Finite Group Databasebeta

← #20278539

Subgroups — #20278539

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×#13133196C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D53C211yesmaximal, characteristic
3#16810564C211yesmaximal, characteristic, complemented
4#16810557C211yesmaximal, characteristic, complemented
5#16810547C211yesmaximal, characteristic, complemented
6#16810536C211yesmaximal, characteristic, complemented
7#16810527C211yesmaximal, characteristic, complemented
8#16810505C211yesmaximal, characteristic, complemented
9#16810502C211yesmaximal, characteristic
10#16810502C211yesmaximal, characteristic
11#16810497C211yesmaximal, characteristic, complemented
12#16810496C211yesmaximal, characteristic, complemented
13#16810496C211yesmaximal, characteristic, complemented
14#16810482C211yesmaximal, characteristic, complemented
15C2×#95×C53C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133196—24no—
20C4×C22⋊D53—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C23×C4×C53C2211yescharacteristic
47C23×C4×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C2×C22⋊D53—22no—
71C2×C22⋊D53—22no—
72C2×C22⋊D53Q814yescomplemented
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—41no—
83C2×Dic106—22no—
84C2×Dic106—22no—
85C2×Dic106—41no—
86C22⋊Dic53—22no—
87C22⋊Dic53—22no—
88C22⋊Dic53D414yes—
89#11719921—22no—
90#11719921—22no—
91#11719921—41no—
92#11719921—41no—
93#11719921—41no—
94C4⋊Dic53—22no—
95#11719919—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×C53C2×D412yes—
108C23×C53C2×Q812yes—
109C23×C53#4112yes—
110C23×C53C2411yescharacteristic
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53C2×Q812yes—
118C2×C4×C53—21no—
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—21no—
134C2×C4×C53—44no—
135C2×C4×C53—44no—
136C2×C4×C53—44no—
137C2×C4×C53#4112yes—
138C2×C4×C53#4112yes—
139C22⋊D53—28no—
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—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53C2×D412yes—
159C2×Dic53C2×D412yes—
160C2×Dic53C2×Q812yes—
161C4×D53—44no—
162Dic106—44no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53C22⋊Q814yes—
170C22×C53C22×D411yescharacteristic
171C22×C53C22×Q811yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53C2×#4111yescharacteristic
174C22×C53C2×#4111yescharacteristic
175C22×C53H(5,𝔽2)11yescharacteristic
176C22×C53H(5,𝔽2)11yescharacteristic
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190Dic53—22no—
191Dic53—22no—
192Dic53—22no—
193Dic53—41no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#2450—531nomaximal, Sylow
198C2×C53—24no—
199C2×C53C2×C22⋊Q811yescharacteristic
200C2×C53#52911yescharacteristic
201C2×C53#53811yescharacteristic
202C2×C53#54611yescharacteristic
203C2×C53#54611yescharacteristic
204C2×C53#55411yescharacteristic
205C2×C53#55411yescharacteristic
206D53—42no—
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#386—531no—
217#385—531no—
218#385—531no—
219#385—531no—
220#380—531no—
221C2×#95D5311yescharacteristic, complemented
222C53#245011yescharacteristic, complemented, Sylow
223C22×Q8—531no—
224C22×D4—531no—
225C23×C4—531no—
226C23×C4C2×D5311yescharacteristic
227C23×C4C2×D5311yescharacteristic
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⋊C4—531no—
245C2×C22⋊C4—531no—
246C2×C22⋊C4—531no—
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#95C2×D5311yescharacteristic
257#95C2×D5311yescharacteristic
258#95C2×D5311yescharacteristic
259#95C2×D5311yescharacteristic
260C24—531no—
261C24C22×D5311yescharacteristic
262C2×Q8—1062no—
263C2×Q8—1062no—
264C2×Q8—2121no—
265C2×D4—534no—
266C2×D4—1062no—
267C2×D4—1062no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—2121no—
287C22×C4C22×D5311yescharacteristic
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—534no—
302C22⋊C4—1062no—
303C22⋊C4—1062no—
304C22⋊C4—1062no—
305C22⋊C4—1062no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—1062no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—532no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23#1171994512yes—
322C23D4×D5312yes—
323C23Q8×D5312yes—
324C23C23×D5311yescharacteristic
325Q8—2124no—
326D4—1068no—
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—21no—
342C2×C4—21no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—532no—
347C2×C4—532no—
348C2×C4—532no—
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—1061no—
367C2×C4—2124no—
368C2×C4#1171994512yes—
369C2×C4#1171994512yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—1062no—
377C22—1062no—
378C22—1062no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22#1313319514yes—
384C22C2×D4×D5311yescharacteristic
385C22C2×Q8×D5311yescharacteristic
386C22C2×#1171994511yescharacteristic
387C22C2×#1171994511yescharacteristic
388C22C2×#1171994511yescharacteristic
389C22#1313326211yescharacteristic
390C22#1313326211yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—1062no—
398C4—1062no—
399C4—1062no—
400C4—2121no—
401C4—2121no—
402C4—2121no—
403C4—2121no—
404C2—24no—
405C2—2122no—
406C2C2×#1313319511yescharacteristic
407C2#1681115711yescharacteristic
408C2#1681121111yescharacteristic
409C2#1681126311yescharacteristic
410C2#1681126311yescharacteristic
411C2#1681132811yescharacteristic
412C2#1681132811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#95×C53C2
hypercenterC2×#95D53
hypocenterC53#2450
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C2×#13133196C2semidirect
3. #16810564C2semidirect
4. #16810557C2semidirect
5. #16810547C2semidirect
6. #16810536C2semidirect
7. #16810527C2semidirect
8. #16810505C2semidirect
11. #16810497C2semidirect
12. #16810496C2semidirect
13. #16810496C2semidirect
14. #16810482C2semidirect
15. C2×#95×C53C2semidirect
72. C2×C22⋊D53Q8semidirect
221. C2×#95D53semidirect
222. C53#2450semidirect