Finite Group Databasebeta

← #20280316

Subgroups — #20280316

Order 6784.

518 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133212C211yesmaximal, characteristic, complemented
2C2×#13133176C211yesmaximal, characteristic, complemented
3#16810557C211yesmaximal, characteristic, complemented
4#16810546C211yesmaximal, characteristic, complemented
5#16810545C211yesmaximal, characteristic, complemented
6#16810538C211yesmaximal, characteristic
7#16810523C211yesmaximal, characteristic
8#16810522C211yesmaximal, characteristic, complemented
9#16810516C211yesmaximal, characteristic, complemented
10#16810515C211yesmaximal, characteristic, complemented
11#16810515C211yesmaximal, characteristic, complemented
12#16810504C211yesmaximal, characteristic, complemented
13#16810502C211yesmaximal, characteristic
14#16810474C211yesmaximal, characteristic, complemented
15C53×#385C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17#13133212—24no—
18#13133176—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C23×C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×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×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—22no—
65C2×C22⋊D53—22no—
66C2×C22⋊D53—41no—
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C4⋊C4×C53—22no—
75C4⋊C4×C53—22no—
76C22⋊C4×C53—22no—
77C22⋊C4×C53—22no—
78C22⋊Dic53—22no—
79C22⋊Dic53—22no—
80C22⋊Dic53—41no—
81#11719921—22no—
82#11719921—22no—
83#11719921—22no—
84#11719921—22no—
85#11719921—22no—
86#11719921—41no—
87C4⋊Dic53—22no—
88C4⋊Dic53—22no—
89C4⋊Dic53—22no—
90C4⋊Dic53—41no—
91#11719919—22no—
92#11719919—22no—
93#11719919—22no—
94#11719919—22no—
95C4×Dic53—22no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C4×Dic53D414yescomplemented
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53#4112yes—
106C23×C53C2411yescharacteristic
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C2×C4×C53—28no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
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×C53C2×D412yes—
131C2×C4×C53#4112yes—
132C2×C4×C53#4112yes—
133C22⋊D53—44no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
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×Dic53C2×D412yes—
153C2×Dic53C2×D412yes—
154C2×Dic53#4112yes—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53C22×D411yescharacteristic
163C22×C53C2×#4111yescharacteristic
164C22×C53C2×#4111yescharacteristic
165C22×C53C2×#4111yescharacteristic
166C22×C53C2×#4111yescharacteristic
167C22×C53H(5,𝔽2)11yescharacteristic
168C22×C532−1+411yescharacteristic
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C4×C53—22no—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182Dic53—22no—
183Dic53—22no—
184Dic53—22no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189#2601—531nomaximal, Sylow
190C2×C53—42no—
191C2×C53#52511yescharacteristic
192C2×C53#53211yescharacteristic
193C2×C53#53711yescharacteristic
194C2×C53#55111yescharacteristic
195C2×C53#55211yescharacteristic
196C2×C53#55311yescharacteristic
197C2×C53#55611yescharacteristic
198D53—81no—
199C2×#126—531no—
200C2×#117—531no—
201#398—531no—
202#396—531no—
203#396—531no—
204#394—531no—
205#390—531no—
206#389—531no—
207#388—531no—
208#388—531no—
209#388—531no—
210#387—531no—
211#386—531no—
212#385D5311yescharacteristic, complemented
213C4×C4⋊C4—531no—
214C53#260111yescharacteristic, complemented, Sylow
215C22×D4—531no—
216C23×C4C2×D5311yescharacteristic
217#126—1064no—
218#117—1064no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4C2×D5311yescharacteristic
228C2×C4⋊C4C2×D5311yescharacteristic
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C22⋊C4—531no—
234C2×C22⋊C4—531no—
235C2×C22⋊C4—531no—
236C2×C22⋊C4—531no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4C2×D5311yescharacteristic
239C2×C22⋊C4C2×D5311yescharacteristic
240C2×C42—531no—
241C2×C42—531no—
242C2×C42—531no—
243C2×C42—531no—
244#95—531no—
245#95—531no—
246#95—531no—
247#95—531no—
248#95—531no—
249#95—531no—
250#95C2×D5311yescharacteristic
251#95C2×D5311yescharacteristic
252C24—531no—
253C24C22×D5311yescharacteristic
254C2×D4—1062no—
255C2×D4—1062no—
256C2×D4—2121no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4C22×D5311yescharacteristic
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—1062no—
276C4⋊C4—1062no—
277C4⋊C4—1062no—
278C4⋊C4—1062no—
279C4⋊C4—1062no—
280C4⋊C4—1062no—
281C4⋊C4—1062no—
282C4⋊C4—2121no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—1062no—
286C22⋊C4—1062no—
287C22⋊C4—1062no—
288C22⋊C4—1062no—
289C22⋊C4—1062no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C42—534no—
295C42—1062no—
296C42—2121no—
297C42—2121no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23#1171994712yes—
312C23C23×D5311yescharacteristic
313D4—2124no—
314C2×C4—28no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—532no—
331C2×C4—532no—
332C2×C4—532no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4#1171994512yes—
352C2×C4D4×D5312yes—
353C2×C4#1171994712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—2121no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22C2×D4×D5311yescharacteristic
369C22#41×D5311yescharacteristic
370C22C2×#1171994711yescharacteristic
371C22C2×#1171994711yescharacteristic
372C22C2×#1171994511yescharacteristic
373C22#1313326311yescharacteristic
374C22#1313326511yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1062no—
382C4—1062no—
383C4—1062no—
384C4—2121no—
385C4—2121no—
386C4—2121no—
387C4—2121no—
388C2—42no—
389C2—4241no—
390C2#1681112911yescharacteristic
391C2#1681117011yescharacteristic
392C2#1681120311yescharacteristic
393C2#1681129611yescharacteristic
394C2#1681130811yescharacteristic
395C2#1681131911yescharacteristic
396C2#1681134411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2601
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#13133212C2semidirect
2. C2×#13133176C2semidirect
3. #16810557C2semidirect
4. #16810546C2semidirect
5. #16810545C2semidirect
8. #16810522C2semidirect
9. #16810516C2semidirect
10. #16810515C2semidirect
11. #16810515C2semidirect
12. #16810504C2semidirect
14. #16810474C2semidirect
15. C53×#385C2semidirect
98. C4×Dic53D4semidirect
212. #385D53semidirect
214. C53#2601semidirect