Finite Group Databasebeta

← #20286809

Subgroups — #20286809

Order 6784.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811312C211yesmaximal, characteristic, complemented
2#16811275C211yesmaximal, characteristic
3#16810990C211yesmaximal, characteristic, complemented
4#16810960C211yesmaximal, characteristic, complemented
5#16810922C211yesmaximal, characteristic, complemented
6#16810882C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810832C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810722C211yesmaximal, characteristic, complemented
12#16810691C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14#16810617C211yesmaximal, characteristic, complemented
15C53×#485C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133210—21no—
25#13133209—21no—
26#13133208C2211yescharacteristic
27#13133202—21no—
28#13133199—21no—
29#13133199—21no—
30#13133197C2211yescharacteristic
31#13133197C2211yescharacteristic
32#13133196C2211yescharacteristic
33#13133190—21no—
34#13133182C2211yescharacteristic
35#13133179C2211yescharacteristic
36C4×C22⋊D53C2211yescharacteristic
37#13133174—21no—
38C4⋊C4×D53—21no—
39C4⋊C4×D53C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C2×C22⋊D53—21no—
62C2×C22⋊D53D411yescharacteristic
63#11719949—22no—
64#11719949—41no—
65#11719945—22no—
66#11719945—41no—
67C2×D212—21no—
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×Dic106C2311yescharacteristic
75C2×Dic106C2311yescharacteristic
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C42×C53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53D411yescharacteristic
87#11719925—41no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921D411yescharacteristic
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C2×#88322—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108Dic212—41no—
109#11719913—22no—
110#11719912—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53—21no—
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C22⋊D53—41no—
125C22⋊D53—42no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132D212—21no—
133D212—42no—
134D212#4112yes—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—42no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—42no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—21no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53C22×D411yescharacteristic
153C2×D53—21no—
154C2×D53—22no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C2×#4111yescharacteristic
165C4×C53H(5,𝔽2)11yescharacteristic
166Dic53—21no—
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#3315—531nomaximal, Sylow
173C2×C53—41no—
174C2×C53#54611yescharacteristic
175C2×C53#57511yescharacteristic
176C2×C53#57811yescharacteristic
177D53—41no—
178D53—42no—
179#552—531no—
180#547—531no—
181#491—531no—
182#485D5311yescharacteristic, complemented
183#484—531no—
184#478—531no—
185#470—531no—
186#467—531no—
187#462—531no—
188#452—531no—
189#452—531no—
190#440—531no—
191#435—531no—
192#417—531no—
193#416—531no—
194C53#331511yescharacteristic, complemented, Sylow
195C2×#41—531no—
196C2×#41—531no—
197C2×Q16—531no—
198C2×Q16—531no—
199C2×SD16—531no—
200C2×M16—531no—
201C22×C8—531no—
202C4⋊Q8—531no—
203#126—1061no—
204#125—531no—
205#125—1061no—
206#123—1061no—
207#123—1061no—
208#123—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—531no—
212C22⋊Q8C2×D5311yescharacteristic
213#121—1061no—
214C4×Q8—531no—
215C4×D4—531no—
216C4×D4—531no—
217#117—1061no—
218#117C2×D5311yescharacteristic
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221#107—531no—
222#107C2×D5311yescharacteristic
223#106C2×D5311yescharacteristic
224C4⋊C8—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103C2×D5311yescharacteristic
231#103C2×D5311yescharacteristic
232#102—531no—
233#102—531no—
234#102—531no—
235C22⋊C8—531no—
236C22⋊C8C2×D5311yescharacteristic
237#97—531no—
238#41—1062no—
239#41—1062no—
240#41—2121no—
241#41—2121no—
242C2×Q8—531no—
243C2×Q8—531no—
244C2×Q8C22×D5311yescharacteristic
245C2×D4—531no—
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1061no—
249C22×C4—531no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4C22×D5311yescharacteristic
255Q16—2121no—
256Q16—2121no—
257SD16—1062no—
258M16—2121no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—2121no—
262C2×C8C22×D5311yescharacteristic
263C2×C8C22×D5311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4C22×D5311yescharacteristic
277C4⋊C4C22×D5311yescharacteristic
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—531no—
282C22⋊C4—531no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C42—21no—
288C42—531no—
289C42—1061no—
290C23—531no—
291C23—1061no—
292C23D4×D5311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1061no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299D4—532no—
300D4—1061no—
301D4—2122no—
302D4—2121no—
303D4—2122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4D4×D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C8—21no—
325C8—21no—
326C8—1061no—
327C8—1061no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—1061no—
332C22—1062no—
333C22—1061no—
334C22—1061no—
335C22—2121no—
336C22—2121no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—1061no—
344C4—1061no—
345C4—1061no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4#41×D5311yescharacteristic
350C4#1313326311yescharacteristic
351C2—41no—
352C2—2121no—
353C2—2122no—
354C2#1681126511yescharacteristic
355C2#1681142011yescharacteristic
356C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3315
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #16811312C2semidirect
3. #16810990C2semidirect
4. #16810960C2semidirect
5. #16810922C2semidirect
6. #16810882C2semidirect
7. #16810862C2semidirect
8. #16810832C2semidirect
9. #16810788C2semidirect
10. #16810787C2semidirect
11. #16810722C2semidirect
12. #16810691C2semidirect
13. #16810623C2semidirect
14. #16810617C2semidirect
15. C53×#485C2semidirect
182. #485D53semidirect
194. C53#3315semidirect