Finite Group Databasebeta

← #20280687

Subgroups — #20280687

Order 6784.

436 conjugacy classes of subgroups.

The subgroup lattice has 362 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#16810563C211yesmaximal, characteristic, complemented
4#16810561C211yesmaximal, characteristic, complemented
5#16810557C211yesmaximal, characteristic, complemented
6#16810556C211yesmaximal, characteristic, complemented
7#16810554C211yesmaximal, characteristic, complemented
8#16810553C211yesmaximal, characteristic, complemented
9#16810498C211yesmaximal, characteristic
10#16810496C211yesmaximal, characteristic
11#16810495C211yesmaximal, characteristic
12#16810481C211yesmaximal, characteristic
13#16810481C211yesmaximal, characteristic
14C2×#13133124C211yesmaximal, characteristic
15C53×#385C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊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×C4⋊C4×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C22×C4×C53—22no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53Q814yes—
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—22no—
78C22⋊C4×C53—22no—
79C22⋊Dic53—22no—
80C22⋊Dic53—22no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83#11719919—41no—
84#11719919—41no—
85#11719919—41no—
86#11719919—41no—
87#11719919—41no—
88#11719919—41no—
89#11719919—41no—
90#11719919—41no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53C2×Q812yes—
96C23×C53#4112yes—
97C23×C53#4112yes—
98C23×C53C2411yescharacteristic
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—21no—
116C2×C4×C53—21no—
117C2×C4×C53—44no—
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—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—44no—
141C2×Dic53—44no—
142C2×Dic53C2×Q812yes—
143C2×Dic53C2×Q812yes—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53#12514yes—
151C22×C53C22×Q811yescharacteristic
152C22×C53C2×#4111yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C53H(5,𝔽2)11yescharacteristic
156C22×C53H(5,𝔽2)11yescharacteristic
157C22×C532−1+411yescharacteristic
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164Dic53—22no—
165Dic53—22no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53—41no—
172#2646—531nomaximal, Sylow
173C2×C53—24no—
174C2×C53C2×#12511yescharacteristic
175C2×C53#53811yescharacteristic
176C2×C53#53811yescharacteristic
177C2×C53#53911yescharacteristic
178C2×C53#53911yescharacteristic
179C2×C53#54311yescharacteristic
180C2×C53#54411yescharacteristic
181#400—531no—
182#400—531no—
183#400—531no—
184#400—531no—
185#398—531no—
186#398—531no—
187#398—531no—
188#398—531no—
189#385—531no—
190#385—531no—
191#385—531no—
192#385D5311yescharacteristic
193#380—531no—
194#380—531no—
195C2×#95—531no—
196C53#264611yescharacteristic, complemented, Sylow
197C23×C4—531no—
198C23×C4—531no—
199C23×C4C2×D5311yescharacteristic
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4C2×D5311yescharacteristic
211C2×C4⋊C4C2×D5311yescharacteristic
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4C2×D5311yescharacteristic
215C2×C22⋊C4C2×D5311yescharacteristic
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#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95C2×D5311yescharacteristic
231#95C2×D5311yescharacteristic
232C24C22×D5311yescharacteristic
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—534no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—1062no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C22×C4C22×D5311yescharacteristic
253C22×C4C22×D5311yescharacteristic
254C22×C4C22×D5311yescharacteristic
255C22×C4C22×D5311yescharacteristic
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—1062no—
271C22⋊C4—1062no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23#1171994512yes—
277C23#1171994712yes—
278C23Q8×D5312yes—
279C23C23×D5311yescharacteristic
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—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—44no—
299C2×C4—532no—
300C2×C4—532no—
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—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2124no—
324C2×C4—2124no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22#1313320714yes—
332C22C2×Q8×D5311yescharacteristic
333C22C2×#1171994711yescharacteristic
334C22C2×#1171994511yescharacteristic
335C22#1313326211yescharacteristic
336C22#1313326211yescharacteristic
337C22#1313326311yescharacteristic
338C22#1313326411yescharacteristic
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—1062no—
346C4—1062no—
347C4—2121no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C2—24no—
354C2C2×#1313320711yescharacteristic
355C2#1681120611yescharacteristic
356C2#1681120811yescharacteristic
357C2#1681121511yescharacteristic
358C2#1681121711yescharacteristic
359C2#1681124811yescharacteristic
360C2#1681125211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2646
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
2. #16810564C2semidirect
3. #16810563C2semidirect
4. #16810561C2semidirect
5. #16810557C2semidirect
6. #16810556C2semidirect
7. #16810554C2semidirect
8. #16810553C2semidirect
196. C53#2646semidirect