Finite Group Databasebeta

← #20278545

Subgroups — #20278545

Order 6784.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133194C211yesmaximal, characteristic, complemented
2C2×D4×Dic53C211yesmaximal, characteristic
3#16810565C211yesmaximal, characteristic, complemented
4#16810557C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810527C211yesmaximal, characteristic, complemented
8#16810510C211yesmaximal, characteristic, complemented
9#16810503C211yesmaximal, characteristic
10#16810496C211yesmaximal, characteristic, complemented
11#16810495C211yesmaximal, characteristic, complemented
12#16810495C211yesmaximal, characteristic, complemented
13#16810481C211yesmaximal, characteristic, complemented
14C2×#13133124C211yesmaximal, characteristic, complemented
15C53×#386C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20#13133194—24no—
21D4×Dic53—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C23×C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53Q814yescomplemented
58C22×C4×C53—22no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C2×Dic106—22no—
84C2×Dic106—22no—
85C2×Dic106—41no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊Dic53—22no—
90C22⋊Dic53—22no—
91C22⋊Dic53—22no—
92C22⋊Dic53—41no—
93C22⋊Dic53D414yes—
94C4⋊Dic53—22no—
95C4⋊Dic53—22no—
96#11719919—22no—
97#11719919—22no—
98#11719919—22no—
99#11719919—41no—
100#11719919—41no—
101#11719919—41no—
102C4×Dic53—22no—
103C23×C53—21no—
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×C53C2×D412yes—
114C23×C53C2×Q812yes—
115C23×C53C2×Q812yes—
116C23×C53#4112yes—
117C23×C53C2411yescharacteristic
118D4×C53—28no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
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—44no—
134C2×C4×C53C2×Q812yes—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
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—44no—
156C2×Dic53—44no—
157C2×Dic53—44no—
158C2×Dic53C2×D412yes—
159C2×Dic53C2×D412yes—
160C2×Dic53#4112yes—
161C2×Dic53#4112yes—
162Dic106—44no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53C22⋊Q814yes—
177C22×C53C22×D411yescharacteristic
178C22×C53C22×Q811yescharacteristic
179C22×C53C2×#4111yescharacteristic
180C22×C53C2×#4111yescharacteristic
181C22×C53C2×#4111yescharacteristic
182C22×C53H(5,𝔽2)11yescharacteristic
183C22×C53H(5,𝔽2)11yescharacteristic
184C4×C53—22no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189Dic53—22no—
190Dic53—22no—
191Dic53—22no—
192Dic53—22no—
193Dic53—41no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#2450—531nomaximal, Sylow
198C2×C53—24no—
199C2×C53—42no—
200C2×C53C2×C22⋊Q811yescharacteristic
201C2×C53#52911yescharacteristic
202C2×C53#53811yescharacteristic
203C2×C53#54611yescharacteristic
204C2×C53#54611yescharacteristic
205C2×C53#55411yescharacteristic
206C2×C53#55411yescharacteristic
207C2×C22⋊Q8—531no—
208C2×C4×D4—531no—
209#400—531no—
210#398—531no—
211#396—531no—
212#393—531no—
213#391—531no—
214#387—531no—
215#386—531no—
216#386D5311yescharacteristic
217#385—531no—
218#385—531no—
219#385—531no—
220#380—531no—
221C2×#95—531no—
222C53#245011yescharacteristic, complemented, Sylow
223C22×Q8—531no—
224C22×D4C2×D5311yescharacteristic
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—531no—
228C23×C4C2×D5311yescharacteristic
229C22⋊Q8—1064no—
230C4×D4—1064no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—531no—
244C2×C22⋊C4C2×D5311yescharacteristic
245C2×C22⋊C4C2×D5311yescharacteristic
246C2×C22⋊C4C2×D5311yescharacteristic
247C2×C42—531no—
248#95—531no—
249#95—531no—
250#95—531no—
251#95—531no—
252#95—531no—
253#95—531no—
254#95—531no—
255#95—531no—
256#95—531no—
257#95—531no—
258#95C2×D5311yescharacteristic
259#95C2×D5311yescharacteristic
260C24C22×D5311yescharacteristic
261C24C22×D5311yescharacteristic
262C2×Q8—1062no—
263C2×Q8—1062no—
264C2×Q8—2121no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4Dic10614yescomplemented
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—1062no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4C22×D5311yescharacteristic
289C22×C4C22×D5311yescharacteristic
290C22×C4C22×D5311yescharacteristic
291C22×C4C22×D5311yescharacteristic
292C22×C4C22×D5311yescharacteristic
293C4⋊C4—1062no—
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—1062no—
297C4⋊C4—1062no—
298C4⋊C4—2121no—
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—534no—
305C22⋊C4—1062no—
306C22⋊C4—1062no—
307C22⋊C4—1062no—
308C22⋊C4—2121no—
309C42—1062no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×Dic10612yes—
321C23C2×Dic10612yes—
322C23D4×D5312yes—
323C23#1171994712yes—
324C23C23×D5311yescharacteristic
325Q8—2124no—
326D4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—532no—
343C2×C4—532no—
344C2×C4—532no—
345C2×C4—532no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—2124no—
367C2×C4—2124no—
368C2×C4—2124no—
369C2×C4C2×Dic10612yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22#1313319414yes—
384C22C2×D4×D5311yescharacteristic
385C22C22×Dic10611yescharacteristic
386C22#41×D5311yescharacteristic
387C22#41×D5311yescharacteristic
388C22C2×#1171994711yescharacteristic
389C22#1313326211yescharacteristic
390C22#1313326211yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—1062no—
397C4—1062no—
398C4—1062no—
399C4—1062no—
400C4—2121no—
401C4—2121no—
402C4—2121no—
403C4—2121no—
404C2—24no—
405C2—42no—
406C2C2×#1313319411yescharacteristic
407C2#1681115011yescharacteristic
408C2#1681120811yescharacteristic
409C2#1681126211yescharacteristic
410C2#1681126211yescharacteristic
411C2#1681133211yescharacteristic
412C2#1681133211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2450
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#13133194C2semidirect
3. #16810565C2semidirect
4. #16810557C2semidirect
5. #16810546C2semidirect
6. #16810535C2semidirect
7. #16810527C2semidirect
8. #16810510C2semidirect
10. #16810496C2semidirect
11. #16810495C2semidirect
12. #16810495C2semidirect
13. #16810481C2semidirect
14. C2×#13133124C2semidirect
57. C2×D4×C53Q8semidirect
222. C53#2450semidirect
267. C2×D4Dic106semidirect