Finite Group Databasebeta

← #21438634

Subgroups — #21438634

Order 7552.

1024 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16838152C212yesmaximal, complemented
2#16838146C212yesmaximal, complemented
3#16838111C212yesmaximal, complemented
4#16838104C212yesmaximal, complemented
5#16838102C212yesmaximal, complemented
6#16838052C212yesmaximal, complemented
7#16838051C214yesmaximal, complemented
8#16838029C212yesmaximal, complemented
9#16838006C212yesmaximal, complemented
10#16837997C212yesmaximal, complemented
11#16837994C214yesmaximal, complemented
12#16837979C212yesmaximal, complemented
13#16837936C211yesmaximal, characteristic, complemented
14C2×#14273719C211yesmaximal, characteristic, complemented
15C59×#532C211yesmaximal, characteristic, complemented
16C2×#11745687C2214yes—
17#14273726C2212yes—
18#14273725C2214yes—
19#14273721C2212yes—
20#14273720C2214yes—
21#14273720C2214yes—
22#14273719C2212yes—
23#14273719C2211yescharacteristic
24#14273719C2212yes—
25#14273719C2212yes—
26#14273719C2212yes—
27#14273718C2212yes—
28#14273717C2214yes—
29#14273715C2211yescharacteristic
30#14273711C2214yes—
31#14273710C2212yes—
32#14273710C2214yes—
33#14273709C2214yes—
34#14273708C2214yes—
35#14273708C2214yes—
36#14273708C2214yes—
37#14273706C2211yescharacteristic
38#14273706C2214yes—
39#14273705C2214yes—
40#14273704C2214yes—
41#14273704C2214yes—
42#14273704C2214yes—
43#14273703C2214yes—
44#14273701C2211yescharacteristic
45#14273700C2214yes—
46#14273699C2214yes—
47#14273699C2214yes—
48#14273699C2214yes—
49#14273698C2212yes—
50#14273698C2212yes—
51C4×Dic118C2211yescharacteristic
52C4×D236C2211yescharacteristic
53D4×Dic59C2214yes—
54D4×Dic59C2212yes—
55D4×Dic59C2212yes—
56C4×C22⋊D59C2212yes—
57C4×C22⋊D59C2214yes—
58#14273685C2212yes—
59#14273684C2212yes—
60C2×C4⋊Dic59C2211yescharacteristic
61C2×C4⋊Dic59C2211yescharacteristic
62C2×Dic59⋊C4C2212yes—
63C2×Dic59⋊C4C2212yes—
64C2×Dic59⋊C4C2214yes—
65C4⋊C4×D59C2212yes—
66C2×#41×C59C2212yes—
67C4⋊Q8×C59C2211yescharacteristic
68#127×C59C2211yescharacteristic
69#124×C59C2212yes—
70#121×C59C2212yes—
71#121×C59C2212yes—
72C4×D4×C59C2212yes—
73C4×D4×C59C2212yes—
74C2×C42×C59C2211yescharacteristic
75#41×C59—44no—
76C2×Q8×C59C2312yes—
77C2×D4×C59C2312yes—
78C2×D4×C59C2314yes—
79C2×D4×C59C2312yes—
80C2×D4×C59C2312yes—
81C22×C4×C59C2311yescharacteristic
82C22×C4×C59C2311yescharacteristic
83C22×C4×C59C2311yescharacteristic
84C22×C4×C59C2314yes—
85C2×C22⋊D59C2312yes—
86C2×C22⋊D59C2314yes—
87C2×C22⋊D59C2314yes—
88C22×Dic59C2312yes—
89C22×Dic59C2312yes—
90C22×Dic59C2314yes—
91C22×Dic59C2314yes—
92#11745687—48no—
93C2×D236C2311yescharacteristic
94C2×C4×D59C2312yes—
95C2×C4×D59C2314yes—
96C2×Dic118C2311yescharacteristic
97C2×Dic118C2314yes—
98C4⋊C4×C59C2312yes—
99C4⋊C4×C59C2312yes—
100C22⋊C4×C59C2314yes—
101C22⋊C4×C59C2314yes—
102C42×C59C2311yescharacteristic
103C42×C59C2311yescharacteristic
104C42×C59C2311yescharacteristic
105C42×C59C2311yescharacteristic
106C22⋊Dic59C2312yes—
107C22⋊Dic59C2314yes—
108C22⋊Dic59C2314yes—
109C22⋊Dic59C2314yes—
110C22⋊Dic59C2314yes—
