Finite Group Databasebeta

← #21433723

Subgroups — #21433723

Order 7552.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273759C211yesmaximal, characteristic, complemented
2C22×#11745659C211yesmaximal, characteristic, complemented
3#16837915C211yesmaximal, characteristic
4C22⋊Q8×D59C211yesmaximal, characteristic, complemented
5#16837659C211yesmaximal, characteristic, complemented
6#16837657C211yesmaximal, characteristic, complemented
7#16837651C211yesmaximal, characteristic, complemented
8#16837649C211yesmaximal, characteristic, complemented
9#16837641C211yesmaximal, characteristic, complemented
10#16837633C211yesmaximal, characteristic, complemented
11#16837631C211yesmaximal, characteristic, complemented
12#16837443C211yesmaximal, characteristic, complemented
13#16837409C211yesmaximal, characteristic, complemented
14#16837402C211yesmaximal, characteristic, complemented
15C59×#467C211yesmaximal, characteristic, complemented
16C2×#11745689C2211yescharacteristic
17C2×Q8×D59C2211yescharacteristic
18C22×C4×D59C2211yescharacteristic
19#14273759—24no—
20C2×#11745672C2211yescharacteristic, complemented
21C2×#11745660C2211yescharacteristic, complemented
22C2×#11745671C2211yescharacteristic, complemented
23C2×#11745659—24no—
24C2×#11745659C2211yescharacteristic
25C2×#11745659C2211yescharacteristic
26C2×#11745659C2211yescharacteristic
27C2×#11745659C2211yescharacteristic
28C22×C59⋊C8C2211yescharacteristic
29#14273712—21no—
30#14273707C2211yescharacteristic
31#14273706C2211yescharacteristic
32#14273705—21no—
33#14273704—21no—
34#14273703C2211yescharacteristic
35#14273703C2211yescharacteristic
36#14273702C2211yescharacteristic
37#14273701C2211yescharacteristic
38#14273697—21no—
39#14273689—21no—
40(C22⋊D59)⋊C4—21no—
41#14273685C2211yescharacteristic
42C4⋊C4×D59—21no—
43C4⋊C4×D59C2211yescharacteristic
44C22⋊C4×D59—21no—
45#14273660C2211yescharacteristic, complemented
46#14273659C2211yescharacteristic, complemented
47#14273658C2211yescharacteristic, complemented
48#14273652C2211yescharacteristic, complemented
49#14273651C2211yescharacteristic, complemented
50#14273651C2211yescharacteristic, complemented
51#14273650C2211yescharacteristic, complemented
52#14273649C2211yescharacteristic, complemented
53#14273647C2211yescharacteristic, complemented
54C2×Q16×C59C2211yescharacteristic, complemented
55C22×C8×C59C2211yescharacteristic
56C22⋊Q8×C59C2211yescharacteristic
57C22⋊Q8×C59C2211yescharacteristic
58#107×C59C2211yescharacteristic, complemented
59#103×C59C2211yescharacteristic, complemented
60#103×C59C2211yescharacteristic, complemented
61C23×D59D411yescharacteristic
62C2×Q8×C59C2311yescharacteristic
63C2×Q8×C59C2311yescharacteristic
64C22×C4×C59C2311yescharacteristic
65C2×C22⋊D59—21no—
66C22×Dic59D411yescharacteristic
67#11745689—22no—
68#11745689—22no—
69Q8×D59—22no—
70Q8×D59—22no—
71C2×D236—21no—
72C2×D236—21no—
73C2×D236C2311yescharacteristic
74C2×C4×D59—22no—
75C2×C4×D59—22no—
76C2×C4×D59—21no—
77C2×C4×D59—21no—
78C2×C4×D59—21no—
79C2×C4×D59D411yescharacteristic
80C2×C4×D59D411yescharacteristic
81C2×C4×D59C2311yescharacteristic
82C2×C4×D59C2311yescharacteristic
83C2×Dic118—21no—
84C2×Dic118C2311yescharacteristic
85Q16×C59—22no—
86C2×C8×C59—22no—
87C2×C8×C59C2311yescharacteristic
88C2×C8×C59C2311yescharacteristic
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59C2311yescharacteristic
92C4⋊C4×C59C2311yescharacteristic
93C22⋊C4×C59—21no—
94C22⋊C4×C59—21no—
