Finite Group Databasebeta

← #20280474

Subgroups — #20280474

Order 6784.

612 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133203C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×#13133190C211yesmaximal, characteristic
4C2×#13133175C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810541C211yesmaximal, characteristic, complemented
9#16810540C211yesmaximal, characteristic, complemented
10#16810539C211yesmaximal, characteristic, complemented
11#16810538C211yesmaximal, characteristic, complemented
12#16810527C211yesmaximal, characteristic, complemented
13#16810523C211yesmaximal, characteristic, complemented
14#16810492C211yesmaximal, characteristic, complemented
15C53×#396C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133203—24no—
22#13133195—24no—
23#13133190—24no—
24#13133175—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
44C2×C4×Dic53C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—22no—
65C2×C22⋊D53—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—41no—
69C2×C22⋊D53—41no—
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C2×D212—22no—
77C2×D212—41no—
78C2×D212—41no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53D414yes—
82C2×Dic106—22no—
83C2×Dic106—22no—
84C2×Dic106—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—41no—
91C22⋊Dic53—22no—
92C22⋊Dic53—41no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—22no—
100#11719921—41no—
101#11719921—41no—
102C4⋊Dic53—22no—
103C4⋊Dic53—41no—
104#11719919—22no—
105#11719919—22no—
106#11719919—22no—
107#11719919—41no—
108#11719919—41no—
109C4×Dic53—22no—
110C4×Dic53D414yescomplemented
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
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×C53C2×D412yes—
150C2×C4×C53C2×D412yes—
151C22⋊D53—44no—
152C22⋊D53—44no—
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—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53C2×D412yes—
168C2×Dic53C2×D412yes—
169C2×Dic53C2×D412yes—
170C2×Dic53#4112yes—
171D212—44no—
172C4×D53—28no—
173Dic106—44no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53C22×D411yescharacteristic
182C22×C53C22×D411yescharacteristic
183C22×C53C2×#4111yescharacteristic
184C22×C53H(5,𝔽2)11yescharacteristic
185C22×C53H(5,𝔽2)11yescharacteristic
186C22×C532−1+411yescharacteristic
187C22×C532−1+411yescharacteristic
188C2×D53—22no—
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—41no—
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—
202C4×C53—22no—
203C4×C53—22no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—41no—
208Dic53—22no—
209Dic53—41no—
210Dic53—41no—
211Dic53—41no—
212Dic53—41no—
213Dic53#12114yes—
214#2620—531nomaximal, Sylow
215C2×C53—81no—
216C2×C53C2×#12111yescharacteristic
217C2×C53#53511yescharacteristic
218C2×C53#53611yescharacteristic
219C2×C53#53711yescharacteristic
220C2×C53#53711yescharacteristic
221C2×C53#54211yescharacteristic
222C2×C53#54211yescharacteristic
223D53—42no—
224D53—81no—
225C2×#124—531no—
226C2×C22⋊Q8—531no—
227C2×#121—531no—
228C2×#117—531no—
229#396—531no—
230#396—531no—
231#396—531no—
232#396D5311yescharacteristic, complemented
233#394—531no—
234#394—531no—
235#394—531no—
236#394—531no—
237#391—531no—
238#390—531no—
239#384—531no—
240C53#262011yescharacteristic, complemented, Sylow
241C22×Q8—531no—
242C22×D4—531no—
243C22×D4—531no—
244C22×D4—531no—
245C23×C4—531no—
246#124—1064no—
247C22⋊Q8—1064no—
248#121—1064no—
249#117—1064no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—531no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4C2×D5311yescharacteristic
258C2×C4⋊C4C2×D5311yescharacteristic
259C2×C4⋊C4C2×D5311yescharacteristic
260C2×C22⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
272C2×C22⋊C4C2×D5311yescharacteristic
273C2×C22⋊C4C2×D5311yescharacteristic
274C2×C42—531no—
275C2×C42—531no—
276#95—531no—
277#95—531no—
278#95—531no—
279#95C2×D5311yescharacteristic
280C24—531no—
281C24—531no—
282C24C22×D5311yescharacteristic
283C2×Q8—1062no—
284C2×Q8—1062no—
285C2×Q8—1062no—
286C2×D4—1062no—
287C2×D4—1062no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—534no—
298C22×C4—531no—
299C22×C4—531no—
300C22×C4—531no—
301C22×C4—531no—
302C22×C4—1062no—
303C22×C4—1062no—
304C22×C4C22×D5311yescharacteristic
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307C22×C4C22×D5311yescharacteristic
308C22×C4C22×D5311yescharacteristic
309C22×C4C22×D5311yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—1062no—
314C4⋊C4—1062no—
315C4⋊C4—1062no—
316C4⋊C4—1062no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—41no—
323C22⋊C4—1062no—
324C22⋊C4—1062no—
325C22⋊C4—1062no—
326C22⋊C4—1062no—
327C22⋊C4—1062no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C22⋊C4—1062no—
331C22⋊C4—2121no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C42—534no—
335C42—1062no—
336C23—21no—
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—
357C23C23×D5311yescharacteristic
358Q8—2124no—
359D4—2124no—
360D4—2124no—
361D4—2124no—
362C2×C4—21no—
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—532no—
379C2×C4—532no—
380C2×C4—532no—
381C2×C4—532no—
382C2×C4—1068no—
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×C4D4×D5312yes—
398C2×C4D4×D5312yes—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—1062no—
407C22—1062no—
408C22—1062no—
409C22—2121no—
410C22—2121no—
411C22—2121no—
412C22—2121no—
413C22—2121no—
414C22—2121no—
415C22—2121no—
416C22—2121no—
417C22—2121no—
418C22—2121no—
419C22—2121no—
420C22C2×D4×D5311yescharacteristic
421C22C2×D4×D5311yescharacteristic
422C22#41×D5311yescharacteristic
423C22#1313326211yescharacteristic
424C22#1313326311yescharacteristic
425C22#1313326411yescharacteristic
426C22#1313326511yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—534no—
434C4—1062no—
435C4—2121no—
436C4—2121no—
437C4—2121no—
438C4—2121no—
439C2—81no—
440C2—4241no—
441C2—2122no—
442C2#121×D5311yescharacteristic
443C2#1681118911yescharacteristic
444C2#1681119611yescharacteristic
445C2#1681120011yescharacteristic
446C2#1681120311yescharacteristic
447C2#1681123911yescharacteristic
448C2#1681124311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2620
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133195C2semidirect
4. C2×#13133175C2semidirect
5. #16810546C2semidirect
6. #16810545C2semidirect
7. #16810545C2semidirect
8. #16810541C2semidirect
9. #16810540C2semidirect
10. #16810539C2semidirect
11. #16810538C2semidirect
12. #16810527C2semidirect
13. #16810523C2semidirect
14. #16810492C2semidirect
15. C53×#396C2semidirect
110. C4×Dic53D4semidirect
232. #396D53semidirect
240. C53#2620semidirect