Finite Group Databasebeta

← #20280732

Subgroups — #20280732

Order 6784.

386 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810565C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810552C211yesmaximal, characteristic
4#16810550C211yesmaximal, characteristic
5#16810545C211yesmaximal, characteristic
6#16810544C211yesmaximal, characteristic, complemented
7#16810536C211yesmaximal, characteristic
8#16810528C211yesmaximal, characteristic, complemented
9#16810515C211yesmaximal, characteristic
10#16810513C211yesmaximal, characteristic
11#16810512C211yesmaximal, characteristic
12#16810510C211yesmaximal, characteristic, complemented
13#16810506C211yesmaximal, characteristic, complemented
14#16810474C211yesmaximal, characteristic, complemented
15C53×#400C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44C2×C4⋊C4×C53C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46#95×C53C2211yescharacteristic
47#95×C53C2211yescharacteristic
48#95×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C23×D53C2311yescharacteristic
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C2×Dic106—22no—
67C2×Dic106—41no—
68C2×Dic106—41no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71#11719921—22no—
72#11719921—22no—
73#11719921—22no—
74#11719921—41no—
75#11719921—41no—
76#11719921—41no—
77#11719921—41no—
78C4⋊Dic53—22no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81#11719919—22no—
82#11719919—22no—
83#11719919—22no—
84C4×Dic53—22no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C23×C53C2411yescharacteristic
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53#4112yes—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53#4112yes—
136C2×Dic53#4112yes—
137Dic106—44no—
138C22×C53C2×#4111yescharacteristic
139C22×C53C2×#4111yescharacteristic
140C22×C53C2×#4111yescharacteristic
141C22×C53H(5,𝔽2)11yescharacteristic
142C22×C53H(5,𝔽2)11yescharacteristic
143C22×C532−1+411yescharacteristic
144C22×C532−1+411yescharacteristic
145C2×D53—41no—
146C2×D53—41no—
147C2×D53—41no—
148C2×D53—41no—
149C2×D53—41no—
150C2×D53—41no—
151C2×D53—41no—
152C4×C53—22no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159Dic53—22no—
160Dic53—22no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164Dic53—41no—
165Dic53—41no—
166#2650—531nomaximal, Sylow
167C2×C53#52911yescharacteristic
168C2×C53#53811yescharacteristic
169C2×C53#53911yescharacteristic
170C2×C53#54111yescharacteristic
171C2×C53#54211yescharacteristic
172C2×C53#54211yescharacteristic
173C2×C53#54211yescharacteristic
174D53—81no—
175#400—531no—
176#400—531no—
177#400D5311yescharacteristic, complemented
178#397—531no—
179#397—531no—
180#396—531no—
181#395—531no—
182#393—531no—
183#391—531no—
184#388—531no—
185#388—531no—
186#388—531no—
187#387—531no—
188#387—531no—
189C4×C4⋊C4—531no—
190C53#265011yescharacteristic, complemented, Sylow
191C22×Q8—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4C2×D5311yescharacteristic
199C2×C4⋊C4C2×D5311yescharacteristic
200C2×C22⋊C4—531no—
201C2×C22⋊C4—531no—
202C2×C22⋊C4—531no—
203C2×C22⋊C4—531no—
204C2×C22⋊C4—531no—
205C2×C22⋊C4—531no—
206C2×C22⋊C4—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210#95—531no—
211#95—531no—
212#95—531no—
213#95—531no—
214#95—531no—
215#95—531no—
216#95—531no—
217#95—531no—
218#95—531no—
219#95—531no—
220#95—531no—
221#95C2×D5311yescharacteristic
222#95C2×D5311yescharacteristic
223#95C2×D5311yescharacteristic
224#95C2×D5311yescharacteristic
225#95C2×D5311yescharacteristic
226C24—531no—
227C2×Q8—1062no—
228C2×Q8—2121no—
229C2×Q8—2121no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4C22×D5311yescharacteristic
238C22×C4C22×D5311yescharacteristic
239C22×C4C22×D5311yescharacteristic
240C22×C4C22×D5311yescharacteristic
241C22×C4C22×D5311yescharacteristic
242C22×C4C22×D5311yescharacteristic
243C22×C4C22×D5311yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—1062no—
247C4⋊C4—1062no—
248C4⋊C4—1062no—
249C4⋊C4—1062no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C22⋊C4—1062no—
253C22⋊C4—1062no—
254C22⋊C4—1062no—
255C22⋊C4—2121no—
256C22⋊C4—2121no—
257C22⋊C4—2121no—
258C22⋊C4—2121no—
259C42—1062no—
260C42—2121no—
261C42—2121no—
262C23—1061no—
263C23—1061no—
264C23—1061no—
265C23—1061no—
266C23—1061no—
267C23—1061no—
268C23—1061no—
269C23C23×D5311yescharacteristic
270Q8—2124no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—532no—
292C2×C4—532no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4#1171994912yes—
313C22—2121no—
314C22—2121no—
315C22—2121no—
316C22—2121no—
317C22—2121no—
318C22—2121no—
319C22—2121no—
320C22#41×D5311yescharacteristic
321C22#41×D5311yescharacteristic
322C22C2×#1171994911yescharacteristic
323C22#1313326211yescharacteristic
324C22#1313326311yescharacteristic
325C22#1313326411yescharacteristic
326C22#1313326411yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—1062no—
335C4—1062no—
336C4—2121no—
337C4—2121no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C2—4241no—
342C2#1681115811yescharacteristic
343C2#1681121311yescharacteristic
344C2#1681121411yescharacteristic
345C2#1681123611yescharacteristic
346C2#1681123711yescharacteristic
347C2#1681123911yescharacteristic
348C2#1681124111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2650
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #16810565C2semidirect
2. #16810562C2semidirect
6. #16810544C2semidirect
8. #16810528C2semidirect
12. #16810510C2semidirect
13. #16810506C2semidirect
14. #16810474C2semidirect
15. C53×#400C2semidirect
177. #400D53semidirect
190. C53#2650semidirect