Finite Group Databasebeta

← #20280967

Subgroups — #20280967

Order 6784.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133203C211yesmaximal, characteristic, complemented
2C2×#13133199C211yesmaximal, characteristic, complemented
3C2×#13133195C211yesmaximal, characteristic, complemented
4C2×#13133191C211yesmaximal, characteristic
5#16810558C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810544C211yesmaximal, characteristic, complemented
8#16810541C211yesmaximal, characteristic, complemented
9#16810538C211yesmaximal, characteristic, complemented
10#16810538C211yesmaximal, characteristic, complemented
11#16810520C211yesmaximal, characteristic, complemented
12#16810513C211yesmaximal, characteristic, complemented
13#16810501C211yesmaximal, characteristic, complemented
14#16810496C211yesmaximal, characteristic, complemented
15C53×#380C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133203—24no—
22#13133199—24no—
23#13133195—24no—
24#13133191—24no—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—22no—
68C2×C22⋊D53—22no—
69C2×C22⋊D53—22no—
70C2×C22⋊D53—22no—
71C2×C22⋊D53—41no—
72C2×C22⋊D53—41no—
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C2×D212—22no—
80C2×D212—22no—
81C2×D212—41no—
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—41no—
85C2×Dic106—22no—
86C2×Dic106—22no—
87C2×Dic106—41no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—22no—
92C22⋊Dic53—41no—
93C22⋊Dic53D414yescomplemented
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—41no—
100#11719921—41no—
101#11719921—41no—
102#11719921D414yes—
103C4⋊Dic53—22no—
104C4⋊Dic53—22no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—41no—
110C4×Dic53—22no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53C2×D412yes—
118C23×C53C2411yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
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×D53C2×D412yes—
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×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—44no—
150C2×C4×C53C2×D412yes—
151C2×C4×C53#4112yes—
152C22⋊D53—44no—
153C22⋊D53—44no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
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×Dic53C2×D412yes—
170C2×Dic53C2×D412yes—
171C2×Dic53C2×D412yes—
172D212—44no—
173C4×D53—44no—
174Dic106—44no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53C22×D411yescharacteristic
183C22×C53C22×D411yescharacteristic
184C22×C53C2×#4111yescharacteristic
185C22×C53H(5,𝔽2)11yescharacteristic
186C22×C53H(5,𝔽2)11yescharacteristic
187C22×C532−1+411yescharacteristic
188C22×C532−1+411yescharacteristic
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C2×D53—41no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—41no—
203C4×C53—22no—
204C4×C53—22no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—41no—
208C4×C53—41no—
209Dic53—22no—
210Dic53—22no—
211Dic53—22no—
212Dic53—41no—
213Dic53—41no—
214Dic53—41no—
215#2672—531nomaximal, Sylow
216C2×C53—42no—
217C2×C53#53611yescharacteristic
218C2×C53#53711yescharacteristic
219C2×C53#54211yescharacteristic
220C2×C53D4211yescharacteristic
221C2×C53#54611yescharacteristic
222C2×C53#54711yescharacteristic
223C2×C53#56211yescharacteristic
224D53—42no—
225D53—81no—
226C2×#124—531no—
227C2×#123—531no—
228C2×C22⋊Q8—531no—
229C2×#121—531no—
230#399—531no—
231#396—531no—
232#395—531no—
233#394—531no—
234#394—531no—
235#394—531no—
236#389—531no—
237#388—531no—
238#386—531no—
239#385—531no—
240#380D5311yescharacteristic, complemented
241C53#267211yescharacteristic, complemented, Sylow
242C22×Q8—531no—
243C22×D4—531no—
244C22×D4—531no—
245C22×D4—531no—
246C23×C4—531no—
