Finite Group Databasebeta

← #21430438

Subgroups — #21430438

Order 7552.

568 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273710C211yesmaximal, characteristic
2C2×#14273709C211yesmaximal, characteristic
3C2×#14273706C211yesmaximal, characteristic, complemented
4C2×#14273705C211yesmaximal, characteristic, complemented
5#16837336C211yesmaximal, characteristic, complemented
6#16837335C211yesmaximal, characteristic, complemented
7#16837331C211yesmaximal, characteristic, complemented
8#16837330C211yesmaximal, characteristic, complemented
9#16837323C211yesmaximal, characteristic, complemented
10#16837318C211yesmaximal, characteristic
11#16837294C211yesmaximal, characteristic, complemented
12#16837290C211yesmaximal, characteristic, complemented
13#16837273C211yesmaximal, characteristic, complemented
14#16837271C211yesmaximal, characteristic, complemented
15C59×#385C211yesmaximal, characteristic, complemented
16C22×Dic118C2211yescharacteristic
17C22×C22⋊D59C2211yescharacteristic
18C22×C4×D59C2211yescharacteristic
19#14273710—24no—
20#14273709—24no—
21#14273706—24no—
22#14273705—24no—
23C2×C4⋊Dic59C2211yescharacteristic
24C2×C4⋊Dic59C2211yescharacteristic
25C2×C4⋊Dic59C2211yescharacteristic
26C2×Dic59⋊C4C2211yescharacteristic
27C2×Dic59⋊C4C2211yescharacteristic
28C2×Dic59⋊C4C2211yescharacteristic
29C2×Dic59⋊C4C2211yescharacteristic
30C2×Dic59⋊C4C2211yescharacteristic
31C2×Dic59⋊C4C2211yescharacteristic
32C2×Dic59⋊C4C2211yescharacteristic
33C2×C22⋊Dic59C2211yescharacteristic
34C2×C22⋊Dic59C2211yescharacteristic
35C2×C22⋊Dic59C2211yescharacteristic
36C2×(C2×D59)⋊C4C2211yescharacteristic
37C2×(C2×D59)⋊C4C2211yescharacteristic
38C2×(C2×D59)⋊C4C2211yescharacteristic
39C2×(C2×D59)⋊C4C2211yescharacteristic
40C2×(C2×D59)⋊C4C2211yescharacteristic
41C2×C4×Dic59C2211yescharacteristic
42C59⋊#95C2211yescharacteristic
43C59⋊#95C2211yescharacteristic
44C59⋊#95C2211yescharacteristic
45C59⋊#95C2211yescharacteristic
46C59⋊#95C2211yescharacteristic
47C59⋊#95C2211yescharacteristic
48C23×C4×C59C2211yescharacteristic
49C2×C4⋊C4×C59C2211yescharacteristic
50C2×C4⋊C4×C59C2211yescharacteristic
51C2×C22⋊C4×C59C2211yescharacteristic
52C2×C22⋊C4×C59C2211yescharacteristic
53#95×C59C2211yescharacteristic
54#95×C59C2211yescharacteristic
55C24×C59C2311yescharacteristic
56C23×D59C2311yescharacteristic
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—41no—
60C22×C4×C59C2311yescharacteristic
61C22×C4×C59C2311yescharacteristic
62C22×C4×C59C2311yescharacteristic
63C22×C4×C59C2311yescharacteristic
64C22×C4×C59C2311yescharacteristic
65C22×C4×C59C2311yescharacteristic
66C2×C22⋊D59—22no—
67C2×C22⋊D59—22no—
68C2×C22⋊D59—41no—
69C22×Dic59C2311yescharacteristic
70C22×Dic59C2311yescharacteristic
71C22×Dic59C2311yescharacteristic
72C22×Dic59C2311yescharacteristic
73C22×Dic59C2311yescharacteristic
74C22×Dic59C2311yescharacteristic
75C22×Dic59C2311yescharacteristic
76C2×C4×D59—22no—
77C2×C4×D59—22no—
