Finite Group Databasebeta

← #20291581

Subgroups — #20291581

Order 6784.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811658C211yesmaximal, characteristic, complemented
2#16811656C211yesmaximal, characteristic, complemented
3#16811655C211yesmaximal, characteristic, complemented
4#16811654C211yesmaximal, characteristic, complemented
5#16811654C211yesmaximal, characteristic, complemented
6#16811652C211yesmaximal, characteristic, complemented
7#16811650C211yesmaximal, characteristic, complemented
8D4×#88324C211yesmaximal, characteristic, complemented
9#16811647C211yesmaximal, characteristic, complemented
10#16811647C211yesmaximal, characteristic, complemented
11#16811597C211yesmaximal, characteristic, complemented
12#16811596C211yesmaximal, characteristic, complemented
13#16811593C211yesmaximal, characteristic, complemented
14C2×C22⋊#88324C211yesmaximal, characteristic, complemented
15#16811129C211yesmaximal, characteristic
16C2×#11719936C2211yescharacteristic, complemented
17C2×#11719940C2211yescharacteristic
18C2×#11719940C2211yescharacteristic
19C2×#11719940C2211yescharacteristic, complemented
20C2×D53⋊C8C2211yescharacteristic, complemented
21C22×#88324C2211yescharacteristic
22C22×#88324C2211yescharacteristic
23C22×#88324C2211yescharacteristic, complemented
24#13133284C2211yescharacteristic, complemented
25#13133284C2211yescharacteristic, complemented
26C4⋊#88324C2211yescharacteristic, complemented
27C4⋊#88324C2211yescharacteristic, complemented
28#13133274C2211yescharacteristic, complemented
29#13133274C2211yescharacteristic, complemented
30C22⋊#88324C2211yescharacteristic
31C22⋊#88324C2211yescharacteristic
32C22⋊#88324C2211yescharacteristic
33C22⋊#88324C2211yescharacteristic
34C22⋊#88324C2211yescharacteristic
35C22⋊#88324C2211yescharacteristic
36C22⋊#88324C2211yescharacteristic
37C22⋊#88324C2211yescharacteristic
38C22⋊#88324C2211yescharacteristic, complemented
39C22⋊#88324C2211yescharacteristic, complemented
40#13133272C2211yescharacteristic, complemented
41#13133272C2211yescharacteristic, complemented
42C4×#88324C2211yescharacteristic, complemented
43C4×#88324C2211yescharacteristic, complemented
44C2×#11719947C2211yescharacteristic
45C23×Dic53C2211yescharacteristic
46#13133201C411yescharacteristic
47#13133198C411yescharacteristic
48#13133194C411yescharacteristic
49#13133194C411yescharacteristic
50#13133189C411yescharacteristic
51#13133189C411yescharacteristic
52#13133186C411yescharacteristic
53D4×Dic53C2211yescharacteristic
54D4×Dic53C2211yescharacteristic
55#13133177C2211yescharacteristic
56#13133177C2211yescharacteristic
57#13133173C2211yescharacteristic
58#120×C53C411yescharacteristic
59C24×C53C2×C411yescharacteristic
60C2×D4×C53C2×C411yescharacteristic
61C2×D4×C53C2×C411yescharacteristic
62C2×D4×C53C2×C411yescharacteristic
63C2×C22⋊D53C2×C411yescharacteristic
64C2×C22⋊D53C2×C411yescharacteristic
65C2×C22⋊D53C2×C411yescharacteristic
66C2×C22⋊D53C2×C411yescharacteristic
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53C2×C411yescharacteristic
70C22×Dic53C2×C411yescharacteristic
71C22×Dic53C2×C411yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76#11719947—41no—
77#11719947—41no—
78C2×C4×D53C2311yescharacteristic
79C2×Dic106C2×C411yescharacteristic
