Finite Group Databasebeta

← #20286831

Subgroups — #20286831

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#16811333C211yesmaximal, characteristic, complemented
2#16811285C211yesmaximal, characteristic
3#16810983C211yesmaximal, characteristic, complemented
4#16810955C211yesmaximal, characteristic, complemented
5#16810943C211yesmaximal, characteristic, complemented
6#16810908C211yesmaximal, characteristic, complemented
7#16810880C211yesmaximal, characteristic, complemented
8#16810855C211yesmaximal, characteristic, complemented
9#16810845C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810777C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#441C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133217C2211yescharacteristic
23#13133209—21no—
24#13133205—21no—
25#13133205C2211yescharacteristic
26#13133197—21no—
27#13133197C2211yescharacteristic
28#13133195—21no—
29#13133195—21no—
30#13133193C2211yescharacteristic
31#13133193C2211yescharacteristic
32#13133192—21no—
33#13133192C2211yescharacteristic
34C4×D212—21no—
35C4×D212—21no—
36C4×D212C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133148C2211yescharacteristic
40#13133148C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×Q8×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61#11719949—22no—
62#11719949—41no—
63Q8×D53—22no—
64Q8×D53—41no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212D411yescharacteristic
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78Q16×C53—22no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C42×C53—21no—
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88#11719925—41no—
89#11719924—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921D411yescharacteristic
97#11719921D411yescharacteristic
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
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×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53C2×D411yescharacteristic
129C2×Dic53C2×D411yescharacteristic
130D212—22no—
131D212—21no—
132D212—21no—
133D212—42no—
134D212—42no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53—81no—
142C4×D53—81no—
143Dic106—21no—
144Dic106—21no—
145Dic106—42no—
146Dic106—81no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53C22×D411yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53#13712yes—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C53H(5,𝔽2)11yescharacteristic
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3316—531nomaximal, Sylow
175C2×C53#54611yescharacteristic
176C2×C53C2×#13711yescharacteristic
177C2×C53#57711yescharacteristic
178D53—22no—
179D53—41no—
180D53—81no—
181#554—531no—
182#548—531no—
183#489—531no—
184#483—531no—
185#481—531no—
186#475—531no—
187#470—531no—
188#465—531no—
189#464—531no—
190#452—531no—
191#451—531no—
192#441D5311yescharacteristic, complemented
193#435—531no—
194#417—531no—
195C2×#103—531no—
196C53#331611yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C22×Q8—531no—
199C2×Q16—531no—
200C2×Q16C2×D5311yescharacteristic
201C2×SD16—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205#125—1061no—
206#124—531no—
207#124—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213C22⋊Q8—1061no—
214#121—531no—
215#121—1061no—
216C4×Q8C2×D5311yescharacteristic
217C4×Q8C2×D5311yescharacteristic
218C4×D4—531no—
219C4×D4—1061no—
220C4×D4—1061no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223#106—531no—
224#106—531no—
225#106C2×D5311yescharacteristic
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103C2×D5311yescharacteristic
233#103C2×D5311yescharacteristic
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8—531no—
239#97C2×D5311yescharacteristic
240#41—1062no—
241#41—2121no—
242C2×Q8—531no—
243C2×Q8—1061no—
244C2×Q8—1061no—
245C2×Q8—1062no—
246C2×Q8—2121no—
247C2×Q8C22×D5311yescharacteristic
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1061no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259Q16—22no—
260Q16—2121no—
261SD16—2121no—
262M16—2121no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—2121no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
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⋊C4C22×D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—531no—
282C22⋊C4—531no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—21no—
290C42—21no—
291C42C22×D5311yescharacteristic
292C23—531no—
293C23—531no—
294C23—1061no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—1061no—
299Q8—1061no—
300Q8—4241no—
301Q8—2122no—
302Q8#1171994512yes—
303D4—1062no—
304D4—1061no—
305D4—1061no—
306D4—2122no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4D4×D5311yescharacteristic
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—532no—
333C22—1061no—
334C22—1061no—
335C22—1061no—
336C22—1061no—
337C22—1061no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22C2×D4×D5311yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—1061no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994511yescharacteristic
354C4#1313326311yescharacteristic
355C2—1062no—
356C2—4241no—
357C2—2121no—
358C2#1681126911yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#441C2
socleC22×C53C22×D4
hypercenter#441D53
hypocenterC53#3316
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #16811333C2semidirect
3. #16810983C2semidirect
4. #16810955C2semidirect
5. #16810943C2semidirect
6. #16810908C2semidirect
7. #16810880C2semidirect
8. #16810855C2semidirect
9. #16810845C2semidirect
10. #16810784C2semidirect
11. #16810777C2semidirect
12. #16810696C2semidirect
13. #16810623C2semidirect
14. #103×D53C2semidirect
15. C53×#441C2semidirect
192. #441D53semidirect
196. C53#3316semidirect