Finite Group Databasebeta

← #20283788

Subgroups — #20283788

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811169C211yesmaximal, characteristic
2#16810699C211yesmaximal, characteristic, complemented
3#16810698C211yesmaximal, characteristic, complemented
4#16810697C211yesmaximal, characteristic, complemented
5#16810694C211yesmaximal, characteristic, complemented
6#16810692C211yesmaximal, characteristic, complemented
7#16810691C211yesmaximal, characteristic, complemented
8C8×C22⋊D53C211yesmaximal, characteristic, complemented
9#16810686C211yesmaximal, characteristic, complemented
10#16810677C211yesmaximal, characteristic, complemented
11#16810673C211yesmaximal, characteristic, complemented
12#16810590C211yesmaximal, characteristic, complemented
13#16810588C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#408C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719917C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C22×#88322C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133215C411yescharacteristic
24#13133213C411yescharacteristic
25#13133206C411yescharacteristic
26#13133204C411yescharacteristic
27#13133189C411yescharacteristic
28#13133189C411yescharacteristic
29#13133189C411yescharacteristic
30#13133189C411yescharacteristic
31C4×C22⋊D53C2211yescharacteristic
32C4×C22⋊D53C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36#13133146C2211yescharacteristic
37#13133146C2211yescharacteristic
38#13133146C2211yescharacteristic
39#13133146C2211yescharacteristic
40C22⋊#88322C2211yescharacteristic
41C22⋊#88322C2211yescharacteristic
42C22⋊#88322C2211yescharacteristic
43C22⋊#88322C2211yescharacteristic
44#13133129C2211yescharacteristic
45#13133129C2211yescharacteristic
46#13133129C2211yescharacteristic
47#13133129C2211yescharacteristic
48#13133127C2211yescharacteristic
49#13133127C2211yescharacteristic
50C8×Dic53C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C22×C8×C53C2211yescharacteristic
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61C2×Q8×C53C2×C411yescharacteristic
62C2×D4×C53C2×C411yescharacteristic
63C2×D4×C53C2×C411yescharacteristic
64C2×D4×C53C2×C411yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53C2×C411yescharacteristic
69C2×C22⋊D53C2×C411yescharacteristic
70C2×C22⋊D53C2×C411yescharacteristic
71C2×C22⋊D53C2×C411yescharacteristic
72C2×C22⋊D53C2×C411yescharacteristic
73C2×C22⋊D53C2×C411yescharacteristic
74C22×Dic53C2×C411yescharacteristic
75C22×Dic53C2×C411yescharacteristic
76#11719945—41no—
77#11719945—41no—
78C2×D212C2×C411yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×Dic106C2×C411yescharacteristic
82M16×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C22⋊Dic53C2×C411yescharacteristic
89C22⋊Dic53C2×C411yescharacteristic
90C22⋊Dic53C2×C411yescharacteristic
91C22⋊Dic53C2×C411yescharacteristic
92#11719921C2×C411yescharacteristic
93#11719921C2×C411yescharacteristic
94#11719921C2×C411yescharacteristic
95#11719921C2×C411yescharacteristic
96#11719919C2×C411yescharacteristic
97#11719919C2×C411yescharacteristic
98#11719919C2×C411yescharacteristic
99#11719919C2×C411yescharacteristic
100C4×Dic53C2×C411yescharacteristic
101C4×Dic53C2×C411yescharacteristic
102C4×Dic53C2311yescharacteristic
103C4×Dic53C2311yescharacteristic
104#11719917—41no—
105C2×#88322—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111#11719912—41no—
112C8×D53—41no—
113C8×D53—41no—
114C23×C53C22×C411yescharacteristic
115C23×C53C22×C411yescharacteristic
116C23×C53C22×C411yescharacteristic
117C22×D53C22×C411yescharacteristic
118C22×D53C22×C411yescharacteristic
119Q8×C53—41no—
120D4×C53—41no—
121D4×C53—41no—
122D4×C53—41no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53C22×C411yescharacteristic
130C2×C4×C53C22×C411yescharacteristic
131C2×C4×C53C22×C411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—22no—
134C22⋊D53—22no—
135C22⋊D53—41no—
136C22⋊D53—41no—
137C22⋊D53—41no—
138C22⋊D53—41no—
139C2×Dic53—22no—
140C2×Dic53—41no—
141C2×Dic53C22×C411yescharacteristic
142C2×Dic53C22×C411yescharacteristic
143C2×Dic53C22×C411yescharacteristic
144C2×Dic53C22×C411yescharacteristic
145C2×Dic53C22×C411yescharacteristic
146C2×Dic53C22×C411yescharacteristic
147D212—41no—
148C4×D53—21no—
149C4×D53—21no—
150C4×D53—21no—
151C4×D53—21no—
152Dic106—41no—
153C8×C53—21no—
154C8×C53—21no—
155C8×C53—21no—
156C8×C53—21no—
157#88322—21no—
158#88322—21no—
159#88322—21no—
160#88322—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×C53C23×C411yescharacteristic
