Finite Group Databasebeta

← #20280410

Subgroups — #20280410

Order 6784.

476 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×#13133207C211yesmaximal, characteristic, complemented
2C2×#13133173C211yesmaximal, characteristic
3#16810564C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic
5#16810557C211yesmaximal, characteristic, complemented
6#16810551C211yesmaximal, characteristic
7#16810521C211yesmaximal, characteristic, complemented
8#16810518C211yesmaximal, characteristic, complemented
9#16810517C211yesmaximal, characteristic
10#16810510C211yesmaximal, characteristic, complemented
11#16810506C211yesmaximal, characteristic, complemented
12#16810498C211yesmaximal, characteristic
13#16810495C211yesmaximal, characteristic
14#16810469C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17#13133207—24no—
18#13133173—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C42×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C4⋊C4×C53—22no—
72C22⋊C4×C53—22no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—22no—
75C42×C53—22no—
76C22⋊Dic53—22no—
77C22⋊Dic53—22no—
78C22⋊Dic53—22no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82#11719919—22no—
83#11719919—22no—
84#11719919—22no—
85#11719919—22no—
86#11719919—22no—
87#11719919—22no—
88#11719919—22no—
89#11719919—22no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53D414yes—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#4112yes—
100C23×C53C2411yescharacteristic
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×C53C2×D412yes—
118C2×C4×C53#4112yes—
119C2×Dic53—28no—
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×Dic53C2×D412yes—
141C2×Dic53C2×D412yes—
142C2×Dic53#4112yes—
143C2×Dic53#4112yes—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C22×D411yescharacteristic
152C22×C53C2×#4111yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53C2×#4111yescharacteristic
155C22×C53C2×#4111yescharacteristic
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#2611—531nomaximal, Sylow
173C2×C53—42no—
174C2×C53#52511yescharacteristic
175C2×C53#53211yescharacteristic
176C2×C53#53611yescharacteristic
177C2×C53#55211yescharacteristic
178C2×C53#55211yescharacteristic
179C2×C53#55211yescharacteristic
180C2×C53#55211yescharacteristic
181C2×#125—531no—
182C2×#117—531no—
183#400—531no—
184#399—531no—
185#398—531no—
186#397—531no—
187#389—531no—
188#389—531no—
189#388—531no—
190#388D5311yescharacteristic
191#387—531no—
192#387—531no—
193#385—531no—
194#385—531no—
195#376—531no—
196C53#261111yescharacteristic, complemented, Sylow
197C23×C4—531no—
198#125—1064no—
199#117—1064no—
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⋊C4—531no—
211C2×C4⋊C4C2×D5311yescharacteristic
212C2×C22⋊C4—531no—
213C2×C22⋊C4—531no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4C2×D5311yescharacteristic
216C2×C22⋊C4C2×D5311yescharacteristic
217C2×C22⋊C4C2×D5311yescharacteristic
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42C2×D5311yescharacteristic
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#95—531no—
232#95C2×D5311yescharacteristic
233#95C2×D5311yescharacteristic
234C24C22×D5311yescharacteristic
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5311yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—1062no—
254C4⋊C4—1062no—
255C4⋊C4—1062no—
256C4⋊C4—1062no—
257C4⋊C4—1062no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—1062no—
261C4⋊C4—1062no—
262C4⋊C4—1062no—
263C4⋊C4—1062no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—1062no—
268C22⋊C4—1062no—
269C22⋊C4—1062no—
270C42—22no—
271C42—534no—
272C42—1062no—
273C42—1062no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1171994512yes—
281C23C23×D5311yescharacteristic
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—1068no—
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×C4D4×D5312yes—
324C2×C4#1171994712yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C2×D4×D5311yescharacteristic
333C22#41×D5311yescharacteristic
334C22#41×D5311yescharacteristic
335C22C2×#1171994711yescharacteristic
336C22C2×#1171994511yescharacteristic
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#1681113211yescharacteristic
355C2#1681116811yescharacteristic
356C2#1681119311yescharacteristic
357C2#1681130611yescharacteristic
358C2#1681130911yescharacteristic
359C2#1681131011yescharacteristic
360C2#1681131311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2611
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133207C2semidirect
3. #16810564C2semidirect
5. #16810557C2semidirect
7. #16810521C2semidirect
8. #16810518C2semidirect
10. #16810510C2semidirect
11. #16810506C2semidirect
14. #16810469C2semidirect
196. C53#2611semidirect