80#11719940—41no—
81#11719940—41no—
82#11719940—41no—
83C2×#88324—41no—
84C2×#88324—41no—
85C2×#88324—41no—
86C2×#88324C2311yescharacteristic
87C2×#88324C2311yescharacteristic
88C2×#88324C2311yescharacteristic
89C2×#88324C2311yescharacteristic
90C2×#88324C2311yescharacteristic
91C2×#88324C2311yescharacteristic
92C2×#88324C2311yescharacteristic
93C2×#88324C2311yescharacteristic
94#11719936—41no—
95D53⋊C8—41no—
96C22⋊C4×C53C2×C411yescharacteristic
97C22⋊C4×C53C2×C411yescharacteristic
98C22⋊C4×C53C2×C411yescharacteristic
99C22⋊Dic53C2×C411yescharacteristic
100C22⋊Dic53C2×C411yescharacteristic
101C22⋊Dic53C2×C411yescharacteristic
102C22⋊Dic53C2×C411yescharacteristic
103C22⋊Dic53C2×C411yescharacteristic
104#11719921C2×C411yescharacteristic
105#11719921C2×C411yescharacteristic
106C4⋊Dic53C2×C411yescharacteristic
107C4⋊Dic53C2×C411yescharacteristic
108#11719919C2×C411yescharacteristic
109#11719919C2×C411yescharacteristic
110#11719919C2×C411yescharacteristic
111#11719919C2×C411yescharacteristic
112C4×Dic53C2311yescharacteristic
113C4×Dic53C2311yescharacteristic
114C23×C53—22no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53C22×C411yescharacteristic
118C23×C53C22×C411yescharacteristic
119C23×C53C22×C411yescharacteristic
120C23×C53C22×C411yescharacteristic
121C22×D53C22×C411yescharacteristic
122D4×C53—22no—
123D4×C53—41no—
124D4×C53—41no—
125C2×C4×C53C22×C411yescharacteristic
126C2×C4×C53C22×C411yescharacteristic
127C2×C4×C53C22×C411yescharacteristic
128C22⋊D53—22no—
129C22⋊D53—41no—
130C22⋊D53—41no—
131C22⋊D53—41no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53C22×C411yescharacteristic
144C2×Dic53C22×C411yescharacteristic
145C2×Dic53C22×C411yescharacteristic
146C2×Dic53C22×C411yescharacteristic
147C2×Dic53C22×C411yescharacteristic
148C2×Dic53C22×C411yescharacteristic
149C2×Dic53C2411yescharacteristic
150C4×D53—21no—
151C4×D53—21no—
152Dic106—41no—
153#88324—21no—
154#88324—21no—
155#88324—21no—
156#88324—21no—
157#88324—21no—
158#88324—21no—
159#88324—21no—
160#88324—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—44no—
173C22×C53#13112yes—
174C22×C53C23×C411yescharacteristic
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181Dic53—21no—
182Dic53—21no—
183Dic53—21no—
184Dic53—21no—
185Dic53—21no—
186Dic53—21no—
187Dic53H(5,𝔽2)11yescharacteristic
188Dic53H(5,𝔽2)11yescharacteristic
189#2976—531nomaximal, Sylow
190C2×C53—22no—
191C2×C53—41no—
192C2×C53—41no—
193C2×C53—41no—
194C2×C53#51811yescharacteristic
195C2×C53C2×#13111yescharacteristic
196C2×C53#56811yescharacteristic
197D53—41no—
198#525—531no—
199#436—531no—
200#436—531no—
201#435—531no—
202#435—531no—
203#435—531no—
204#435—531no—
205C8×D4—531no—
206C8×D4—531no—
207#433—531no—
208#433—531no—
209#408—531no—
210#408—531no—
211#407—531no—
212C2×C22⋊C8—531no—
213C53#297611yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C23×C4—531no—
216C2×M16—531no—
217C2×M16—531no—
218C2×M16—531no—
219C2×M16—531no—
220C22×C8—531no—
221C22×C8—531no—
222C22×C8—531no—
223C22×C8—531no—
224#123—531no—
225#123—531no—
226C22⋊Q8—531no—
227C22⋊Q8—531no—
228#121—531no—
229#121—531no—
230#120—531no—
231#120#615711yescharacteristic
232C4×D4—531no—
233C4×D4—531no—
234C4×D4—531no—
235C4×D4—531no—
236#117—531no—
237C4⋊C8—531no—
238C4⋊C8—531no—
