Finite Group Databasebeta

← #20278384

Subgroups — #20278384

Order 6784.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133193C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic
3#16810565C211yesmaximal, characteristic, complemented
4#16810556C211yesmaximal, characteristic, complemented
5#16810555C211yesmaximal, characteristic, complemented
6#16810530C211yesmaximal, characteristic, complemented
7#16810518C211yesmaximal, characteristic, complemented
8#16810509C211yesmaximal, characteristic, complemented
9#16810508C211yesmaximal, characteristic, complemented
10#16810497C211yesmaximal, characteristic
11#16810494C211yesmaximal, characteristic
12#16810485C211yesmaximal, characteristic
13#16810480C211yesmaximal, characteristic
14#95×D53C211yesmaximal, characteristic
15C53×#395C211yesmaximal, characteristic, complemented
16C22×C4×D53C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133193—24no—
20#13133175—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×Q8×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53Q814yescomplemented
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—41no—
83#11719921—22no—
84#11719921—22no—
85#11719921—22no—
86#11719921D414yes—
87C4⋊Dic53—22no—
88C4⋊Dic53—22no—
89C4⋊Dic53—22no—
90C4⋊Dic53—41no—
91#11719919—22no—
92#11719919—22no—
93#11719919—22no—
94#11719919—41no—
95C4×Dic53—22no—
96C4×Dic53—22no—
97C23×C53C2411yescharacteristic
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53C2×D412yes—
103C22×D53C2×Q812yes—
104C22×D53#4112yes—
105Q8×C53—44no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
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×C53C2×D412yes—
124C2×C4×C53C2×Q812yes—
125C2×C4×C53#4112yes—
126C2×C4×C53#4112yes—
127C2×Dic53—21no—
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×Dic53C2×D412yes—
147C2×Dic53C2×Q812yes—
148C4×D53—28no—
149C4×D53—44no—
150C4×D53—44no—
151C22×C53C22×D411yescharacteristic
152C22×C53C22×Q811yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53C2×#4111yescharacteristic
155C22×C53C2×#4111yescharacteristic
156C22×C53H(5,𝔽2)11yescharacteristic
157C22×C532−1+411yescharacteristic
158C2×D53—22no—
159C2×D53—22no—
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—22no—
163C2×D53—22no—
164C2×D53C22⋊Q814yes—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172Dic53—22no—
173Dic53—22no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179#2438—531nomaximal, Sylow
180C2×C53C2×C22⋊Q811yescharacteristic
181C2×C53#53311yescharacteristic
182C2×C53#54411yescharacteristic
183C2×C53#54611yescharacteristic
184C2×C53#54711yescharacteristic
185C2×C53#55111yescharacteristic
186C2×C53#55211yescharacteristic
187D53—24no—
188C2×C22⋊Q8—531no—
189C2×#117—531no—
190#400—531no—
191#398—531no—
192#398—531no—
193#395D5311yescharacteristic, complemented
194#391—531no—
195#389—531no—
196#387—531no—
197#387—531no—
198#385—531no—
199#385—531no—
200#381—531no—
201#380—531no—
202C2×#95—531no—
203C53#243811yescharacteristic, complemented, Sylow
204C22×Q8C2×D5311yescharacteristic
205C23×C4—531no—
206C23×C4—531no—
207C23×C4—531no—
208C22⋊Q8—1064no—
209#117—1064no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4C2×D5311yescharacteristic
219C2×C4⋊C4C2×D5311yescharacteristic
220C2×C4⋊C4C2×D5311yescharacteristic
221C2×C22⋊C4—531no—
222C2×C22⋊C4—531no—
223C2×C22⋊C4—531no—
224C2×C22⋊C4—531no—
225C2×C42—531no—
226C2×C42—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95—531no—
231#95—531no—
232#95—531no—
233#95—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95—531no—
238#95C2×D5311yescharacteristic
239#95C2×D5311yescharacteristic
240#95C2×D5311yescharacteristic
241C24—531no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—534no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—1062no—
254C22×C4—1062no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—2121no—
260C22×C4—2121no—
261C22×C4C22×D5311yescharacteristic
262C22×C4C22×D5311yescharacteristic
263C22×C4C22×D5311yescharacteristic
264C22×C4C22×D5311yescharacteristic
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267C22×C4C22×D5311yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—1062no—
272C4⋊C4—1062no—
273C4⋊C4—1062no—
274C4⋊C4—1062no—
275C4⋊C4—1062no—
276C4⋊C4—1062no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C22⋊C4—534no—
280C22⋊C4—1062no—
281C22⋊C4—1062no—
282C22⋊C4—1062no—
283C42—1062no—
284C42—1062no—
285C23—532no—
286C23—532no—
287C23—532no—
288C23—1061no—
289C23—1061no—
290C23—1061no—
291C23—1061no—
292C23C23×D5311yescharacteristic
293Q8—44no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—1068no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
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—2124no—
334C2×C4—2124no—
335C2×C4D4×D5312yes—
336C2×C4#1171994712yes—
337C2×C4Q8×D5312yes—
338C2×C4#1171994912yes—
339C22—534no—
340C22—1062no—
341C22—1062no—
342C22—1062no—
343C22—1062no—
344C22—1062no—
345C22—1062no—
346C22C2×D4×D5311yescharacteristic
347C22C2×Q8×D5311yescharacteristic
348C22#41×D5311yescharacteristic
349C22C2×#1171994711yescharacteristic
350C22C2×#1171994911yescharacteristic
351C22#1313326211yescharacteristic
352C22#1313326511yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—1062no—
361C4—1062no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4—2121no—
367C2—1064no—
368C2C22⋊Q8×D5311yescharacteristic
369C2#1681117811yescharacteristic
370C2#1681125011yescharacteristic
371C2#1681126011yescharacteristic
372C2#1681127611yescharacteristic
373C2#1681130011yescharacteristic
374C2#1681130711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2438
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #16810565C2semidirect
4. #16810556C2semidirect
5. #16810555C2semidirect
6. #16810530C2semidirect
7. #16810518C2semidirect
8. #16810509C2semidirect
9. #16810508C2semidirect
15. C53×#395C2semidirect
79. C2×C4×D53Q8semidirect
193. #395D53semidirect
203. C53#2438semidirect