Finite Group Databasebeta

← #20282156

Subgroups — #20282156

Order 6784.

392 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
1C53⋊#401C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810556C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic
6#16810522C211yesmaximal, characteristic, complemented
7#16810517C211yesmaximal, characteristic
8#16810511C211yesmaximal, characteristic
9#16810509C211yesmaximal, characteristic, complemented
10#16810504C211yesmaximal, characteristic, complemented
11#16810498C211yesmaximal, characteristic
12#16810498C211yesmaximal, characteristic
13#16810488C211yesmaximal, characteristic, complemented
14#16810484C211yesmaximal, characteristic
15C53×#396C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2211yescharacteristic
30#13133124C2211yescharacteristic
31#13133124C2211yescharacteristic
32#13133124C2211yescharacteristic
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
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—22no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—41no—
75C22⋊Dic53—22no—
76C22⋊Dic53—22no—
77C22⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83#11719919—22no—
84#11719919—41no—
85C4×Dic53—22no—
86C4×Dic53—41no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53#4112yes—
94C23×C53C2411yescharacteristic
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#4112yes—
112C2×C4×C53#4112yes—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
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—21no—
136C2×Dic53—21no—
137C2×Dic53—44no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53C2×#4111yescharacteristic
146C22×C53C2×#4111yescharacteristic
147C22×C53C2×#4111yescharacteristic
148C22×C53H(5,𝔽2)11yescharacteristic
149C22×C53H(5,𝔽2)11yescharacteristic
150C22×C532−1+411yescharacteristic
151C22×C532−1+411yescharacteristic
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158Dic53—41no—
159Dic53—41no—
160Dic53—41no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164Dic53—41no—
165Dic53—41no—
166#2765—531nomaximal, Sylow
167C2×C53—42no—
168C2×C53#53811yescharacteristic
169C2×C53#53911yescharacteristic
170C2×C53#54211yescharacteristic
171C2×C53#55211yescharacteristic
172C2×C53#55211yescharacteristic
173C2×C53#55311yescharacteristic
174C2×C53#56311yescharacteristic
175#401—531no—
176#400—531no—
177#398—531no—
178#398—531no—
179#397—531no—
180#396D5311yescharacteristic
181#389—531no—
182#388—531no—
183#388—531no—
184#387—531no—
185#387—531no—
186#385—531no—
187#385—531no—
188#382—531no—
189#381—531no—
190C53#276511yescharacteristic, complemented, Sylow
191C23×C4—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⋊C4—531no—
199C2×C4⋊C4C2×D5311yescharacteristic
200C2×C4⋊C4C2×D5311yescharacteristic
201C2×C4⋊C4C2×D5311yescharacteristic
202C2×C22⋊C4—531no—
203C2×C22⋊C4—531no—
204C2×C22⋊C4—531no—
205C2×C22⋊C4C2×D5311yescharacteristic
206C2×C22⋊C4C2×D5311yescharacteristic
207C2×C22⋊C4C2×D5311yescharacteristic
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#95—531no—
222#95—531no—
223#95—531no—
224#95—531no—
225#95C2×D5311yescharacteristic
226C24C22×D5311yescharacteristic
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1062no—
236C22×C4—2121no—
237C22×C4—2121no—
238C22×C4C22×D5311yescharacteristic
239C22×C4C22×D5311yescharacteristic
240C22×C4C22×D5311yescharacteristic
241C22×C4C22×D5311yescharacteristic
242C22×C4C22×D5311yescharacteristic
243C22×C4C22×D5311yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—1062no—
248C4⋊C4—2121no—
249C4⋊C4—2121no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C4⋊C4—2121no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—41no—
257C22⋊C4—1062no—
258C22⋊C4—1062no—
259C22⋊C4—2121no—
260C42—1062no—
261C42—2121no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1171994912yes—
269C23C23×D5311yescharacteristic
270C2×C4—21no—
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—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
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—2124no—
311C2×C4#1171994712yes—
312C2×C4#1171994712yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#1171994711yescharacteristic
321C22C2×#1171994711yescharacteristic
322C22C2×#1171994911yescharacteristic
323C22#1313326211yescharacteristic
324C22#1313326311yescharacteristic
325C22#1313326411yescharacteristic
326C22#1313326411yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—2121no—
334C4—2121no—
335C4—2121no—
336C4—2121no—
337C4—2121no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C2—42no—
342C2#1681121311yescharacteristic
343C2#1681121811yescharacteristic
344C2#1681124011yescharacteristic
345C2#1681131511yescharacteristic
346C2#1681131511yescharacteristic
347C2#1681132511yescharacteristic
348C2#1681137511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2765
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C53⋊#401C2semidirect
2. #16810562C2semidirect
3. #16810556C2semidirect
4. #16810555C2semidirect
6. #16810522C2semidirect
9. #16810509C2semidirect
10. #16810504C2semidirect
13. #16810488C2semidirect
190. C53#2765semidirect