Finite Group Databasebeta

← #20287315

Subgroups — #20287315

Order 6784.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811319C211yesmaximal, characteristic, complemented
2#16811022C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810936C211yesmaximal, characteristic, complemented
6#16810934C211yesmaximal, characteristic, complemented
7#16810923C211yesmaximal, characteristic, complemented
8#16810920C211yesmaximal, characteristic, complemented
9#16810874C211yesmaximal, characteristic, complemented
10#16810874C211yesmaximal, characteristic, complemented
11C4×#11719922C211yesmaximal, characteristic, complemented
12#16810701C211yesmaximal, characteristic, complemented
13#16810666C211yesmaximal, characteristic, complemented
14#16810663C211yesmaximal, characteristic, complemented
15C53×#553C211yesmaximal, characteristic, complemented
16C2×#11719922C2211yescharacteristic
17C2×#11719917C2211yescharacteristic
18C2×#11719917C2211yescharacteristic
19#13133212—21no—
20#13133209C2211yescharacteristic
21#13133199—21no—
22#13133191C2211yescharacteristic
23#13133191C2211yescharacteristic
24#13133189—21no—
25C4×D212C2211yescharacteristic
26#13133177—21no—
27#13133176C2211yescharacteristic
28C2×C4⋊Dic53C2211yescharacteristic
29#13133150C2211yescharacteristic
30#13133150C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133150C2211yescharacteristic
34#13133148C2211yescharacteristic
35#13133148C2211yescharacteristic
36#13133148C2211yescharacteristic
37#13133148C2211yescharacteristic
38C4⋊#88322C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48#126×C53—21no—
49#125×C53C2211yescharacteristic
50#123×C53—21no—
51#121×C53—21no—
52#121×C53C2211yescharacteristic
53#121×C53C2211yescharacteristic
54C4×D4×C53—21no—
55C4×D4×C53C2211yescharacteristic
56C4×D4×C53C2211yescharacteristic
57#117×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53D411yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C22×Dic53—21no—
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C4⋊C4×C53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53D411yescharacteristic
76C4⋊C4×C53C2311yescharacteristic
77C4⋊C4×C53C2311yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C22⋊C4×C53—21no—
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53D411yescharacteristic
84C22⋊C4×C53D411yescharacteristic
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88#11719922—22no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98C4×Dic53—21no—
99#11719917—22no—
100#11719917—22no—
101C2×#88322C2311yescharacteristic
102C2×#88322C2311yescharacteristic
103C2×#88322C2311yescharacteristic
104C2×#88322C2311yescharacteristic
105C23×C53—21no—
106C23×C53C2×D411yescharacteristic
107C23×C53C2×D411yescharacteristic
108C22×D53—21no—
109D4×C53—21no—
110D4×C53—21no—
111D4×C53—41no—
112D4×C53—42no—
113D4×C53—81no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—81no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—81no—
134D212—21no—
135D212—21no—
136C4×D53—81no—
137#88322—21no—
138#88322—21no—
139#88322—21no—
140#88322#4112yes—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53C22×D411yescharacteristic
151C2×D53—41no—
152C2×D53—41no—
153C2×D53—41no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C2×#4111yescharacteristic
162C4×C532−1+411yescharacteristic
163Dic53—41no—
164Dic53—41no—
165Dic53—41no—
166#3355—531nomaximal, Sylow
167C2×C53—41no—
168C2×C53—41no—
169C2×C53—81no—
170C2×C53#54711yescharacteristic
171C2×C53#57511yescharacteristic
