Finite Group Databasebeta

← #20286830

Subgroups — #20286830

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#16811331C211yesmaximal, characteristic, complemented
2#16811284C211yesmaximal, characteristic
3#16810984C211yesmaximal, characteristic, complemented
4#16810952C211yesmaximal, characteristic, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810906C211yesmaximal, characteristic, complemented
7#16810878C211yesmaximal, characteristic, complemented
8#16810854C211yesmaximal, characteristic, complemented
9#16810847C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810730C211yesmaximal, characteristic, complemented
12#16810691C211yesmaximal, characteristic, complemented
13#16810623C211yesmaximal, characteristic, complemented
14C2×#13133141C211yesmaximal, characteristic, complemented
15C53×#451C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C22×#88322C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133207—21no—
22#13133206—21no—
23#13133206C2211yescharacteristic
24#13133197—21no—
25#13133194—21no—
26#13133193—21no—
27#13133193C2211yescharacteristic
28#13133193C2211yescharacteristic
29#13133191—21no—
30#13133191C2211yescharacteristic
31Q8×Dic53C2211yescharacteristic
32Q8×Dic53C2211yescharacteristic
33C4×C22⋊D53—21no—
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36C2×C4⋊Dic53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133149C2211yescharacteristic
39#13133149C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×Q16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69C22×Dic53—21no—
70#11719949—22no—
71#11719949—41no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53C2311yescharacteristic
77C2×Dic106—21no—
78Q16×C53—22no—
79Q16×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊Dic53D411yescharacteristic
86#11719924—41no—
87#11719921—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921D411yescharacteristic
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53—21no—
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—42no—
120Q8×C53#4112yes—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—22no—
131C22⋊D53—42no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—81no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—42no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—42no—
145C4×D53—81no—
146Dic106—81no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53#13712yes—
154C22×C53C22×D411yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53H(5,𝔽2)11yescharacteristic
168Dic53—21no—
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3316—531nomaximal, Sylow
175C2×C53—22no—
176C2×C53#54611yescharacteristic
177C2×C53C2×#13711yescharacteristic
178C2×C53#57711yescharacteristic
179D53—41no—
180D53—81no—
181#554—531no—
182#548—531no—
183#489—531no—
184#483—531no—
185#481—531no—
186#475—531no—
187#470—531no—
188#465—531no—
189#464—531no—
190#452—531no—
191#451D5311yescharacteristic, complemented
192#441—531no—
193#435—531no—
194#417—531no—
195C2×#103—531no—
196C53#331611yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C22×Q8C2×D5311yescharacteristic
199C2×Q16C2×D5311yescharacteristic
200C2×Q16C2×D5311yescharacteristic
201C2×SD16—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205#125—1061no—
206#124—531no—
207#124—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—531no—
210C22⋊Q8—1061no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213C22⋊Q8C2×D5311yescharacteristic
214#121—531no—
215#121—1061no—
216C4×Q8—531no—
217C4×Q8—531no—
218C4×D4—531no—
219C4×D4—1061no—
220C4×D4—1061no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223#106—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103C2×D5311yescharacteristic
233#103C2×D5311yescharacteristic
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239#97—531no—
240#41—1062no—
241#41—2121no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—1061no—
246C2×Q8C22×D5311yescharacteristic
247C2×Q8C22×D5311yescharacteristic
248C2×Q8C22×D5311yescharacteristic
249C2×D4—531no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1061no—
253C22×C4—21no—
254C22×C4—531no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4C22×D5311yescharacteristic
259Q16—22no—
260Q16—41no—
261SD16—2121no—
262M16—2121no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—2121no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—1061no—
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—531no—
290C42—1061no—
291C42—1061no—
292C23—531no—
293C23—1061no—
294C23D4×D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—4241no—
302Q8#1171994712yes—
303D4—1062no—
304D4—1061no—
305D4—1061no—
306D4—2122no—
307D4—2122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—2122no—
326C2×C4D4×D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—1061no—
335C22—1061no—
336C22—1061no—
337C22—2121no—
338C22—2121no—
339C22—2121no—
340C22#1313324912yes—
341C22C2×D4×D5311yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—1061no—
348C4—1061no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994711yescharacteristic
354C4#1313326211yescharacteristic
355C2—22no—
356C2—4241no—
357C2—2121no—
358C2#1681126011yescharacteristic
359C2C2×#1313324911yescharacteristic
360C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53#3316
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #16811331C2semidirect
3. #16810984C2semidirect
4. #16810952C2semidirect
5. #16810942C2semidirect
6. #16810906C2semidirect
7. #16810878C2semidirect
8. #16810854C2semidirect
9. #16810847C2semidirect
10. #16810784C2semidirect
11. #16810730C2semidirect
12. #16810691C2semidirect
13. #16810623C2semidirect
14. C2×#13133141C2semidirect
15. C53×#451C2semidirect
191. #451D53semidirect
196. C53#3316semidirect