Finite Group Databasebeta

← #21433718

Subgroups — #21433718

Order 7552.

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

normal subgroups only

This group has 548 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
1C2×#14273752C211yesmaximal, characteristic, complemented
2C2×M16×D59C211yesmaximal, characteristic, complemented
3#16837909C211yesmaximal, characteristic, complemented
4C2×#14273702C211yesmaximal, characteristic, complemented
5#16837654C212yesmaximal, complemented
6#16837651C212yesmaximal, complemented
7#16837637C212yesmaximal, complemented
8#16837634C212yesmaximal, complemented
9#16837442C211yesmaximal, characteristic, complemented
10#16837408C211yesmaximal, characteristic, complemented
11C59×#417C211yesmaximal, characteristic, complemented
12C22×D236—21no—
13C22×D236C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15#14273752—24no—
16C2×#11745671C2212yescomplemented
17C2×#11745669C2212yescomplemented
18C2×#11745664C2211yescharacteristic
19C2×#11745659C2212yescomplemented
20M16×D59—24no—
21C2×C8×D59C2212yescomplemented
22#14273708—21no—
23#14273703C2211yescharacteristic, complemented
24#14273703C2211yescharacteristic, complemented
25#14273702—21no—
26#14273702—21no—
27#14273702C2212yes—
28#14273702C2212yes—
29#14273698C2211yescharacteristic, complemented
30#14273698C2211yescharacteristic, complemented
31#14273695—21no—
32C4×C22⋊D59—21no—
33C4×C22⋊D59—21no—
34#14273686C2211yescharacteristic
35C2×(C2×D59)⋊C4—21no—
36#14273660C2212yescomplemented
37#14273658C2212yescomplemented
38#14273649C2212yescomplemented
39#14273649C2212yescomplemented
40C2×#41×C59C2211yescharacteristic
41C2×M16×C59C2211yescharacteristic
42C2×C4⋊C4×C59C2211yescharacteristic
43#103×C59C2212yescomplemented
44#102×C59C2212yescomplemented
45C23×D59—21no—
46C23×D59D411yescharacteristic
47#41×C59—22no—
48#41×C59—22no—
49C2×Q8×C59C2311yescharacteristic
50C2×D4×C59—21no—
51C2×D4×C59C2311yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59C2311yescharacteristic
55C2×C22⋊D59—21no—
56C2×C22⋊D59—21no—
57C22×Dic59D411yescharacteristic
58C2×D236—22no—
59C2×D236—22no—
60C2×D236—22no—
61C2×D236—22no—
62C2×D236—21no—
63C2×D236—44no—
64C2×D236C2312yes—
65C2×C4×D59—22no—
66C2×C4×D59—22no—
67C2×C4×D59D411yescharacteristic
68C2×C4×D59D411yescharacteristic
69C2×C4×D59C2311yescharacteristic
70C2×C4×D59C2311yescharacteristic
71M16×C59—22no—
72C2×C8×C59C2312yes—
73C4⋊C4×C59—21no—
74C4⋊C4×C59C2312yes—
75C22⋊Dic59—21no—
76C22⋊Dic59—21no—
77C22⋊Dic59—21no—
78#11745671—24no—
79#11745669—24no—
80(C2×D59)⋊C4—21no—
81(C2×D59)⋊C4—21no—
82(C2×D59)⋊C4—22no—
83(C2×D59)⋊C4—22no—
84C4⋊Dic59C2311yescharacteristic
85C4⋊Dic59C2311yescharacteristic
86Dic59⋊C4—21no—
87Dic59⋊C4—21no—
88C4×Dic59—21no—
89#11745664D414yescomplemented
90C2×C59⋊C8C2312yes—
91#11745659—24no—
92C8×D59—24no—