239C4⋊C8—531no—
240C4⋊C8—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243C22⋊C8—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246C22⋊C8—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249C22⋊C8—531no—
250C22⋊C8—531no—
251C22⋊C8—531no—
252C22⋊C8—531no—
253#97—531no—
254#97—531no—
255C4×C8—531no—
256C4×C8—531no—
257C24C2×#615711yescharacteristic
258#41—2121no—
259#41—2121no—
260C2×Q8—531no—
261C2×D4—531no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4C2×#615711yescharacteristic
266C2×D4C2×#615711yescharacteristic
267C2×D4C2×#615711yescharacteristic
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—2121no—
277C22×C4—2121no—
278M16—2121no—
279M16—2121no—
280M16—2121no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—531no—
289C2×C8—531no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C4⋊C4—531no—
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—531no—
299C4⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—531no—
302C22⋊C4—531no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—531no—
306C22⋊C4—531no—
307C22⋊C4C2×#615711yescharacteristic
308C22⋊C4C2×#615711yescharacteristic
309C22⋊C4C2×#615711yescharacteristic
310C42—531no—
311C42—531no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—531no—
316C23C22×#615711yescharacteristic
317C23C22×#615711yescharacteristic
318C23C22×#615711yescharacteristic
319C23C22×#615711yescharacteristic
320Q8—2121no—
321D4—22no—
322D4—41no—
323D4—41no—
324D4—1062no—
325D4—2121no—
326D4—2121no—
327D4—2121no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4C22×#615711yescharacteristic
349C2×C4C22×#615711yescharacteristic
350C2×C4C22×#615711yescharacteristic
351C8—1061no—
352C8—1061no—
353C8—1061no—
354C8—1061no—
355C8—1061no—
356C8—1061no—
357C8—1061no—
358C8—1061no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22—1061no—
372C22—1061no—
373C22—1061no—
374C22#1313330412yes—
375C22C23×#615711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—531no—
380C4—531no—
381C4—1061no—
382C4—1061no—
383C4—1061no—
384C4—1061no—
385C4—1061no—
386C4—1061no—
387C2—22no—
388C2—41no—
389C2—41no—
390C2—41no—
391C2—2121no—
392C2#1681169011yescharacteristic
393C2C2×#1313330411yescharacteristic
394C2#1681170111yescharacteristic
subgroupG/H
center, FrattiniC22C23×#6157
commutator, socleC22×C53C23×C4
Fitting#120×C53C4
hypercenter#120#6157
hypocenterC53#2976
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #16811658C2semidirect
2. #16811656C2semidirect
3. #16811655C2semidirect
4. #16811654C2semidirect
5. #16811654C2semidirect
6. #16811652C2semidirect
7. #16811650C2semidirect
8. D4×#88324C2semidirect
9. #16811647C2semidirect
10. #16811647C2semidirect
11. #16811597C2semidirect
12. #16811596C2semidirect
13. #16811593C2semidirect
14. C2×C22⋊#88324C2semidirect
16. C2×#11719936C22semidirect
19. C2×#11719940C22semidirect
20. C2×D53⋊C8C22semidirect
23. C22×#88324C22semidirect
24. #13133284C22semidirect
25. #13133284C22semidirect
26. C4⋊#88324C22semidirect
27. C4⋊#88324C22semidirect
28. #13133274C22semidirect
29. #13133274C22semidirect
38. C22⋊#88324C22semidirect
39. C22⋊#88324C22semidirect
40. #13133272C22semidirect
41. #13133272C22semidirect
42. C4×#88324C22semidirect
43. C4×#88324C22semidirect
213. C53#2976semidirect