Finite Group Databasebeta

← #20286718

Subgroups — #20286718

Order 6784.

404 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#16811326C211yesmaximal, characteristic, complemented
2#16811295C211yesmaximal, characteristic, complemented
3#16810989C211yesmaximal, characteristic, complemented
4#16810949C211yesmaximal, characteristic, complemented
5#16810938C211yesmaximal, characteristic, complemented
6#16810877C211yesmaximal, characteristic, complemented
7#16810860C211yesmaximal, characteristic, complemented
8#16810819C211yesmaximal, characteristic, complemented
9#16810771C211yesmaximal, characteristic, complemented
10#16810771C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#479C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133214—21no—
24#13133214C2211yescharacteristic
25#13133212—21no—
26#13133202—21no—
27#13133192—21no—
28#13133192—21no—
29#13133192—21no—
30#13133192C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33#13133192C2211yescharacteristic
34C4×D212—21no—
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic
37#13133179—21no—
38#13133175C2211yescharacteristic
39C4⋊C4×D53C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic
44#13133137C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52#125×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60#11719949—22no—
61#11719949—22no—
62#11719949—41no—
63#11719949—41no—
64C2×D212—21no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212D411yescharacteristic
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×C4×D53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C42×C53—21no—
91C42×C53C2311yescharacteristic
92#11719924—22no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921D411yescharacteristic
100#11719921D411yescharacteristic
101C4⋊Dic53D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104C4×Dic53—21no—
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107D424—41no—
108D424—41no—
109#11719912—41no—
110C8×D53—41no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53C2×D411yescharacteristic
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53#4112yes—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C2×Dic53—21no—
126C2×Dic53C2×D411yescharacteristic
127C2×Dic53C2×D411yescharacteristic
128D212—21no—
129D212—21no—
130D212—21no—
131D212—21no—
132D212—21no—
133D212—21no—
134D212—41no—
135D212—42no—
136D212—42no—
137D212—42no—
138D212—42no—
139D212—81no—
140D212—81no—
141D212—81no—
142D212—81no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—42no—
148C4×D53—42no—
149C4×D53—81no—
150C4×D53—81no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53C2×#4111yescharacteristic
178C4×C53H(5,𝔽2)11yescharacteristic
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182#3309—531nomaximal, Sylow
183C2×C53#54611yescharacteristic
184C2×C53#57511yescharacteristic
185C2×C53#57611yescharacteristic
186D53—41no—
187D53—41no—
188D53—81no—
189D53—81no—
190D53—81no—
191#553—531no—
192#550—531no—
193#490—531no—
194#482—531no—
195#480—531no—
196#479D5311yescharacteristic, complemented
197#469—531no—
198#466—531no—
199#459—531no—
200#449—531no—
201#449—531no—
202#440—531no—
203#435—531no—
204#417—531no—
205#416—531no—
206C53#330911yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41—531no—
209C2×SD16—531no—
210C2×D8—531no—
211C2×D8—531no—
212C2×M16—531no—
213C22×C8—531no—
214#127—531no—
215#127—1061no—
216#126—1061no—
217#125C2×D5311yescharacteristic
218#123—1061no—
219#121—531no—
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224#121—1061no—
225#121—1061no—
226C4×Q8C2×D5311yescharacteristic
227C4×D4—531no—
228C4×D4—531no—
229C4×D4—1061no—
230C4×D4—1061no—
231#117—531no—
232C2×C4⋊C4—531no—
233#107—531no—
234#107C2×D5311yescharacteristic
235#106C2×D5311yescharacteristic
236C4⋊C8C2×D5311yescharacteristic
237#103—531no—
238#103C2×D5311yescharacteristic
239#103C2×D5311yescharacteristic
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249#97—531no—
250#41—1062no—
251#41—1062no—
252#41—2121no—
253#41—2121no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—531no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—1061no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273SD16—1062no—
274D8—2121no—
275D8—2121no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—2121no—
280C2×C8C22×D5311yescharacteristic
281C2×C8C22×D5311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—531no—
286C4⋊C4—531no—
287C4⋊C4—1061no—
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C4⋊C4C22×D5311yescharacteristic
291C22⋊C4—531no—
292C22⋊C4—531no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C42—21no—
300C42—1061no—
301C42C22×D5311yescharacteristic
302C23—531no—
303C23—531no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307Q8—21no—
308Q8#1171994912yes—
309D4—1061no—
310D4—1061no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—4241no—
316D4—4241no—
317D4—4241no—
318D4—4241no—
319D4—2121no—
320D4—2122no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4D4×D5311yescharacteristic
340C2×C4D4×D5311yescharacteristic
341C2×C4C23×D5311yescharacteristic
342C8—21no—
343C8—21no—
344C8—1061no—
345C8—1061no—
346C22—1061no—
347C22—1061no—
348C22—1061no—
349C22—1061no—
350C22—1061no—
351C22—1061no—
352C22—2121no—
353C22—2121no—
354C22—2121no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C2×D4×D5311yescharacteristic
362C4—22no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—1061no—
369C4—1061no—
370C4—2121no—
371C4C2×#1171994911yescharacteristic
372C4#1313326311yescharacteristic
373C2—4241no—
374C2—4241no—
375C2—4241no—
376C2—2121no—
377C2—2121no—
378C2#1681126411yescharacteristic
379C2#1681142211yescharacteristic
380C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#479C2
socleC22×C53C22×D4
hypercenter#479D53
hypocenterC53#3309
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #16811326C2semidirect
2. #16811295C2semidirect
3. #16810989C2semidirect
4. #16810949C2semidirect
5. #16810938C2semidirect
6. #16810877C2semidirect
7. #16810860C2semidirect
8. #16810819C2semidirect
9. #16810771C2semidirect
10. #16810771C2semidirect
11. #16810721C2semidirect
12. #16810693C2semidirect
13. #16810623C2semidirect
14. #16810618C2semidirect
15. C53×#479C2semidirect
196. #479D53semidirect
206. C53#3309semidirect