Finite Group Databasebeta

← #20287024

Subgroups — #20287024

Order 6784.

388 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#16811331C211yesmaximal, characteristic, complemented
2#16811272C211yesmaximal, characteristic
3#16810972C211yesmaximal, characteristic, complemented
4#16810954C211yesmaximal, characteristic, complemented
5#16810952C211yesmaximal, characteristic, complemented
6#16810911C211yesmaximal, characteristic, complemented
7#16810879C211yesmaximal, characteristic, complemented
8#16810878C211yesmaximal, characteristic, complemented
9#16810831C211yesmaximal, characteristic, complemented
10#16810782C211yesmaximal, characteristic, complemented
11SD16×Dic53C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719917C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133215C2211yescharacteristic
22#13133207—21no—
23#13133200—21no—
24#13133198—21no—
25#13133197—21no—
26#13133196C2211yescharacteristic
27#13133194—21no—
28#13133193C2211yescharacteristic
29#13133193C2211yescharacteristic
30#13133189—21no—
31Q8×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34#13133177—21no—
35C2×C4⋊Dic53C2211yescharacteristic
36C2×#11719919—21no—
37C4⋊C4×D53C2211yescharacteristic
38#13133149C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—22no—
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⋊D53D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72#11719947—22no—
73#11719947—41no—
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78Q16×C53—41no—
79SD16×C53—22no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88#11719925—41no—
89#11719921—21no—
90#11719921—21no—
91#11719921D411yescharacteristic
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107#11719917—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—42no—
120D4×C53#4112yes—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—22no—
130C22⋊D53—42no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—81no—
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—81no—
144Dic106—21no—
145Dic106—21no—
146Dic106—81no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—22no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53C2×#4111yescharacteristic
166C4×C53H(5,𝔽2)11yescharacteristic
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53#13712yes—
174#3331—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—42no—
177C2×C53#54611yescharacteristic
178C2×C53C2×#13711yescharacteristic
179C2×C53#57711yescharacteristic
180D53—41no—
181#554—531no—
182#547—531no—
183#487—531no—
184#483—531no—
185#483—531no—
186#476—531no—
187#470—531no—
188#470—531no—
189#462—531no—
190#452—531no—
191#452D5311yescharacteristic, complemented
192C4×SD16—531no—
193#436—531no—
194#417—531no—
195#417—531no—
196C53#333111yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×#41C2×D5311yescharacteristic
199C2×Q16—531no—
200C2×Q16C2×D5311yescharacteristic
201C2×SD16C2×D5311yescharacteristic
202C2×M16—531no—
203C2×M16—531no—
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206#125—1061no—
207#123—1061no—
208#123—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—531no—
212C22⋊Q8—1061no—
213C22⋊Q8—1061no—
214C22⋊Q8C2×D5311yescharacteristic
215#121—1061no—
216C4×Q8—531no—
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—1061no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—1061no—
223#106—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103C2×D5311yescharacteristic
234#102—531no—
235#102—531no—
236#102C2×D5311yescharacteristic
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239C4×C8—531no—
240#41—22no—
241#41—41no—
242#41—1062no—
243#41—2121no—
244C2×Q8—531no—
245C2×Q8—1061no—
246C2×Q8C22×D5311yescharacteristic
247C2×Q8C22×D5311yescharacteristic
248C2×D4—21no—
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4C22×D5311yescharacteristic
252C22×C4—21no—
253C22×C4—531no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4C22×D5311yescharacteristic
259Q16—41no—
260Q16—2121no—
261SD16—22no—
262M16—2121no—
263M16—2121no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4C22×D5311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—531no—
285C22⋊C4—531no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C42—531no—
291C42—1061no—
292C23—21no—
293C23—531no—
294C23D4×D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—1061no—
300Q8—1061no—
301Q8—4241no—
302D4—21no—
303D4—42no—
304D4—1062no—
305D4—2122no—
306D4#1171994712yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—531no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—532no—
347C4—1061no—
348C4—1061no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994711yescharacteristic
354C4#1313326211yescharacteristic
355C2—41no—
356C2—42no—
357C2—2121no—
358C2#1681126011yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3331
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
1. #16811331C2semidirect
3. #16810972C2semidirect
4. #16810954C2semidirect
5. #16810952C2semidirect
6. #16810911C2semidirect
7. #16810879C2semidirect
8. #16810878C2semidirect
9. #16810831C2semidirect
10. #16810782C2semidirect
11. SD16×Dic53C2semidirect
12. #16810698C2semidirect
13. #16810624C2semidirect
14. #16810621C2semidirect
15. C53×#452C2semidirect
191. #452D53semidirect
196. C53#3331semidirect