Finite Group Databasebeta

← #20277882

Subgroups — #20277882

Order 6784.

464 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
1C2×#13133194C211yesmaximal, characteristic
2#16810529C211yesmaximal, characteristic, complemented
3#16810527C211yesmaximal, characteristic, complemented
4#16810520C211yesmaximal, characteristic, complemented
5#16810519C211yesmaximal, characteristic, complemented
6#16810517C211yesmaximal, characteristic
7#16810517C211yesmaximal, characteristic
8#16810516C211yesmaximal, characteristic
9#16810508C211yesmaximal, characteristic, complemented
10#16810507C211yesmaximal, characteristic, complemented
11#16810505C211yesmaximal, characteristic, complemented
12#16810504C211yesmaximal, characteristic, complemented
13#16810484C211yesmaximal, characteristic
14#16810481C211yesmaximal, characteristic
15C53×#388C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18#13133194C418yescomplemented
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52C24×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53C2×C414yes—
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106C2×C414yes—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53C2×C414yes—
77C42×C53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C22⋊Dic53C2×C414yes—
81C4⋊Dic53—41no—
82C4⋊Dic53C2×C414yes—
83#11719919—41no—
84#11719919—41no—
85#11719919—41no—
86#11719919—41no—
87#11719919C2×C414yes—
88#11719919C2×C414yes—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53C22×C412yes—
99C23×C53C2411yescharacteristic
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×C53C22×C412yes—
117C2×C4×C53C22×C412yes—
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—44no—
139C2×Dic53C22×C412yes—
140C2×Dic53C22×C412yes—
141C2×Dic53C22×C412yes—
142C2×Dic53C22×C412yes—
143Dic106—44no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C23×C411yescharacteristic
152C22×C53H(5,𝔽2)11yescharacteristic
153C22×C53H(5,𝔽2)11yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C532−1+411yescharacteristic
156C22×C532−1+411yescharacteristic
157C22×C532−1+411yescharacteristic
158C4×C53—22no—
159C4×C53—22no—
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—
171Dic53—41no—
172#2378—531nomaximal, Sylow
173C2×C53—42no—
174C2×C53#51811yescharacteristic
175C2×C53#51911yescharacteristic
176C2×C53#52011yescharacteristic
177C2×C53#56211yescharacteristic
178C2×C53#56211yescharacteristic
179C2×C53#56311yescharacteristic
180C2×C53#56311yescharacteristic
181C2×C22⋊Q8—531no—
182#391—531no—
183#391—531no—
184#389—531no—
185#389—531no—
186#388—531no—
187#388—531no—
188#388—531no—
189#388D5311yescharacteristic
190#387—531no—
191#387—531no—
192#387—531no—
193#387—531no—
194#381—531no—
195#380—531no—
196C53#237811yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198C23×C4—531no—
199C22⋊Q8—538no—
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⋊C4C2×D5311yescharacteristic
209C2×C22⋊C4—531no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4C2×D5311yescharacteristic
213C2×C22⋊C4C2×D5311yescharacteristic
214C2×C22⋊C4C2×D5311yescharacteristic
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42C2×D5311yescharacteristic
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#95—531no—
231#95C2×D5311yescharacteristic
232#95C2×D5311yescharacteristic
233C24C22×D5311yescharacteristic
234C2×Q8—534no—
235C2×Q8—2121no—
236C2×Q8—2121no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—534no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C22×C4C22×D5311yescharacteristic
253C22×C4C22×D5311yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—534no—
256C4⋊C4—534no—
257C4⋊C4—534no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—534no—
266C22⋊C4—2121no—
267C22⋊C4—2121no—
268C22⋊C4C4×D5314yes—
269C42—41no—
270C42—2121no—
271C42—2121no—
272C42—2121no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D5312yes—
280C23C23×D5311yescharacteristic
281Q8—2124no—
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—532no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—532no—
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—2124no—
323C2×C4C2×C4×D5312yes—
324C2×C4C2×C4×D5312yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D5311yescharacteristic
333C22#1313326211yescharacteristic
334C22#1313326211yescharacteristic
335C22#1313326311yescharacteristic
336C22#1313326411yescharacteristic
337C22#1313326411yescharacteristic
338C22#1313326411yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—1062no—
346C4—1062no—
347C4—1062no—
348C4—1062no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C2—42no—
354C2#1681109511yescharacteristic
355C2#1681110011yescharacteristic
356C2#1681110211yescharacteristic
357C2#1681136711yescharacteristic
358C2#1681136811yescharacteristic
359C2#1681137511yescharacteristic
360C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2378
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #16810529C2semidirect
3. #16810527C2semidirect
4. #16810520C2semidirect
5. #16810519C2semidirect
9. #16810508C2semidirect
10. #16810507C2semidirect
11. #16810505C2semidirect
12. #16810504C2semidirect
18. #13133194C4semidirect
196. C53#2378semidirect