111(C2×D59)⋊C4C2312yes—
112(C2×D59)⋊C4C2312yes—
113(C2×D59)⋊C4C2314yes—
114(C2×D59)⋊C4C2312yes—
115(C2×D59)⋊C4C2312yes—
116C4⋊Dic59C2312yes—
117C4⋊Dic59C2312yes—
118C4⋊Dic59C2312yes—
119C4⋊Dic59C2312yes—
120C4⋊Dic59C2314yes—
121C4⋊Dic59C2311yescharacteristic
122C4⋊Dic59C2311yescharacteristic
123Dic59⋊C4C2314yes—
124Dic59⋊C4C2314yes—
125Dic59⋊C4C2312yes—
126Dic59⋊C4C2314yes—
127Dic59⋊C4C2314yes—
128Dic59⋊C4C2314yes—
129Dic59⋊C4C2314yes—
130Dic59⋊C4C2314yes—
131Dic59⋊C4C2312yes—
132Dic59⋊C4C2312yes—
133Dic59⋊C4C2312yes—
134C4×Dic59C2312yes—
135C4×Dic59C2314yes—
136C23×C59C2414yes—
137C23×C59C2411yescharacteristic
138C22×D59C2412yes—
139Q8×C59—42no—
140D4×C59—28no—
141D4×C59—24no—
142D4×C59—42no—
143D4×C59—42no—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—44no—
148C2×C4×C59C2414yes—
149C2×C4×C59C2411yescharacteristic
150C2×C4×C59C2411yescharacteristic
151C2×C4×C59C2411yescharacteristic
152C2×C4×C59C2411yescharacteristic
153C2×C4×C59C2411yescharacteristic
154C2×C4×C59C2411yescharacteristic
155C22⋊D59—24no—
156C22⋊D59—44no—
157C22⋊D59—44no—
158C2×Dic59—24no—
159C2×Dic59—44no—
160C2×Dic59—44no—
161C2×Dic59—42no—
162C2×Dic59C2412yes—
163C2×Dic59C2414yes—
164C2×Dic59C2412yes—
165C2×Dic59C2412yes—
166C2×Dic59C2414yes—
167D236—41no—
168C4×D59—24no—
169C4×D59—44no—
170Dic118—41no—
171Dic118—44no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—24no—
175C22×C59—24no—
176C22×C59—24no—
177C22×C59H(5,𝔽2)12yes—
178C22×C59C2511yescharacteristic
179C2×D59—22no—
180C2×D59—22no—
181C2×D59—22no—
182C4×C59—24no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C592−1+412yes—
189Dic59—22no—
190Dic59—24no—
191Dic59—22no—
192Dic59—22no—
193Dic59—24no—
194#3559—591nomaximal, Sylow
195C2×C59—22no—
196C2×C59—44no—
197C2×C59C2×H(5,𝔽2)11yescharacteristic
198C2×C59C2×2−1+411yescharacteristic
199C2×C59#58511yescharacteristic
200D59—42no—
201#563—592no—
202#562—592no—
203#554—592no—
204#553—592no—
205#553—592no—
206#547—594no—
207#547—592no—
208#544—592no—
209#540—592no—
210#539—592no—
211#539—594no—
212#537—592no—
213#532D5911yescharacteristic, complemented
214#529—591no—
215C2×#125—591no—
216C59#355911yescharacteristic, complemented, Sylow
217C2×#41—594no—
218C2×#41C2×D5912yes—
219C4⋊Q8—594no—
220C4⋊Q8—592no—
221C4⋊Q8C2×D5911yescharacteristic
222#127C2×D5911yescharacteristic
223#126—594no—
224#126—594no—
225#126—592no—
226#125—594no—
227#125—592no—
228#125—592no—
229#125—592no—
230#125—592no—
231#125—592no—
232#125—591no—
233#124—591no—
234#124C2×D5912yes—
235#123—594no—
236#123—594no—
237#123—594no—
238#123—592no—
239#123—594no—
240#123—594no—
241#123—594no—
242C22⋊Q8—594no—
243C22⋊Q8—591no—
244C22⋊Q8—594no—
245C22⋊Q8—594no—
246C22⋊Q8—594no—
247C22⋊Q8—594no—
248C22⋊Q8—594no—
249#121—594no—
250#121—594no—
251#121—592no—
252#121—594no—
253#121—592no—
254#121—591no—
255#121—594no—
256#121C2×D5912yes—
257#121C2×D5912yes—
258C4×Q8—591no—
259C4×D4—594no—
260C4×D4—592no—
261C4×D4—592no—
262C4×D4—592no—
263C4×D4—594no—
264C4×D4—591no—
265C4×D4C2×D5912yes—
266C4×D4C2×D5912yes—
267#117—592no—
268#117—592no—
269C2×C4⋊C4—592no—
