Finite Group Databasebeta

← #20286578

Subgroups — #20286578

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#16811274C211yesmaximal, characteristic
3#16810972C211yesmaximal, characteristic, complemented
4#16810955C211yesmaximal, characteristic, complemented
5#16810907C211yesmaximal, characteristic, complemented
6#16810854C211yesmaximal, characteristic, complemented
7#16810853C211yesmaximal, characteristic, complemented
8#16810827C211yesmaximal, characteristic, complemented
9#16810764C211yesmaximal, characteristic, complemented
10#16810763C211yesmaximal, characteristic, complemented
11#16810710C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13C2×#13133140C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#483C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C22×Dic106C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23C2×C8×D53C2211yescharacteristic
24#13133216—21no—
25#13133216C2211yescharacteristic
26#13133215C2211yescharacteristic
27#13133202—21no—
28#13133200—21no—
29#13133197C2211yescharacteristic
30#13133195—21no—
31#13133195—21no—
32#13133194—21no—
33#13133192C2211yescharacteristic
34#13133191C2211yescharacteristic
35#13133182C2211yescharacteristic
36#13133179C2211yescharacteristic
37C4×C22⋊D53C2211yescharacteristic
38#13133177—21no—
39C4⋊C4×D53C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133140C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#106×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53D411yescharacteristic
64C22×Dic53—21no—
65Q8×D53—22no—
66Q8×D53—41no—
67#11719945—44no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
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
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53—21no—
89C22⋊Dic53D411yescharacteristic
90#11719924—41no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719913—22no—
110#11719913—41no—
111C8×D53—41no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—42no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—42no—
127C22⋊D53SD1614yescomplemented
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—42no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—42no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—42no—
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×C53C2×SD1612yes—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53C2×SD1612yes—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—22no—
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53C2×SD1612yes—
177#3298—531nomaximal, Sylow
178C2×C53—22no—
179C2×C53#54611yescharacteristic
180C2×C53C22×SD1611yescharacteristic
181C2×C53#57811yescharacteristic
182D53—22no—
183D53—81no—
184D4×Q8—531no—
185#547—531no—
186#487—531no—
187#483—531no—
188#483D5311yescharacteristic, complemented
189#475—531no—
190#465—531no—
191#465—531no—
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×SD16—531no—
206C22×C8—531no—
207C22×C8—531no—
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#121—531no—
220C4×Q8—531no—
221C4×D4—531no—
222C4×D4—531no—
223C4×D4—1061no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4C2×D5311yescharacteristic
226#106—531no—
227#106C2×D5311yescharacteristic
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#103C2×D5311yescharacteristic
238#103C2×D5311yescharacteristic
239#102—531no—
240C22⋊C8—531no—
241C22⋊C8C2×D5311yescharacteristic
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—531no—
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4—1061no—
258C22×C4—21no—
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4C22×D5311yescharacteristic
265SD16—1062no—
266SD16—2121no—
267SD16—2121no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—2121no—
271C2×C8—2121no—
272C2×C8C22×D5311yescharacteristic
273C2×C8C22×D5311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
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⋊C4C22×D5311yescharacteristic
285C4⋊C4C22×D5311yescharacteristic
286C4⋊C4C22×D5311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—531no—
289C22⋊C4—531no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—531no—
295C42—1061no—
296C23—531no—
297C23—1061no—
298C23D4×D5311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—532no—
302Q8—1061no—
303Q8—1061no—
304Q8—1061no—
305Q8—2122no—
306Q8—2122no—
307Q8—2122no—
308D4—534no—
309D4—1061no—
310D4—1061no—
311D4—2122no—
312D4—2122no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
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—532no—
341C22—1061no—
342C22—1061no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22SD16×D5312yes—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—532no—
354C4—1061no—
355C4—1062no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4#41×D5311yescharacteristic
360C4#1313326311yescharacteristic
361C2—22no—
362C2—1062no—
363C2—4241no—
364C2#1681126511yescharacteristic
365C2C2×SD16×D5311yescharacteristic
366C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#483C2
socleC22×C53C22×D4
hypercenter#483D53
hypocenterC53#3298
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #16811286C2semidirect
3. #16810972C2semidirect
4. #16810955C2semidirect
5. #16810907C2semidirect
6. #16810854C2semidirect
7. #16810853C2semidirect
8. #16810827C2semidirect
9. #16810764C2semidirect
10. #16810763C2semidirect
11. #16810710C2semidirect
12. #16810686C2semidirect
13. C2×#13133140C2semidirect
14. #103×D53C2semidirect
15. C53×#483C2semidirect
127. C22⋊D53SD16semidirect
149. Dic106#41semidirect
188. #483D53semidirect
199. C53#3298semidirect