Finite Group Databasebeta

← #20280004

Subgroups — #20280004

Order 6784.

412 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#16810564C211yesmaximal, characteristic, complemented
2#16810564C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic
4#16810548C211yesmaximal, characteristic
5#16810517C211yesmaximal, characteristic
6#16810511C211yesmaximal, characteristic
7#16810508C211yesmaximal, characteristic, complemented
8#16810507C211yesmaximal, characteristic, complemented
9#16810507C211yesmaximal, characteristic, complemented
10#16810506C211yesmaximal, characteristic, complemented
11#16810496C211yesmaximal, characteristic
12#16810489C211yesmaximal, characteristic, complemented
13#16810486C211yesmaximal, characteristic
14C4×#11719919C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C53C211yesmaximal, characteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
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
44C23×C4×C53C2211yescharacteristic
45C2×C22⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C42×C53C2211yescharacteristic
48C2×C42×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—41no—
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22⋊C4×C53—22no—
70C22⋊C4×C53—22no—
71C42×C53—22no—
72C42×C53—41no—
73C22⋊Dic53—22no—
74C22⋊Dic53—22no—
75C22⋊Dic53—22no—
76C22⋊Dic53—41no—
77C4⋊Dic53—22no—
78C4⋊Dic53—22no—
79#11719919—22no—
80#11719919—22no—
81#11719919—22no—
82#11719919—22no—
83#11719919—41no—
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—44no—
111C2×C4×C53#4112yes—
112C2×C4×C53#4112yes—
113C2×C4×C53#4112yes—
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#4112yes—
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×C53C2×#4111yescharacteristic
149C22×C53C2×#4111yescharacteristic
150C22×C53H(5,𝔽2)11yescharacteristic
151C22×C532−1+411yescharacteristic
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158Dic53—22no—
159Dic53—41no—
160Dic53—41no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164Dic53—41no—
165Dic53—41no—
166#2575—531nomaximal, Sylow
167C2×C53—42no—
168C2×C53#52911yescharacteristic
169C2×C53#52911yescharacteristic
170C2×C53#53911yescharacteristic
171C2×C53#55111yescharacteristic
172C2×C53#55211yescharacteristic
173C2×C53#55211yescharacteristic
174C2×C53#55911yescharacteristic
175#400—531no—
176#400—531no—
177#399—531no—
178#397—531no—
179#388—531no—
180#388—531no—
181#387—531no—
182#387—531no—
183#387—531no—
184#387—531no—
185#385—531no—
186#382—531no—
187#381—531no—
188C4×C4⋊C4—531no—
189C4×C22⋊C4D5311yescharacteristic
190C53#257511yescharacteristic, complemented, Sylow
191C23×C4C2×D5311yescharacteristic
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⋊C4—531no—
200C2×C22⋊C4—531no—
201C2×C22⋊C4—531no—
202C2×C22⋊C4—531no—
203C2×C22⋊C4—531no—
204C2×C22⋊C4C2×D5311yescharacteristic
205C2×C22⋊C4C2×D5311yescharacteristic
206C2×C42—531no—
207C2×C42—531no—
208C2×C42C2×D5311yescharacteristic
209C2×C42C2×D5311yescharacteristic
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#95C2×D5311yescharacteristic
225#95C2×D5311yescharacteristic
226C24C22×D5311yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4C22×D5311yescharacteristic
239C22×C4C22×D5311yescharacteristic
240C22×C4C22×D5311yescharacteristic
241C22×C4C22×D5311yescharacteristic
242C22×C4C22×D5311yescharacteristic
243C22×C4C22×D5311yescharacteristic
244C4⋊C4—1062no—
245C4⋊C4—1062no—
246C4⋊C4—1062no—
247C4⋊C4—1062no—
248C4⋊C4—1062no—
249C4⋊C4—1062no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—1062no—
255C22⋊C4—1062no—
256C22⋊C4—1062no—
257C22⋊C4—2121no—
258C42—22no—
259C42—41no—
260C42—1062no—
261C42—2121no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1171994512yes—
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—44no—
286C2×C4—532no—
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#1171994512yes—
311C2×C4#1171994512yes—
312C2×C4#1171994712yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#41×D5311yescharacteristic
321C22C2×#1171994711yescharacteristic
322C22C2×#1171994511yescharacteristic
323C22C2×#1171994511yescharacteristic
324C22C2×#1171994511yescharacteristic
325C22#1313326211yescharacteristic
326C22#1313326411yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—1062no—
334C4—2121no—
335C4—2121no—
336C4—2121no—
337C4—2121no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C2—42no—
342C2#1681115511yescharacteristic
343C2#1681115711yescharacteristic
344C2#1681121511yescharacteristic
345C2#1681129911yescharacteristic
346C2#1681130611yescharacteristic
347C2#1681131311yescharacteristic
348C2#1681135611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC4×C22⋊C4×C53C2
hypercenterC4×C22⋊C4D53
hypocenterC53#2575
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
2. #16810564C2semidirect
7. #16810508C2semidirect
8. #16810507C2semidirect
9. #16810507C2semidirect
10. #16810506C2semidirect
12. #16810489C2semidirect
14. C4×#11719919C2semidirect
190. C53#2575semidirect