Finite Group Databasebeta

← #21432032

Subgroups — #21432032

Order 7552.

470 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16837331C211yesmaximal, characteristic, complemented
2#16837319C211yesmaximal, characteristic, complemented
3#16837318C211yesmaximal, characteristic, complemented
4#16837318C211yesmaximal, characteristic, complemented
5#16837317C211yesmaximal, characteristic, complemented
6#16837311C211yesmaximal, characteristic
7#16837304C211yesmaximal, characteristic, complemented
8#16837302C211yesmaximal, characteristic, complemented
9#16837300C211yesmaximal, characteristic, complemented
10#16837295C211yesmaximal, characteristic, complemented
11#16837294C211yesmaximal, characteristic, complemented
12#16837294C211yesmaximal, characteristic, complemented
13#16837291C211yesmaximal, characteristic, complemented
14#16837288C211yesmaximal, characteristic, complemented
15C59×#388C211yesmaximal, characteristic, complemented
16C22×D236C2211yescharacteristic
17C22×C22⋊D59C2211yescharacteristic
18C22×C22⋊D59C2211yescharacteristic
19C2×C4⋊Dic59C2211yescharacteristic
20C2×C4⋊Dic59C2211yescharacteristic
21C2×C4⋊Dic59C2211yescharacteristic
22C2×Dic59⋊C4C2211yescharacteristic
23C2×Dic59⋊C4C2211yescharacteristic
24C2×C22⋊Dic59C2211yescharacteristic
25C2×C22⋊Dic59C2211yescharacteristic
26C2×(C2×D59)⋊C4C2211yescharacteristic
27C2×(C2×D59)⋊C4C2211yescharacteristic
28C2×(C2×D59)⋊C4C2211yescharacteristic
29C2×(C2×D59)⋊C4C2211yescharacteristic
30C2×(C2×D59)⋊C4C2211yescharacteristic
31C2×(C2×D59)⋊C4C2211yescharacteristic
32C2×(C2×D59)⋊C4C2211yescharacteristic
33C2×(C2×D59)⋊C4C2211yescharacteristic
34C2×(C2×D59)⋊C4C2211yescharacteristic
35C2×(C2×D59)⋊C4C2211yescharacteristic
36C2×C4×Dic59C2211yescharacteristic
37C2×C4×Dic59C2211yescharacteristic
38C59⋊#95C2211yescharacteristic
39C59⋊#95C2211yescharacteristic
40C59⋊#95C2211yescharacteristic
41C59⋊#95C2211yescharacteristic
42C59⋊#95C2211yescharacteristic
43C59⋊#95C2211yescharacteristic
44C2×C4⋊C4×C59C2211yescharacteristic
45C2×C22⋊C4×C59C2211yescharacteristic
46C2×C22⋊C4×C59C2211yescharacteristic
47C2×C22⋊C4×C59C2211yescharacteristic
48C2×C42×C59C2211yescharacteristic
49#95×C59C2211yescharacteristic
50#95×C59C2211yescharacteristic
51C24×C59C2311yescharacteristic
52C23×D59C2311yescharacteristic
53C23×D59C2311yescharacteristic
54C22×C4×C59C2311yescharacteristic
55C22×C4×C59C2311yescharacteristic
56C22×C4×C59C2311yescharacteristic
57C22×C4×C59C2311yescharacteristic
58C22×C4×C59C2311yescharacteristic
59C22×C4×C59C2311yescharacteristic
60C2×C22⋊D59—22no—
61C2×C22⋊D59—41no—
62C2×C22⋊D59—41no—
63C2×C22⋊D59—41no—
64C2×C22⋊D59—41no—
65C2×C22⋊D59—41no—
66C22×Dic59C2311yescharacteristic
67C22×Dic59C2311yescharacteristic
68C22×Dic59C2311yescharacteristic
69C22×Dic59C2311yescharacteristic
70C22×Dic59C2311yescharacteristic
71C22×Dic59C2311yescharacteristic
72C2×D236—22no—
73C2×D236—41no—
74C2×D236—41no—
75C4⋊C4×C59—22no—
76C22⋊C4×C59—22no—
77C22⋊C4×C59—22no—
78C22⋊C4×C59—41no—
79C42×C59—41no—
80C22⋊Dic59—41no—
81C22⋊Dic59—41no—
82(C2×D59)⋊C4—22no—
83(C2×D59)⋊C4—22no—
84(C2×D59)⋊C4—22no—
85(C2×D59)⋊C4—41no—
86(C2×D59)⋊C4—41no—
