Finite Group Databasebeta

← #21435013

Subgroups — #21435013

Order 7552.

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

normal subgroups only

This group has 532 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
1#16837949C211yesmaximal, characteristic, complemented
2#127×D59C211yesmaximal, characteristic, complemented
3#16837777C212yesmaximal, complemented
4#16837747C212yesmaximal, complemented
5#16837598C211yesmaximal, characteristic, complemented
6#16837597C211yesmaximal, characteristic, complemented
7#16837594C211yesmaximal, characteristic, complemented
8#16837537C212yesmaximal, complemented
9#16837534C212yesmaximal, complemented
10#16837454C211yesmaximal, characteristic, complemented
11C59×#459C211yesmaximal, characteristic, complemented
12C2×#11745685—21no—
13C2×D4×D59—21no—
14C2×D4×D59—21no—
15C2×D4×D59C2212yes—
16C2×D472C2212yescomplemented
17C2×#11745669C2212yescomplemented
18C2×#11745669C2212yescomplemented
19#14273726C2211yescharacteristic, complemented
20#14273724C2211yescharacteristic
21#14273723—21no—
22#14273723—21no—
23#14273723C2212yes—
24#14273723C2211yescharacteristic, complemented
25#14273713—21no—
26#14273702C2211yescharacteristic
27#14273700—21no—
28#14273698C2211yescharacteristic
29C4×D236C2211yescharacteristic, complemented
30C4×C22⋊D59—21no—
31C42×D59C2211yescharacteristic
32C4⋊(C59⋊C8)C2211yescharacteristic, complemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273647C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37C8×Dic59C2212yescomplemented
38C2×D8×C59C2212yescomplemented
39#127×C59C2211yescharacteristic
40C4×D4×C59C2211yescharacteristic, complemented
41C4⋊C8×C59C2211yescharacteristic, complemented
42#102×C59C2212yescomplemented
43C23×D59—22no—
44C2×D4×C59—21no—
45C2×D4×C59—21no—
46C2×D4×C59C2312yes—
47C2×D4×C59C2311yescharacteristic
48C22×C4×C59—21no—
49C2×C22⋊D59—21no—
50C2×C22⋊D59—21no—
51C2×C22⋊D59—21no—
52C2×C22⋊D59—22no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—22no—
55D4×D59—42no—
56D4×D59—42no—
57D4×D59—44no—
58D4×D59—44no—
59#11745685—44no—
60C2×D236—21no—
61C2×D236—21no—
62C2×D236—21no—
63C2×D236C2312yes—
64C2×D236C2311yescharacteristic
65C2×C4×D59—21no—
66C2×C4×D59D411yescharacteristic
67C2×C4×D59D411yescharacteristic
68C2×C4×D59C2311yescharacteristic
69C2×Dic118—21no—
70D8×C59—42no—
71C2×C8×C59C2312yes—
72C4⋊C4×C59C2311yescharacteristic
73C22⋊C4×C59—21no—
74C42×C59C2311yescharacteristic
75C22⋊Dic59—21no—
76#11745669—42no—
77#11745669—42no—
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—21no—
80C4⋊Dic59C2311yescharacteristic
81Dic59⋊C4—21no—
82C4×Dic59D411yescharacteristic
83C4×Dic59D411yescharacteristic
84C4×Dic59C2311yescharacteristic
85C2×C59⋊C8C2312yes—
86D472—42no—
87C23×C59—21no—
88C23×C59—22no—
89C22×D59—21no—
90C22×D59—22no—
91C22×D59—44no—
92C22×D59—44no—
93C22×D59—44no—
94C22×D59C2×D411yescharacteristic
95D4×C59—21no—
96D4×C59—22no—
97D4×C59—21no—
98D4×C59—22no—
99D4×C59—42no—
100D4×C59—42no—
101C2×C4×C59—21no—
102C2×C4×C59—42no—
103C2×C4×C59C2×D411yescharacteristic
104C2×C4×C59C2×D411yescharacteristic
105C2×C4×C59C2411yescharacteristic
106C22⋊D59—44no—
107C22⋊D59—42no—
108C22⋊D59—42no—
109C22⋊D59—44no—
110C22⋊D59—44no—
111C22⋊D59—42no—
112C2×Dic59—21no—
113C2×Dic59C2×D411yescharacteristic
114C2×Dic59C2×D411yescharacteristic
115C2×Dic59C2×D411yescharacteristic
116D236—21no—
117D236—22no—
118D236—21no—
119D236—22no—
120D236—42no—
121D236—42no—
122D236—42no—
123C4×D59—21no—
124C4×D59—21no—
125C4×D59—22no—
126C4×D59—42no—
127C4×D59D814yescomplemented
128Dic118—42no—
129C8×C59—22no—
130C59⋊C8—22no—
131C22×C59—42no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—41no—
137C22×C59C22×D411yescharacteristic
138C2×D59—21no—
139C2×D59—21no—
140C2×D59—42no—
141C2×D59—41no—
142C2×D59—41no—
143C2×D59—42no—
144C2×D59—42no—
145C2×D59—41no—
146C2×D59—88no—
147C2×D59C2×D812yes—
148C4×C59—21no—
149C4×C59—41no—
150C4×C59C2×D812yes—
151C4×C59H(5,𝔽2)11yescharacteristic
152C4×C59H(5,𝔽2)11yescharacteristic
153Dic59—21no—
154Dic59—41no—
155Dic59C2×D812yes—
156Dic59AGL(1,ℤ/8)12yes—
157#3205—591nomaximal, Sylow
158C2×C59—82no—
159C2×C59—81no—