95C22⋊Dic59—21no—
96#11745672—22no—
97#11745671—22no—
98(C2×D59)⋊C4—21no—
99(C2×D59)⋊C4—21no—
100(C2×D59)⋊C4—21no—
101(C2×D59)⋊C4—21no—
102(C2×D59)⋊C4—21no—
103(C2×D59)⋊C4—21no—
104C4⋊Dic59—21no—
105C4⋊Dic59C2311yescharacteristic
106C4⋊Dic59C2311yescharacteristic
107Dic59⋊C4—21no—
108Dic59⋊C4—21no—
109Dic59⋊C4—21no—
110Dic59⋊C4—21no—
111C4×Dic59—21no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8C2311yescharacteristic
114C2×C59⋊C8C2311yescharacteristic
115#11745660—22no—
116#11745659—22no—
117#11745659—22no—
118#11745659—22no—
119#11745659D414yescomplemented
120C23×C59C2×D411yescharacteristic
121C22×D59—22no—
122C22×D59—22no—
123C22×D59—21no—
124C22×D59—41no—
125C22×D59C2×D411yescharacteristic
126C22×D59C2×D411yescharacteristic
127Q8×C59—21no—
128Q8×C59—21no—
129Q8×C59—21no—
130Q8×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59C2×D411yescharacteristic
138C2×C4×C59C2411yescharacteristic
139C22⋊D59—42no—
140C2×Dic59—22no—
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59C2×D411yescharacteristic
145C2×Dic59C2×D411yescharacteristic
146D236—21no—
147D236—21no—
148D236—42no—
149D236—42no—
150C4×D59—22no—
151C4×D59—21no—
152C4×D59—21no—
153C4×D59—22no—
154C4×D59—42no—
155C4×D59—42no—
156C4×D59—42no—
157C4×D59C2×D412yes—
158C4×D59#4112yes—
159Dic118—21no—
160Dic118—21no—
161Dic118—42no—
162C8×C59—21no—
163C8×C59C2×D412yes—
164C59⋊C8—21no—
165C59⋊C8C2×D412yes—
166C22×C59—21no—
167C22×C59—21no—
168C22×C59#13712yescomplemented
169C22×C59C22×D411yescharacteristic
170C2×D59—21no—
171C2×D59—21no—
172C2×D59—21no—
173C2×D59—21no—
174C2×D59—21no—
175C2×D59—44no—
176C2×D59—41no—
177C2×D59—41no—
178C2×D59—41no—
179C2×D59#12112yes—
180C4×C59—21no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—41no—
185C4×C59C22×D411yescharacteristic
186C4×C59C2×#4111yescharacteristic
187Dic59—21no—
188Dic59—41no—
189Dic59—41no—
190Dic59—41no—
191Dic59#12112yes—
192#3056—591nomaximal, Sylow
193C2×C59—22no—
194C2×C59C2×#12111yescharacteristic
195C2×C59C2×#13711yescharacteristic
196C2×C59#57511yescharacteristic
197D59—22no—
198D59—41no—
199D59—81no—
200C2×#137—591no—
201C22×M16—591no—
202#525—591no—
203C2×C22⋊Q8—591no—
204#470—591no—
205#470—591no—
206#468—591no—
207#468—591no—
208#467—591no—
209#467D5911yescharacteristic, complemented
210#465—591no—
211#465—591no—
212#428—591no—
213#419—591no—
214#417—591no—
215C59#305611yescharacteristic, complemented, Sylow
216C2×#41—591no—
217C22×Q8—591no—
218C23×C4—591no—
219#137—1184no—
220C2×Q16—591no—
221C2×Q16C2×D5911yescharacteristic, complemented
222C2×SD16—591no—
223C2×SD16—591no—
224C2×M16—591no—
225C2×M16—591no—
226C2×M16—591no—
227C2×M16—591no—
228C2×M16—1184no—
229C22×C8—591no—
230C22×C8C2×D5911yescharacteristic
231#123—1181no—
232C22⋊Q8—591no—
233C22⋊Q8—591no—
234C22⋊Q8—591no—
235C22⋊Q8—591no—
236C22⋊Q8—1181no—
237C22⋊Q8—1181no—
238C22⋊Q8C2×D5911yescharacteristic
239C22⋊Q8C2×D5911yescharacteristic
240#121—591no—
241#121—591no—
242#120—1181no—
243C4×D4—1181no—
244C4×D4—1181no—
245#117—591no—
246C2×C4⋊C4—591no—
247C2×C4⋊C4—1181no—
248C2×C22⋊C4—1181no—
249#107—591no—
250#107C2×D5911yescharacteristic, complemented
251#106—591no—
252#106—591no—
253#103—591no—
254#103—591no—
255#103—591no—
256#103—591no—
257#103C2×D5911yescharacteristic, complemented
258#103C2×D5911yescharacteristic, complemented
259#102—591no—
260#102—591no—
261C24—591no—
262#41—1182no—
263#41—1182no—
264C2×Q8—591no—
265C2×Q8—1181no—
266C2×Q8—1182no—
