Finite Group Databasebeta

← #20284688

Subgroups — #20284688

Order 6784.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811202C211yesmaximal, characteristic
2#16811104C211yesmaximal, characteristic, complemented
3#16810966C211yesmaximal, characteristic, complemented
4#16810965C211yesmaximal, characteristic, complemented
5#16810960C211yesmaximal, characteristic, complemented
6#16810958C211yesmaximal, characteristic, complemented
7#16810946C211yesmaximal, characteristic, complemented
8#16810944C211yesmaximal, characteristic, complemented
9#16810935C211yesmaximal, characteristic, complemented
10#16810664C211yesmaximal, characteristic, complemented
11#107×D53C211yesmaximal, characteristic, complemented
12#16810621C211yesmaximal, characteristic, complemented
13#16810616C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#480C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133197C2211yescharacteristic
21#13133196C2211yescharacteristic
22#13133194—21no—
23#13133190—21no—
24#13133189—21no—
25#13133188—21no—
26#13133188C2211yescharacteristic
27#13133182—21no—
28#13133179—21no—
29C4×C22⋊D53—21no—
30C4×C22⋊D53—21no—
31#13133176C2211yescharacteristic
32#13133175C2211yescharacteristic
33#13133175C2211yescharacteristic
34C4⋊C4×D53C2211yescharacteristic
35C4⋊C4×D53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic
37#13133151C2211yescharacteristic
38#13133150C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52#121×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53D411yescharacteristic
66C2×C22⋊D53D411yescharacteristic
67C22×Dic53—21no—
68#11719947—22no—
69#11719947—22no—
70#11719945—22no—
71#11719945—22no—
72C2×D212D411yescharacteristic
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53C2311yescharacteristic
77C2×C4×D53C2311yescharacteristic
78C2×Dic106D411yescharacteristic
79C2×Dic106C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107C8×D53—22no—
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53C2×D411yescharacteristic
111C22×D53C2×D411yescharacteristic
112D4×C53—21no—
113D4×C53—21no—
114D4×C53—81no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—81no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C22⋊D53—22no—
124C22⋊D53—22no—
125C22⋊D53—42no—
126C22⋊D53—81no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—42no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134D212—22no—
135C4×D53—21no—
136C4×D53—21no—
137C4×D53—81no—
138C4×D53—81no—
139C4×D53—81no—
140C4×D53#4112yes—
141C4×D53#4112yes—
142Dic106—22no—
143Dic106—21no—
144Dic106—21no—
145C8×C53—21no—
146C8×C53—21no—
147#88322—21no—
148#88322—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53#12312yes—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53C2×#4111yescharacteristic
167C4×C53C2×#4111yescharacteristic
168Dic53—21no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53#12312yes—
174#3098—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—81no—
177C2×C53C2×#12311yescharacteristic
178C2×C53#57611yescharacteristic
179C2×C53#57811yescharacteristic
180D53—22no—
181D53—41no—
182#537—531no—
183#520—531no—
184#485—531no—
185#485—531no—
186#484—531no—
187#484—531no—
188#482—531no—
189#482—531no—
190#480—531no—
191#480D5311yescharacteristic, complemented
192#428—531no—
193C2×#107—531no—
194#417—531no—
195#416—531no—
196#408—531no—
197C53#309811yescharacteristic, complemented, Sylow
198C2×#41—531no—
199C2×#41—531no—
200C2×M16—531no—
201C22×C8—531no—
202C22⋊Q8—531no—
203C22⋊Q8—531no—
204C22⋊Q8—1061no—
205#121—531no—
206#121—1061no—
207#121—1061no—
208#121—1061no—
209#121C2×D5311yescharacteristic
210C4×Q8—1061no—
211C4×D4—1061no—
212C4×D4—1061no—
213C4×D4—1061no—
214#117—531no—
215#117—531no—
216#117—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4C2×D5311yescharacteristic
221#107—531no—
222#107—531no—
223#107—531no—
224#107—531no—
225#107C2×D5311yescharacteristic
226#107C2×D5311yescharacteristic
227#106—531no—
228#106—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#102—531no—
234#102—531no—
235#102C2×D5311yescharacteristic
236#102C2×D5311yescharacteristic
237C22⋊C8—531no—
238C22⋊C8—531no—
239C22⋊C8—531no—
240C22⋊C8C2×D5311yescharacteristic
241#41—1062no—
242#41—1062no—
243#41—1062no—
244#41—1062no—
245C2×Q8—531no—
246C2×Q8—531no—
247C2×D4—21no—
248C2×D4—531no—
249C2×D4—531no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4C22×D5311yescharacteristic
254C22×C4—21no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4C22×D5311yescharacteristic
262M16—1062no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—1062no—
266C2×C8C22×D5311yescharacteristic
267C2×C8C22×D5311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—531no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4C22×D5311yescharacteristic
278C4⋊C4C22×D5311yescharacteristic
279C4⋊C4C22×D5311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C42—1061no—
288C42—1061no—
289C42—1061no—
290C23—21no—
291C23—531no—
292C23—531no—
293C23D4×D5311yescharacteristic
294Q8—1062no—
295Q8—1061no—
296Q8—1061no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—1062no—
301D4—1062no—
302D4—1062no—
303D4—4241no—
304D4—2122no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—531no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—531no—
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—4241no—
324C2×C4—2122no—
325C2×C4D4×D5311yescharacteristic
326C2×C4C23×D5311yescharacteristic
327C8—21no—
328C8—21no—
329C8—1061no—
330C8—1061no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—532no—
338C22—1061no—
339C22—1061no—
340C22—1061no—
341C22—1061no—
342C22—1061no—
343C22C2×D4×D5311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—532no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4—2121no—
354C4#41×D5311yescharacteristic
355C4#41×D5311yescharacteristic
356C2—41no—
357C2—81no—
358C2—1062no—
359C2—2121no—
360C2#123×D5311yescharacteristic
361C2#1681142411yescharacteristic
362C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#480C2
socleC22×C53C22×D4
hypercenter#480D53
hypocenterC53#3098
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #16811104C2semidirect
3. #16810966C2semidirect
4. #16810965C2semidirect
5. #16810960C2semidirect
6. #16810958C2semidirect
7. #16810946C2semidirect
8. #16810944C2semidirect
9. #16810935C2semidirect
10. #16810664C2semidirect
11. #107×D53C2semidirect
12. #16810621C2semidirect
13. #16810616C2semidirect
14. #16810588C2semidirect
15. C53×#480C2semidirect
191. #480D53semidirect
197. C53#3098semidirect