87(C2×D59)⋊C4—41no—
88(C2×D59)⋊C4—41no—
89(C2×D59)⋊C4—41no—
90(C2×D59)⋊C4—41no—
91(C2×D59)⋊C4—41no—
92C4⋊Dic59—22no—
93C4⋊Dic59—41no—
94C4⋊Dic59—41no—
95Dic59⋊C4—41no—
96Dic59⋊C4—41no—
97C4×Dic59—41no—
98C4×Dic59—41no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59C2411yescharacteristic
107C22×D59—21no—
108C22×D59—21no—
109C22×D59—21no—
110C22×D59—21no—
111C22×D59—21no—
112C22×D59—21no—
113C22×D59—21no—
114C22×D59—21no—
115C22×D59—21no—
116C22×D59—21no—
117C22×D59—21no—
118C22×D59—21no—
119C22×D59—21no—
120C22×D59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×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×C59#4112yes—
138C2×C4×C59#4112yes—
139C22⋊D59—44no—
140C22⋊D59—44no—
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145C2×Dic59—21no—
146C2×Dic59—21no—
147C2×Dic59—21no—
148C2×Dic59—21no—
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—21no—
152C2×Dic59—21no—
153C2×Dic59—21no—
154C2×Dic59—21no—
155C2×Dic59—21no—
156C2×Dic59—21no—
157C2×Dic59—21no—
158C2×Dic59#4112yes—
159D236—44no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59C2×#4111yescharacteristic
168C22×C59C2×#4111yescharacteristic
169C22×C59C2×#4111yescharacteristic
170C22×C59H(5,𝔽2)11yescharacteristic
171C22×C59H(5,𝔽2)11yescharacteristic
172C22×C59H(5,𝔽2)11yescharacteristic
173C22×C59H(5,𝔽2)11yescharacteristic
174C2×D59—41no—
175C2×D59—41no—
176C2×D59—41no—
177C2×D59—41no—
178C2×D59—41no—
179C2×D59—41no—
180C2×D59—41no—
181C2×D59—41no—
182C2×D59—41no—
183C2×D59—41no—
184C2×D59—41no—
185C2×D59—41no—
186C2×D59—41no—
187C2×D59—41no—
188C4×C59—22no—
189C4×C59—22no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—41no—
194Dic59—22no—
195Dic59—41no—
196Dic59—41no—
197Dic59—41no—
198Dic59—41no—
199Dic59—41no—
200#2803—591nomaximal, Sylow
201C2×C59—81no—
202C2×C59#53811yescharacteristic
203C2×C59#53811yescharacteristic
204C2×C59#54011yescharacteristic
205C2×C59#55311yescharacteristic
206C2×C59#55511yescharacteristic
207C2×C59#55911yescharacteristic
208C2×C59#56011yescharacteristic
209D59—81no—
210D59—81no—
211#397—591no—
212#394—591no—
213#394—591no—
214#394—591no—
215#394—591no—
216#392—591no—
217#390—591no—
218#390—591no—
219#389—591no—
220#388—591no—
221#388—591no—
222#388—591no—
223#388—591no—
224#388D5911yescharacteristic, complemented
225#387—591no—
226C59#280311yescharacteristic, complemented, Sylow
227C22×D4—591no—
228C22×D4—591no—
229C22×D4—591no—
230C2×C4⋊C4—591no—
231C2×C4⋊C4—591no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—591no—
235C2×C4⋊C4C2×D5911yescharacteristic
236C2×C22⋊C4—591no—
237C2×C22⋊C4—591no—
238C2×C22⋊C4—591no—
239C2×C22⋊C4—591no—
240C2×C22⋊C4—591no—
241C2×C22⋊C4—591no—
242C2×C22⋊C4—591no—
243C2×C22⋊C4—591no—
244C2×C22⋊C4—591no—
245C2×C22⋊C4—591no—
246C2×C22⋊C4—591no—
247C2×C22⋊C4—591no—
248C2×C22⋊C4C2×D5911yescharacteristic
249C2×C22⋊C4C2×D5911yescharacteristic
250C2×C22⋊C4C2×D5911yescharacteristic
251C2×C42—591no—
252C2×C42—591no—
253C2×C42C2×D5911yescharacteristic
254#95—591no—
255#95—591no—
256#95—591no—
257#95—591no—
258#95—591no—
259#95—591no—
260#95C2×D5911yescharacteristic
261#95C2×D5911yescharacteristic
262C24—591no—
