Finite Group Databasebeta

← #20277474

Subgroups — #20277474

Order 6784.

596 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133189C211yesmaximal, characteristic
2#16810523C211yesmaximal, characteristic, complemented
3#16810513C211yesmaximal, characteristic, complemented
4#16810511C211yesmaximal, characteristic, complemented
5#16810507C211yesmaximal, characteristic, complemented
6#16810504C211yesmaximal, characteristic, complemented
7#16810503C211yesmaximal, characteristic, complemented
8#16810500C211yesmaximal, characteristic, complemented
9#16810499C211yesmaximal, characteristic, complemented
10#16810499C211yesmaximal, characteristic, complemented
11#16810484C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#390C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133189C418yescomplemented
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C22×D4×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C42×C53C2211yescharacteristic
52C24×C53C2311yescharacteristic
53C24×C53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C2×C414yescomplemented
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—41no—
64C2×C22⋊D53—41no—
65C2×C22⋊D53—41no—
66C2×C22⋊D53—41no—
67C2×C22⋊D53C2×C414yes—
68C2×C22⋊D53C2×C414yescomplemented
69C22×Dic53—22no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53C2×C414yescomplemented
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C2×C4×D53—22no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—41no—
91C22⋊Dic53—22no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C22⋊Dic53C2×C414yescomplemented
95#11719921—22no—
96#11719921—41no—
97#11719921—41no—
98#11719921C2×C414yescomplemented
99#11719919—41no—
100#11719919C2×C414yescomplemented
101C4×Dic53—41no—
102C4×Dic53—41no—
103C4×Dic53—41no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53C22×C412yes—
117C23×C53C22×C412yes—
118C23×C53C2411yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53C22×C412yes—
126D4×C53—44no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53C22×C412yes—
141C2×C4×C53#4112yes—
142C22⋊D53—44no—
143C22⋊D53—44no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—44no—
162C2×Dic53—44no—
163C2×Dic53C22×C412yes—
164C2×Dic53C22×C412yes—
165C2×Dic53C22×C412yes—
166C2×Dic53#4112yes—
167C4×D53—44no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53C23×C411yescharacteristic
183C22×C53C2×#4111yescharacteristic
184C22×C53C2×#4111yescharacteristic
185C22×C53H(5,𝔽2)11yescharacteristic
186C22×C53H(5,𝔽2)11yescharacteristic
187C22×C53H(5,𝔽2)11yescharacteristic
188C22×C53H(5,𝔽2)11yescharacteristic
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C4×C53—22no—
197C4×C53—22no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201Dic53—22no—
202Dic53—22no—
203Dic53—22no—
204Dic53—22no—
205Dic53—41no—
206Dic53—41no—
207Dic53—41no—
208#2333—531nomaximal, Sylow
209C2×C53—42no—
210C2×C53—42no—
211C2×C53C4×#4111yescharacteristic
212C2×C53#51811yescharacteristic
213C2×C53#51811yescharacteristic
214C2×C53#53811yescharacteristic
215C2×C53#53811yescharacteristic
216C2×C53#54011yescharacteristic
217C2×C53#54011yescharacteristic
218D53—42no—
219C2×#121—531no—
220#390—531no—
221#390D5311yescharacteristic, complemented
222#388—531no—
223#388—531no—
224#387—531no—
225#387—531no—
226#386—531no—
227#386—531no—
228#386—531no—
229#386—531no—
230#381—531no—
231C4×C22⋊C4—531no—
232C4×C22⋊C4—531no—
233C4×C22⋊C4—531no—
234C53#233311yescharacteristic, complemented, Sylow
235C22×D4—531no—
236C22×D4—531no—
237C22×D4C2×D5311yescharacteristic
238C23×C4—531no—
239C23×C4—531no—
240C23×C4—531no—
241#121—538no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4C2×D5311yescharacteristic
245C2×C22⋊C4—531no—
246C2×C22⋊C4—531no—
247C2×C22⋊C4—531no—
248C2×C22⋊C4—531no—
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4C2×D5311yescharacteristic
254C2×C22⋊C4C2×D5311yescharacteristic
255C2×C22⋊C4C2×D5311yescharacteristic
256C2×C22⋊C4C2×D5311yescharacteristic
257C2×C42—531no—
258C2×C42—531no—
259C2×C42—531no—
260C2×C42C2×D5311yescharacteristic
261#95—531no—
262#95—531no—
263#95—531no—
264#95—531no—
265#95—531no—
266#95—531no—
267#95—531no—
268#95—531no—
269#95—531no—
270#95—531no—
271C24—531no—
272C24C22×D5311yescharacteristic
273C24C22×D5311yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—534no—
277C2×D4—534no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C2×D4C4×D5314yescomplemented
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—534no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—1062no—
292C22×C4—1062no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4C22×D5311yescharacteristic
300C22×C4C22×D5311yescharacteristic
301C22×C4C22×D5311yescharacteristic
302C22×C4C22×D5311yescharacteristic
303C22×C4C22×D5311yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—534no—
306C4⋊C4—2121no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—534no—
312C22⋊C4—534no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C42—41no—
320C42—2121no—
321C42—2121no—
322C42—2121no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—532no—
336C23—1061no—
337C23—1061no—
338C23—1061no—
339C23—1061no—
340C23—1061no—
341C23—1061no—
342C23C2×C4×D5312yes—
343C23C2×C4×D5312yes—
344C23C23×D5311yescharacteristic
345D4—44no—
346D4—2124no—
347D4—2124no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—532no—
362C2×C4—532no—
363C2×C4—532no—
364C2×C4—532no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—2124no—
383C2×C4—2124no—
384C2×C4—2124no—
385C2×C4C2×C4×D5312yes—
386C2×C4#1171994712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—1062no—
402C22—1062no—
403C22—1062no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22C22×C4×D5311yescharacteristic
409C22#41×D5311yescharacteristic
410C22C2×#1171994711yescharacteristic
411C22#1313326211yescharacteristic
412C22#1313326211yescharacteristic
413C22#1313326211yescharacteristic
414C22#1313326311yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—1062no—
421C4—1062no—
422C4—1062no—
423C4—1062no—
424C4—2121no—
425C4—2121no—
426C4—2121no—
427C2—42no—
428C2—42no—
429C2—2122no—
430C2C4×#1171994711yescharacteristic
431C2#1681109511yescharacteristic
432C2#1681109611yescharacteristic
433C2#1681120711yescharacteristic
434C2#1681121011yescharacteristic
435C2#1681122311yescharacteristic
436C2#1681122711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#390C2
socleC23×C53C24
hypercenter#390D53
hypocenterC53#2333
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #16810523C2semidirect
3. #16810513C2semidirect
4. #16810511C2semidirect
5. #16810507C2semidirect
6. #16810504C2semidirect
7. #16810503C2semidirect
8. #16810500C2semidirect
9. #16810499C2semidirect
10. #16810499C2semidirect
11. #16810484C2semidirect
12. C22⋊C4×Dic53C2semidirect
13. C22⋊C4×Dic53C2semidirect
14. C4×#11719921C2semidirect
15. C53×#390C2semidirect
21. #13133189C4semidirect
57. C2×D4×C53C2×C4semidirect
68. C2×C22⋊D53C2×C4semidirect
74. C22×Dic53C2×C4semidirect
94. C22⋊Dic53C2×C4semidirect
98. #11719921C2×C4semidirect
100. #11719919C2×C4semidirect
221. #390D53semidirect
234. C53#2333semidirect
282. C2×D4C4×D53semidirect