78C2×C4×D59—41no—
79C2×Dic118—22no—
80C2×Dic118—22no—
81C2×Dic118—41no—
82C4⋊C4×C59—22no—
83C4⋊C4×C59—41no—
84C22⋊C4×C59—22no—
85C22⋊C4×C59—22no—
86C22⋊Dic59—22no—
87C22⋊Dic59—22no—
88C22⋊Dic59D414yes—
89(C2×D59)⋊C4—22no—
90(C2×D59)⋊C4—22no—
91(C2×D59)⋊C4—22no—
92(C2×D59)⋊C4—22no—
93(C2×D59)⋊C4D414yes—
94C4⋊Dic59—22no—
95C4⋊Dic59—22no—
96C4⋊Dic59—41no—
97Dic59⋊C4—22no—
98Dic59⋊C4—22no—
99Dic59⋊C4—22no—
100Dic59⋊C4—22no—
101Dic59⋊C4—22no—
102Dic59⋊C4—22no—
103Dic59⋊C4—22no—
104C4×Dic59—22no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59C2×D412yes—
112C23×C59C2411yescharacteristic
113C22×D59—21no—
114C22×D59—21no—
115C22×D59—21no—
116C22×D59—21no—
117C22×D59—21no—
118C22×D59—21no—
119C22×D59C2×D412yes—
120C2×C4×C59—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—44no—
137C2×C4×C59C2×D412yes—
138C2×C4×C59#4112yes—
139C22⋊D59—44no—
140C2×Dic59—21no—
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×Dic59C2×D412yes—
159C2×Dic59C2×D412yes—
160C2×Dic59C2×D412yes—
161C4×D59—44no—
162Dic118—44no—
163C22×C59—22no—
164C22×C59—22no—
165C22×C59—22no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59C22×D411yescharacteristic
171C22×C59C22×D411yescharacteristic
172C22×C59C2×#4111yescharacteristic
173C22×C59H(5,𝔽2)11yescharacteristic
174C22×C592−1+411yescharacteristic
175C22×C592−1+411yescharacteristic
176C22×C592−1+411yescharacteristic
177C2×D59—22no—
178C2×D59—22no—
179C2×D59—22no—
180C2×D59—41no—
181C2×D59—41no—
182C2×D59—41no—
183C2×D59—41no—
184C4×C59—22no—
185C4×C59—22no—
186C4×C59—41no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—41no—
190Dic59—22no—
191Dic59—22no—
192Dic59—22no—
193Dic59—41no—
194Dic59—41no—
195Dic59—41no—
196Dic59—41no—
197#2680—591nomaximal, Sylow
198C2×C59—42no—
199C2×C59#53611yescharacteristic
200C2×C59#53611yescharacteristic
201C2×C59#53911yescharacteristic
202C2×C59D4211yescharacteristic
203C2×C59#54711yescharacteristic
204C2×C59#54711yescharacteristic
205C2×C59#56311yescharacteristic
206D59—42no—
207C2×#123—591no—
208C2×#123—591no—
209C2×C22⋊Q8—591no—
210C2×C22⋊Q8—591no—
211#398—591no—
212#398—591no—
213#397—591no—
214#397—591no—
215#395—591no—
216#394—591no—
217#388—591no—
218#388—591no—
219#385—591no—
220#385D5911yescharacteristic, complemented
221#384—591no—
222C59#268011yescharacteristic, complemented, Sylow
223C22×Q8—591no—
224C22×D4—591no—
225C23×C4—591no—
226C23×C4C2×D5911yescharacteristic
227#123—1184no—
228#123—1184no—
229C22⋊Q8—1184no—
230C22⋊Q8—1184no—
231C2×C4⋊C4—591no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—591no—
235C2×C4⋊C4—591no—
236C2×C4⋊C4—591no—
237C2×C4⋊C4—591no—
238C2×C4⋊C4—591no—
239C2×C4⋊C4—591no—
240C2×C4⋊C4—591no—
241C2×C4⋊C4C2×D5911yescharacteristic
242C2×C4⋊C4C2×D5911yescharacteristic
243C2×C22⋊C4—591no—
244C2×C22⋊C4—591no—
245C2×C22⋊C4—591no—
246C2×C22⋊C4—591no—
247C2×C22⋊C4—591no—
248C2×C22⋊C4—591no—
249C2×C22⋊C4—591no—
250C2×C22⋊C4—591no—
251C2×C22⋊C4C2×D5911yescharacteristic
