Finite Group Databasebeta

← #21431111

Subgroups — #21431111

Order 7552.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273703C211yesmaximal, characteristic, complemented
2#16837338C211yesmaximal, characteristic, complemented
3#16837337C211yesmaximal, characteristic, complemented
4#16837333C211yesmaximal, characteristic, complemented
5#16837332C211yesmaximal, characteristic, complemented
6#16837325C211yesmaximal, characteristic, complemented
7#16837324C211yesmaximal, characteristic, complemented
8#16837309C211yesmaximal, characteristic, complemented
9#16837302C211yesmaximal, characteristic
10#16837297C211yesmaximal, characteristic, complemented
11#16837281C211yesmaximal, characteristic
12#16837278C211yesmaximal, characteristic
13#16837275C211yesmaximal, characteristic, complemented
14#16837273C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
16C22×C22⋊D59C2211yescharacteristic
17C22×C4×D59C2211yescharacteristic
18#14273703—24no—
19C2×C4⋊Dic59C2211yescharacteristic
20C2×C4⋊Dic59C2211yescharacteristic
21C2×Dic59⋊C4C2211yescharacteristic
22C2×Dic59⋊C4C2211yescharacteristic
23C2×Dic59⋊C4C2211yescharacteristic
24C2×Dic59⋊C4C2211yescharacteristic
25C2×Dic59⋊C4C2211yescharacteristic
26C2×Dic59⋊C4C2211yescharacteristic
27C2×Dic59⋊C4C2211yescharacteristic
28C2×C22⋊Dic59C2211yescharacteristic
29C2×C22⋊Dic59C2211yescharacteristic
30C2×C22⋊Dic59C2211yescharacteristic
31C2×(C2×D59)⋊C4C2211yescharacteristic
32C2×(C2×D59)⋊C4C2211yescharacteristic
33C2×(C2×D59)⋊C4C2211yescharacteristic
34C2×(C2×D59)⋊C4C2211yescharacteristic
35C2×(C2×D59)⋊C4C2211yescharacteristic
36C2×C4×Dic59C2211yescharacteristic
37C59⋊#95C2211yescharacteristic
38C59⋊#95C2211yescharacteristic
39C59⋊#95C2211yescharacteristic
40C59⋊#95C2211yescharacteristic
41C59⋊#95C2211yescharacteristic
42C59⋊#95C2211yescharacteristic
43C59⋊#95C2211yescharacteristic
44C59⋊#95C2211yescharacteristic
45C22×Q8×C59C2211yescharacteristic
46C23×C4×C59C2211yescharacteristic
47C22⋊Q8×C59—24no—
48C2×C4⋊C4×C59C2211yescharacteristic
49C2×C4⋊C4×C59C2211yescharacteristic
50C2×C4⋊C4×C59C2211yescharacteristic
51C2×C22⋊C4×C59C2211yescharacteristic
52C2×C22⋊C4×C59C2211yescharacteristic
53C24×C59C2311yescharacteristic
54C23×D59C2311yescharacteristic
55C2×Q8×C59—41no—
56C2×Q8×C59—41no—
57C2×Q8×C59D414yescomplemented
58C22×C4×C59—22no—
59C22×C4×C59—41no—
60C22×C4×C59—41no—
61C22×C4×C59C2311yescharacteristic
62C22×C4×C59C2311yescharacteristic
63C22×C4×C59C2311yescharacteristic
64C22×C4×C59C2311yescharacteristic
65C22×C4×C59C2311yescharacteristic
66C22×C4×C59C2311yescharacteristic
67C2×C22⋊D59—41no—
68C2×C22⋊D59—41no—
69C2×C22⋊D59Q814yescomplemented
70C22×Dic59C2311yescharacteristic
71C22×Dic59C2311yescharacteristic
72C22×Dic59C2311yescharacteristic
73C22×Dic59C2311yescharacteristic
74C22×Dic59C2311yescharacteristic
75C22×Dic59C2311yescharacteristic
76C22×Dic59C2311yescharacteristic
77C2×C4×D59—22no—
78C2×C4×D59—41no—
