Finite Group Databasebeta

← #20279728

Subgroups — #20279728

Order 6784.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133196C211yesmaximal, characteristic
2C2×#13133194C211yesmaximal, characteristic
3#16810557C211yesmaximal, characteristic, complemented
4#16810556C211yesmaximal, characteristic, complemented
5#16810544C211yesmaximal, characteristic, complemented
6#16810543C211yesmaximal, characteristic, complemented
7#16810527C211yesmaximal, characteristic, complemented
8#16810527C211yesmaximal, characteristic, complemented
9#16810522C211yesmaximal, characteristic, complemented
10#16810519C211yesmaximal, characteristic, complemented
11#16810498C211yesmaximal, characteristic
12#16810481C211yesmaximal, characteristic
13C22⋊C4×Dic53C211yesmaximal, characteristic
14C2×#13133124C211yesmaximal, characteristic
15C53×#385C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20#13133196—24no—
21#13133194—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
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
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53—41no—
68C22×Dic53Q814yescomplemented
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C2×Dic106—22no—
78C2×Dic106—22no—
79C2×Dic106—22no—
80C2×Dic106—22no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—41no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53Q814yescomplemented
87C22⋊Dic53—22no—
88C22⋊Dic53D414yes—
89C4⋊Dic53—22no—
90C4⋊Dic53—22no—
91C4⋊Dic53—22no—
92C4⋊Dic53—41no—
93#11719919—22no—
94#11719919—22no—
95#11719919—22no—
96#11719919—22no—
97#11719919—41no—
98#11719919—41no—
99C4×Dic53—41no—
100C4×Dic53—41no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53C2×D412yes—
106C23×C53C2×Q812yes—
107C23×C53C2×Q812yes—
108C23×C53C2411yescharacteristic
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
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—44no—
125C2×C4×C53C2×Q812yes—
126C2×C4×C53C2×Q812yes—
127C2×C4×C53#4112yes—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
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—44no—
148C2×Dic53—44no—
149C2×Dic53C2×D412yes—
150C2×Dic53C2×D412yes—
151C2×Dic53C2×Q812yes—
152C2×Dic53C2×Q812yes—
153C2×Dic53#4112yes—
154Dic106—44no—
155Dic106—44no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53C4⋊Q814yes—
163C22×C53C22×D411yescharacteristic
164C22×C53C22×Q811yescharacteristic
165C22×C53C22×Q811yescharacteristic
166C22×C53C2×#4111yescharacteristic
167C22×C53C2×#4111yescharacteristic
168C22×C53H(5,𝔽2)11yescharacteristic
169C22×C53H(5,𝔽2)11yescharacteristic
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176Dic53—22no—
177Dic53—22no—
178Dic53—22no—
179Dic53—22no—
180Dic53—22no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184#2549—531nomaximal, Sylow
185C2×C53—24no—
186C2×C53C2×C4⋊Q811yescharacteristic
187C2×C53#53311yescharacteristic
188C2×C53#54311yescharacteristic
189C2×C53#54611yescharacteristic
190C2×C53#54611yescharacteristic
191C2×C53#55411yescharacteristic
192C2×C53#55411yescharacteristic
193C2×C22⋊Q8—531no—
194C2×C22⋊Q8—531no—
195#398—531no—
196#398—531no—
197#395—531no—
198#395—531no—
199#391—531no—
200#391—531no—
201#389—531no—
202#389—531no—
203#385—531no—
204#385D5311yescharacteristic
205#380—531no—
206C4×C22⋊C4—531no—
207C2×#95—531no—
208C53#254911yescharacteristic, complemented, Sylow
209C22×Q8—531no—
210C22×Q8—531no—
211C23×C4—531no—
212C23×C4—531no—
213C23×C4C2×D5311yescharacteristic
214C22⋊Q8—1064no—
215C22⋊Q8—1064no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
227C2×C4⋊C4C2×D5311yescharacteristic
228C2×C22⋊C4—531no—
229C2×C22⋊C4—531no—
230C2×C22⋊C4C2×D5311yescharacteristic
231C2×C22⋊C4C2×D5311yescharacteristic
232C2×C42—531no—
233C2×C42—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95—531no—
238#95—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95—531no—
243#95—531no—
244#95C2×D5311yescharacteristic
245#95C2×D5311yescharacteristic
246C24C22×D5311yescharacteristic
247C2×Q8—1062no—
248C2×Q8—1062no—
249C2×Q8—1062no—
250C2×Q8—1062no—
251C2×Q8—2121no—
252C2×Q8—2121no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—534no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C22×C4C22×D5311yescharacteristic
274C22×C4C22×D5311yescharacteristic
275C22×C4C22×D5311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—1062no—
279C4⋊C4—1062no—
280C4⋊C4—1062no—
281C4⋊C4—1062no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4—1062no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C22⋊C4—22no—
289C22⋊C4—534no—
290C22⋊C4—1062no—
291C22⋊C4Dic10614yescomplemented
292C42—2121no—
293C42—2121no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C2×Dic10612yes—
299C23D4×D5312yes—
300C23Q8×D5312yes—
301C23C23×D5311yescharacteristic
302Q8—2124no—
303Q8—2124no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—532no—
321C2×C4—532no—
322C2×C4—532no—
323C2×C4—532no—
324C2×C4—532no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
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—2124no—
345C2×C4—2124no—
346C2×C4C2×Dic10612yes—
347C2×C4C2×Dic10612yes—
348C2×C4#1171994712yes—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22#1313321614yes—
356C22C2×D4×D5311yescharacteristic
357C22C2×Q8×D5311yescharacteristic
358C22C22×Dic10611yescharacteristic
359C22#41×D5311yescharacteristic
360C22C2×#1171994711yescharacteristic
361C22#1313326211yescharacteristic
362C22#1313326311yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—1062no—
370C4—1062no—
371C4—1062no—
372C4—1062no—
373C4—1062no—
374C4—2121no—
375C4—2121no—
376C4—2121no—
377C2—24no—
378C2C2×#1313321611yescharacteristic
379C2#1681117511yescharacteristic
380C2#1681124811yescharacteristic
381C2#1681126011yescharacteristic
382C2#1681126511yescharacteristic
383C2#1681133011yescharacteristic
384C2#1681133211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2549
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #16810557C2semidirect
4. #16810556C2semidirect
5. #16810544C2semidirect
6. #16810543C2semidirect
7. #16810527C2semidirect
8. #16810527C2semidirect
9. #16810522C2semidirect
10. #16810519C2semidirect
68. C22×Dic53Q8semidirect
86. C22⋊C4×C53Q8semidirect
208. C53#2549semidirect
291. C22⋊C4Dic106semidirect