247C23×C4C2×D5311yescharacteristic
248#124—1064no—
249#123—1064no—
250C22⋊Q8—1064no—
251#121—1064no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4C2×D5311yescharacteristic
260C2×C4⋊C4C2×D5311yescharacteristic
261C2×C22⋊C4—531no—
262C2×C22⋊C4—531no—
263C2×C22⋊C4—531no—
264C2×C22⋊C4—531no—
265C2×C22⋊C4—531no—
266C2×C22⋊C4—531no—
267C2×C22⋊C4—531no—
268C2×C22⋊C4—531no—
269C2×C22⋊C4—531no—
270C2×C22⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4C2×D5311yescharacteristic
273C2×C22⋊C4C2×D5311yescharacteristic
274C2×C42—531no—
275#95—531no—
276#95—531no—
277#95—531no—
278#95—531no—
279#95C2×D5311yescharacteristic
280#95C2×D5311yescharacteristic
281C24—531no—
282C24—531no—
283C24C22×D5311yescharacteristic
284C2×Q8—1062no—
285C2×Q8—1062no—
286C2×Q8—2121no—
287C2×D4—1062no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—531no—
300C22×C4—531no—
301C22×C4—531no—
302C22×C4—531no—
303C22×C4—531no—
304C22×C4—531no—
305C22×C4—1062no—
306C22×C4—1062no—
307C22×C4—2121no—
308C22×C4C22×D5311yescharacteristic
309C22×C4C22×D5311yescharacteristic
310C22×C4C22×D5311yescharacteristic
311C22×C4C22×D5311yescharacteristic
312C22×C4C22×D5311yescharacteristic
313C22×C4C22×D5311yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C4⋊C4—1062no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1062no—
322C4⋊C4—2121no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—534no—
326C22⋊C4—534no—
327C22⋊C4—1062no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C22⋊C4—1062no—
331C22⋊C4—1062no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C22⋊C4—2121no—
336C42—1062no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—532no—
344C23—1061no—
345C23—1061no—
346C23—1061no—
347C23—1061no—
348C23—1061no—
349C23—1061no—
350C23—1061no—
351C23—1061no—
352C23—1061no—
353C23—1061no—
354C23—1061no—
355C23—1061no—
356C23—1061no—
357C23D4×D5312yes—
358C23C23×D5311yescharacteristic
359Q8—2124no—
360D4—2124no—
361D4—2124no—
362D4—2124no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—44no—
380C2×C4—532no—
381C2×C4—532no—
382C2×C4—532no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—2124no—
399C2×C4#1171994512yes—
400C2×C4D4×D5312yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—1062no—
409C22—1062no—
410C22—1062no—
411C22—2121no—
412C22—2121no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22—2121no—
417C22—2121no—
418C22—2121no—
419C22—2121no—
420C22—2121no—
421C22—2121no—
422C22C2×D4×D5311yescharacteristic
423C22C2×D4×D5311yescharacteristic
424C22C2×#1171994511yescharacteristic
425C22#1313326211yescharacteristic
426C22#1313326311yescharacteristic
427C22#1313326411yescharacteristic
428C22#1313326511yescharacteristic
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—1062no—
436C4—1062no—
437C4—1062no—
438C4—2121no—
439C4—2121no—
440C4—2121no—
441C2—42no—
442C2—4241no—
443C2—2122no—
444C2#1681119611yescharacteristic
445C2#1681120011yescharacteristic
446C2#1681124411yescharacteristic
447C2#1681125711yescharacteristic
448C2#1681126311yescharacteristic
449C2#1681127411yescharacteristic
450C2#1681136911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#380C2
hypercenter#380D53
hypocenterC53#2672
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133199C2semidirect
3. C2×#13133195C2semidirect
5. #16810558C2semidirect
6. #16810545C2semidirect
7. #16810544C2semidirect
8. #16810541C2semidirect
9. #16810538C2semidirect
10. #16810538C2semidirect
11. #16810520C2semidirect
12. #16810513C2semidirect
13. #16810501C2semidirect
14. #16810496C2semidirect
15. C53×#380C2semidirect
93. C22⋊Dic53D4semidirect
240. #380D53semidirect
241. C53#2672semidirect