172C2×C53#57611yescharacteristic
173D53—81no—
174#553—531no—
175#553D5311yescharacteristic, complemented
176#499—531no—
177#482—531no—
178#482—531no—
179#480—531no—
180#480—531no—
181#478—531no—
182#478—531no—
183#469—531no—
184#469—531no—
185C4×D8—531no—
186#436—531no—
187#428—531no—
188#428—531no—
189C53#335511yescharacteristic, complemented, Sylow
190C2×D8—531no—
191C2×M16—531no—
192C2×M16—531no—
193#126—21no—
194#126—1061no—
195#125—531no—
196#125C2×D5311yescharacteristic
197#123—21no—
198#123—1061no—
199#121—21no—
200#121—531no—
201#121—531no—
202#121—1061no—
203#121C2×D5311yescharacteristic
204#121C2×D5311yescharacteristic
205C4×D4—21no—
206C4×D4—531no—
207C4×D4—1061no—
208C4×D4C2×D5311yescharacteristic
209C4×D4C2×D5311yescharacteristic
210#117—531no—
211#117C2×D5311yescharacteristic
212C2×C4⋊C4—531no—
213C2×C4⋊C4C2×D5311yescharacteristic
214#107—531no—
215#107—531no—
216#107—531no—
217#107—531no—
218#107—531no—
219#106—531no—
220#106—531no—
221#106—531no—
222#106—531no—
223C4⋊C8—531no—
224#102—531no—
225#102—531no—
226#102—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230C22⋊C8—531no—
231C22⋊C8—531no—
232C4×C8—531no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—531no—
236C2×D4—1061no—
237C2×D4—1061no—
238C2×D4C22⋊D5311yescharacteristic
239C2×D4C22×D5311yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4C22×D5311yescharacteristic
245C22×C4C22×D5311yescharacteristic
246D8—1062no—
247M16—1062no—
248M16—1062no—
249C2×C8—531no—
250C2×C8—531no—
251C2×C8—531no—
252C2×C8—531no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—531no—
257C4⋊C4—531no—
258C4⋊C4—531no—
259C4⋊C4—1061no—
260C4⋊C4—1061no—
261C4⋊C4—1061no—
262C4⋊C4C22⋊D5311yescharacteristic
263C4⋊C4C22×D5311yescharacteristic
264C4⋊C4C22×D5311yescharacteristic
265C4⋊C4C22×D5311yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1061no—
271C22⋊C4—1061no—
272C22⋊C4—1061no—
273C22⋊C4—1061no—
274C22⋊C4C22⋊D5311yescharacteristic
275C22⋊C4C22⋊D5311yescharacteristic
276C42—21no—
277C42—1061no—
278C42C22×D5311yescharacteristic
279C23—21no—
280C23—1061no—
281C23C2×C22⋊D5311yescharacteristic
282C23C2×C22⋊D5311yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—1061no—
289D4—1061no—
290D4—4241no—
291D4—2122no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—4241no—
305C2×C4—4241no—
306C2×C4C2×C22⋊D5311yescharacteristic
307C2×C4C2×C22⋊D5311yescharacteristic
308C2×C4C2×C22⋊D5311yescharacteristic
309C2×C4C2×C22⋊D5311yescharacteristic
310C2×C4C23×D5311yescharacteristic
311C8—532no—
312C8—1061no—
313C8—1061no—
314C8—1061no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—2121no—
325C22—2121no—
326C22—2121no—
327C22C22×C22⋊D5311yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—2121no—
336C4—2121no—
337C4—2121no—
338C4#41×D5311yescharacteristic
339C4#1313326411yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—4241no—
344C2#1681127211yescharacteristic
345C2#1681142111yescharacteristic
346C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#553C2
socleC22×C53C22×D4
hypercenter#553D53
hypocenterC53#3355
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. #16811022C2semidirect
3. #16810947C2semidirect
4. #16810944C2semidirect
5. #16810936C2semidirect
6. #16810934C2semidirect
7. #16810923C2semidirect
8. #16810920C2semidirect
9. #16810874C2semidirect
10. #16810874C2semidirect
11. C4×#11719922C2semidirect
12. #16810701C2semidirect
13. #16810666C2semidirect
14. #16810663C2semidirect
15. C53×#553C2semidirect
175. #553D53semidirect
189. C53#3355semidirect