79C2×C4×D59—41no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—22no—
82C4⋊C4×C59—22no—
83C22⋊C4×C59—22no—
84C22⋊C4×C59—22no—
85C22⋊Dic59—22no—
86C22⋊Dic59—41no—
87C22⋊Dic59—41no—
88(C2×D59)⋊C4—22no—
89(C2×D59)⋊C4—22no—
90(C2×D59)⋊C4—22no—
91(C2×D59)⋊C4—41no—
92(C2×D59)⋊C4—41no—
93C4⋊Dic59—22no—
94C4⋊Dic59—41no—
95Dic59⋊C4—22no—
96Dic59⋊C4—22no—
97Dic59⋊C4—22no—
98Dic59⋊C4—41no—
99Dic59⋊C4—41no—
100Dic59⋊C4—41no—
101Dic59⋊C4—41no—
102C4×Dic59—41no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59C2×Q812yes—
110C23×C59C2411yescharacteristic
111C22×D59—21no—
112C22×D59—21no—
113C22×D59—21no—
114C22×D59—21no—
115C22×D59—21no—
116C22×D59—21no—
117C22×D59C2×Q812yes—
118Q8×C59—44no—
119C2×C4×C59—21no—
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—44no—
135C2×C4×C59C2×D412yes—
136C2×C4×C59C2×D412yes—
137C2×C4×C59C2×D412yes—
138C22⋊D59—44no—
139C2×Dic59—21no—
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×Q812yes—
159C2×Dic59#4112yes—
160C4×D59—44no—
161C22×C59—22no—
162C22×C59—22no—
163C22×C59—22no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59C22×D411yescharacteristic
169C22×C59C22×Q811yescharacteristic
170C22×C59C2×#4111yescharacteristic
171C22×C59H(5,𝔽2)11yescharacteristic
172C22×C59H(5,𝔽2)11yescharacteristic
173C22×C59H(5,𝔽2)11yescharacteristic
174C22×C592−1+411yescharacteristic
175C2×D59—22no—
176C2×D59—22no—
177C2×D59—22no—
178C2×D59—41no—
179C2×D59—41no—
180C2×D59—41no—
181C2×D59—41no—
182C4×C59—22no—
183C4×C59—22no—
184C4×C59—22no—
185C4×C59—41no—
186C4×C59—41no—
187C4×C59—41no—
188Dic59—22no—
189Dic59—22no—
190Dic59—41no—
191Dic59—41no—
192Dic59—41no—
193Dic59—41no—
194Dic59—41no—
195#2732—591nomaximal, Sylow
196C2×C59—42no—
197C2×C59#53411yescharacteristic
198C2×C59#53911yescharacteristic
199C2×C59#54311yescharacteristic
200C2×C59D4×Q811yescharacteristic
201C2×C59#55011yescharacteristic
202C2×C59#55411yescharacteristic
203C2×C59#56211yescharacteristic
204D59—42no—
205C2×C22⋊Q8—591no—
206C2×C22⋊Q8D5911yescharacteristic, complemented
207#399—591no—
208#399—591no—
209#398—591no—
210#398—591no—
211#396—591no—
212#396—591no—
213#391—591no—
214#390—591no—
215#389—591no—
216#386—591no—
217#386—591no—
218#385—591no—
219#385—591no—
220C59#273211yescharacteristic, complemented, Sylow
221C22×Q8C2×D5911yescharacteristic
222C22×D4—591no—
223C23×C4—591no—
224C23×C4C2×D5911yescharacteristic
225C22⋊Q8—24no—
226C22⋊Q8—1184no—
227C2×C4⋊C4—591no—
228C2×C4⋊C4—591no—
229C2×C4⋊C4—591no—
230C2×C4⋊C4—591no—
231C2×C4⋊C4—591no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—591no—
235C2×C4⋊C4—591no—
236C2×C4⋊C4C2×D5911yescharacteristic
237C2×C4⋊C4C2×D5911yescharacteristic
238C2×C4⋊C4C2×D5911yescharacteristic
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⋊C4C2×D5911yescharacteristic
248C2×C22⋊C4C2×D5911yescharacteristic
249C2×C42—591no—
250#95—591no—
251#95—591no—
252#95—591no—
253#95—591no—
254#95—591no—
255#95—591no—
256#95—591no—
257#95—591no—
258C24—591no—
259C24C22×D5911yescharacteristic
