Finite Group Databasebeta

← #20281729

Subgroups — #20281729

Order 6784.

528 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×#13133195C211yesmaximal, characteristic, complemented
2C2×#13133194C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810558C211yesmaximal, characteristic, complemented
5#16810557C211yesmaximal, characteristic, complemented
6#16810556C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810528C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic
10#16810518C211yesmaximal, characteristic, complemented
11#16810500C211yesmaximal, characteristic
12#16810499C211yesmaximal, characteristic
13#16810497C211yesmaximal, characteristic, complemented
14#16810495C211yesmaximal, characteristic, complemented
15C53×#396C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133195—24no—
21#13133194—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
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
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C23×D53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C2×C22⋊D53—41no—
62C2×C22⋊D53—41no—
63C2×C22⋊D53Q814yescomplemented
64C22×Dic53—22no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C2×C4×D53—22no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106D414yescomplemented
80C4⋊C4×C53—22no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊Dic53—22no—
87C22⋊Dic53—22no—
88#11719921—22no—
89#11719921—22no—
90#11719921—22no—
91#11719921—41no—
92#11719921—41no—
93C4⋊Dic53—22no—
94C4⋊Dic53—41no—
95#11719919—22no—
96#11719919—22no—
97#11719919—22no—
98#11719919—22no—
99#11719919—22no—
100#11719919—41no—
101#11719919—41no—
102C4×Dic53—41no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53C2×Q812yes—
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—21no—
135C2×C4×C53C2×D412yes—
136C22⋊D53—44no—
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—44no—
155C2×Dic53C2×D412yes—
156C2×Dic53C2×D412yes—
157C2×Dic53C2×Q812yes—
158C2×Dic53#4112yes—
159C4×D53—44no—
160Dic106—44no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53C22×D411yescharacteristic
169C22×C53C22×Q811yescharacteristic
170C22×C53C2×#4111yescharacteristic
171C22×C53H(5,𝔽2)11yescharacteristic
172C22×C53H(5,𝔽2)11yescharacteristic
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—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188Dic53—22no—
189Dic53—22no—
190Dic53—22no—
191Dic53—22no—
192Dic53—41no—
193Dic53—41no—
194Dic53—41no—
195#2732—531nomaximal, Sylow
196C2×C53—42no—
197C2×C53#53411yescharacteristic
198C2×C53#53911yescharacteristic
199C2×C53#54311yescharacteristic
200C2×C53D4×Q811yescharacteristic
201C2×C53#55011yescharacteristic
202C2×C53#55411yescharacteristic
203C2×C53#56211yescharacteristic
204D53—42no—
205C2×C22⋊Q8—531no—
206C2×C22⋊Q8—531no—
207#399—531no—
208#399—531no—
209#398—531no—
210#398—531no—
211#396—531no—
212#396D5311yescharacteristic, complemented
213#391—531no—
214#390—531no—
215#389—531no—
216#386—531no—
217#386—531no—
218#385—531no—
219#385—531no—
220C53#273211yescharacteristic, complemented, Sylow
221C22×Q8—531no—
222C22×D4—531no—
223C23×C4—531no—
224C23×C4—531no—
225C22⋊Q8—1064no—
226C22⋊Q8—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⋊C4—531no—
236C2×C4⋊C4C2×D5311yescharacteristic
237C2×C4⋊C4C2×D5311yescharacteristic
238C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4C2×D5311yescharacteristic
247C2×C22⋊C4C2×D5311yescharacteristic
248C2×C22⋊C4C2×D5311yescharacteristic
249C2×C42—531no—
250#95—531no—
251#95—531no—
252#95—531no—
253#95—531no—
254#95—531no—
255#95—531no—
256#95—531no—
257#95C2×D5311yescharacteristic
258C24—531no—
259C24C22×D5311yescharacteristic
260C2×Q8—534no—
261C2×Q8—2121no—
262C2×Q8—2121no—
263C2×D4—534no—
264C2×D4—2121no—
265C2×D4—2121no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—1062no—
274C22×C4—1062no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4C22×D5311yescharacteristic
280C22×C4C22×D5311yescharacteristic
281C22×C4C22×D5311yescharacteristic
282C22×C4C22×D5311yescharacteristic
283C22×C4C22×D5311yescharacteristic
284C22×C4C22×D5311yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—1062no—
289C4⋊C4—1062no—
290C4⋊C4—1062no—
291C4⋊C4—1062no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C4⋊C4—2121no—
297C22⋊C4—22no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—1062no—
301C22⋊C4—1062no—
302C22⋊C4—1062no—
303C22⋊C4—1062no—
304C22⋊C4—1062no—
305C22⋊C4—2121no—
306C22⋊C4—2121no—
307C42—2121no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—532no—
315C23—1061no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23Q8×D5312yes—
322C23C23×D5311yescharacteristic
323Q8—2124no—
324D4—2124no—
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—21no—
341C2×C4—21no—
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—2124no—
364C2×C4—2124no—
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—
380C22C2×D4×D5311yescharacteristic
381C22C2×Q8×D5311yescharacteristic
382C22#41×D5311yescharacteristic
383C22#1313326211yescharacteristic
384C22#1313326211yescharacteristic
385C22#1313326311yescharacteristic
386C22#1313326411yescharacteristic
387C4—22no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—1062no—
394C4—1062no—
395C4—1062no—
396C4—1062no—
397C4—2121no—
398C4—2121no—
399C4—2121no—
400C2—42no—
401C2—2122no—
402C2#1681118111yescharacteristic
403C2#1681121411yescharacteristic
404C2#1681124811yescharacteristic
405C2#1681128611yescharacteristic
406C2#1681129111yescharacteristic
407C2#1681132711yescharacteristic
408C2#1681136811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2732
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
2. C2×#13133194C2semidirect
3. #16810559C2semidirect
4. #16810558C2semidirect
5. #16810557C2semidirect
6. #16810556C2semidirect
7. #16810545C2semidirect
8. #16810528C2semidirect
10. #16810518C2semidirect
13. #16810497C2semidirect
14. #16810495C2semidirect
15. C53×#396C2semidirect
63. C2×C22⋊D53Q8semidirect
79. C2×Dic106D4semidirect
212. #396D53semidirect
220. C53#2732semidirect