Finite Group Databasebeta

← #20286706

Subgroups — #20286706

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811277C211yesmaximal, characteristic, complemented
2#16810982C211yesmaximal, characteristic, complemented
3#16810956C211yesmaximal, characteristic, complemented
4#16810934C211yesmaximal, characteristic, complemented
5#16810918C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810762C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14C2×#13133141C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16C2×#11719947—21no—
17C2×#11719924C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719917C2211yescharacteristic
21C22×#88322C2211yescharacteristic
22#13133217C2211yescharacteristic
23#13133201—21no—
24#13133201—21no—
25#13133197—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic
28C4×D212C2211yescharacteristic
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133150C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133141C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C22×Q8×C53C2211yescharacteristic
50#124×C53—21no—
51#124×C53C2211yescharacteristic
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78#11719947—44no—
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×Dic106—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53D411yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53D411yescharacteristic
91C22⋊C4×C53D411yescharacteristic
92C42×C53—21no—
93C42×C53C2311yescharacteristic
94#11719924—22no—
95#11719924—41no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—42no—
120Q8×C53#4112yescomplemented
121D4×C53—22no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—42no—
141C22⋊D53—42no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147D212—21no—
148D212—21no—
149C4×D53—42no—
150Dic106—42no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53AGL(1,ℤ/8)12yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53C2×#4111yescharacteristic
176C4×C53H(5,𝔽2)11yescharacteristic
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3308—531nomaximal, Sylow
181C2×C53—22no—
182C2×C53—41no—
183C2×C53—81no—
184C2×C53#54611yescharacteristic
185C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
186C2×C53#57811yescharacteristic
187D53—81no—
188#548D5311yescharacteristic, complemented
189#547—531no—
190#489—531no—
191#484—531no—
192#480—531no—
193#477—531no—
194#468—531no—
195#466—531no—
196#463—531no—
197#449—531no—
198#448—531no—
199#440—531no—
200#435—531no—
201#417—531no—
202C2×#103—531no—
203C53#330811yescharacteristic, complemented, Sylow
204C2×#41—1061no—
205C2×#41C2×D5311yescharacteristic
206C22×Q8C2×D5311yescharacteristic
207C2×SD16—531no—
208C2×SD16—531no—
209C2×D8—531no—
210C2×M16—531no—
211C22×C8—531no—
212C4⋊Q8—531no—
213#124—21no—
214#124C2×D5311yescharacteristic
215#123—1061no—
216#123—1061no—
217C22⋊Q8—21no—
218C22⋊Q8—1061no—
219C22⋊Q8C2×D5311yescharacteristic
220#121—21no—
221#121—531no—
222#121—531no—
223#121C2×D5311yescharacteristic
224C4×Q8C2×D5311yescharacteristic
225C4×D4—21no—
226C4×D4—531no—
227C4×D4C2×D5311yescharacteristic
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#103—531no—
239#103—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244C22⋊C8—531no—
245C22⋊C8—531no—
246#97—531no—
247#41—22no—
248#41—41no—
249#41—2124no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—1061no—
254C2×Q8C22×D5311yescharacteristic
255C2×Q8C22×D5311yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4C22⋊D5311yescharacteristic
262C2×D4C22×D5311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270SD16—1062no—
271SD16—2121no—
272D8—2121no—
273M16—2121no—
274C2×C8—531no—
275C2×C8—531no—
276C2×C8—531no—
277C2×C8—531no—
278C2×C8—2121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22⋊D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4C22⋊D5311yescharacteristic
297C22⋊C4C22⋊D5311yescharacteristic
298C42—21no—
299C42C22×D5311yescharacteristic
300C23—21no—
301C23—1061no—
302C23C2×C22⋊D5311yescharacteristic
303C23C2×C22⋊D5311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—2122no—
309Q8#1171994712yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—1061no—
316D4—1061no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4C2×C22⋊D5311yescharacteristic
335C2×C4C2×C22⋊D5311yescharacteristic
336C2×C4C2×C22⋊D5311yescharacteristic
337C2×C4C2×C22⋊D5311yescharacteristic
338C2×C4C23×D5311yescharacteristic
339C8—1061no—
340C8—1061no—
341C8—1061no—
342C8—1061no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22#1313324212yes—
355C22C22×C22⋊D5311yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4C2×#1171994711yescharacteristic
367C4#1313326211yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—4241no—
372C2#1681126711yescharacteristic
373C2C2×#1313324211yescharacteristic
374C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#548C2
socleC22×C53C22×D4
hypercenter#548D53
hypocenterC53#3308
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811277C2semidirect
2. #16810982C2semidirect
3. #16810956C2semidirect
4. #16810934C2semidirect
5. #16810918C2semidirect
6. #16810871C2semidirect
7. #16810857C2semidirect
8. #16810835C2semidirect
9. #16810768C2semidirect
10. #16810762C2semidirect
11. #16810721C2semidirect
12. #16810695C2semidirect
13. #16810624C2semidirect
14. C2×#13133141C2semidirect
15. C53×#548C2semidirect
120. Q8×C53#41semidirect
188. #548D53semidirect
203. C53#3308semidirect
309. Q8#11719947semidirect