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8C22⋊D5914yescomplemented
263C2×D4—594no—
264C2×D4—2361no—
265C2×D4—2361no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—591no—
270C22×C4—591no—
271C22×C4—591no—
272C22×C4—591no—
273C22×C4—591no—
274C22×C4—591no—
275C22×C4—591no—
276C22×C4—1182no—
277C22×C4—2361no—
278C22×C4—2361no—
279C22×C4C22×D5911yescharacteristic
280C22×C4C22×D5911yescharacteristic
281C22×C4C22×D5911yescharacteristic
282C22×C4C22×D5911yescharacteristic
283C22×C4C22×D5911yescharacteristic
284C22×C4C22×D5911yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—1182no—
289C4⋊C4—1182no—
290C4⋊C4—1182no—
291C4⋊C4—1182no—
292C4⋊C4—2361no—
293C4⋊C4—2361no—
294C4⋊C4—2361no—
295C4⋊C4—2361no—
296C4⋊C4—2361no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—1182no—
300C22⋊C4—1182no—
301C22⋊C4—1182no—
302C22⋊C4—1182no—
303C22⋊C4—2361no—
304C22⋊C4—2361no—
305C22⋊C4—2361no—
306C22⋊C4—2361no—
307C42—2361no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—592no—
315C23—1181no—
316C23—1181no—
317C23—1181no—
318C23—1181no—
319C23—1181no—
320C23—1181no—
321C23Q8×D5912yes—
322C23C23×D5911yescharacteristic
323Q8—44no—
324D4—2364no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
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—44no—
341C2×C4—592no—
342C2×C4—592no—
343C2×C4—1181no—
344C2×C4—1181no—
345C2×C4—1181no—
346C2×C4—1181no—
347C2×C4—1181no—
348C2×C4—1181no—
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—2364no—
363C2×C4C2×C22⋊D5912yes—
364C2×C4C2×C22⋊D5912yes—
365C2×C4C2×C22⋊D5912yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—1182no—
374C22—1182no—
375C22—1182no—
376C22—2361no—
377C22—2361no—
378C22—2361no—
379C22—2361no—
380C22C22×C22⋊D5911yescharacteristic
381C22C2×Q8×D5911yescharacteristic
382C22#41×D5911yescharacteristic
383C22#1427377211yescharacteristic
384C22#1427377211yescharacteristic
385C22#1427377311yescharacteristic
386C22#1427377511yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—1182no—
394C4—1182no—
395C4—2361no—
396C4—2361no—
397C4—2361no—
398C4—2361no—
399C4—2361no—
400C2—42no—
401C2—2362no—
402C2#1683796111yescharacteristic
403C2#1683799811yescharacteristic
404C2#1683802711yescharacteristic
405C2Q8×C22⋊D5911yescharacteristic
406C2#1683807111yescharacteristic
407C2#1683810611yescharacteristic
408C2#1683814811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×C22⋊Q8×C59C2
hypercenterC2×C22⋊Q8D59
hypocenterC59#2732
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14273703C2semidirect
2. #16837338C2semidirect
3. #16837337C2semidirect
4. #16837333C2semidirect
5. #16837332C2semidirect
6. #16837325C2semidirect
7. #16837324C2semidirect
8. #16837309C2semidirect
10. #16837297C2semidirect
13. #16837275C2semidirect
14. #16837273C2semidirect
15. C2×C22⋊Q8×C59C2semidirect
57. C2×Q8×C59D4semidirect
69. C2×C22⋊D59Q8semidirect
206. C2×C22⋊Q8D59semidirect
220. C59#2732semidirect
262. C2×Q8C22⋊D59semidirect