Finite Group Databasebeta

← #20287799

Subgroups — #20287799

Order 6784.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811358C211yesmaximal, characteristic, complemented
2#16811013C211yesmaximal, characteristic, complemented
3#16811010C211yesmaximal, characteristic, complemented
4#16811005C211yesmaximal, characteristic, complemented
5#16810845C211yesmaximal, characteristic, complemented
6#16810842C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810818C211yesmaximal, characteristic, complemented
10#16810815C211yesmaximal, characteristic, complemented
11#16810754C211yesmaximal, characteristic, complemented
12#16810723C211yesmaximal, characteristic, complemented
13#16810721C211yesmaximal, characteristic, complemented
14C4×#11719922C211yesmaximal, characteristic, complemented
15C53×#559C211yesmaximal, characteristic, complemented
16C2×#11719925C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×#11719922C2211yescharacteristic
25#13133214C2211yescharacteristic
26#13133206—21no—
27#13133205C2211yescharacteristic
28#13133205C2211yescharacteristic
29#13133202—21no—
30#13133202—21no—
31C4×Dic106C2211yescharacteristic
32C4×D212C2211yescharacteristic
33C4×D212C2211yescharacteristic
34#13133179—21no—
35#13133150C2211yescharacteristic
36C4⋊#88322C2211yescharacteristic
37C4⋊#88322C2211yescharacteristic
38C4⋊#88322C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133140C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45#13133128C2211yescharacteristic
46#13133128C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48#127×C53C2211yescharacteristic
49#124×C53—21no—
50#124×C53C2211yescharacteristic
51#124×C53C2211yescharacteristic
52#123×C53—21no—
53#123×C53—21no—
54C4×Q8×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4×D4×C53—21no—
57C4×D4×C53C2211yescharacteristic
58C4×D4×C53C2211yescharacteristic
59C2×Q8×C53D411yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×Dic106C2311yescharacteristic
74C4⋊C4×C53—21no—
75C4⋊C4×C53D411yescharacteristic
76C4⋊C4×C53D411yescharacteristic
77C4⋊C4×C53D411yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C22⋊C4×C53—21no—
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87C42×C53C2311yescharacteristic
88C42×C53C2311yescharacteristic
89#11719925—41no—
90#11719924—22no—
91#11719924—41no—
92#11719923—22no—
93#11719923—41no—
94#11719922—22no—
95#11719922—41no—
96#11719922—41no—
97#11719922—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53—21no—
113C22×D53—21no—
114C22×D53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—41no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—42no—
123D4×C53—81no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C2×Dic53—21no—
136C2×Dic53—21no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—42no—
142D212—81no—
143C4×D53—81no—
144C4×D53—81no—
145Dic106—21no—
146Dic106—21no—
147#88322—21no—
148#88322—21no—
149#88322—21no—
150#88322#4112yes—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—41no—
175Dic53—41no—
176#3403—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53—81no—
179C2×C53#55011yescharacteristic
180C2×C53#57511yescharacteristic
181C2×C53#57611yescharacteristic
182D53—81no—
183D53—81no—
184#559—531no—
185#559D5311yescharacteristic, complemented
186#496—531no—
187#496—531no—
188#495—531no—
189#464—531no—
190#464—531no—
191#463—531no—
192#463—531no—
193#459—531no—
194#459—531no—
195#446—531no—
196#440—531no—
197#440—531no—
198C4×D8—531no—
199C53#340311yescharacteristic, complemented, Sylow
200C2×Q16—531no—
201C2×SD16—531no—
202C2×SD16—531no—
203C2×SD16—531no—
204C2×SD16—531no—
205C2×D8—531no—
206C2×D8—531no—
207C2×D8—531no—
208C2×D8—531no—
209#127—531no—
210#127C2×D5311yescharacteristic
211#124—21no—
212#124—531no—
213#124—531no—
214#124—1061no—
215#124C2×D5311yescharacteristic
216#124C2×D5311yescharacteristic
217#123—21no—
218#123—21no—
219#123—1061no—
220#123—1061no—
221C4×Q8—531no—
222C4×Q8C2×D5311yescharacteristic
223C4×Q8C2×D5311yescharacteristic
224C4×D4—21no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C4×D4C2×D5311yescharacteristic
229C4×D4C2×D5311yescharacteristic
230#107—531no—
231C4⋊C8—531no—
232C4⋊C8—531no—
233C4⋊C8—531no—
234#103—531no—
235#103—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#97—531no—
241#97—531no—
242C4×C8—531no—
243C2×Q8—531no—
244C2×Q8C22⋊D5311yescharacteristic
245C2×Q8C22×D5311yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—531no—
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4C22×D5311yescharacteristic
253C2×D4C22×D5311yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—1061no—
257C22×C4—1061no—
258Q16—2121no—
259SD16—1062no—
260SD16—1062no—
261SD16—2121no—
262SD16—2121no—
263D8—1062no—
264D8—2121no—
265D8—2121no—
266D8—2121no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—531no—
271C4⋊C4—21no—
272C4⋊C4—531no—
273C4⋊C4—1061no—
274C4⋊C4C22⋊D5311yescharacteristic
275C4⋊C4C22⋊D5311yescharacteristic
276C4⋊C4C22⋊D5311yescharacteristic
277C4⋊C4C22×D5311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C42—21no—
291C42—1061no—
292C42C22×D5311yescharacteristic
293C42C22×D5311yescharacteristic
294C42C22×D5311yescharacteristic
295C23—21no—
296C23—21no—
297C23—1061no—
298C23—1061no—
299Q8—21no—
300Q8—21no—
301Q8—41no—
302Q8—1061no—
303Q8—1061no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—1061no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—4241no—
315D4—2122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4C2×C22⋊D5311yescharacteristic
325C2×C4C2×C22⋊D5311yescharacteristic
326C2×C4C2×C22⋊D5311yescharacteristic
327C2×C4C2×C22⋊D5311yescharacteristic
328C2×C4C2×C22⋊D5311yescharacteristic
329C2×C4C2×C22⋊D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—532no—
332C8—1061no—
333C8—1061no—
334C8—1061no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C22×C22⋊D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—2121no—
357C4—2121no—
358C4#41×D5311yescharacteristic
359C4#1313326311yescharacteristic
360C2—81no—
361C2—81no—
362C2—4241no—
363C2—4241no—
364C2#1681129211yescharacteristic
365C2#1681142111yescharacteristic
366C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#559C2
socleC22×C53C22×D4
hypercenter#559D53
hypocenterC53#3403
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #16811358C2semidirect
2. #16811013C2semidirect
3. #16811010C2semidirect
4. #16811005C2semidirect
5. #16810845C2semidirect
6. #16810842C2semidirect
7. #16810838C2semidirect
8. #16810835C2semidirect
9. #16810818C2semidirect
10. #16810815C2semidirect
11. #16810754C2semidirect
12. #16810723C2semidirect
13. #16810721C2semidirect
14. C4×#11719922C2semidirect
15. C53×#559C2semidirect
185. #559D53semidirect
199. C53#3403semidirect