Finite Group Databasebeta

← #20286893

Subgroups — #20286893

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#16811323C211yesmaximal, characteristic
2#16811288C211yesmaximal, characteristic, complemented
3#16810991C211yesmaximal, characteristic, complemented
4#16810962C211yesmaximal, characteristic, complemented
5#16810957C211yesmaximal, characteristic, complemented
6#16810931C211yesmaximal, characteristic, complemented
7#16810866C211yesmaximal, characteristic, complemented
8#16810849C211yesmaximal, characteristic, complemented
9#16810760C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#461C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20C2×C8×D53C2211yescharacteristic
21#13133216—21no—
22#13133210—21no—
23#13133209C2211yescharacteristic
24#13133200—21no—
25#13133195—21no—
26#13133195—21no—
27#13133193C2211yescharacteristic
28#13133193C2211yescharacteristic
29#13133190—21no—
30#13133188C2211yescharacteristic
31#13133188C2211yescharacteristic
32C4×Dic106C2211yescharacteristic
33C4×D212C2211yescharacteristic
34#13133179—21no—
35C4×C22⋊D53—21no—
36#13133175C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C4⋊Q8×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65Q8×D53—22no—
66Q8×D53—22no—
67Q8×D53—41no—
68Q8×D53—41no—
69C2×D212D411yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78SD16×C53—41no—
79SD16×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719923—22no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94#11719921D411yescharacteristic
95C4⋊Dic53D411yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C23×C53—21no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—81no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53—81no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—22no—
136C4×D53—21no—
137C4×D53—21no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—42no—
141C4×D53—42no—
142C4×D53—81no—
143Dic106—21no—
144Dic106—42no—
145Dic106—42no—
146Dic106#4112yes—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53#13712yes—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53H(5,𝔽2)11yescharacteristic
168Dic53—22no—
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3320—531nomaximal, Sylow
175C2×C53—81no—
176C2×C53#54611yescharacteristic
177C2×C53C2×#13711yescharacteristic
178C2×C53#57711yescharacteristic
179D53—22no—
180D53—41no—
181#553—531no—
182D4×Q8—531no—
183#491—531no—
184#485—531no—
185#484—531no—
186#479—531no—
187#468—531no—
188#465—531no—
189#461D5311yescharacteristic, complemented
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×SD16C2×D5311yescharacteristic
201C2×SD16C2×D5311yescharacteristic
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—1061no—
205C4⋊Q8C2×D5311yescharacteristic
206#126—1061no—
207#125—531no—
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×Q8—531no—
217C4×D4—531no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222C2×C4⋊C4—531no—
223#107—531no—
224#107—531no—
225#106—531no—
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—1061no—
242C2×Q8—1061no—
243C2×Q8—1062no—
244C2×Q8—1062no—
245C2×Q8—2121no—
246C2×Q8—2121no—
247C2×Q8C22×D5311yescharacteristic
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4C22×D5311yescharacteristic
253C22×C4—21no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259SD16—41no—
260SD16—41no—
261SD16—1062no—
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—531no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—531no—
283C22⋊C4—531no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C42—1061no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—21no—
293C23—531no—
294C23—531no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—532no—
300Q8—1061no—
301Q8—2122no—
302Q8—2122no—
303D4—21no—
304D4—21no—
305D4—1062no—
306D4—4241no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
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—1061no—
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—41no—
333C22—41no—
334C22—41no—
335C22—532no—
336C22—1061no—
337C22—1061no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—1061no—
348C4—1061no—
349C4—1062no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4#41×D5311yescharacteristic
354C4#1313326211yescharacteristic
355C2—81no—
356C2—1062no—
357C2—2121no—
358C2#1681127011yescharacteristic
359C2#137×D5311yescharacteristic
360C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#461C2
socleC22×C53C22×D4
hypercenter#461D53
hypocenterC53#3320
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #16811288C2semidirect
3. #16810991C2semidirect
4. #16810962C2semidirect
5. #16810957C2semidirect
6. #16810931C2semidirect
7. #16810866C2semidirect
8. #16810849C2semidirect
9. #16810760C2semidirect
10. #16810760C2semidirect
11. #16810723C2semidirect
12. #16810693C2semidirect
13. #16810630C2semidirect
14. #103×D53C2semidirect
15. C53×#461C2semidirect
189. #461D53semidirect
196. C53#3320semidirect