Finite Group Databasebeta

← #20280168

Subgroups — #20280168

Order 6784.

436 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810564C211yesmaximal, characteristic, complemented
2#16810558C211yesmaximal, characteristic
3#16810553C211yesmaximal, characteristic, complemented
4#16810536C211yesmaximal, characteristic
5#16810527C211yesmaximal, characteristic, complemented
6#16810527C211yesmaximal, characteristic, complemented
7#16810518C211yesmaximal, characteristic, complemented
8#16810512C211yesmaximal, characteristic
9#16810512C211yesmaximal, characteristic
10#16810510C211yesmaximal, characteristic, complemented
11#16810497C211yesmaximal, characteristic
12#16810494C211yesmaximal, characteristic
13C4×#11719919C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic
15C53×#387C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
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×C42×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×C4×D53—22no—
67C2×C4×D53—41no—
68C2×C4×D53—41no—
69C2×Dic106—22no—
70C2×Dic106—22no—
71C2×Dic106—41no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—41no—
74C42×C53—41no—
75#11719921—22no—
76#11719921—22no—
77#11719921—22no—
78#11719921—41no—
79#11719921—41no—
80#11719921Q814yescomplemented
81C4⋊Dic53—41no—
82#11719919—22no—
83#11719919—22no—
84#11719919—22no—
85#11719919—22no—
86#11719919—41no—
87#11719919—41no—
88C4×Dic53—22no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C23×C53C2411yescharacteristic
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53C2×Q812yes—
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—21no—
118C2×C4×C53C2×Q812yes—
119C2×C4×C53#4112yes—
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×Dic53C2×Q812yes—
138C2×Dic53#4112yes—
139C2×Dic53#4112yes—
140C2×Dic53#4112yes—
141C4×D53—44no—
142Dic106—44no—
143C22×C53C22×Q811yescharacteristic
144C22×C53C2×#4111yescharacteristic
145C22×C53C2×#4111yescharacteristic
146C22×C53C2×#4111yescharacteristic
147C22×C53C2×#4111yescharacteristic
148C22×C53H(5,𝔽2)11yescharacteristic
149C22×C53H(5,𝔽2)11yescharacteristic
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164Dic53—22no—
165Dic53—22no—
166Dic53—22no—
167Dic53—22no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#2589—531nomaximal, Sylow
172C2×C53#52911yescharacteristic
173C2×C53#53311yescharacteristic
174C2×C53#53811yescharacteristic
175C2×C53#55111yescharacteristic
176C2×C53#55411yescharacteristic
177C2×C53#55411yescharacteristic
178C2×C53#55911yescharacteristic
179D53—42no—
180#400—531no—
181#399—531no—
182#398—531no—
183#393—531no—
184#391—531no—
185#391—531no—
186#389—531no—
187#388—531no—
188#388—531no—
189#387—531no—
190#387D5311yescharacteristic, complemented
191#385—531no—
192#385—531no—
193C4×C4⋊C4—531no—
194C4×C22⋊C4—531no—
195C53#258911yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C23×C4—531no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4C2×D5311yescharacteristic
206C2×C4⋊C4C2×D5311yescharacteristic
207C2×C22⋊C4—531no—
208C2×C22⋊C4—531no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42C2×D5311yescharacteristic
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#95C2×D5311yescharacteristic
228#95C2×D5311yescharacteristic
229#95C2×D5311yescharacteristic
230#95C2×D5311yescharacteristic
231C24—531no—
232C2×Q8—1062no—
233C2×Q8—1062no—
234C2×Q8—2121no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1062no—
243C22×C4—2121no—
244C22×C4—2121no—
245C22×C4C22×D5311yescharacteristic
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C4⋊C4—1062no—
257C4⋊C4—1062no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C22⋊C4—534no—
262C22⋊C4—1062no—
263C22⋊C4—1062no—
264C22⋊C4—1062no—
265C22⋊C4—2121no—
266C22⋊C4—2121no—
267C42—41no—
268C42—1062no—
269C42—2121no—
270C42—2121no—
271C23—532no—
272C23—1061no—
273C23—1061no—
274C23—1061no—
275C23—1061no—
276C23—1061no—
277C23—1061no—
278C23C23×D5311yescharacteristic
279Q8—2124no—
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—21no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—532no—
302C2×C4—532no—
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—2124no—
321C2×C4#1171994512yes—
322C2×C4Q8×D5312yes—
323C22—1062no—
324C22—1062no—
325C22—1062no—
326C22—2121no—
327C22—2121no—
328C22—2121no—
329C22—2121no—
330C22C2×Q8×D5311yescharacteristic
331C22#41×D5311yescharacteristic
332C22#41×D5311yescharacteristic
333C22#41×D5311yescharacteristic
334C22C2×#1171994511yescharacteristic
335C22#1313326211yescharacteristic
336C22#1313326311yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—1062no—
345C4—1062no—
346C4—1062no—
347C4—1062no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C2—2122no—
352C2#1681115111yescharacteristic
353C2#1681117711yescharacteristic
354C2#1681120611yescharacteristic
355C2#1681129811yescharacteristic
356C2#1681132711yescharacteristic
357C2#1681132911yescharacteristic
358C2#1681135711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#387C2
hypercenter#387D53
hypocenterC53#2589
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
3. #16810553C2semidirect
5. #16810527C2semidirect
6. #16810527C2semidirect
7. #16810518C2semidirect
10. #16810510C2semidirect
13. C4×#11719919C2semidirect
15. C53×#387C2semidirect
80. #11719921Q8semidirect
190. #387D53semidirect
195. C53#2589semidirect