Finite Group Databasebeta

← #20277661

Subgroups — #20277661

Order 6784.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133177C211yesmaximal, characteristic
2C2×#13133174C211yesmaximal, characteristic, complemented
3C2×#13133173C211yesmaximal, characteristic, complemented
4#16810523C211yesmaximal, characteristic
5#16810523C211yesmaximal, characteristic
6#16810516C211yesmaximal, characteristic, complemented
7#16810515C211yesmaximal, characteristic, complemented
8#16810513C211yesmaximal, characteristic, complemented
9#16810492C211yesmaximal, characteristic, complemented
10#16810482C211yesmaximal, characteristic, complemented
11#16810481C211yesmaximal, characteristic, complemented
12C4×#11719919C211yesmaximal, characteristic, complemented
13#16810474C211yesmaximal, characteristic, complemented
14#16810474C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133177C418yescomplemented
20#13133174—24no—
21#13133173—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41C2×C4×Dic53C2211yescharacteristic
42C2×C4×Dic53C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C42×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54C24×C53C2311yescharacteristic
55C23×D53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—22no—
63C2×C22⋊D53—22no—
64C2×C22⋊D53C2×C414yes—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53C2×C414yes—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53C2×C414yes—
78C4⋊C4×C53—41no—
79C22⋊C4×C53—22no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53C2×C414yes—
82C42×C53—22no—
83C22⋊Dic53—22no—
84C22⋊Dic53—41no—
85#11719921—22no—
86#11719921—22no—
87#11719921—41no—
88#11719921—41no—
89#11719921C2×C414yes—
90C4⋊Dic53—41no—
91#11719919—22no—
92#11719919—22no—
93#11719919—22no—
94#11719919—22no—
95#11719919—22no—
96#11719919C2×C414yes—
97#11719919D414yescomplemented
98C4×Dic53—22no—
99C4×Dic53—22no—
100C4×Dic53—22no—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C4×Dic53C2×C414yes—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53C22×C412yes—
111C23×C53C2411yescharacteristic
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53C22×C412yes—
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×C53C22×C412yes—
135C2×C4×C53C22×C412yes—
136C2×C4×C53C2×D412yes—
137C22⋊D53—28no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—28no—
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×Dic53C22×C412yes—
153C2×Dic53C22×C412yes—
154C2×Dic53C22×C412yes—
155C2×Dic53C2×D412yes—
156C2×Dic53C2×D412yes—
157C2×Dic53#4112yes—
158C2×Dic53#4112yes—
159C2×Dic53#4112yes—
160C4×D53—28no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53C23×C411yescharacteristic
169C22×C53C22×D411yescharacteristic
170C22×C53C2×#4111yescharacteristic
171C22×C53C2×#4111yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53H(5,𝔽2)11yescharacteristic
174C22×C532−1+411yescharacteristic
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C4×C53—22no—
183C4×C53—22no—
184C4×C53—22no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188Dic53—22no—
189Dic53—22no—
190Dic53—41no—
191Dic53—41no—
192Dic53—41no—
193Dic53#11714yescomplemented
194Dic53C4×D414yes—
195#2354—531nomaximal, Sylow
196C2×C53—42no—
197C2×C53C2×#11711yescharacteristic
198C2×C53C2×C4×D411yescharacteristic
199C2×C53#52011yescharacteristic
200C2×C53#54711yescharacteristic
201C2×C53#55011yescharacteristic
202C2×C53#55511yescharacteristic
203C2×C53#55611yescharacteristic
204D53—42no—
205C2×C4×D4—531no—
206C2×#117—531no—
207C2×#117—531no—
208#390—531no—
209#390—531no—
210#388—531no—
211#388—531no—
212#388—531no—
213#388D5311yescharacteristic, complemented
214#384—531no—
215#380—531no—
216#380—531no—
217C4×C4⋊C4—531no—
218C4×C4⋊C4—531no—
219C4×C4⋊C4—531no—
220C53#235411yescharacteristic, complemented, Sylow
221C22×D4—531no—
222C23×C4—531no—
223C23×C4—531no—
224C4×D4—538no—
225#117—1064no—
226#117—1064no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4C2×D5311yescharacteristic
236C2×C22⋊C4—531no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4C2×D5311yescharacteristic
244C2×C22⋊C4C2×D5311yescharacteristic
245C2×C22⋊C4C2×D5311yescharacteristic
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42C2×D5311yescharacteristic
253#95—531no—
254#95—531no—
255#95—531no—
256#95—531no—
257#95C2×D5311yescharacteristic
258#95C2×D5311yescharacteristic
259C24—531no—
260C24C22×D5311yescharacteristic
261C2×D4—534no—
262C2×D4—1062no—
263C2×D4—1062no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—534no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—534no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—1062no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4C22×D5311yescharacteristic
278C22×C4C22×D5311yescharacteristic
279C22×C4C22×D5311yescharacteristic
280C22×C4C22×D5311yescharacteristic
281C22×C4C22×D5311yescharacteristic
282C22×C4C22×D5311yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—534no—
285C4⋊C4—534no—
286C4⋊C4—1062no—
287C4⋊C4—1062no—
288C4⋊C4—1062no—
289C4⋊C4—1062no—
290C4⋊C4—1062no—
291C4⋊C4—2121no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—534no—
295C22⋊C4—1062no—
296C22⋊C4—1062no—
297C22⋊C4—1062no—
298C22⋊C4—2121no—
299C22⋊C4—2121no—
300C22⋊C4—2121no—
301C22⋊C4C4×D5314yes—
302C42—22no—
303C42—534no—
304C42—1062no—
305C42—1062no—
306C42—1062no—
307C42—1062no—
308C42—1062no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—532no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23C2×C4×D5312yes—
323C23C23×D5311yescharacteristic
324D4—1068no—
325C2×C4—21no—
326C2×C4—21no—
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—532no—
341C2×C4—532no—
342C2×C4—532no—
343C2×C4—532no—
344C2×C4—532no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—532no—
348C2×C4—1068no—
349C2×C4—1068no—
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×C4C2×C4×D5312yes—
364C2×C4C2×C4×D5312yes—
365C2×C4D4×D5312yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—1062no—
374C22—1062no—
375C22—1062no—
376C22—2121no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22C22×C4×D5311yescharacteristic
381C22C2×D4×D5311yescharacteristic
382C22#41×D5311yescharacteristic
383C22#41×D5311yescharacteristic
384C22#41×D5311yescharacteristic
385C22#1313326211yescharacteristic
386C22#1313326411yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—534no—
394C4—534no—
395C4—1062no—
396C4—1062no—
397C4—2121no—
398C4—2121no—
399C4—2121no—
400C2—42no—
401C2—2122no—
402C2#117×D5311yescharacteristic
403C2C4×D4×D5311yescharacteristic
404C2#1681110211yescharacteristic
405C2#1681127111yescharacteristic
406C2#1681129111yescharacteristic
407C2#1681133511yescharacteristic
408C2#1681134011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2354
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#13133174C2semidirect
3. C2×#13133173C2semidirect
6. #16810516C2semidirect
7. #16810515C2semidirect
8. #16810513C2semidirect
9. #16810492C2semidirect
10. #16810482C2semidirect
11. #16810481C2semidirect
12. C4×#11719919C2semidirect
13. #16810474C2semidirect
14. #16810474C2semidirect
15. C53×#388C2semidirect
19. #13133177C4semidirect
97. #11719919D4semidirect
193. Dic53#117semidirect
213. #388D53semidirect
220. C53#2354semidirect