Finite Group Databasebeta

← #20286707

Subgroups — #20286707

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#16811280C211yesmaximal, characteristic, complemented
2#16811274C211yesmaximal, characteristic, complemented
3#16810981C211yesmaximal, characteristic, complemented
4#16810958C211yesmaximal, characteristic, complemented
5#16810935C211yesmaximal, characteristic, complemented
6#16810916C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810858C211yesmaximal, characteristic, complemented
9#16810837C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810763C211yesmaximal, characteristic, complemented
12#16810726C211yesmaximal, characteristic, complemented
13#16810692C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#417C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×Q8×D53C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719924C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133216C2211yescharacteristic
23#13133204—21no—
24#13133204C2211yescharacteristic
25#13133202—21no—
26#13133200—21no—
27#13133196—21no—
28#13133196C2211yescharacteristic
29#13133195—21no—
30#13133192C2211yescharacteristic
31#13133191C2211yescharacteristic
32#13133188—21no—
33#13133188C2211yescharacteristic
34Q8×Dic53C2211yescharacteristic
35#13133179C2211yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C2×C4⋊C4×C53C2211yescharacteristic
55#103×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70Q8×D53—22no—
71Q8×D53—41no—
72#11719945—44no—
73C2×D212—21no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106C2311yescharacteristic
82M16×C53—41no—
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53D411yescharacteristic
93#11719924—22no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919D411yescharacteristic
106C4×Dic53—21no—
107C4×Dic53C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719913—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53#4112yescomplemented
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—42no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—22no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
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—42no—
146C4×D53—42no—
147Dic106—21no—
148Dic106—21no—
149Dic106—42no—
150Dic106—42no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53AGL(1,ℤ/8)12yes—
168C4×C53—22no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—21no—
175Dic53—21no—
176Dic53—21no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3308—531nomaximal, Sylow
181C2×C53—41no—
182C2×C53—81no—
183C2×C53#54611yescharacteristic
184C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C53#57811yescharacteristic
186D53—22no—
187D53—81no—
188#548—531no—
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#417D5311yescharacteristic, complemented
202C2×#103—531no—
203C53#330811yescharacteristic, complemented, Sylow
204C2×#41—1061no—
205C2×#41C2×D5311yescharacteristic
206C22×Q8—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×D8—531no—
210C2×M16C2×D5311yescharacteristic
211C22×C8—531no—
212C4⋊Q8—531no—
213#124—531no—
214#124—1061no—
215#123—1061no—
216#123—1061no—
217C22⋊Q8—531no—
218C22⋊Q8—1061no—
219C22⋊Q8—1061no—
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224C4×Q8—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4C2×D5311yescharacteristic
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#103C2×D5311yescharacteristic
239#103C2×D5311yescharacteristic
240#102—531no—
241#102—531no—
242#102C2×D5311yescharacteristic
243#102C2×D5311yescharacteristic
244C22⋊C8—531no—
245C22⋊C8—531no—
246#97—531no—
247#41—22no—
248#41—41no—
249#41—2124no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8—1061no—
253C2×Q8—1062no—
254C2×Q8—2121no—
255C2×Q8C22×D5311yescharacteristic
256C2×D4—21no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4C22×D5311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—531no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4C22×D5311yescharacteristic
270SD16—1062no—
271SD16—2121no—
272D8—2121no—
273M16—41no—
274C2×C8—531no—
275C2×C8—531no—
276C2×C8—2121no—
277C2×C8C22×D5311yescharacteristic
278C2×C8C22×D5311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—531no—
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C22⋊C4—531no—
290C22⋊C4—531no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C42—531no—
299C42—1061no—
300C23—21no—
301C23—531no—
302C23—1061no—
303C23D4×D5311yescharacteristic
304Q8—21no—
305Q8—1061no—
306Q8—1061no—
307Q8—2122no—
308Q8—2122no—
309Q8#1171994512yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—1062no—
314D4—1061no—
315D4—1061no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—531no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4D4×D5311yescharacteristic
338C2×C4C23×D5311yescharacteristic
339C8—21no—
340C8—21no—
341C8—1061no—
342C8—1061no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—532no—
350C22—1061no—
351C22—1061no—
352C22—2121no—
353C22—2121no—
354C22—2121no—
355C22C2×D4×D5311yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—1061no—
361C4—1061no—
362C4—1061no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4C2×#1171994511yescharacteristic
367C4#1313326211yescharacteristic
368C2—41no—
369C2—81no—
370C2—1062no—
371C2—4241no—
372C2#1681126311yescharacteristic
373C2AGL(1,ℤ/8)×D5311yescharacteristic
374C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#417C2
socleC22×C53C22×D4
hypercenter#417D53
hypocenterC53#3308
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #16811280C2semidirect
2. #16811274C2semidirect
3. #16810981C2semidirect
4. #16810958C2semidirect
5. #16810935C2semidirect
6. #16810916C2semidirect
7. #16810869C2semidirect
8. #16810858C2semidirect
9. #16810837C2semidirect
10. #16810768C2semidirect
11. #16810763C2semidirect
12. #16810726C2semidirect
13. #16810692C2semidirect
14. #103×D53C2semidirect
15. C53×#417C2semidirect
117. Q8×C53#41semidirect
201. #417D53semidirect
203. C53#3308semidirect
309. Q8#11719945semidirect