Finite Group Databasebeta

← #20282970

Subgroups — #20282970

Order 6784.

396 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#16810565C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic
4#16810557C211yesmaximal, characteristic, complemented
5#16810556C211yesmaximal, characteristic, complemented
6#16810556C211yesmaximal, characteristic, complemented
7#16810555C211yesmaximal, characteristic, complemented
8#16810546C211yesmaximal, characteristic
9#16810546C211yesmaximal, characteristic
10#16810544C211yesmaximal, characteristic, complemented
11#16810543C211yesmaximal, characteristic, complemented
12#16810535C211yesmaximal, characteristic
13#16810486C211yesmaximal, characteristic
14#16810483C211yesmaximal, characteristic
15C53×#380C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
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—41no—
53C22×C4×C53—41no—
54C22×C4×C53Q814yescomplemented
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
69C2×Dic106—41no—
70C2×Dic106—41no—
71C2×Dic106—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊Dic53—41no—
77C22⋊Dic53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—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×C53—21no—
96C23×C53—21no—
97C23×C53C2×Q812yes—
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—44no—
116C2×C4×C53C2×Q812yes—
117C2×C4×C53C2×Q812yes—
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—21no—
141C2×Dic53—21no—
142Dic106—44no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53C22×Q811yescharacteristic
151C22×C53H(5,𝔽2)11yescharacteristic
152C22×C53H(5,𝔽2)11yescharacteristic
153C22×C53H(5,𝔽2)11yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C532−1+411yescharacteristic
156C22×C532−1+411yescharacteristic
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163Dic53—41no—
164Dic53—41no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#2841—531nomaximal, Sylow
172C2×C53—42no—
173C2×C53#54311yescharacteristic
174C2×C53#54411yescharacteristic
175C2×C53#54411yescharacteristic
176C2×C53#56011yescharacteristic
177C2×C53#56211yescharacteristic
178C2×C53#56211yescharacteristic
179C2×C53#56311yescharacteristic
180#400—531no—
181#400—531no—
182#399—531no—
183#398—531no—
184#398—531no—
185#398—531no—
186#398—531no—
187#396—531no—
188#396—531no—
189#395—531no—
190#395—531no—
191#393—531no—
192#381—531no—
193#380—531no—
194#380D5311yescharacteristic
195C53#284111yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C23×C4C2×D5311yescharacteristic
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⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4C2×D5311yescharacteristic
210C2×C4⋊C4C2×D5311yescharacteristic
211C2×C22⋊C4—531no—
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4C2×D5311yescharacteristic
216C2×C22⋊C4C2×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#95—531no—
228#95—531no—
229#95C2×D5311yescharacteristic
230#95C2×D5311yescharacteristic
231C24C22×D5311yescharacteristic
232C2×Q8—2121no—
233C2×Q8—2121no—
234C2×Q8—2121no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4Dic10614yescomplemented
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—2121no—
268C22⋊C4—2121no—
269C22⋊C4—2121no—
270C22⋊C4—2121no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23C2×Dic10612yes—
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—44no—
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—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×C4C2×Dic10612yes—
322C2×C4C2×Dic10612yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×Dic10611yescharacteristic
331C22#1313326211yescharacteristic
332C22#1313326211yescharacteristic
333C22#1313326211yescharacteristic
334C22#1313326311yescharacteristic
335C22#1313326411yescharacteristic
336C22#1313326511yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2121no—
344C4—2121no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C2—42no—
352C2#1681124911yescharacteristic
353C2#1681125111yescharacteristic
354C2#1681125411yescharacteristic
355C2#1681136111yescharacteristic
356C2#1681136811yescharacteristic
357C2#1681136911yescharacteristic
358C2#1681137411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#380C2
hypercenter#380D53
hypocenterC53#2841
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. #16810565C2semidirect
2. #16810562C2semidirect
4. #16810557C2semidirect
5. #16810556C2semidirect
6. #16810556C2semidirect
7. #16810555C2semidirect
10. #16810544C2semidirect
11. #16810543C2semidirect
54. C22×C4×C53Q8semidirect
195. C53#2841semidirect
245. C22×C4Dic106semidirect