemented
30#14273716C2211yescharacteristic, complemented
31#14273714C2211yescharacteristic, complemented
32#14273699—22no—
33C4×C22⋊D59—22no—
34C2×C4×Dic59C2211yescharacteristic
35#14273637C2212yescomplemented
36C8×Dic59C2212yescomplemented
37C2×#41×C59C2211yescharacteristic
38C22×D4×C59C2211yescharacteristic
39AGL(1,ℤ/8)×C59—24no—
40C2×SD16×C59C2212yescomplemented
41C2×D8×C59C2212yescomplemented
42C2×M16×C59C2211yescharacteristic
43C24×C59—21no—
44C23×D59—21no—
45#41×C59—22no—
46#41×C59—22no—
47C2×Q8×C59C2311yescharacteristic
48C2×D4×C59—21no—
49C2×D4×C59—22no—
50C2×D4×C59—22no—
51C2×D4×C59—21no—
52C2×D4×C59C2311yescharacteristic
53C2×D4×C59C2312yes—
54C22×C4×C59—21no—
55C22×C4×C59C2311yescharacteristic
56C2×C22⋊D59—21no—
57C2×C22⋊D59—22no—
58C2×C22⋊D59—24no—
59C2×C22⋊D59—24no—
60C2×C22⋊D59—48no—
61C22×Dic59D412yes—
62#11745685—22no—
63#11745685—22no—
64C2×D236—22no—
65C2×D236—22no—
66C2×D236—21no—
67C2×D236C2311yescharacteristic
68C2×D236C2312yes—
69C2×C4×D59—21no—
70C2×Dic118C2311yescharacteristic
71SD16×C59—24no—
72D8×C59—24no—
73M16×C59D414yescomplemented
74C2×C8×C59C2312yes—
75C22⋊Dic59—22no—
76#11745671—24no—
77#11745670—24no—
78#11745669—24no—
79#11745669—24no—
80(C2×D59)⋊C4—22no—
81Dic59⋊C4—22no—
82C4×Dic59D412yes—
83C4×Dic59C2311yescharacteristic
84C4×Dic59C2311yescharacteristic
85#11745664D414yescomplemented
86C2×C59⋊C8C2312yes—
87D472—24no—
88#11745660—24no—
89C23×C59—21no—
90C23×C59—22no—
91C23×C59—44no—
92C23×C59—42no—
93C23×C59C2×D411yescharacteristic
94C22×D59—21no—
95C22×D59—22no—
96C22×D59—44no—
97C22×D59—42no—
98Q8×C59—21no—
99Q8×C59—21no—
100D4×C59—21no—
101D4×C59—22no—
102D4×C59—21no—
103D4×C59—22no—
104D4×C59—42no—
105D4×C59—42no—
106C2×C4×C59—21no—
107C2×C4×C59—42no—
108C2×C4×C59C2×D411yescharacteristic
109C2×C4×C59C2×D412yes—
110C2×C4×C59C2×D412yes—
111C2×C4×C59C2411yescharacteristic
112C22⋊D59—48no—
113C22⋊D59—42no—
114C22⋊D59—44no—
115C22⋊D59—48no—
116C2×Dic59—24no—
117C2×Dic59—21no—
118C2×Dic59C2×D412yes—
119C2×Dic59C2×D412yes—
120D236—22no—
121D236—21no—
122D236—21no—
123D236—22no—
124D236—42no—
125C4×D59—42no—
126Dic118—21no—
127Dic118—21no—
128C8×C59C2×D414yescomplemented
129C59⋊C8C2×D414yescomplemented
130C22×C59—22no—
131C22×C59—41no—
132C22×C59—42no—
133C22×C59—44no—
134C22×C59—42no—
135C22×C59—84no—
136C22×C59#12712yes—
137C22×C59C22×D411yescharacteristic
138C2×D59—41no—
139C2×D59—42no—
140C2×D59—44no—
141C2×D59—42no—
142C2×D59—84no—
143C4×C59—41no—
144C4×C59#12712yes—
145C4×C59C22×D411yescharacteristic
146C4×C59C22×D411yescharacteristic
147Dic59—22no—
148Dic59—41no—
149Dic59AGL(1,ℤ/8)14yescomplemented
150#3151—591nomaximal, Sylow
151C2×C59—22no—
152C2×C59—81no—
153C2×C59—82no—
154C2×C59C2×#12711yescharacteristic
155C2×C59C2×AGL(1,ℤ/8)12yes—
156D59—81no—
157D59—82no—
158C2×AGL(1,ℤ/8)—591no—
159C2×AGL(1,ℤ/8)—591no—
160C2×AGL(1,ℤ/8)—591no—
161C2×AGL(1,ℤ/8)D5911yescharacteristic, complemented
162#532—591no—
163C2×#127—591no—
164#496—592no—
165#496—592no—
166#493—592no—
167#492—592no—
168C4×M16—591no—
169C59#315111yescharacteristic, complemented, Sylow
170C2×#41—591no—
171C2×#41C2×D5911yescharacteristic
172C22×D4—591no—
173C22×D4—1182no—
174C22×D4C2×D5911yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—1184no—
177AGL(1,ℤ/8)—1184no—
178AGL(1,ℤ/8)—1184no—
179C2×SD16—592no—
180C2×SD16—592no—
181C2×SD16—592no—
182C2×SD16C2×D5912yescomplemented
