Finite Group Databasebeta

← #20287786

Subgroups — #20287786

Order 6784.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811356C211yesmaximal, characteristic, complemented
2#16811343C211yesmaximal, characteristic, complemented
3#16811017C211yesmaximal, characteristic, complemented
4#16811011C211yesmaximal, characteristic
5#16811006C211yesmaximal, characteristic, complemented
6#16810844C211yesmaximal, characteristic, complemented
7#16810844C211yesmaximal, characteristic, complemented
8#16810838C211yesmaximal, characteristic, complemented
9#16810837C211yesmaximal, characteristic, complemented
10#16810828C211yesmaximal, characteristic, complemented
11#16810822C211yesmaximal, characteristic, complemented
12#16810751C211yesmaximal, characteristic, complemented
13#16810729C211yesmaximal, characteristic, complemented
14#16810710C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×Dic212C2211yescharacteristic
17C2×Dic212C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24#13133217C2211yescharacteristic
25#13133213C2211yescharacteristic
26#13133210—21no—
27#13133210—21no—
28#13133205C2211yescharacteristic
29#13133204C2211yescharacteristic
30#13133204C2211yescharacteristic
31#13133204C2211yescharacteristic
32#13133203—21no—
33#13133200—21no—
34#13133199—21no—
35#13133196—21no—
36C4×Dic106C2211yescharacteristic
37#13133182C2211yescharacteristic
38#13133182C2211yescharacteristic
39#13133182C2211yescharacteristic
40#13133179C2211yescharacteristic
41C4×C22⋊D53—21no—
42#13133176—21no—
43C4⋊#88322C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133146C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133140C2211yescharacteristic
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×D212C2311yescharacteristic
66C2×C4×D53—21no—
67C2×Dic106D411yescharacteristic
68C2×Dic106C2311yescharacteristic
69C2×Dic106C2311yescharacteristic
70C2×Dic106C2311yescharacteristic
71D8×C53—22no—
72C2×C8×C53C2311yescharacteristic
73C2×C8×C53C2311yescharacteristic
74C4⋊C4×C53C2311yescharacteristic
75C4⋊C4×C53C2311yescharacteristic
76C22⋊C4×C53—21no—
77C22⋊C4×C53—21no—
78C42×C53C2311yescharacteristic
79C22⋊Dic53—21no—
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85#11719923—41no—
86#11719923—41no—
87#11719923—41no—
88#11719922—41no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53D411yescharacteristic
95#11719919—21no—
96#11719919—21no—
97#11719919—21no—
98#11719919D411yescharacteristic
99#11719919D411yescharacteristic
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53C2311yescharacteristic
104C4×Dic53C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107Dic212—22no—
108Dic212—41no—
109#11719913—22no—
110#11719913—41no—
111C23×C53—21no—
112C23×C53—21no—
113C22×D53—21no—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—42no—
126C22⋊D53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C2×D411yescharacteristic
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134D212—21no—
135D212—21no—
136C4×D53—81no—
137Dic106—21no—
138Dic106—21no—
139Dic106—21no—
140Dic106—21no—
141Dic106—21no—
142Dic106—21no—
143Dic106—41no—
144C8×C53—21no—
145C8×C53#4112yes—
146#88322—21no—
147#88322—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53C2×#4111yescharacteristic
163C4×C53H(5,𝔽2)11yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—21no—
167Dic53—21no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#3401—531nomaximal, Sylow
172C2×C53—81no—
173C2×C53—81no—
174C2×C53#55011yescharacteristic
175C2×C53#57511yescharacteristic
176C2×C53#57711yescharacteristic
177D53—81no—
178#559—531no—
179#556—531no—
180#497—531no—
181#496—531no—
182#495—531no—
183#464—531no—
184#464—531no—
185#463—531no—
186#463—531no—
187#461—531no—
188#460—531no—
189#446—531no—
190#442D5311yescharacteristic, complemented
191#441—531no—
192C4×SD16—531no—
193C53#340111yescharacteristic, complemented, Sylow
194C2×Q16—531no—
195C2×Q16—531no—
196C2×SD16—531no—
197C2×SD16—531no—
198C2×SD16—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×D8—531no—
202C2×D8C2×D5311yescharacteristic
203C4⋊Q8—531no—
204#127—531no—
205#126—1061no—
206#126—1061no—
207#124—531no—
208#124—531no—
209#124—531no—
210#124—531no—
211#124—1061no—
212#123—1061no—
213#123—1061no—
214C22⋊Q8—1061no—
215C4×Q8—531no—
216C4×Q8—531no—
217C4×Q8—531no—
218C4×Q8—531no—
219C4×D4—531no—
220C4×D4—1061no—
221C4×D4C2×D5311yescharacteristic
222C4×D4C2×D5311yescharacteristic
223#117—1061no—
224#106C2×D5311yescharacteristic
225C4⋊C8—531no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#102—531no—
232#102C2×D5311yescharacteristic
233#102C2×D5311yescharacteristic
234#97—531no—
235#97C2×D5311yescharacteristic
236C4×C8—531no—
237C2×Q8—531no—
238C2×Q8—531no—
239C2×Q8—531no—
240C2×Q8—531no—
241C2×D4—531no—
242C2×D4—1061no—
243C2×D4—1061no—
244C2×D4C22×D5311yescharacteristic
245C2×D4C22×D5311yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—1061no—
249Q16—1062no—
250Q16—2121no—
251SD16—1062no—
252SD16—2121no—
253SD16—2121no—
254SD16—2121no—
255SD16—2121no—
256D8—22no—
257D8—2121no—
258C2×C8—531no—
259C2×C8—531no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—531no—
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4C22×D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—531no—
284C42—531no—
285C42—1061no—
286C42—1061no—
287C42—1061no—
288C42C22×D5311yescharacteristic
289C23—21no—
290C23—21no—
291C23—1061no—
292Q8—1061no—
293Q8—1061no—
294Q8—1061no—
295Q8—1061no—
296Q8—1061no—
297Q8—1061no—
298Q8—2121no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—1061no—
304D4—1061no—
305D4—4241no—
306D4—2122no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—4241no—
319C2×C4D4×D5311yescharacteristic
320C2×C4D4×D5311yescharacteristic
321C2×C4C23×D5311yescharacteristic
322C8—21no—
323C8—1061no—
324C8—1061no—
325C8#1171994512yes—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22C2×D4×D5311yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—1061no—
341C4—1061no—
342C4—1061no—
343C4—1061no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4C2×#1171994511yescharacteristic
348C4#1313326311yescharacteristic
349C2—81no—
350C2—81no—
351C2—4241no—
352C2#1681129411yescharacteristic
353C2#1681141911yescharacteristic
354C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3401
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. #16811356C2semidirect
2. #16811343C2semidirect
3. #16811017C2semidirect
5. #16811006C2semidirect
6. #16810844C2semidirect
7. #16810844C2semidirect
8. #16810838C2semidirect
9. #16810837C2semidirect
10. #16810828C2semidirect
11. #16810822C2semidirect
12. #16810751C2semidirect
13. #16810729C2semidirect
14. #16810710C2semidirect
15. C53×#442C2semidirect
190. #442D53semidirect
193. C53#3401semidirect