Finite Group Databasebeta

← #20284918

Subgroups — #20284918

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811224C211yesmaximal, characteristic, complemented
2#16811102C211yesmaximal, characteristic, complemented
3#16810943C211yesmaximal, characteristic, complemented
4#16810935C211yesmaximal, characteristic, complemented
5#16810933C211yesmaximal, characteristic, complemented
6#16810921C211yesmaximal, characteristic, complemented
7#16810822C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810769C211yesmaximal, characteristic, complemented
10#16810765C211yesmaximal, characteristic, complemented
11#16810725C211yesmaximal, characteristic, complemented
12C4×#11719923C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810685C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20#13133213C2211yescharacteristic
21#13133208C2211yescharacteristic
22#13133200—21no—
23#13133199—21no—
24#13133192C2211yescharacteristic
25#13133191—21no—
26#13133190—21no—
27#13133188C2211yescharacteristic
28C4×Dic106C2211yescharacteristic
29#13133182C2211yescharacteristic
30C4×D212C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32C4×C22⋊D53—21no—
33#13133177—21no—
34#13133174C2211yescharacteristic
35#13133173—21no—
36C2×#11719919—21no—
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51C4×#88322C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69#11719947—22no—
70#11719947—22no—
71C2×D212—21no—
72C2×D212C2311yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77D8×C53—22no—
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87#11719923—22no—
88#11719922—22no—
89#11719921—21no—
90#11719921—21no—
91#11719921D411yescharacteristic
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919D411yescharacteristic
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107#11719913—22no—
108C23×C53—21no—
109C23×C53—21no—
110C22×D53—21no—
111C22×D53C2×D411yescharacteristic
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—21no—
115D4×C53#4112yes—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—24no—
124C22⋊D53—81no—
125C22⋊D53—81no—
126C22⋊D53—81no—
127C2×Dic53—24no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—81no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—81no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—81no—
141C4×D53—81no—
142Dic106—21no—
143Dic106#4112yes—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—22no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53C2×#4111yescharacteristic
167Dic53—22no—
168Dic53—21no—
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173#3119—531nomaximal, Sylow
174C2×C53—42no—
175C2×C53—81no—
176C2×C53#52511yescharacteristic
177C2×C53#57611yescharacteristic
178C2×C53#57711yescharacteristic
179D53—41no—
180D53—81no—
181#540—531no—
182#520—531no—
183#481—531no—
184#480—531no—
185#479—531no—
186#478—531no—
187#460—531no—
188#459—531no—
189#449—531no—
190#449—531no—
191#442D5311yescharacteristic, complemented
192#440—531no—
193C4×SD16—531no—
194C4×D8—531no—
195#433—531no—
196C53#311911yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×SD16—531no—
199C2×SD16—531no—
200C2×D8—531no—
201C2×D8C2×D5311yescharacteristic
202#127—531no—
203#125—531no—
204#123—1061no—
205#123—1061no—
206#121—531no—
207#121—531no—
208#121—1061no—
209#121—1061no—
210C4×Q8—531no—
211C4×Q8—531no—
212C4×D4—531no—
213C4×D4—531no—
214C4×D4—1061no—
215C4×D4—1061no—
216C4×D4C2×D5311yescharacteristic
217C4×D4C2×D5311yescharacteristic
218#117—531no—
219#117—1061no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222#107—531no—
223#107—531no—
224#106—531no—
225#106C2×D5311yescharacteristic
226C4⋊C8—531no—
227C4⋊C8—531no—
228#103—531no—
229#103—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235#102C2×D5311yescharacteristic
236C22⋊C8—531no—
237C22⋊C8—531no—
238#97C2×D5311yescharacteristic
239C4×C8—531no—
240#41—1062no—
241#41—1062no—
242C2×Q8—531no—
243C2×D4—531no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C2×D4C22×D5311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—531no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258SD16—1062no—
259SD16—1062no—
260D8—22no—
261D8—1062no—
262C2×C8—531no—
263C2×C8—531no—
264C2×C8C22×D5311yescharacteristic
265C2×C8C22×D5311yescharacteristic
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—531no—
280C22⋊C4—531no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—531no—
286C42—1061no—
287C42—1061no—
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—21no—
291C23—531no—
292C23—1061no—
293Q8—532no—
294Q8—1061no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—1064no—
299D4—1061no—
300D4—1061no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305D4#1171994512yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—1064no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4D4×D5311yescharacteristic
323C2×C4D4×D5311yescharacteristic
324C2×C4C23×D5311yescharacteristic
325C8—21no—
326C8—21no—
327C8—1061no—
328C8—1061no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—1061no—
336C22—1061no—
337C22—1061no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—1061no—
347C4—1062no—
348C4—1061no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4#41×D5311yescharacteristic
353C4C2×#1171994511yescharacteristic
354C2—42no—
355C2—81no—
356C2—4241no—
357C2—2121no—
358C2#1681113211yescharacteristic
359C2#1681142711yescharacteristic
360C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3119
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811224C2semidirect
2. #16811102C2semidirect
3. #16810943C2semidirect
4. #16810935C2semidirect
5. #16810933C2semidirect
6. #16810921C2semidirect
7. #16810822C2semidirect
8. #16810816C2semidirect
9. #16810769C2semidirect
10. #16810765C2semidirect
11. #16810725C2semidirect
12. C4×#11719923C2semidirect
13. C4×#11719922C2semidirect
14. #16810685C2semidirect
15. C53×#442C2semidirect
191. #442D53semidirect
196. C53#3119semidirect