183C2×D8—592no—
184C2×D8—592no—
185C2×D8—592no—
186C2×D8C2×D5912yescomplemented
187C2×M16—591no—
188C2×M16C2×D5911yescharacteristic
189C4⋊Q8—591no—
190#127—592no—
191#127—592no—
192#127—591no—
193#127—1182no—
194#124—591no—
195#124—591no—
196#121—1182no—
197C4×D4—1182no—
198C2×C42—591no—
199#97—592no—
200C4×C8—592no—
201C24—21no—
202C24—1181no—
203#41—22no—
204#41—22no—
205#41—1182no—
206#41—1182no—
207C2×Q8—591no—
208C2×Q8C22×D5911yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—591no—
214C2×D4—592no—
215C2×D4—1182no—
216C2×D4—1184no—
217C2×D4—1184no—
218C2×D4—1181no—
219C2×D4—1182no—
220C2×D4—1181no—
221C2×D4—1182no—
222C2×D4—2368no—
223C2×D4C22×D5911yescharacteristic
224C2×D4C22×D5912yes—
225C22×C4—21no—
226C22×C4—592no—
227C22×C4—1181no—
228C22×C4C22×D5911yescharacteristic
229SD16—24no—
230SD16—1184no—
231SD16—1184no—
232SD16—1184no—
233D8—24no—
234D8—1184no—
235D8—1184no—
236D8—1184no—
237M16—594no—
238M16C22⋊D5914yescomplemented
239C2×C8—592no—
240C2×C8C22×D5912yes—
241C4⋊C4—1182no—
242C22⋊C4—1182no—
243C22⋊C4—1182no—
244C42—592no—
245C42—591no—
246C42—591no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—1181no—
252C23—1182no—
253C23—2364no—
254C23—2362no—
255C23D4×D5911yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—1181no—
259Q8—1181no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—1181no—
267D4—1182no—
268D4—1181no—
269D4—1182no—
270D4—2364no—
271D4—2368no—
272D4—2368no—
273D4—2362no—
274D4—2362no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—592no—
278C2×C4—592no—
279C2×C4—1184no—
280C2×C4—1181no—
281C2×C4—2362no—
282C2×C4D4×D5911yescharacteristic
283C2×C4D4×D5912yes—
284C2×C4C2×C22⋊D5912yes—
285C2×C4C23×D5911yescharacteristic
286C8—594no—
287C8C2×C22⋊D5914yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—4724no—
295C22—2364no—
296C22—2362no—
297C22—2361no—
298C22—2362no—
299C22#1427372312yes—
300C22C2×D4×D5911yescharacteristic
301C4—41no—
302C4—594no—
303C4—1182no—
304C4—2361no—
305C4#1427372312yes—
306C4C2×D4×D5911yescharacteristic
307C4C22×C22⋊D5911yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—4722no—
312C2—4721no—
313C2C2×#1427372311yescharacteristic
314C2AGL(1,ℤ/8)×D5912yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×AGL(1,ℤ/8)×C59C2
socleC22×C59C22×D4
hypercenterC2×AGL(1,ℤ/8)D59
hypocenterC59#3151
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14273755C2semidirect
2. C2×#14273753C2semidirect
3. C2×#14273752C2semidirect
4. #16837948C2semidirect
5. C2×#14273723C2semidirect
6. #16837790C2semidirect
7. #16837788C2semidirect
8. #16837777C2semidirect
9. #16837773C2semidirect
10. M16×Dic59C2semidirect
11. C2×AGL(1,ℤ/8)×C59C2semidirect
18. C2×#11745660C22semidirect
19. C2×#11745670C22semidirect
20. C2×#11745671C22semidirect
21. C2×D472C22semidirect
22. C2×#11745669C22semidirect
23. C2×#11745669C22semidirect
25. #14273725C22semidirect
29. #14273723C22semidirect
30. #14273716C22semidirect
31. #14273714C22semidirect
35. #14273637C22semidirect
36. C8×Dic59C22semidirect
40. C2×SD16×C59C22semidirect
41. C2×D8×C59C22semidirect
73. M16×C59D4semidirect
85. #11745664D4semidirect
128. C8×C59C2×D4semidirect
129. C59⋊C8C2×D4semidirect
149. Dic59AGL(1,ℤ/8)semidirect
161. C2×AGL(1,ℤ/8)D59semidirect
169. C59#3151semidirect
182. C2×SD16C2×D59semidirect
186. C2×D8C2×D59semidirect
238. M16C22⋊D59semidirect
287. C8C2×C22⋊D59semidirect