Finite Group Databasebeta

← #20285670

Subgroups — #20285670

Order 6784.

394 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811177C211yesmaximal, characteristic, complemented
2#16811172C211yesmaximal, characteristic
3#16811001C211yesmaximal, characteristic, complemented
4#16810994C211yesmaximal, characteristic, complemented
5#16810972C211yesmaximal, characteristic, complemented
6#16810968C211yesmaximal, characteristic, complemented
7#16810846C211yesmaximal, characteristic, complemented
8#16810845C211yesmaximal, characteristic, complemented
9#16810842C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810786C211yesmaximal, characteristic, complemented
12#16810784C211yesmaximal, characteristic, complemented
13#16810782C211yesmaximal, characteristic, complemented
14#16810675C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719925C2211yescharacteristic
21C2×#11719913C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719924C2211yescharacteristic
24#13133216C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133215C2211yescharacteristic
27#13133213C2211yescharacteristic
28#13133207—21no—
29#13133205C2211yescharacteristic
30#13133197C2211yescharacteristic
31#13133195—21no—
32#13133193C2211yescharacteristic
33#13133190—21no—
34#13133190—21no—
35C4×Dic106—21no—
36C4×Dic106C2211yescharacteristic
37#13133177—21no—
38#13133177—21no—
39#13133174—21no—
40C42×D53C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50C8×Dic53C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C22×Dic53—21no—
66#11719949—41no—
67#11719949—41no—
68#11719947—41no—
69#11719947—41no—
70C2×D212—21no—
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53D411yescharacteristic
74C2×C4×D53D411yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79Q16×C53—41no—
80SD16×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C42×C53—21no—
88C42×C53C2311yescharacteristic
89#11719925—41no—
90#11719925—41no—
91#11719924—41no—
92#11719923—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53D411yescharacteristic
108C4×Dic53D411yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112Dic212—41no—
113#11719913—41no—
114C23×C53—21no—
115C22×D53—21no—
116C22×D53C2×D411yescharacteristic
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—42no—
130C22⋊D53—42no—
131C22⋊D53—42no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—42no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—42no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—22no—
145C4×D53—22no—
146C4×D53—42no—
147Dic106—21no—
148Dic106—21no—
149Dic106—21no—
150Dic106—21no—
151Dic106—42no—
152C8×C53—21no—
153C8×C53—21no—
154#88322—21no—
155#88322—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53C22×D411yescharacteristic
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53H(5,𝔽2)11yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53#13512yes—
178Dic53#13712yes—
179#3210—531nomaximal, Sylow
180C2×C53—81no—
181C2×C53#53511yescharacteristic
182C2×C53C2×#13511yescharacteristic
183C2×C53C2×#13711yescharacteristic
184D53—41no—
185D53—81no—
186#533—531no—
187#532—531no—
188#494—531no—
189#492—531no—
190#487—531no—
191#486—531no—
192#464—531no—
193#464—531no—
194#464—531no—
195#464D5311yescharacteristic, complemented
196#452—531no—
197#452—531no—
198#452—531no—
199#452—531no—
200#431—531no—
201C53#321011yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C2×#41—531no—
204C2×Q16—531no—
205C2×Q16—531no—
206C2×Q16—531no—
207C2×Q16C2×D5311yescharacteristic
208C2×SD16—531no—
209C2×SD16—531no—
210C2×SD16—531no—
211C2×SD16C2×D5311yescharacteristic
212C4⋊Q8—531no—
213C4⋊Q8—531no—
214C4⋊Q8—531no—
215#127—531no—
216#125—1061no—
217#124—531no—
218#124C2×D5311yescharacteristic
219C22⋊Q8—531no—
220C22⋊Q8—531no—
221C22⋊Q8—1061no—
222#121—1061no—
223#121—1061no—
224C4×Q8—531no—
225C4×Q8—1061no—
226C4×Q8C2×D5311yescharacteristic
227C4×D4—1061no—
228C4×D4—1061no—
229#117—1061no—
230C2×C42—531no—
231C4⋊C8—531no—
232C4⋊C8C2×D5311yescharacteristic
233#103—531no—
234#103—531no—
235#103—531no—
236#103C2×D5311yescharacteristic
237#102—531no—
238#102—531no—
239#102—531no—
240#102C2×D5311yescharacteristic
241C22⋊C8—531no—
242C22⋊C8—531no—
243C4×C8—531no—
244C4×C8—531no—
245#41—2121no—
246#41—2121no—
247#41—2121no—
248#41—2121no—
249C2×Q8—531no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8C22×D5311yescharacteristic
253C2×Q8C22×D5311yescharacteristic
254C2×D4—531no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4C22×D5311yescharacteristic
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4—1061no—
265Q16—41no—
266Q16—2121no—
267Q16—2121no—
268Q16—2121no—
269SD16—41no—
270SD16—2121no—
271SD16—2121no—
272SD16—2121no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8C22×D5311yescharacteristic
276C2×C8C22×D5311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—21no—
295C42—531no—
296C42—531no—
297C42—531no—
298C42—1061no—
299C42C22×D5311yescharacteristic
300C23—21no—
301C23—531no—
302C23—1061no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—1061no—
308Q8—1061no—
309Q8—1061no—
310Q8—1061no—
311Q8—2122no—
312D4—21no—
313D4—21no—
314D4—1061no—
315D4—1061no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319D4—2122no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—1062no—
327C2×C4—1062no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4D4×D5311yescharacteristic
336C2×C4D4×D5311yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—21no—
339C8—21no—
340C8—1061no—
341C8—1061no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—1061no—
346C22—1061no—
347C22—1061no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22C2×D4×D5311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—532no—
358C4—532no—
359C4—1061no—
360C4—2121no—
361C4—2121no—
362C4—2121no—
363C4#1313326211yescharacteristic
364C4#1313326311yescharacteristic
365C2—81no—
366C2—4241no—
367C2—2121no—
368C2#1681118911yescharacteristic
369C2#135×D5311yescharacteristic
370C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3210
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #16811177C2semidirect
3. #16811001C2semidirect
4. #16810994C2semidirect
5. #16810972C2semidirect
6. #16810968C2semidirect
7. #16810846C2semidirect
8. #16810845C2semidirect
9. #16810842C2semidirect
10. #16810788C2semidirect
11. #16810786C2semidirect
12. #16810784C2semidirect
13. #16810782C2semidirect
14. #16810675C2semidirect
15. C53×#464C2semidirect
195. #464D53semidirect
201. C53#3210semidirect