93C23×C59—21no—
94C23×C59C2×D411yescharacteristic
95C22×D59—22no—
96C22×D59—22no—
97C22×D59—22no—
98C22×D59—41no—
99C22×D59—42no—
100C22×D59—42no—
101C22×D59—42no—
102C22×D59C2×D411yescharacteristic
103C22×D59C2×D411yescharacteristic
104Q8×C59—21no—
105Q8×C59—21no—
106D4×C59—21no—
107D4×C59—21no—
108D4×C59—42no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59C2×D411yescharacteristic
114C2×C4×C59C2×D412yes—
115C2×C4×C59#4112yes—
116C2×C4×C59C2411yescharacteristic
117C22⋊D59—42no—
118C22⋊D59—42no—
119C2×Dic59—22no—
120C2×Dic59—21no—
121C2×Dic59—21no—
122C2×Dic59C2×D411yescharacteristic
123C2×Dic59C2×D411yescharacteristic
124D236—22no—
125D236—22no—
126D236—44no—
127D236—44no—
128D236—42no—
129C4×D59—22no—
130C4×D59—22no—
131C4×D59—21no—
132C4×D59—21no—
133C4×D59—21no—
134C4×D59—21no—
135C8×C59—22no—
136C59⋊C8C2×D414yescomplemented
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59#12112yes—
143C22×C59C22×D411yescharacteristic
144C2×D59—21no—
145C2×D59—21no—
146C2×D59—21no—
147C2×D59—21no—
148C2×D59—21no—
149C2×D59—44no—
150C2×D59—42no—
151C2×D59—42no—
152C2×D59—42no—
153C2×D59—84no—
154C2×D59AGL(1,ℤ/8)12yescomplemented
155C4×C59—41no—
156C4×C59—42no—
157C4×C59#12112yes—
158C4×C59C22×D411yescharacteristic
159C4×C59C2×#4111yescharacteristic
160Dic59—21no—
161Dic59—21no—
162Dic59—41no—
163Dic59—41no—
164#3055—591nomaximal, Sylow
165C2×C59—22no—
166C2×C59—81no—
167C2×C59C2×#12111yescharacteristic
168C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C59#57511yescharacteristic
170D59—22no—
171D59—41no—
172D59—82no—
173C2×AGL(1,ℤ/8)—591no—
174C22×M16—591no—
175#525—591no—
176C2×#121—591no—
177#469—592no—
178#468—592no—
179#466—592no—
180#465—592no—
181#428—591no—
182#418—591no—
183#417D5911yescharacteristic, complemented
184C59#305511yescharacteristic, complemented, Sylow
185C2×#41C2×D5911yescharacteristic
186C22×D4—591no—
187C22×D4—1181no—
188C23×C4—591no—
189AGL(1,ℤ/8)—1184no—
190C2×SD16—592no—
191C2×D8—592no—
192C2×M16—591no—
193C2×M16—592no—
194C2×M16—1184no—
195C2×M16C2×D5911yescharacteristic
196C22×C8—592no—
197#123—1181no—
198C22⋊Q8—591no—
199C22⋊Q8—591no—
200#121—591no—
201#121—591no—
202#121—592no—
203#121—592no—
204#121—1181no—
205#121—1181no—
206#120—1181no—
207C4×D4—1181no—
208C4×D4—1181no—
209#117—591no—
210C2×C4⋊C4C2×D5911yescharacteristic
211C2×C22⋊C4—1181no—
212#107—592no—
213#106—592no—
214#103C2×D5912yescomplemented
215#102—592no—
216#102—592no—
217#102C2×D5912yescomplemented
218C24—591no—
219C24—1181no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D5911yescharacteristic
223C2×D4—21no—
224C2×D4—592no—
225C2×D4—1182no—
226C2×D4—1182no—
227C2×D4—1181no—
228C2×D4—1181no—
229C2×D4—1182no—
230C2×D4—1182no—
231C2×D4—1181no—
232C2×D4—2364no—
233C2×D4C22×D5911yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—591no—
