Finite Group Databasebeta

← #20280991

Subgroups — #20280991

Order 6784.

562 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×#13133206C211yesmaximal, characteristic
2C2×#13133202C211yesmaximal, characteristic
3C2×#13133193C211yesmaximal, characteristic, complemented
4#16810558C211yesmaximal, characteristic, complemented
5#16810549C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810542C211yesmaximal, characteristic, complemented
8#16810540C211yesmaximal, characteristic
9#16810534C211yesmaximal, characteristic, complemented
10#16810518C211yesmaximal, characteristic, complemented
11#16810514C211yesmaximal, characteristic, complemented
12#16810514C211yesmaximal, characteristic, complemented
13#16810509C211yesmaximal, characteristic, complemented
14#16810494C211yesmaximal, characteristic, complemented
15C2×C4⋊Q8×C53C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18#13133206—24no—
19#13133202—24no—
20#13133193—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×Q8×C53C2211yescharacteristic
48C22×Q8×C53C2211yescharacteristic
49C4⋊Q8×C53—24no—
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic
52C2×C4⋊C4×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54C2×C42×C53C2211yescharacteristic
55C23×D53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53—41no—
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C2×D212—22no—
77C2×D212—22no—
78C2×D212—41no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—41no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53D414yes—
86C42×C53—22no—
87#11719921—22no—
88#11719921—22no—
89#11719921—22no—
90#11719921—22no—
91#11719921—22no—
92#11719921—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—41no—
97#11719921D414yes—
98C4⋊Dic53—22no—
99#11719919—22no—
100#11719919—22no—
101#11719919—22no—
102#11719919—22no—
103#11719919—41no—
104C4×Dic53—22no—
105C23×C53C2411yescharacteristic
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53C2×D412yes—
120Q8×C53—44no—
121Q8×C53—44no—
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—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53C2×D412yes—
139C2×C4×C53C2×D412yes—
140C2×C4×C53C2×D412yes—
141C2×C4×C53C2×D412yes—
142C2×C4×C53#4112yes—
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×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53C2×D412yes—
161D212—44no—
162C4×D53—44no—
163C22×C53C22×D411yescharacteristic
164C22×C53C22×D411yescharacteristic
165C22×C53C2×#4111yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C532−1+411yescharacteristic
168C22×C532−1+411yescharacteristic
169C22×C532−1+411yescharacteristic
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—41no—
190C4×C53—41no—
191Dic53—22no—
192Dic53—41no—
193Dic53—41no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#2674—531nomaximal, Sylow
198C2×C53#53611yescharacteristic
199C2×C53#53611yescharacteristic
200C2×C53#54211yescharacteristic
201C2×C53D4211yescharacteristic
202C2×C53#54711yescharacteristic
203C2×C53#54811yescharacteristic
204C2×C53#56311yescharacteristic
205D53—42no—
206D53—81no—
207C2×C4⋊Q8D5311yescharacteristic, complemented
208C2×#124—531no—
209C2×#123—531no—
210C2×C22⋊Q8—531no—
211#399—531no—
212#397—531no—
213#396—531no—
214#395—531no—
215#394—531no—
216#393—531no—
217#389—531no—
218#388—531no—
219#388—531no—
220#387—531no—
221#385—531no—
222C53#267411yescharacteristic, complemented, Sylow
223C22×Q8C2×D5311yescharacteristic
224C22×Q8C2×D5311yescharacteristic
225C22×D4—531no—
226C23×C4—531no—
227C4⋊Q8—24no—
228#124—1064no—
229#123—1064no—
230C22⋊Q8—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⋊C4C2×D5311yescharacteristic
238C2×C4⋊C4C2×D5311yescharacteristic
239C2×C4⋊C4C2×D5311yescharacteristic
240C2×C4⋊C4C2×D5311yescharacteristic
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×C22⋊C4—531no—
248C2×C22⋊C4—531no—
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4—531no—
252C2×C42—531no—
253C2×C42C2×D5311yescharacteristic
254#95—531no—
255#95—531no—
256#95—531no—
257#95—531no—
258#95—531no—
259#95—531no—
260#95—531no—
261#95—531no—
262C24—531no—
263C24—531no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×D4—1062no—
271C2×D4—1062no—
272C2×D4—2121no—
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—2121no—
282C22×C4C22×D5311yescharacteristic
283C22×C4C22×D5311yescharacteristic
284C22×C4C22×D5311yescharacteristic
285C22×C4C22×D5311yescharacteristic
286C22×C4C22×D5311yescharacteristic
287C22×C4C22×D5311yescharacteristic
288C22×C4C22×D5311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—1062no—
297C4⋊C4—2121no—
298C4⋊C4C22⋊D5314yes—
299C22⋊C4—534no—
300C22⋊C4—1062no—
301C22⋊C4—1062no—
302C22⋊C4—1062no—
303C22⋊C4—1062no—
304C22⋊C4—1062no—
305C22⋊C4—1062no—
306C22⋊C4—1062no—
307C22⋊C4—1062no—
308C22⋊C4—1062no—
309C22⋊C4—2121no—
310C42—22no—
311C42—1062no—
312C23—532no—
313C23—1061no—
314C23—1061no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23—1061no—
325C23—1061no—
326C23C23×D5311yescharacteristic
327Q8—44no—
328Q8—44no—
329D4—2124no—
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—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—532no—
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—2124no—
365C2×C4D4×D5312yes—
366C2×C4#1171994912yes—
367C2×C4C2×C22⋊D5312yes—
368C2×C4C2×C22⋊D5312yes—
369C2×C4C2×C22⋊D5312yes—
370C22—1062no—
371C22—1062no—
372C22—1062no—
373C22—2121no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22C2×D4×D5311yescharacteristic
385C22C22×C22⋊D5311yescharacteristic
386C22C2×#1171994911yescharacteristic
387C22#1313326211yescharacteristic
388C22#1313326511yescharacteristic
389C22#1313326511yescharacteristic
390C22#1313326511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—41no—
397C4—41no—
398C4—1062no—
399C4—2121no—
400C4—2121no—
401C4—2121no—
402C4—2121no—
403C4—2121no—
404C2—4241no—
405C2—2122no—
406C2#1681119711yescharacteristic
407C2#1681119811yescharacteristic
408C2#1681124711yescharacteristic
409C2D4×C22⋊D5311yescharacteristic
410C2#1681127611yescharacteristic
411C2#1681128411yescharacteristic
412C2#1681137211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4⋊Q8×C53C2
hypercenterC2×C4⋊Q8D53
hypocenterC53#2674
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#13133193C2semidirect
4. #16810558C2semidirect
5. #16810549C2semidirect
6. #16810545C2semidirect
7. #16810542C2semidirect
9. #16810534C2semidirect
10. #16810518C2semidirect
11. #16810514C2semidirect
12. #16810514C2semidirect
13. #16810509C2semidirect
14. #16810494C2semidirect
15. C2×C4⋊Q8×C53C2semidirect
207. C2×C4⋊Q8D53semidirect
222. C53#2674semidirect