Finite Group Databasebeta

← #20287417

Subgroups — #20287417

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#16811293C211yesmaximal, characteristic, complemented
3#16810987C211yesmaximal, characteristic, complemented
4#16810986C211yesmaximal, characteristic, complemented
5#16810968C211yesmaximal, characteristic, complemented
6#16810924C211yesmaximal, characteristic, complemented
7#16810921C211yesmaximal, characteristic, complemented
8#16810911C211yesmaximal, characteristic, complemented
9#16810816C211yesmaximal, characteristic, complemented
10#16810816C211yesmaximal, characteristic, complemented
11#16810752C211yesmaximal, characteristic, complemented
12#16810735C211yesmaximal, characteristic, complemented
13#16810733C211yesmaximal, characteristic, complemented
14D8×Dic53C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719922C2211yescharacteristic
18#13133216C2211yescharacteristic
19#13133213C2211yescharacteristic
20#13133213C2211yescharacteristic
21#13133209C2211yescharacteristic
22#13133208C2211yescharacteristic
23#13133196—21no—
24#13133194—21no—
25#13133194—21no—
26#13133189—21no—
27#13133189—21no—
28#13133188—21no—
29C4×Dic106C2211yescharacteristic
30C4×D212C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic
33D4×Dic53C2211yescharacteristic
34D4×Dic53C2211yescharacteristic
35C2×#11719919—21no—
36#13133151C2211yescharacteristic
37#13133149C2211yescharacteristic
38#13133148C2211yescharacteristic
39#13133146C2211yescharacteristic
40#13133146C2211yescharacteristic
41#13133139C2211yescharacteristic
42#13133139C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133128C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#127×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73#11719947—44no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×Dic106—21no—
77C2×Dic106D411yescharacteristic
78D8×C53—22no—
79D8×C53—22no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719922—22no—
90#11719921—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53D411yescharacteristic
93C4⋊Dic53C2311yescharacteristic
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—
108C23×C53—21no—
109C22×D53—21no—
110D4×C53—21no—
111D4×C53—21no—
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—21no—
115D4×C53—42no—
116D4×C53—81no—
117D4×C53#4112yescomplemented
118C2×C4×C53—21no—
119C2×C4×C53—81no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—42no—
124C22⋊D53—42no—
125C22⋊D53—81no—
126C22⋊D53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—81no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
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×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53C22×D411yescharacteristic
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—41no—
163C4×C53C2×#4111yescharacteristic
164C4×C532−1+411yescharacteristic
165Dic53—21no—
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53AGL(1,ℤ/8)12yes—
172#3366—531nomaximal, Sylow
173C2×C53—42no—
174C2×C53—81no—
175C2×C53—81no—
176C2×C53#54811yescharacteristic
177C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C53#57611yescharacteristic
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#459D5311yescharacteristic, complemented
191#446—531no—
192#442—531no—
193#442—531no—
194C4×D8—531no—
195C53#336611yescharacteristic, complemented, Sylow
196C2×#41—1061no—
197C2×D8—531no—
198C2×D8C2×D5311yescharacteristic
199C2×D8C2×D5311yescharacteristic
200C4⋊Q8—531no—
201#127—531no—
202#127—531no—
203#127C2×D5311yescharacteristic
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×D4—531no—
218C4×D4C2×D5311yescharacteristic
219C2×C4⋊C4—1061no—
220#107—531no—
221#106—531no—
222#106—531no—
223C4⋊C8—531no—
224C4⋊C8—531no—
225C4⋊C8C2×D5311yescharacteristic
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#97—531no—
238C4×C8—531no—
239#41—2124no—
240C2×Q8—531no—
241C2×Q8—1061no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—531no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4C22×D5311yescharacteristic
250C2×D4C22×D5311yescharacteristic
251C2×D4C22×D5311yescharacteristic
252C22×C4—21no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258D8—22no—
259D8—22no—
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—531no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—531no—
284C42—531no—
285C42C22×D5311yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—1061no—
290Q8—1062no—
291Q8—4241no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—1061no—
300D4—1061no—
301D4—4241no—
302D4—4241no—
303D4—2122no—
304D4—2122no—
305D4#1171994712yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—4241no—
316C2×C4—2122no—
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—41no—
334C22—41no—
335C22—41no—
336C22—2121no—
337C22—2121no—
338C22—2121no—
339C22C2×D4×D5311yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—532no—
344C4—1061no—
345C4—1061no—
346C4—1061no—
347C4—2121no—
348C4—2121no—
349C4—2121no—
350C4C2×#1171994711yescharacteristic
351C4#1313326411yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—4241no—
356C2#1681127911yescharacteristic
357C2AGL(1,ℤ/8)×D5311yescharacteristic
358C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3366
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16811332C2semidirect
2. #16811293C2semidirect
3. #16810987C2semidirect
4. #16810986C2semidirect
5. #16810968C2semidirect
6. #16810924C2semidirect
7. #16810921C2semidirect
8. #16810911C2semidirect
9. #16810816C2semidirect
10. #16810816C2semidirect
11. #16810752C2semidirect
12. #16810735C2semidirect
13. #16810733C2semidirect
14. D8×Dic53C2semidirect
15. C53×#459C2semidirect
117. D4×C53#41semidirect
190. #459D53semidirect
195. C53#3366semidirect
305. D4#11719947semidirect