237C22×C4—591no—
238C22×C4—591no—
239C22×C4—591no—
240C22×C4—591no—
241C22×C4—1182no—
242C22×C4—1182no—
243C22×C4C22×D5911yescharacteristic
244SD16—1184no—
245D8—1184no—
246M16—22no—
247M16—594no—
248M16—1184no—
249C2×C8—592no—
250C2×C8—1184no—
251C2×C8C22×D5912yes—
252C4⋊C4—21no—
253C4⋊C4—591no—
254C4⋊C4—591no—
255C4⋊C4—1181no—
256C4⋊C4—1181no—
257C4⋊C4C22×D5912yes—
258C22⋊C4—1182no—
259C22⋊C4—1182no—
260C22⋊C4—1181no—
261C22⋊C4—1181no—
262C22⋊C4—1181no—
263C22⋊C4—1181no—
264C22⋊C4—1181no—
265C42—1181no—
266C23—21no—
267C23—591no—
268C23—591no—
269C23—1182no—
270C23—1182no—
271C23—1182no—
272C23—2361no—
273C23—2362no—
274C23—2362no—
275C23—2362no—
276C23D4×D5911yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—1182no—
283D4—1182no—
284D4—2364no—
285D4—2364no—
286D4—2362no—
287D4—2362no—
288D4—2362no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—591no—
294C2×C4—591no—
295C2×C4—1182no—
296C2×C4—1182no—
297C2×C4—1182no—
298C2×C4—1181no—
299C2×C4—1181no—
300C2×C4—1181no—
301C2×C4—1181no—
302C2×C4—1181no—
303C2×C4—1181no—
304C2×C4#1174568512yes—
305C2×C4D4×D5911yescharacteristic
306C2×C4D4×D5912yes—
307C2×C4C23×D5911yescharacteristic
308C8—22no—
309C8—594no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—592no—
316C22—1181no—
317C22—1181no—
318C22—1181no—
319C22—1181no—
320C22—1181no—
321C22—4724no—
322C22—2364no—
323C22—2362no—
324C22—2362no—
325C22—2362no—
326C22#1427370012yes—
327C22C2×D4×D5911yescharacteristic
328C4—41no—
329C4—42no—
330C4—1181no—
331C4—1181no—
332C4—2361no—
333C4—2361no—
334C4#1427370012yes—
335C4C2×D4×D5911yescharacteristic
336C4C2×#1174568511yescharacteristic
337C2—22no—
338C2—81no—
339C2—1182no—
340C2—4722no—
341C2—2361no—
342C2C2×#1427370011yescharacteristic
343C2AGL(1,ℤ/8)×D5911yescharacteristic
344C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#417C2
socleC22×C59C22×D4
hypercenter#417D59
hypocenterC59#3055
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#14273752C2semidirect
2. C2×M16×D59C2semidirect
3. #16837909C2semidirect
4. C2×#14273702C2semidirect
5. #16837654C2semidirect
6. #16837651C2semidirect
7. #16837637C2semidirect
8. #16837634C2semidirect
9. #16837442C2semidirect
10. #16837408C2semidirect
11. C59×#417C2semidirect
16. C2×#11745671C22semidirect
17. C2×#11745669C22semidirect
19. C2×#11745659C22semidirect
21. C2×C8×D59C22semidirect
23. #14273703C22semidirect
24. #14273703C22semidirect
29. #14273698C22semidirect
30. #14273698C22semidirect
36. #14273660C22semidirect
37. #14273658C22semidirect
38. #14273649C22semidirect
39. #14273649C22semidirect
43. #103×C59C22semidirect
44. #102×C59C22semidirect
89. #11745664D4semidirect
136. C59⋊C8C2×D4semidirect
154. C2×D59AGL(1,ℤ/8)semidirect
183. #417D59semidirect
184. C59#3055semidirect
214. #103C2×D59semidirect
217. #102C2×D59semidirect