Finite Group Databasebeta

← #20285594

Subgroups — #20285594

Order 6784.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811201C211yesmaximal, characteristic
2#16811196C211yesmaximal, characteristic, complemented
3#16810962C211yesmaximal, characteristic, complemented
4#16810956C211yesmaximal, characteristic, complemented
5#16810954C211yesmaximal, characteristic, complemented
6#16810946C211yesmaximal, characteristic, complemented
7#16810878C211yesmaximal, characteristic, complemented
8#16810868C211yesmaximal, characteristic, complemented
9#16810863C211yesmaximal, characteristic, complemented
10#16810849C211yesmaximal, characteristic, complemented
11#16810785C211yesmaximal, characteristic, complemented
12#16810777C211yesmaximal, characteristic, complemented
13#16810760C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133216—21no—
23#13133203—21no—
24#13133202—21no—
25#13133199—21no—
26#13133197C2211yescharacteristic
27#13133196—21no—
28#13133196C2211yescharacteristic
29#13133193C2211yescharacteristic
30#13133193C2211yescharacteristic
31#13133193C2211yescharacteristic
32#13133191—21no—
33#13133189—21no—
34#13133189—21no—
35#13133188C2211yescharacteristic
36#13133188C2211yescharacteristic
37#13133175C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133149C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70C2×C22⋊D53D411yescharacteristic
71C22×Dic53—21no—
72Q8×D53—41no—
73Q8×D53—41no—
74#11719945—41no—
75#11719945—41no—
76C2×D212D411yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×Dic106—21no—
81C2×Dic106D411yescharacteristic
82C2×Dic106C2311yescharacteristic
83Q16×C53—41no—
84SD16×C53—41no—
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719925—41no—
94#11719923—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719912—41no—
113C8×D53—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53C2×D411yescharacteristic
117C22×D53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—41no—
134C22⋊D53—41no—
135C22⋊D53—81no—
136C22⋊D53—81no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—81no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—41no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—42no—
150Dic106—21no—
151Dic106—21no—
152Dic106—41no—
153Dic106—42no—
154C8×C53—21no—
155C8×C53—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C4×C53—21no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53H(5,𝔽2)11yescharacteristic
176C4×C53H(5,𝔽2)11yescharacteristic
177Dic53—21no—
178Dic53—21no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3201—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#53411yescharacteristic
187C2×C53#57711yescharacteristic
188C2×C53#57811yescharacteristic
189D53—41no—
190D53—41no—
191#537—531no—
192#536—531no—
193#485—531no—
194#484—531no—
195#483—531no—
196#482—531no—
197#470—531no—
198#468—531no—
199#467—531no—
200#465—531no—
201#452—531no—
202#452D5311yescharacteristic, complemented
203#451—531no—
204#448—531no—
205#408—531no—
206C53#320111yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41C2×D5311yescharacteristic
209C22×Q8—531no—
210C2×Q16—531no—
211C2×Q16C2×D5311yescharacteristic
212C2×SD16—531no—
213C2×SD16C2×D5311yescharacteristic
214C2×M16—531no—
215C22×C8—531no—
216C4⋊Q8—1061no—
217#124—1061no—
218#123—1061no—
219#123—1061no—
220C22⋊Q8—531no—
221C22⋊Q8—531no—
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8—1061no—
226C22⋊Q8C2×D5311yescharacteristic
227#121—531no—
228#121—531no—
229#121—1061no—
230#121—1061no—
231#121—1061no—
232#117—531no—
233C2×C4⋊C4—531no—
234#107—531no—
235#107—531no—
236#106—531no—
237#106—531no—
238#103—531no—
239#103—531no—
240#103—531no—
241#103—531no—
242#103—531no—
243#103C2×D5311yescharacteristic
244#102—531no—
245#102C2×D5311yescharacteristic
246C22⋊C8—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249C22⋊C8C2×D5311yescharacteristic
250#41—41no—
251#41—41no—
252#41—2121no—
253#41—2121no—
254C2×Q8—531no—
255C2×Q8—531no—
256C2×Q8—1061no—
257C2×Q8—2121no—
258C2×Q8—2121no—
259C2×Q8C22×D5311yescharacteristic
260C2×Q8C22×D5311yescharacteristic
261C2×D4—21no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4C22×D5311yescharacteristic
268C22×C4—21no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4C22×D5311yescharacteristic
274Q16—41no—
275Q16—2121no—
276SD16—41no—
277SD16—2121no—
278M16—2121no—
279C2×C8—531no—
280C2×C8—531no—
281C2×C8—2121no—
282C2×C8C22×D5311yescharacteristic
283C2×C8C22×D5311yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—531no—
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4C22×D5311yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C42—1061no—
305C23—21no—
306C23—531no—
307C23—531no—
308C23D4×D5311yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—1061no—
314Q8—1061no—
315Q8—2121no—
316Q8—2122no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—4241no—
321D4—4241no—
322D4—2121no—
323D4—2121no—
324D4—2121no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—4241no—
342C2×C4—2122no—
343C2×C4D4×D5311yescharacteristic
344C2×C4C23×D5311yescharacteristic
345C8—21no—
346C8—21no—
347C8—1061no—
348C8—1061no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—1061no—
356C22—1061no—
357C22—1061no—
358C22—1061no—
359C22—1061no—
360C22—1061no—
361C22C2×D4×D5311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—1061no—
367C4—1061no—
368C4—2121no—
369C4—2121no—
370C4—2121no—
371C4—2121no—
372C4#1313326211yescharacteristic
373C4#1313326211yescharacteristic
374C2—41no—
375C2—81no—
376C2—2121no—
377C2—2121no—
378C2#1681118311yescharacteristic
379C2#1681143011yescharacteristic
380C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3201
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #16811196C2semidirect
3. #16810962C2semidirect
4. #16810956C2semidirect
5. #16810954C2semidirect
6. #16810946C2semidirect
7. #16810878C2semidirect
8. #16810868C2semidirect
9. #16810863C2semidirect
10. #16810849C2semidirect
11. #16810785C2semidirect
12. #16810777C2semidirect
13. #16810760C2semidirect
14. #16810588C2semidirect
15. C53×#452C2semidirect
202. #452D53semidirect
206. C53#3201semidirect