Finite Group Databasebeta

← #20286883

Subgroups — #20286883

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811326C211yesmaximal, characteristic
2Q8×D212C211yesmaximal, characteristic, complemented
3#16810993C211yesmaximal, characteristic, complemented
4#16810967C211yesmaximal, characteristic, complemented
5#16810960C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810853C211yesmaximal, characteristic, complemented
8#16810829C211yesmaximal, characteristic, complemented
9#16810763C211yesmaximal, characteristic, complemented
10#16810763C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#479C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×#11719912C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133217—21no—
24#13133217C2211yescharacteristic
25#13133212—21no—
26#13133202—21no—
27#13133197—21no—
28#13133197—21no—
29#13133197C2211yescharacteristic
30#13133197C2211yescharacteristic
31#13133192—21no—
32#13133192C2211yescharacteristic
33#13133192C2211yescharacteristic
34C4×D212—21no—
35C4×D212C2211yescharacteristic
36C4×D212C2211yescharacteristic
37#13133179—21no—
38#13133175C2211yescharacteristic
39C4⋊C4×D53C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52#125×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60Q8×D53—22no—
61Q8×D53—22no—
62Q8×D53—41no—
63Q8×D53—41no—
64C2×D212—21no—
65C2×D212—21no—
66C2×D212D411yescharacteristic
67C2×D212C2311yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53C2311yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77C2×Dic106C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88#11719924—22no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96#11719921D411yescharacteristic
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104C4×Dic53—21no—
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107#11719913—41no—
108#11719913—41no—
109#11719912—41no—
110C8×D53—41no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53#4112yes—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53C2×D411yescharacteristic
127C2×Dic53C2×D411yescharacteristic
128D212—22no—
129D212—21no—
130D212—21no—
131D212—42no—
132D212—81no—
133C4×D53—21no—
134C4×D53—21no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—42no—
138C4×D53—42no—
139C4×D53—81no—
140C4×D53—81no—
141Dic106—21no—
142Dic106—21no—
143Dic106—21no—
144Dic106—21no—
145Dic106—42no—
146Dic106—42no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53C22×D411yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53#13712yes—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53C2×#4111yescharacteristic
168C4×C53H(5,𝔽2)11yescharacteristic
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3320—531nomaximal, Sylow
175C2×C53#54611yescharacteristic
176C2×C53C2×#13711yescharacteristic
177C2×C53#57711yescharacteristic
178D53—22no—
179D53—41no—
180D53—81no—
181#553—531no—
182D4×Q8—531no—
183#491—531no—
184#485—531no—
185#484—531no—
186#479D5311yescharacteristic, complemented
187#468—531no—
188#465—531no—
189#461—531no—
190#448—531no—
191#448—531no—
192#440—531no—
193#435—531no—
194#419—531no—
195C2×#103—531no—
196C53#332011yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198C22×Q8—531no—
199C2×SD16—531no—
200C2×SD16—531no—
201C2×SD16—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205C4⋊Q8—1061no—
206#126—1061no—
207#125C2×D5311yescharacteristic
208#123—1061no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213#121—531no—
214#121—531no—
215#121—1061no—
216C4×Q8C2×D5311yescharacteristic
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—1061no—
220C4×D4—1061no—
221#117—531no—
222C2×C4⋊C4—531no—
223#107—531no—
224#107C2×D5311yescharacteristic
225#106C2×D5311yescharacteristic
226C4⋊C8C2×D5311yescharacteristic
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103C2×D5311yescharacteristic
235#103C2×D5311yescharacteristic
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8—531no—
239#97—531no—
240C2×Q8—531no—
241C2×Q8—531no—
242C2×Q8—1061no—
243C2×Q8—1061no—
244C2×Q8—1062no—
245C2×Q8—1062no—
246C2×Q8—2121no—
247C2×Q8—2121no—
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1061no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259SD16—1062no—
260SD16—2121no—
261SD16—2121no—
262M16—2121no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—2121no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C4⋊C4C22×D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—531no—
282C22⋊C4—531no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—21no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—531no—
293C23—531no—
294C23—1061no—
295Q8—21no—
296Q8—1061no—
297Q8—1061no—
298Q8—1061no—
299Q8—1061no—
300Q8—2122no—
301Q8—2122no—
302Q8#1171994912yes—
303D4—1062no—
304D4—1061no—
305D4—1061no—
306D4—4241no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4D4×D5311yescharacteristic
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—532no—
333C22—1061no—
334C22—1061no—
335C22—1061no—
336C22—1061no—
337C22—1061no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22C2×D4×D5311yescharacteristic
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—1061no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994911yescharacteristic
354C4#1313326311yescharacteristic
355C2—1062no—
356C2—4241no—
357C2—2121no—
358C2#1681126411yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#479C2
socleC22×C53C22×D4
hypercenter#479D53
hypocenterC53#3320
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
2. Q8×D212C2semidirect
3. #16810993C2semidirect
4. #16810967C2semidirect
5. #16810960C2semidirect
6. #16810870C2semidirect
7. #16810853C2semidirect
8. #16810829C2semidirect
9. #16810763C2semidirect
10. #16810763C2semidirect
11. #16810721C2semidirect
12. #16810693C2semidirect
13. #16810630C2semidirect
14. #103×D53C2semidirect
15. C53×#479C2semidirect
186. #479D53semidirect
196. C53#3320semidirect