Finite Group Databasebeta

← #20287423

Subgroups — #20287423

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811332C211yesmaximal, characteristic, complemented
2#16811294C211yesmaximal, characteristic, complemented
3#16810988C211yesmaximal, characteristic, complemented
4#16810986C211yesmaximal, characteristic, complemented
5#16810968C211yesmaximal, characteristic, complemented
6#16810923C211yesmaximal, characteristic, complemented
7#16810922C211yesmaximal, characteristic, complemented
8#16810912C211yesmaximal, characteristic, complemented
9#16810818C211yesmaximal, characteristic, complemented
10#16810816C211yesmaximal, characteristic, complemented
11#16810816C211yesmaximal, characteristic, complemented
12#16810751C211yesmaximal, characteristic, complemented
13#16810732C211yesmaximal, characteristic, complemented
14#16810703C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719922C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19#13133216C2211yescharacteristic
20#13133214C2211yescharacteristic
21#13133213C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133209C2211yescharacteristic
24#13133208C2211yescharacteristic
25#13133196—21no—
26#13133194—21no—
27#13133194—21no—
28#13133191—21no—
29#13133190—21no—
30#13133190—21no—
31C4×Dic106C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33D4×Dic53C2211yescharacteristic
34#13133179C2211yescharacteristic
35#13133179C2211yescharacteristic
36C2×#11719919—21no—
37#13133150C2211yescharacteristic
38#13133148C2211yescharacteristic
39C4⋊#88322C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69#11719945—44no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×Dic106—21no—
77C2×Dic106D411yescharacteristic
78D8×C53—22no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53C2311yescharacteristic
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88#11719922—22no—
89#11719922—22no—
90#11719921—21no—
91#11719921—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53D411yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919D411yescharacteristic
101#11719919D411yescharacteristic
102C4×Dic53C2311yescharacteristic
103C4×Dic53C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C23×C53—21no—
107C23×C53—21no—
108C22×D53—21no—
109C22×D53—21no—
110D4×C53—21no—
111D4×C53—21no—
112D4×C53—21no—
113D4×C53#4112yescomplemented
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53—81no—
118C2×C4×C53C2×D411yescharacteristic
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2411yescharacteristic
121C22⋊D53—42no—
122C22⋊D53—42no—
123C22⋊D53—81no—
124C22⋊D53—81no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—42no—
128C2×Dic53—81no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—21no—
135D212—21no—
136D212—21no—
137D212—42no—
138D212—81no—
139C4×D53—42no—
140C4×D53—42no—
141Dic106—22no—
142Dic106—81no—
143C8×C53—21no—
144C8×C53—21no—
145#88322—21no—
146#88322—21no—
147C22×C53—22no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53C22×D411yescharacteristic
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C2×#4111yescharacteristic
165C4×C532−1+411yescharacteristic
166Dic53—21no—
167Dic53—21no—
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53AGL(1,ℤ/8)12yes—
172#3366—531nomaximal, Sylow
173C2×C53—42no—
174C2×C53—81no—
175C2×C53#54811yescharacteristic
176C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C53#57611yescharacteristic
178D53—81no—
179D53—81no—
180#554—531no—
181#550—531no—
182#490—531no—
183#490—531no—
184#486—531no—
185#478—531no—
186#478—531no—
187#476—531no—
188#459—531no—
189#459—531no—
190#459—531no—
191#446—531no—
192#442—531no—
193#442D5311yescharacteristic, complemented
194C4×D8—531no—
195C53#336611yescharacteristic, complemented, Sylow
196C2×#41—1061no—
197C2×D8—531no—
198C2×D8—531no—
199C2×D8C2×D5311yescharacteristic
200C4⋊Q8—531no—
201#127—531no—
202#127—531no—
203#127—531no—
204#125—531no—
205#125—531no—
206C22⋊Q8—1061no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209#121—1061no—
210#121—1061no—
211#121—1061no—
212C4×Q8—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4C2×D5311yescharacteristic
218C4×D4C2×D5311yescharacteristic
219C2×C4⋊C4—1061no—
220#107—531no—
221#106—531no—
222#106C2×D5311yescharacteristic
223C4⋊C8—531no—
224C4⋊C8—531no—
225C4⋊C8—531no—
226#102—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230#102—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102C2×D5311yescharacteristic
235#102C2×D5311yescharacteristic
236#97—531no—
237#97C2×D5311yescharacteristic
238C4×C8—531no—
239#41—2124no—
240C2×Q8—531no—
241C2×Q8—1061no—
242C2×D4—531no—
243C2×D4—531no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4C22×D5311yescharacteristic
251C2×D4C22×D5311yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258D8—22no—
259D8—1062no—
260D8—1062no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8C22×D5311yescharacteristic
264C2×C8C22×D5311yescharacteristic
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4C22×D5311yescharacteristic
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—531no—
284C42—531no—
285C42C22×D5311yescharacteristic
286C23—21no—
287C23—21no—
288C23—1061no—
289C23—1061no—
290Q8—1062no—
291Q8—4241no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—1061no—
296D4—1061no—
297D4—1061no—
298D4—1061no—
299D4—4241no—
300D4—4241no—
301D4—4241no—
302D4—2122no—
303D4—2122no—
304D4—2122no—
305D4#1171994512yescomplemented
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—4241no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4D4×D5311yescharacteristic
321C2×C4D4×D5311yescharacteristic
322C2×C4C23×D5311yescharacteristic
323C8—21no—
324C8—21no—
325C8—1061no—
326C8—1061no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22C2×D4×D5311yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—532no—
345C4—1061no—
346C4—1061no—
347C4—1061no—
348C4—2121no—
349C4—2121no—
350C4C2×#1171994511yescharacteristic
351C4#1313326411yescharacteristic
352C2—42no—
353C2—81no—
354C2—4241no—
355C2—4241no—
356C2#1681128211yescharacteristic
357C2AGL(1,ℤ/8)×D5311yescharacteristic
358C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3366
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811332C2semidirect
2. #16811294C2semidirect
3. #16810988C2semidirect
4. #16810986C2semidirect
5. #16810968C2semidirect
6. #16810923C2semidirect
7. #16810922C2semidirect
8. #16810912C2semidirect
9. #16810818C2semidirect
10. #16810816C2semidirect
11. #16810816C2semidirect
12. #16810751C2semidirect
13. #16810732C2semidirect
14. #16810703C2semidirect
15. C53×#442C2semidirect
113. D4×C53#41semidirect
193. #442D53semidirect
195. C53#3366semidirect
305. D4#11719945semidirect