160C2×C59#53511yescharacteristic
161C2×C59C22×D811yescharacteristic
162C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
163D59—22no—
164D59—81no—
165D59—82no—
166#532—591no—
167C2×#127—591no—
168#493—592no—
169#486—592no—
170#459—591no—
171#459—591no—
172#459—591no—
173#459D5911yescharacteristic, complemented
174#447—592no—
175#447—592no—
176#431—591no—
177C59#320511yescharacteristic, complemented, Sylow
178C2×#41—1181no—
179C22×D4—592no—
180C22×D4—1181no—
181C22×D4—1181no—
182C2×D8—592no—
183C2×D8—592no—
184C2×D8—592no—
185C2×D8C2×D5912yescomplemented
186C4⋊Q8—591no—
187#127—592no—
188#127—591no—
189#127—591no—
190#127—1181no—
191#127—1181no—
192#127C2×D5911yescharacteristic
193#124—1181no—
194#121—591no—
195#121—591no—
196#121—1181no—
197C4×D4—591no—
198C4×D4—1181no—
199C4×D4C2×D5911yescharacteristic, complemented
200C2×C42—591no—
201C4⋊C8—591no—
202C4⋊C8C2×D5911yescharacteristic, complemented
203#102—592no—
204#102—592no—
205#102—592no—
206#102C2×D5912yescomplemented
207C22⋊C8—592no—
208C4×C8—592no—
209C24—1182no—
210#41—2364no—
211C2×Q8—1181no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—592no—
215C2×D4—591no—
216C2×D4—1181no—
217C2×D4—1181no—
218C2×D4—1181no—
219C2×D4—1182no—
220C2×D4—1182no—
221C2×D4—1182no—
222C2×D4—1181no—
223C2×D4—1181no—
224C2×D4—1181no—
225C2×D4—2364no—
226C2×D4—2362no—
227C2×D4—2362no—
228C2×D4—2364no—
229C2×D4C22×D5912yes—
230C2×D4C22×D5911yescharacteristic
231C22×C4—21no—
232C22×C4—591no—
233C22×C4—591no—
234C22×C4—591no—
235C22×C4—1181no—
236D8—42no—
237D8—2362no—
238D8—2362no—
239D8—2362no—
240C2×C8—592no—
241C2×C8C22×D5912yes—
242C4⋊C4—591no—
243C4⋊C4—1181no—
244C4⋊C4C22×D5911yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—1181no—
247C22⋊C4—1181no—
248C22⋊C4—1181no—
249C42—591no—
250C42—591no—
251C42—591no—
252C42C22×D5911yescharacteristic
253C23—21no—
254C23—22no—
255C23—591no—
256C23—1181no—
257C23—1182no—
258C23—2364no—
259C23—2364no—
260C23—2364no—
261Q8—2362no—
262D4—21no—
263D4—22no—
264D4—21no—
265D4—22no—
266D4—42no—
267D4—42no—
268D4—1181no—
269D4—1182no—
270D4—1181no—
271D4—1182no—
272D4—2362no—
273D4—2362no—
274D4—2362no—
275D4—2364no—
276D4—2364no—
277D4—2362no—
278D4—2364no—
279D4—2362no—
280D4—2362no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—591no—
284C2×C4—591no—
285C2×C4—594no—
286C2×C4—591no—
287C2×C4—1182no—
288C2×C4—1181no—
289C2×C4—1181no—
290C2×C4—1181no—
291C2×C4—2362no—
292C2×C4D4×D5911yescharacteristic
293C2×C4D4×D5911yescharacteristic
294C2×C4C23×D5911yescharacteristic
295C8—22no—
296C8—1182no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—592no—
304C22—1181no—
305C22—1181no—
306C22—4728no—
307C22—2361no—
308C22—2362no—
309C22—2362no—
310C22—2361no—
311C22—2362no—
312C22—2361no—
313C22C2×D4×D5911yescharacteristic
314C4—21no—
315C4—41no—
316C4—592no—
317C4—592no—
318C4—1181no—
319C4—2361no—
320C4D8×D5912yes—
321C4#1427377211yescharacteristic
322C4#1427377311yescharacteristic
323C2—82no—
324C2—81no—
325C2—1182no—
326C2—4722no—
327C2—4721no—
328C2#1683796811yescharacteristic
329C2C2×D8×D5911yescharacteristic
330C2AGL(1,ℤ/8)×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#459C2
socleC22×C59C22×D4
hypercenter#459D59
hypocenterC59#3205
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16837949C2semidirect
2. #127×D59C2semidirect
3. #16837777C2semidirect
4. #16837747C2semidirect
5. #16837598C2semidirect
6. #16837597C2semidirect
7. #16837594C2semidirect
8. #16837537C2semidirect
9. #16837534C2semidirect
10. #16837454C2semidirect
11. C59×#459C2semidirect
16. C2×D472C22semidirect
17. C2×#11745669C22semidirect
18. C2×#11745669C22semidirect
19. #14273726C22semidirect
24. #14273723C22semidirect
29. C4×D236C22semidirect
32. C4⋊(C59⋊C8)C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273647C22semidirect
37. C8×Dic59C22semidirect
38. C2×D8×C59C22semidirect
40. C4×D4×C59C22semidirect
41. C4⋊C8×C59C22semidirect
42. #102×C59C22semidirect
127. C4×D59D8semidirect
173. #459D59semidirect
177. C59#3205semidirect
185. C2×D8C2×D59semidirect
199. C4×D4C2×D59semidirect
202. C4⋊C8C2×D59semidirect
206. #102C2×D59semidirect