270C2×C4⋊C4—592no—
271C2×C4⋊C4—591no—
272C2×C4⋊C4—591no—
273C2×C4⋊C4—594no—
274C2×C4⋊C4—592no—
275C2×C42C2×D5911yescharacteristic
276#41—44no—
277#41—2368no—
278C2×Q8—591no—
279C2×Q8—594no—
280C2×Q8C22×D5912yes—
281C2×D4—592no—
282C2×D4—594no—
283C2×D4—594no—
284C2×D4—591no—
285C2×D4C22×D5912yes—
286C2×D4C22×D5914yes—
287C2×D4C22×D5912yes—
288C2×D4C22×D5912yes—
289C22×C4—592no—
290C22×C4—592no—
291C22×C4—594no—
292C22×C4—594no—
293C22×C4—592no—
294C22×C4—594no—
295C22×C4C22×D5911yescharacteristic
296C22×C4C22×D5911yescharacteristic
297C22×C4C22×D5911yescharacteristic
298C22×C4C22×D5914yes—
299C4⋊C4—592no—
300C4⋊C4—594no—
301C4⋊C4—592no—
302C4⋊C4—592no—
303C4⋊C4—594no—
304C4⋊C4—594no—
305C4⋊C4—594no—
306C4⋊C4—594no—
307C4⋊C4—592no—
308C4⋊C4—594no—
309C4⋊C4—594no—
310C4⋊C4—592no—
311C4⋊C4—592no—
312C4⋊C4—592no—
313C4⋊C4—594no—
314C4⋊C4—592no—
315C4⋊C4—591no—
316C4⋊C4—591no—
317C4⋊C4C22×D5912yes—
318C4⋊C4C22×D5912yes—
319C22⋊C4—592no—
320C22⋊C4—594no—
321C22⋊C4—594no—
322C22⋊C4—594no—
323C22⋊C4—594no—
324C22⋊C4—592no—
325C22⋊C4—594no—
326C22⋊C4—592no—
327C22⋊C4—592no—
328C22⋊C4—592no—
329C22⋊C4C22×D5914yes—
330C22⋊C4C22×D5914yes—
331C42—592no—
332C42—594no—
333C42C22×D5911yescharacteristic
334C42C22×D5911yescharacteristic
335C42C22×D5911yescharacteristic
336C42C22×D5911yescharacteristic
337C23—592no—
338C23C23×D5914yes—
339C23C23×D5911yescharacteristic
340Q8—42no—
341Q8—2361no—
342Q8—2364no—
343D4—28no—
344D4—24no—
345D4—42no—
346D4—42no—
347D4—1184no—
348D4—2361no—
349D4—2364no—
350D4—2364no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—592no—
356C2×C4—592no—
357C2×C4—594no—
358C2×C4—592no—
359C2×C4—594no—
360C2×C4—1184no—
361C2×C4—1184no—
362C2×C4—2364no—
363C2×C4—2362no—
364C2×C4—2364no—
365C2×C4—2364no—
366C2×C4C23×D5914yes—
367C2×C4C23×D5911yescharacteristic
368C2×C4C23×D5911yescharacteristic
369C2×C4C23×D5911yescharacteristic
370C2×C4C23×D5911yescharacteristic
371C2×C4C23×D5911yescharacteristic
372C2×C4C23×D5911yescharacteristic
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—1182no—
379C22—1182no—
380C22—1182no—
381C22#1427377212yes—
382C22C24×D5911yescharacteristic
383C4—21no—
384C4—21no—
385C4—24no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—1184no—
390C4—1182no—
391C4—1182no—
392C4—1184no—
393C4—1182no—
394C4#1427377412yes—
395C2—22no—
396C2—44no—
397C2—2362no—
398C2C2×#1427377211yescharacteristic
399C2C2×#1427377411yescharacteristic
400C2#1683823811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D59
commutator, socleC22×C59C25
FittingC59×#532C2
hypercenter#532D59
hypocenterC59#3559
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #16838152C2semidirect
2. #16838146C2semidirect
3. #16838111C2semidirect
4. #16838104C2semidirect
5. #16838102C2semidirect
6. #16838052C2semidirect
7. #16838051C2semidirect
8. #16838029C2semidirect
9. #16838006C2semidirect
10. #16837997C2semidirect
11. #16837994C2semidirect
12. #16837979C2semidirect
13. #16837936C2semidirect
14. C2×#14273719C2semidirect
15. C59×#532C2semidirect
213. #532D59semidirect
216. C59#3559semidirect