Finite Group Databasebeta

← #20277728

Subgroups — #20277728

Order 6784.

788 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133190C211yesmaximal, characteristic
2C2×C4×D212C211yesmaximal, characteristic, complemented
3C2×#13133177C211yesmaximal, characteristic, complemented
4C2×#13133177C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
6#16810523C211yesmaximal, characteristic, complemented
7#16810520C211yesmaximal, characteristic, complemented
8#16810515C211yesmaximal, characteristic, complemented
9#16810501C211yesmaximal, characteristic, complemented
10#16810499C211yesmaximal, characteristic, complemented
11#16810498C211yesmaximal, characteristic, complemented
12#16810494C211yesmaximal, characteristic, complemented
13C4×#11719919C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic
24#13133190C418yescomplemented
25C4×D212—24no—
26#13133177—24no—
27#13133177—24no—
28C2×C4⋊Dic53C2211yescharacteristic
29C2×C4⋊Dic53C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C4⋊C4×D53—24no—
35C2×C22⋊Dic53C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×C4×Dic53C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
44C2×C4×Dic53C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48#13133124C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C42×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55#95×C53C2211yescharacteristic
56C24×C53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—22no—
69C2×C22⋊D53C2×C414yescomplemented
70C2×C22⋊D53C2×C414yescomplemented
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—41no—
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C2×D212—22no—
81C2×D212—22no—
82C2×D212C2×C414yescomplemented
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C4×D53C2×C414yescomplemented
95C4⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53C2×C414yescomplemented
99C42×C53—22no—
100C22⋊Dic53—41no—
101#11719921—22no—
102#11719921—22no—
103#11719921—22no—
104#11719921—41no—
105#11719921C2×C414yescomplemented
106#11719921D414yescomplemented
107C4⋊Dic53—22no—
108C4⋊Dic53—22no—
109#11719919—22no—
110#11719919—22no—
111#11719919C2×C414yescomplemented
112#11719919D414yescomplemented
113C4×Dic53—22no—
114C4×Dic53—22no—
115C4×Dic53—22no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53C22×C412yes—
123C23×C53C2411yescharacteristic
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53C22×C412yes—
135C22×D53C22×C412yes—
136C22×D53C2×D412yes—
137C22×D53#4112yes—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53C22×C412yes—
153C2×C4×C53C22×C412yes—
154C2×C4×C53C2×D412yes—
155C2×C4×C53C2×D412yes—
156C22⋊D53—28no—
157C22⋊D53—28no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—44no—
171C2×Dic53C22×C412yes—
172C2×Dic53C22×C412yes—
173C2×Dic53C2×D412yes—
174C2×Dic53C2×D412yes—
175C2×Dic53C2×D412yes—
176C2×Dic53#4112yes—
177D212—28no—
178C4×D53—28no—
179C4×D53—44no—
180C4×D53—44no—
181C4×D53—44no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53C23×C411yescharacteristic
190C22×C53C22×D411yescharacteristic
191C22×C53C22×D411yescharacteristic
192C22×C53C2×#4111yescharacteristic
193C22×C53C2×#4111yescharacteristic
194C22×C53H(5,𝔽2)11yescharacteristic
195C22×C532−1+411yescharacteristic
196C2×D53—22no—
197C2×D53—22no—
198C2×D53—22no—
199C2×D53—22no—
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—41no—
206C2×D53—41no—
207C2×D53—41no—
208C2×D53—41no—
209C2×D53C4×D414yescomplemented
210C4×C53—22no—
211C4×C53—22no—
212C4×C53—22no—
213C4×C53—22no—
214C4×C53—41no—
215C4×C53—41no—
216Dic53—22no—
217Dic53—22no—
218Dic53—22no—
219Dic53—41no—
220Dic53—41no—
221Dic53C4×D414yescomplemented
222#2361—531nomaximal, Sylow
223C2×C53—42no—
224C2×C53C2×C4×D411yescharacteristic
225C2×C53C2×C4×D411yescharacteristic
226C2×C53#52011yescharacteristic
227C2×C53D4211yescharacteristic
228C2×C53#54711yescharacteristic
229C2×C53#54811yescharacteristic
230C2×C53#55311yescharacteristic
231D53—24no—
232D53—42no—
233C2×#121—531no—
234C2×C4×D4—531no—
235C2×C4×D4—531no—
236C2×C4×D4—531no—
237C22×C4⋊C4—531no—
238#390—531no—
239#389—531no—
240#388—531no—
241#388D5311yescharacteristic, complemented
242#386—531no—
243#386—531no—
244#385—531no—
245#385—531no—
246C4×C4⋊C4—531no—
247C4×C22⋊C4—531no—
248C53#236111yescharacteristic, complemented, Sylow
249C22×D4—531no—
250C22×D4—531no—
251C22×D4—531no—
