Finite Group Databasebeta

← #20287037

Subgroups — #20287037

Order 6784.

430 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811291C211yesmaximal, characteristic, complemented
2#16811273C211yesmaximal, characteristic, complemented
3#16810968C211yesmaximal, characteristic, complemented
4#16810943C211yesmaximal, characteristic, complemented
5#16810939C211yesmaximal, characteristic, complemented
6#16810907C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810766C211yesmaximal, characteristic, complemented
10#16810765C211yesmaximal, characteristic, complemented
11#16810714C211yesmaximal, characteristic, complemented
12#16810699C211yesmaximal, characteristic, complemented
13#16810625C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C53×#469C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947—21no—
18C2×#11719947C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22#13133216C2211yescharacteristic
23#13133213—21no—
24#13133213C2211yescharacteristic
25#13133199—21no—
26#13133198—21no—
27#13133194—21no—
28#13133192C2211yescharacteristic
29#13133190—21no—
30#13133189—21no—
31#13133188C2211yescharacteristic
32#13133182C2211yescharacteristic
33D4×Dic53C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36C2×C4⋊Dic53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133148C2211yescharacteristic
39#13133148C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133140C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73#11719947—22no—
74#11719947—41no—
75#11719947—44no—
76#11719945—22no—
77#11719945—41no—
78C2×D212—21no—
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—21no—
84C2×Dic106C2311yescharacteristic
85D8×C53—41no—
86M16×C53—41no—
87C2×C8×C53C2311yescharacteristic
88C2×C8×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C4⋊C4×C53C2311yescharacteristic
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53D411yescharacteristic
96#11719923—22no—
97#11719922—41no—
98#11719921—21no—
99#11719921D411yescharacteristic
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919D411yescharacteristic
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111#11719912—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53C2×D411yescharacteristic
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—42no—
122D4×C53—81no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—22no—
130C22⋊D53—42no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C22⋊D53—42no—
134C22⋊D53—81no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—21no—
145D212—21no—
146D212—81no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—42no—
150C4×D53—42no—
151Dic106—21no—
152Dic106—42no—
153Dic106#4112yescomplemented
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×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C22×D411yescharacteristic
168C2×D53—21no—
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53C2×#4111yescharacteristic
178C4×C53H(5,𝔽2)11yescharacteristic
179Dic53—22no—
180Dic53—21no—
181Dic53—21no—
182Dic53—41no—
183Dic53—41no—
184Dic53AGL(1,ℤ/8)12yes—
185#3332—531nomaximal, Sylow
186C2×C53—41no—
187C2×C53—81no—
188C2×C53—81no—
189C2×C53#54611yescharacteristic
190C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C53#57711yescharacteristic
192D53—41no—
193D53—81no—
194#550—531no—
195#547—531no—
196#486—531no—
197#481—531no—
198#481—531no—
199#475—531no—
200#469—531no—
201#469D5311yescharacteristic, complemented
202#459—531no—
203#449—531no—
204#449—531no—
205C4×SD16—531no—
206#436—531no—
207#417—531no—
208#417—531no—
209C53#333211yescharacteristic, complemented, Sylow
210C2×#41—531no—
211C2×#41—531no—
212C2×#41—1061no—
213C2×SD16—531no—
214C2×D8—531no—
215C2×D8C2×D5311yescharacteristic
216C2×M16—531no—
217C2×M16C2×D5311yescharacteristic
218C4⋊Q8—531no—
219#127—531no—
220#127—1061no—
221#123—1061no—
222#123—1061no—
223C22⋊Q8—1061no—
224#121—531no—
225#121—531no—
226#121—1061no—
227#121—1061no—
228#121C2×D5311yescharacteristic
229#121C2×D5311yescharacteristic
230C4×Q8—531no—
231C4×D4—531no—
232C4×D4—531no—
233C4×D4—1061no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236#106—531no—
237#106—531no—
238#106C2×D5311yescharacteristic
239C4⋊C8—531no—
240#103—531no—
241#103—531no—
242#103—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102C2×D5311yescharacteristic
249#102C2×D5311yescharacteristic
250C22⋊C8—531no—
251C22⋊C8—531no—
252C4×C8—531no—
253#41—1062no—
254#41—1062no—
255#41—2121no—
256#41—2121no—
257#41—2124no—
258C2×Q8—531no—
259C2×Q8—1061no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4C22×D5311yescharacteristic
271C2×D4C22×D5311yescharacteristic
272C22×C4—531no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4C22×D5311yescharacteristic
279SD16—1062no—
280D8—41no—
281D8—2121no—
282M16—41no—
283M16—2121no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8C22×D5311yescharacteristic
287C2×C8C22×D5311yescharacteristic
288C4⋊C4—531no—
289C4⋊C4—531no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4C22×D5311yescharacteristic
296C4⋊C4C22×D5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—531no—
300C22⋊C4—531no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C42—531no—
305C42—1061no—
306C23—21no—
307C23—21no—
308C23—531no—
309C23—1061no—
310C23D4×D5311yescharacteristic
311Q8—532no—
312Q8—1061no—
313Q8—2122no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—81no—
320D4—1062no—
321D4—1061no—
322D4—1061no—
323D4—4241no—
324D4—4241no—
325D4—2122no—
326D4—2122no—
327D4—2122no—
328D4—2122no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4D4×D5311yescharacteristic
347C2×C4C23×D5311yescharacteristic
348C8—21no—
349C8—21no—
350C8—1061no—
351C8—1061no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—1061no—
362C22—1061no—
363C22—1061no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22C2×D4×D5311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—532no—
372C4—1061no—
373C4—1061no—
374C4—1062no—
375C4—2121no—
376C4—2121no—
377C4#41×D5311yescharacteristic
378C4#1313326211yescharacteristic
379C2—41no—
380C2—81no—
381C2—81no—
382C2—4241no—
383C2—2121no—
384C2#1681126211yescharacteristic
385C2AGL(1,ℤ/8)×D5311yescharacteristic
386C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3332
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16811291C2semidirect
2. #16811273C2semidirect
3. #16810968C2semidirect
4. #16810943C2semidirect
5. #16810939C2semidirect
6. #16810907C2semidirect
7. #16810875C2semidirect
8. #16810816C2semidirect
9. #16810766C2semidirect
10. #16810765C2semidirect
11. #16810714C2semidirect
12. #16810699C2semidirect
13. #16810625C2semidirect
14. #16810621C2semidirect
15. C53×#469C2semidirect
153. Dic106#41semidirect
201. #469D53semidirect
209. C53#3332semidirect