171C2×D53—21no—
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53H(5,𝔽2)11yescharacteristic
181C4×C53H(5,𝔽2)11yescharacteristic
182Dic53—21no—
183Dic53—21no—
184Dic53—21no—
185Dic53—21no—
186Dic53—21no—
187Dic53#13112yes—
188#2981—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53—41no—
191C2×C53—41no—
192C2×C53#51811yescharacteristic
193C2×C53C2×#13111yescharacteristic
194C2×C53#56811yescharacteristic
195D53—41no—
196D53—41no—
197#532—531no—
198#436—531no—
199#436—531no—
200#435—531no—
201#435—531no—
202#435—531no—
203#435—531no—
204C8×D4—531no—
205C8×D4—531no—
206#432—531no—
207#431—531no—
208#408—531no—
209#408—531no—
210#408—531no—
211#408D5311yescharacteristic, complemented
212C53#298111yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C2×#41C2×D5311yescharacteristic
215C2×M16—531no—
216C2×M16—531no—
217C2×M16—531no—
218C2×M16C2×D5311yescharacteristic
219C22×C8—531no—
220C22×C8—531no—
221C22×C8—531no—
222C22×C8C2×D5311yescharacteristic
223C4⋊Q8—531no—
224#127—531no—
225#124—531no—
226#124—531no—
227#121—531no—
228#121—531no—
229#121—531no—
230#121—531no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4—531no—
234C4×D4—531no—
235C2×C42—531no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238C4⋊C8—531no—
239C4⋊C8—531no—
240C22⋊C8—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243C22⋊C8—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246C22⋊C8—531no—
247C22⋊C8—531no—
248C22⋊C8C2×D5311yescharacteristic
249C22⋊C8C2×D5311yescharacteristic
250C22⋊C8C2×D5311yescharacteristic
251C22⋊C8C2×D5311yescharacteristic
252#97—531no—
253#97—531no—
254C4×C8—531no—
255C4×C8—531no—
256#41—41no—
257#41—41no—
258#41—2121no—
259#41—2121no—
260C2×Q8—531no—
261C2×Q8C4×D5311yescharacteristic
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—531no—
269C2×D4C4×D5311yescharacteristic
270C2×D4C4×D5311yescharacteristic
271C2×D4C4×D5311yescharacteristic
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4C22×D5311yescharacteristic
277C22×C4C22×D5311yescharacteristic
278C22×C4C22×D5311yescharacteristic
279M16—41no—
280M16—2121no—
281M16—2121no—
282M16—2121no—
283C2×C8—41no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8C22×D5311yescharacteristic
292C2×C8C22×D5311yescharacteristic
293C2×C8C22×D5311yescharacteristic
294C2×C8C22×D5311yescharacteristic
295C4⋊C4—531no—
296C4⋊C4—531no—
297C4⋊C4—531no—
298C4⋊C4—531no—
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—531no—
302C22⋊C4—531no—
303C22⋊C4—531no—
304C22⋊C4—531no—
305C22⋊C4—531no—
306C22⋊C4—531no—
307C42—531no—
308C42—531no—
309C42—531no—
310C42—531no—
311C23—531no—
312C23—531no—
313C23C2×C4×D5311yescharacteristic
314C23C2×C4×D5311yescharacteristic
315C23C2×C4×D5311yescharacteristic
316Q8—41no—
317Q8—2121no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—1062no—
322D4—1062no—
323D4—2121no—
324D4—2121no—
325D4—2121no—
326D4—2121no—
327D4—2121no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—531no—
338C2×C4—531no—
339C2×C4—531no—
340C2×C4—1062no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—2121no—
346C2×C4C2×C4×D5311yescharacteristic
347C2×C4C2×C4×D5311yescharacteristic
348C2×C4C2×C4×D5311yescharacteristic
349C2×C4C23×D5311yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—1061no—
355C8—1061no—
356C8—1061no—
357C8—1061no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—1061no—
368C22—1061no—
369C22—1061no—
370C22—1061no—
371C22—1061no—
372C22—1061no—
373C22C22×C4×D5311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—532no—
378C4—1061no—
379C4—1061no—
380C4—1061no—
381C4—1061no—
382C4—1061no—
383C4#1313326211yescharacteristic
384C4#1313326211yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—2121no—
389C2—2121no—
390C2#1681109611yescharacteristic
391C2#131×D5311yescharacteristic
392C2#1681138811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D53
commutator, socleC22×C53C23×C4
FittingC53×#408C2
hypercenter#408D53
hypocenterC53#2981
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #16810699C2semidirect
3. #16810698C2semidirect
4. #16810697C2semidirect
5. #16810694C2semidirect
6. #16810692C2semidirect
7. #16810691C2semidirect
8. C8×C22⋊D53C2semidirect
9. #16810686C2semidirect
10. #16810677C2semidirect
11. #16810673C2semidirect
12. #16810590C2semidirect
13. #16810588C2semidirect
14. #16810588C2semidirect
15. C53×#408C2semidirect
211. #408D53semidirect
212. C53#2981semidirect