252C23×C4—531no—
253C23×C4—531no—
254C23×C4—531no—
255C23×C4—531no—
256C23×C4—531no—
257#121—538no—
258C4×D4—1064no—
259C4×D4—1064no—
260C4×D4—1064no—
261C2×C4⋊C4—531no—
262C2×C4⋊C4—531no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—1064no—
268C2×C4⋊C4C2×D5311yescharacteristic
269C2×C22⋊C4—531no—
270C2×C22⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4—531no—
273C2×C22⋊C4—531no—
274C2×C22⋊C4—531no—
275C2×C22⋊C4—531no—
276C2×C22⋊C4C2×D5311yescharacteristic
277C2×C22⋊C4C2×D5311yescharacteristic
278C2×C22⋊C4C2×D5311yescharacteristic
279C2×C42—531no—
280C2×C42—531no—
281C2×C42—531no—
282C2×C42C2×D5311yescharacteristic
283#95—531no—
284#95—531no—
285#95—531no—
286#95—531no—
287#95C2×D5311yescharacteristic
288#95C2×D5311yescharacteristic
289C24—531no—
290C24—531no—
291C24C22×D5311yescharacteristic
292C2×D4—534no—
293C2×D4—534no—
294C2×D4—534no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—1062no—
300C2×D4—1062no—
301C22×C4—531no—
302C22×C4—531no—
303C22×C4—531no—
304C22×C4—531no—
305C22×C4—534no—
306C22×C4—531no—
307C22×C4—531no—
308C22×C4—1062no—
309C22×C4—1062no—
310C22×C4—1062no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—1062no—
314C22×C4—1062no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—2121no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4C22×D5311yescharacteristic
323C22×C4C22×D5311yescharacteristic
324C22×C4C22×D5311yescharacteristic
325C22×C4C22×D5311yescharacteristic
326C22×C4C22×D5311yescharacteristic
327C22×C4C22×D5311yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—534no—
330C4⋊C4—534no—
331C4⋊C4—1062no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—534no—
338C22⋊C4—534no—
339C22⋊C4—1062no—
340C22⋊C4—1062no—
341C22⋊C4—1062no—
342C22⋊C4—2121no—
343C22⋊C4—2121no—
344C22⋊C4C4×D5314yescomplemented
345C42—22no—
346C42—1062no—
347C42—1062no—
348C42—1062no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—532no—
356C23—532no—
357C23—532no—
358C23—532no—
359C23—1061no—
360C23—1061no—
361C23—1061no—
362C23—1061no—
363C23—1061no—
364C23—1061no—
365C23—1061no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23C2×C4×D5312yes—
370C23C23×D5311yescharacteristic
371D4—1068no—
372D4—1068no—
373D4—1068no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—532no—
389C2×C4—532no—
390C2×C4—532no—
391C2×C4—532no—
392C2×C4—532no—
393C2×C4—532no—
394C2×C4—1068no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—2124no—
408C2×C4—2124no—
409C2×C4—2124no—
410C2×C4—2124no—
411C2×C4C2×C4×D5312yes—
412C2×C4C2×C4×D5312yes—
413C2×C4D4×D5312yes—
414C2×C4D4×D5312yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—534no—
423C22—1062no—
424C22—1062no—
425C22—1062no—
426C22—1062no—
427C22—1062no—
428C22—1062no—
429C22—1062no—
430C22—1062no—
431C22—1062no—
432C22—2121no—
433C22—2121no—
434C22—2121no—
435C22—2121no—
436C22C22×C4×D5311yescharacteristic
437C22C2×D4×D5311yescharacteristic
438C22C2×D4×D5311yescharacteristic
439C22#41×D5311yescharacteristic
440C22#41×D5311yescharacteristic
441C22#1313326311yescharacteristic
442C22#1313326411yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—534no—
450C4—1062no—
451C4—1062no—
452C4—1062no—
453C4—2121no—
454C4—2121no—
455C2—42no—
456C2—1064no—
457C2—2122no—
458C2C4×D4×D5311yescharacteristic
459C2C4×D4×D5311yescharacteristic
460C2#1681110311yescharacteristic
461C2#1681125711yescharacteristic
462C2#1681127111yescharacteristic
463C2#1681128111yescharacteristic
464C2#1681132411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2361
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D212C2semidirect
3. C2×#13133177C2semidirect
4. C2×#13133177C2semidirect
5. C2×C4⋊C4×D53C2semidirect
6. #16810523C2semidirect
7. #16810520C2semidirect
8. #16810515C2semidirect
9. #16810501C2semidirect
10. #16810499C2semidirect
11. #16810498C2semidirect
12. #16810494C2semidirect
13. C4×#11719919C2semidirect
14. #16810470C2semidirect
15. C53×#388C2semidirect
24. #13133190C4semidirect
69. C2×C22⋊D53C2×C4semidirect
70. C2×C22⋊D53C2×C4semidirect
82. C2×D212C2×C4semidirect
94. C2×C4×D53C2×C4semidirect
98. C22⋊C4×C53C2×C4semidirect
105. #11719921C2×C4semidirect
106. #11719921D4semidirect
111. #11719919C2×C4semidirect
112. #11719919D4semidirect
209. C2×D53C4×D4semidirect
221. Dic53C4×D4semidirect
241. #388D53semidirect
248. C53#2361semidirect
344. C22⋊C4C4×D53semidirect