252C2×C22⋊C4C2×D5911yescharacteristic
253C2×C42—591no—
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—
263C24C22×D5911yescharacteristic
264C2×Q8—1182no—
265C2×Q8—1182no—
266C2×Q8—2361no—
267C2×D4—1182no—
268C2×D4—1182no—
269C2×D4—2361no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—591no—
274C22×C4—591no—
275C22×C4—591no—
276C22×C4—591no—
277C22×C4—591no—
278C22×C4—591no—
279C22×C4—591no—
280C22×C4—1182no—
281C22×C4—1182no—
282C22×C4—2361no—
283C22×C4C22×D5911yescharacteristic
284C22×C4C22×D5911yescharacteristic
285C22×C4C22×D5911yescharacteristic
286C22×C4C22×D5911yescharacteristic
287C22×C4C22×D5911yescharacteristic
288C22×C4C22×D5911yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—1182no—
292C4⋊C4—1182no—
293C4⋊C4—1182no—
294C4⋊C4—1182no—
295C4⋊C4—1182no—
296C4⋊C4—1182no—
297C4⋊C4—1182no—
298C4⋊C4—1182no—
299C4⋊C4—1182no—
300C4⋊C4—2361no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—594no—
304C22⋊C4—594no—
305C22⋊C4—1182no—
306C22⋊C4—1182no—
307C22⋊C4—1182no—
308C22⋊C4—1182no—
309C22⋊C4—1182no—
310C22⋊C4—1182no—
311C42—1182no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—592no—
319C23—1181no—
320C23—1181no—
321C23—1181no—
322C23—1181no—
323C23—1181no—
324C23—1181no—
325C23D4×D5912yes—
326C23C23×D5911yescharacteristic
327Q8—2364no—
328D4—2364no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
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—44no—
346C2×C4—592no—
347C2×C4—592no—
348C2×C4—592no—
349C2×C4—1181no—
350C2×C4—1181no—
351C2×C4—1181no—
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—2364no—
368C2×C4#1174568512yes—
369C2×C4D4×D5912yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—1182no—
378C22—1182no—
379C22—1182no—
380C22—2361no—
381C22—2361no—
382C22—2361no—
383C22—2361no—
384C22C2×D4×D5911yescharacteristic
385C22C2×D4×D5911yescharacteristic
386C22C2×#1174568511yescharacteristic
387C22#1427377211yescharacteristic
388C22#1427377411yescharacteristic
389C22#1427377411yescharacteristic
390C22#1427377511yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—1182no—
398C4—1182no—
399C4—1182no—
400C4—2361no—
401C4—2361no—
402C4—2361no—
403C4—2361no—
404C2—42no—
405C2—2362no—
406C2#1683797411yescharacteristic
407C2#1683797511yescharacteristic
408C2#1683799411yescharacteristic
409C2#1683803411yescharacteristic
410C2#1683805311yescharacteristic
411C2#1683805711yescharacteristic
412C2#1683815311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#385C2
hypercenter#385D59
hypocenterC59#2680
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C2×#14273706C2semidirect
4. C2×#14273705C2semidirect
5. #16837336C2semidirect
6. #16837335C2semidirect
7. #16837331C2semidirect
8. #16837330C2semidirect
9. #16837323C2semidirect
11. #16837294C2semidirect
12. #16837290C2semidirect
13. #16837273C2semidirect
14. #16837271C2semidirect
15. C59×#385C2semidirect
220. #385D59semidirect
222. C59#2680semidirect