Finite Group Databasebeta

← #20278190

Subgroups — #20278190

Order 6784.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133198C211yesmaximal, characteristic
2C2×D4×Dic53C211yesmaximal, characteristic
3#16810563C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic, complemented
7#16810508C211yesmaximal, characteristic, complemented
8#16810508C211yesmaximal, characteristic, complemented
9#16810498C211yesmaximal, characteristic, complemented
10#16810498C211yesmaximal, characteristic, complemented
11#16810484C211yesmaximal, characteristic, complemented
12#16810484C211yesmaximal, characteristic, complemented
13#16810483C211yesmaximal, characteristic, complemented
14C2×#13133124C211yesmaximal, characteristic, complemented
15C53×#386C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19#13133198—24no—
20D4×Dic53—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C23×C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53D414yes—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—41no—
86C22⋊Dic53—22no—
87C22⋊Dic53—22no—
88C22⋊Dic53—22no—
89C22⋊Dic53—22no—
90C22⋊Dic53—22no—
91C4⋊Dic53—22no—
92C4⋊Dic53—22no—
93C4⋊Dic53—22no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96#11719919—22no—
97#11719919—22no—
98C4×Dic53—22no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53C2×D412yes—
110C23×C53C2×D412yes—
111C23×C53#4112yes—
112C23×C53#4112yes—
113C23×C53C2411yescharacteristic
114D4×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—44no—
129C2×C4×C53C2×D412yes—
130C2×C4×C53#4112yes—
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—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—44no—
155C2×Dic53—44no—
156C2×Dic53—44no—
157C2×Dic53#4112yes—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53#12314yes—
172C22×C53C22×D411yescharacteristic
173C22×C53C2×#4111yescharacteristic
174C22×C53C2×#4111yescharacteristic
175C22×C53C2×#4111yescharacteristic
176C22×C53C2×#4111yescharacteristic
177C22×C53H(5,𝔽2)11yescharacteristic
178C22×C532−1+411yescharacteristic
179C4×C53—22no—
180C4×C53—22no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184Dic53—22no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190Dic53—41no—
191Dic53—41no—
192#2419—531nomaximal, Sylow
193C2×C53—24no—
194C2×C53—42no—
195C2×C53C2×#12311yescharacteristic
196C2×C53#52911yescharacteristic
197C2×C53#53911yescharacteristic
198C2×C53#54611yescharacteristic
199C2×C53#54711yescharacteristic
200C2×C53#55111yescharacteristic
201C2×C53#55211yescharacteristic
202C2×#123—531no—
203C2×C4×D4—531no—
204#400—531no—
205#398—531no—
206#397—531no—
207#397—531no—
208#387—531no—
209#387—531no—
210#386D5311yescharacteristic
211#385—531no—
212#385—531no—
213#381—531no—
214#381—531no—
215#380—531no—
216C2×#95—531no—
217C53#241911yescharacteristic, complemented, Sylow
218C22×D4C2×D5311yescharacteristic
219C23×C4—531no—
220C23×C4—531no—
221C23×C4—531no—
222C23×C4C2×D5311yescharacteristic
223#123—1064no—
224C4×D4—1064no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C22⋊C4—531no—
234C2×C22⋊C4—531no—
235C2×C22⋊C4—531no—
236C2×C22⋊C4—531no—
237C2×C22⋊C4C2×D5311yescharacteristic
238C2×C22⋊C4C2×D5311yescharacteristic
239C2×C22⋊C4C2×D5311yescharacteristic
240C2×C42—531no—
241#95—531no—
242#95—531no—
243#95—531no—
244#95—531no—
245#95—531no—
246#95—531no—
247#95—531no—
248#95—531no—
249#95—531no—
250#95—531no—
251#95—531no—
252#95—531no—
253#95C2×D5311yescharacteristic
254#95C2×D5311yescharacteristic
255C24C22×D5311yescharacteristic
256C24C22×D5311yescharacteristic
257C2×D4—22no—
258C2×D4—22no—
259C2×D4D21214yes—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—1062no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—2121no—
279C22×C4—2121no—
280C22×C4C22×D5311yescharacteristic
281C22×C4C22×D5311yescharacteristic
282C22×C4C22×D5311yescharacteristic
283C22×C4C22×D5311yescharacteristic
284C22×C4C22×D5311yescharacteristic
285C4⋊C4—1062no—
286C4⋊C4—1062no—
287C4⋊C4—1062no—
288C4⋊C4—1062no—
289C4⋊C4—1062no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—1062no—
296C22⋊C4—1062no—
297C22⋊C4—1062no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C42—1062no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23C2×D21212yes—
312C23C2×D21212yes—
313C23#1171994712yes—
314C23#1171994712yes—
315C23C23×D5311yescharacteristic
316D4—28no—
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—44no—
331C2×C4—532no—
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—1061no—
345C2×C4—1061no—
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—2124no—
356C2×C4—2124no—
357C2×C4—2124no—
358C2×C4C2×D21212yes—
359C2×C4#1171994712yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#1313320114yes—
374C22C22×D21211yescharacteristic
375C22#41×D5311yescharacteristic
376C22C2×#1171994711yescharacteristic
377C22C2×#1171994711yescharacteristic
378C22C2×#1171994711yescharacteristic
379C22#1313326211yescharacteristic
380C22#1313326411yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—1062no—
387C4—2121no—
388C4—2121no—
389C4—2121no—
390C4—2121no—
391C4—2121no—
392C4—2121no—
393C4—2121no—
394C2—24no—
395C2—42no—
396C2C2×#1313320111yescharacteristic
397C2#1681115411yescharacteristic
398C2#1681121811yescharacteristic
399C2#1681126811yescharacteristic
400C2#1681127711yescharacteristic
401C2#1681130411yescharacteristic
402C2#1681131011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2419
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #16810563C2semidirect
4. #16810555C2semidirect
5. #16810548C2semidirect
6. #16810548C2semidirect
7. #16810508C2semidirect
8. #16810508C2semidirect
9. #16810498C2semidirect
10. #16810498C2semidirect
11. #16810484C2semidirect
12. #16810484C2semidirect
13. #16810483C2semidirect
14. C2×#13133124C2semidirect
217. C53#2419semidirect