263C24—591no—
264C24C22×D5911yescharacteristic
265C2×D4—1182no—
266C2×D4—1182no—
267C2×D4—2361no—
268C2×D4—2361no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2361no—
272C2×D4—2361no—
273C2×D4—2361no—
274C22×C4—591no—
275C22×C4—591no—
276C22×C4—591no—
277C22×C4—591no—
278C22×C4—591no—
279C22×C4—591no—
280C22×C4C22×D5911yescharacteristic
281C22×C4C22×D5911yescharacteristic
282C22×C4C22×D5911yescharacteristic
283C22×C4C22×D5911yescharacteristic
284C22×C4C22×D5911yescharacteristic
285C22×C4C22×D5911yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—1182no—
288C4⋊C4—2361no—
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C4⋊C4—2361no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—1182no—
296C22⋊C4—1182no—
297C22⋊C4—1182no—
298C22⋊C4—2361no—
299C22⋊C4—2361no—
300C22⋊C4—2361no—
301C22⋊C4—2361no—
302C22⋊C4—2361no—
303C22⋊C4—2361no—
304C22⋊C4—2361no—
305C22⋊C4—2361no—
306C22⋊C4—2361no—
307C42—41no—
308C42—2361no—
309C42—2361no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—1181no—
318C23—1181no—
319C23—1181no—
320C23—1181no—
321C23—1181no—
322C23—1181no—
323C23—1181no—
324C23—1181no—
325C23—1181no—
326C23—1181no—
327C23—1181no—
328C23—1181no—
329C23—1181no—
330C23—1181no—
331C23C23×D5911yescharacteristic
332D4—2364no—
333D4—2364no—
334D4—2364no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—592no—
352C2×C4—1181no—
353C2×C4—1181no—
354C2×C4—1181no—
355C2×C4—1181no—
356C2×C4—1181no—
357C2×C4—1181no—
358C2×C4—1181no—
359C2×C4—1181no—
360C2×C4—1181no—
361C2×C4—1181no—
362C2×C4—1181no—
363C2×C4—1181no—
364C2×C4—1181no—
365C2×C4—1181no—
366C2×C4—1181no—
367C2×C4—1181no—
368C2×C4—1181no—
369C2×C4#1174568512yes—
370C2×C4#1174568712yes—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—2361no—
379C22—2361no—
380C22—2361no—
381C22—2361no—
382C22—2361no—
383C22—2361no—
384C22—2361no—
385C22—2361no—
386C22—2361no—
387C22—2361no—
388C22—2361no—
389C22—2361no—
390C22—2361no—
391C22—2361no—
392C22#41×D5911yescharacteristic
393C22C2×#1174568711yescharacteristic
394C22C2×#1174568511yescharacteristic
395C22#1427377211yescharacteristic
396C22#1427377311yescharacteristic
397C22#1427377311yescharacteristic
398C22#1427377311yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—1182no—
406C4—2361no—
407C4—2361no—
408C4—2361no—
409C4—2361no—
410C4—2361no—
411C2—81no—
412C2—4721no—
413C2—4721no—
414C2#1683798911yescharacteristic
415C2#1683799111yescharacteristic
416C2#1683800711yescharacteristic
417C2#1683810111yescharacteristic
418C2#1683811711yescharacteristic
419C2#1683813411yescharacteristic
420C2#1683814211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#388C2
hypercenter#388D59
hypocenterC59#2803
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #16837331C2semidirect
2. #16837319C2semidirect
3. #16837318C2semidirect
4. #16837318C2semidirect
5. #16837317C2semidirect
7. #16837304C2semidirect
8. #16837302C2semidirect
9. #16837300C2semidirect
10. #16837295C2semidirect
11. #16837294C2semidirect
12. #16837294C2semidirect
13. #16837291C2semidirect
14. #16837288C2semidirect
15. C59×#388C2semidirect
224. #388D59semidirect
226. C59#2803semidirect