Finite Group Databasebeta

← #20286575

Subgroups — #20286575

Order 6784.

420 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#16811286C211yesmaximal, characteristic, complemented
2#16811273C211yesmaximal, characteristic
3#16810972C211yesmaximal, characteristic, complemented
4#16810955C211yesmaximal, characteristic, complemented
5#16810951C211yesmaximal, characteristic, complemented
6#16810907C211yesmaximal, characteristic, complemented
7#16810851C211yesmaximal, characteristic, complemented
8#16810826C211yesmaximal, characteristic, complemented
9#16810761C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11#16810714C211yesmaximal, characteristic, complemented
12C8×C22⋊D53C211yesmaximal, characteristic, complemented
13C2×#13133142C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C22×Dic106C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133216—21no—
23#13133216C2211yescharacteristic
24#13133215C2211yescharacteristic
25#13133199—21no—
26#13133198—21no—
27#13133197C2211yescharacteristic
28#13133195—21no—
29#13133194—21no—
30#13133194—21no—
31#13133188C2211yescharacteristic
32#13133182C2211yescharacteristic
33D4×Dic53C2211yescharacteristic
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36C2×C4⋊Dic53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133148C2211yescharacteristic
39#13133148C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133140C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#106×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68Q8×D53—22no—
69Q8×D53—41no—
70#11719947—44no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—22no—
75C2×Dic106—21no—
76C2×Dic106—21no—
77C2×Dic106—21no—
78C2×Dic106—41no—
79C2×Dic106C2311yescharacteristic
80C2×Dic106C2311yescharacteristic
81SD16×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53D411yescharacteristic
93#11719923—22no—
94#11719923—41no—
95#11719921—21no—
96#11719921D411yescharacteristic
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919D411yescharacteristic
107C4×Dic53—21no—
108C4×Dic53C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C8×D53—41no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—42no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53SD1614yescomplemented
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—42no—
142C4×D53—42no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—42no—
147Dic106—42no—
148Dic106—42no—
149Dic106#4112yescomplemented
150C8×C53—21no—
151C8×C53—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53C2×SD1612yes—
160C22×C53C22×D411yescharacteristic
161C2×D53—21no—
162C2×D53—21no—
163C2×D53C2×SD1612yes—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53C2×#4111yescharacteristic
169C4×C53H(5,𝔽2)11yescharacteristic
170Dic53—22no—
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174Dic53—41no—
175Dic53—41no—
176Dic53C2×SD1612yes—
177#3298—531nomaximal, Sylow
178C2×C53—22no—
179C2×C53—81no—
180C2×C53#54611yescharacteristic
181C2×C53C22×SD1611yescharacteristic
182C2×C53#57811yescharacteristic
183D53—22no—
184D4×Q8—531no—
185#547—531no—
186#487—531no—
187#483—531no—
188#483—531no—
189#475—531no—
190#465—531no—
191#465D5311yescharacteristic, complemented
192#461—531no—
193#448—531no—
194#448—531no—
195C4×SD16—531no—
196C8×D4—531no—
197C2×#103—531no—
198C2×#103—531no—
199C53#329811yescharacteristic, complemented, Sylow
200C2×#41—1061no—
201C22×Q8—531no—
202C22×Q8—531no—
203C2×SD16—531no—
204C2×SD16—531no—
205C2×SD16C2×D5311yescharacteristic
206C22×C8—531no—
207C22×C8C2×D5311yescharacteristic
208C4⋊Q8—531no—
209C4⋊Q8—531no—
210C4⋊Q8—1061no—
211#123—1061no—
212#123—1061no—
213C22⋊Q8—531no—
214C22⋊Q8—1061no—
215C22⋊Q8—1061no—
216C22⋊Q8—1061no—
217C22⋊Q8C2×D5311yescharacteristic
218#121—531no—
219#121C2×D5311yescharacteristic
220C4×Q8—531no—
221C4×D4—531no—
222C4×D4—531no—
223C4×D4—1061no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226#106—531no—
227#106—531no—
228#106C2×D5311yescharacteristic
229C4⋊C8—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#103C2×D5311yescharacteristic
239#102C2×D5311yescharacteristic
240C22⋊C8—531no—
241C22⋊C8—531no—
242C4×C8—531no—
243#41—2124no—
244C2×Q8—531no—
245C2×Q8—531no—
246C2×Q8—1061no—
247C2×Q8—1061no—
248C2×Q8—1062no—
249C2×Q8—1061no—
250C2×Q8—1062no—
251C2×Q8—2121no—
252C2×Q8—2121no—
253C2×Q8C22×D5311yescharacteristic
254C2×D4—21no—
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4C22×D5311yescharacteristic
258C22×C4—531no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4C22×D5311yescharacteristic
265SD16—41no—
266SD16—1062no—
267SD16—2121no—
268C2×C8—41no—
269C2×C8—531no—
270C2×C8—531no—
271C2×C8—2121no—
272C2×C8C22×D5311yescharacteristic
273C2×C8C22×D5311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4C22×D5311yescharacteristic
286C4⋊C4C22×D5311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—531no—
290C22⋊C4—531no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—531no—
295C42—1061no—
296C23—21no—
297C23—531no—
298C23D4×D5311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—532no—
302Q8—1061no—
303Q8—1061no—
304Q8—1061no—
305Q8—2122no—
306Q8—2122no—
307Q8—2122no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—534no—
312D4—2122no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4D4×D5311yescharacteristic
333C2×C4C23×D5311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1061no—
337C8—1061no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—532no—
344C22—1061no—
345C22—1061no—
346C22SD16×D5312yes—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—532no—
353C4—1061no—
354C4—1061no—
355C4—1062no—
356C4—2121no—
357C4—2121no—
358C4—2121no—
359C4#41×D5311yescharacteristic
360C4#1313326211yescharacteristic
361C2—22no—
362C2—81no—
363C2—1062no—
364C2#1681126211yescharacteristic
365C2C2×SD16×D5311yescharacteristic
366C2#1681143711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3298
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16811286C2semidirect
3. #16810972C2semidirect
4. #16810955C2semidirect
5. #16810951C2semidirect
6. #16810907C2semidirect
7. #16810851C2semidirect
8. #16810826C2semidirect
9. #16810761C2semidirect
10. #16810760C2semidirect
11. #16810714C2semidirect
12. C8×C22⋊D53C2semidirect
13. C2×#13133142C2semidirect
14. #103×D53C2semidirect
15. C53×#465C2semidirect
128. C22⋊D53SD16semidirect
149. Dic106#41semidirect
191. #465D53semidirect
199. C53#3298semidirect