267C2×Q8—1182no—
268C2×Q8C22×D5911yescharacteristic
269C2×Q8C22×D5911yescharacteristic
270C2×D4—591no—
271C2×D4—1181no—
272C2×D4—1181no—
273C2×D4—1181no—
274C22×C4—591no—
275C22×C4—591no—
276C22×C4—591no—
277C22×C4—591no—
278C22×C4—591no—
279C22×C4—1182no—
280C22×C4—1182no—
281C22×C4—1181no—
282C22×C4—1181no—
283C22×C4—1181no—
284C22×C4C22×D5911yescharacteristic
285Q16—22no—
286Q16—1182no—
287SD16—1182no—
288SD16—1182no—
289M16—594no—
290M16—1182no—
291M16—1182no—
292M16—1182no—
293C2×C8—22no—
294C2×C8—591no—
295C2×C8—591no—
296C2×C8—1182no—
297C2×C8C22×D5911yescharacteristic
298C2×C8C22×D5911yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—591no—
302C4⋊C4—591no—
303C4⋊C4—1181no—
304C4⋊C4—1181no—
305C4⋊C4—1181no—
306C4⋊C4—1181no—
307C4⋊C4—1181no—
308C4⋊C4C22×D5911yescharacteristic
309C4⋊C4C22×D5911yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1181no—
313C22⋊C4—1181no—
314C22⋊C4—1181no—
315C22⋊C4—1181no—
316C22⋊C4—1181no—
317C22⋊C4—1181no—
318C22⋊C4—1181no—
319C42—1181no—
320C23—591no—
321C23—591no—
322C23—1182no—
323C23—1182no—
324C23—1181no—
325C23—2361no—
326C23D4×D5911yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—1181no—
332Q8—1181no—
333Q8—2362no—
334D4—1181no—
335D4—1181no—
336D4—2362no—
337D4—2362no—
338D4—2362no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—591no—
346C2×C4—592no—
347C2×C4—592no—
348C2×C4—591no—
349C2×C4—1182no—
350C2×C4—1182no—
351C2×C4—1182no—
352C2×C4—1181no—
353C2×C4—1181no—
354C2×C4—1181no—
355C2×C4—1181no—
356C2×C4—1181no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4D4×D5911yescharacteristic
361C2×C4C23×D5911yescharacteristic
362C8—21no—
363C8—592no—
364C8—1181no—
365C8D4×D5912yes—
366C22—21no—
367C22—21no—
368C22—592no—
369C22—1181no—
370C22—1181no—
371C22—1181no—
372C22—1181no—
373C22—1181no—
374C22—2364no—
375C22—2361no—
376C22—2361no—
377C22—2361no—
378C22#1427375912yescomplemented
379C22C2×D4×D5911yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—592no—
386C4—1181no—
387C4—2361no—
388C4—2361no—
389C4—2361no—
390C4C2×D4×D5911yescharacteristic
391C4#41×D5911yescharacteristic
392C2—22no—
393C2—1182no—
394C2—4721no—
395C2—2361no—
396C2#121×D5911yescharacteristic
397C2C2×#1427375911yescharacteristic
398C2#1683819911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#467C2
socleC22×C59C22×D4
hypercenter#467D59
hypocenterC59#3056
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#14273759C2semidirect
2. C22×#11745659C2semidirect
4. C22⋊Q8×D59C2semidirect
5. #16837659C2semidirect
6. #16837657C2semidirect
7. #16837651C2semidirect
8. #16837649C2semidirect
9. #16837641C2semidirect
10. #16837633C2semidirect
11. #16837631C2semidirect
12. #16837443C2semidirect
13. #16837409C2semidirect
14. #16837402C2semidirect
15. C59×#467C2semidirect
20. C2×#11745672C22semidirect
21. C2×#11745660C22semidirect
22. C2×#11745671C22semidirect
45. #14273660C22semidirect
46. #14273659C22semidirect
47. #14273658C22semidirect
48. #14273652C22semidirect
49. #14273651C22semidirect
50. #14273651C22semidirect
51. #14273650C22semidirect
52. #14273649C22semidirect
53. #14273647C22semidirect
54. C2×Q16×C59C22semidirect
58. #107×C59C22semidirect
59. #103×C59C22semidirect
60. #103×C59C22semidirect
119. #11745659D4semidirect
168. C22×C59#137semidirect
209. #467D59semidirect
215. C59#3056semidirect
221. C2×Q16C2×D59semidirect
250. #107C2×D59semidirect
257. #103C2×D59semidirect
258. #103C2×D59semidirect
378. C22#14273759semidirect