Finite Group Databasebeta

← #20278117

Subgroups — #20278117

Order 6784.

610 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×#13133199C211yesmaximal, characteristic
2C2×#13133199C211yesmaximal, characteristic
3C2×#13133175C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
5#16810556C211yesmaximal, characteristic, complemented
6#16810553C211yesmaximal, characteristic, complemented
7#16810546C211yesmaximal, characteristic, complemented
8#16810522C211yesmaximal, characteristic, complemented
9#16810520C211yesmaximal, characteristic, complemented
10#16810516C211yesmaximal, characteristic, complemented
11#16810511C211yesmaximal, characteristic, complemented
12#16810499C211yesmaximal, characteristic
13#16810494C211yesmaximal, characteristic, complemented
14#16810485C211yesmaximal, characteristic, complemented
15C53×#396C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133199—24no—
21#13133199—24no—
22#13133175—24no—
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C4⋊C4×D53—24no—
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—22no—
64C2×C22⋊D53—22no—
65C2×C22⋊D53—22no—
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53D414yes—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C22⋊Dic53—22no—
89C22⋊Dic53—22no—
90C22⋊Dic53—22no—
91#11719921—22no—
92#11719921—22no—
93#11719921D414yes—
94C4⋊Dic53—22no—
95C4⋊Dic53—22no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98#11719919—22no—
99#11719919—22no—
100#11719919—22no—
101#11719919—22no—
102#11719919—22no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53C2411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53C2×D412yes—
118C22×D53C2×D412yes—
119C22×D53#4112yes—
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—
136C2×C4×C53C2×D412yes—
137C2×C4×C53#4112yes—
138C22⋊D53—44no—
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—21no—
156C2×Dic53—21no—
157C2×Dic53C2×D412yes—
158C2×Dic53C2×D412yes—
159C2×Dic53#4112yes—
160C4×D53—28no—
161C4×D53—28no—
162C4×D53—44no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53C22×D411yescharacteristic
171C22×C53C22×D411yescharacteristic
172C22×C53C2×#4111yescharacteristic
173C22×C53C2×#4111yescharacteristic
174C22×C53C2×#4111yescharacteristic
175C22×C532−1+411yescharacteristic
176C22×C532−1+411yescharacteristic
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—22no—
183C2×D53#12114yes—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—41no—
190Dic53—22no—
191Dic53—22no—
192Dic53—22no—
193Dic53—41no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#2413—531nomaximal, Sylow
198C2×C53—81no—
199C2×C53C2×#12111yescharacteristic
200C2×C53#52511yescharacteristic
201C2×C53#53611yescharacteristic
202C2×C53#54711yescharacteristic
203C2×C53#54711yescharacteristic
204C2×C53#55211yescharacteristic
205C2×C53#55211yescharacteristic
206D53—24no—
207C2×#123—531no—
208C2×#123—531no—
209C2×#117—531no—
210C22×C4⋊C4—531no—
211#398—531no—
212#398—531no—
213#396—531no—
214#396D5311yescharacteristic, complemented
215#389—531no—
216#389—531no—
217#388—531no—
218#388—531no—
219#386—531no—
220#385—531no—
221#381—531no—
222C53#241311yescharacteristic, complemented, Sylow
223C22×D4—531no—
224C23×C4—531no—
225C23×C4—531no—
226C23×C4—531no—
227#123—1064no—
228#123—1064no—
229#117—1064no—
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⋊C4—531no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—1064no—
240C2×C4⋊C4C2×D5311yescharacteristic
241C2×C4⋊C4C2×D5311yescharacteristic
242C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4C2×D5311yescharacteristic
250C2×C22⋊C4C2×D5311yescharacteristic
251C2×C22⋊C4C2×D5311yescharacteristic
252C2×C42—531no—
253C2×C42—531no—
254#95—531no—
255#95—531no—
256#95—531no—
257#95—531no—
258#95—531no—
259#95—531no—
260#95—531no—
261#95C2×D5311yescharacteristic
262C24—531no—
263C24C22×D5311yescharacteristic
264C2×D4—1062no—
265C2×D4—1062no—
266C2×D4—1062no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—534no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—1062no—
279C22×C4—1062no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1062no—
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—1062no—
298C4⋊C4—1062no—
299C4⋊C4—1062no—
300C4⋊C4—1062no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—534no—
305C22⋊C4—1062no—
306C22⋊C4—1062no—
307C22⋊C4—1062no—
308C22⋊C4—1062no—
309C22⋊C4—1062no—
310C42—1062no—
311C42—1062no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—532no—
320C23—532no—
321C23—532no—
322C23—1061no—
323C23—1061no—
324C23—1061no—
325C23—1061no—
326C23C23×D5311yescharacteristic
327D4—2124no—
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—532no—
344C2×C4—532no—
345C2×C4—532no—
346C2×C4—1068no—
347C2×C4—1068no—
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×C4D4×D5312yes—
368C2×C4D4×D5312yes—
369C2×C4#1171994712yes—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—534no—
378C22—1062no—
379C22—1062no—
380C22—1062no—
381C22—1062no—
382C22—1062no—
383C22—1062no—
384C22C2×D4×D5311yescharacteristic
385C22C2×D4×D5311yescharacteristic
386C22#41×D5311yescharacteristic
387C22#41×D5311yescharacteristic
388C22C2×#1171994711yescharacteristic
389C22#1313326411yescharacteristic
390C22#1313326511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—1062no—
398C4—1062no—
399C4—1062no—
400C4—2121no—
401C4—2121no—
402C4—2121no—
403C4—2121no—
404C2—81no—
405C2—1064no—
406C2#121×D5311yescharacteristic
407C2#1681113311yescharacteristic
408C2#1681119611yescharacteristic
409C2#1681127311yescharacteristic
410C2#1681127511yescharacteristic
411C2#1681130711yescharacteristic
412C2#1681130911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2413
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
3. C2×#13133175C2semidirect
4. C2×C4⋊C4×D53C2semidirect
5. #16810556C2semidirect
6. #16810553C2semidirect
7. #16810546C2semidirect
8. #16810522C2semidirect
9. #16810520C2semidirect
10. #16810516C2semidirect
11. #16810511C2semidirect
13. #16810494C2semidirect
14. #16810485C2semidirect
15. C53×#396C2semidirect
214. #396D53semidirect
222. C53#2413semidirect