Finite Group Databasebeta

← #20285608

Subgroups — #20285608

Order 6784.

378 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811197C211yesmaximal, characteristic
2#16811196C211yesmaximal, characteristic
3#16810965C211yesmaximal, characteristic, complemented
4#16810963C211yesmaximal, characteristic, complemented
5#16810962C211yesmaximal, characteristic, complemented
6#16810961C211yesmaximal, characteristic, complemented
7#16810880C211yesmaximal, characteristic, complemented
8#16810878C211yesmaximal, characteristic, complemented
9#16810863C211yesmaximal, characteristic, complemented
10#16810861C211yesmaximal, characteristic, complemented
11#16810779C211yesmaximal, characteristic, complemented
12#16810777C211yesmaximal, characteristic, complemented
13#16810777C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic
15C53×#451C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133216—21no—
23#13133215—21no—
24#13133206—21no—
25#13133203—21no—
26#13133202—21no—
27#13133200—21no—
28#13133200—21no—
29#13133199—21no—
30#13133197C2211yescharacteristic
31#13133196C2211yescharacteristic
32#13133193C2211yescharacteristic
33#13133193C2211yescharacteristic
34#13133193C2211yescharacteristic
35#13133193C2211yescharacteristic
36#13133193C2211yescharacteristic
37#13133176C2211yescharacteristic
38#13133175C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133148C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133141C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×Q16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53D411yescharacteristic
68C2×C22⋊D53D411yescharacteristic
69Q8×D53—41no—
70Q8×D53—41no—
71#11719945—41no—
72#11719945—41no—
73C2×D212D411yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×C4×D53C2311yescharacteristic
77C2×Dic106—21no—
78C2×Dic106D411yescharacteristic
79C2×Dic106C2311yescharacteristic
80Q16×C53—41no—
81Q16×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719925—41no—
90#11719925—41no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108C4×Dic53—21no—
109C4×Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719912—41no—
113C8×D53—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53C2×D411yescharacteristic
116C22×D53C2×D411yescharacteristic
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122Q8×C53—21no—
123Q8×C53—42no—
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—42no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—41no—
134C22⋊D53—41no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—41no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—42no—
147Dic106—21no—
148Dic106—21no—
149Dic106—41no—
150Dic106—42no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53H(5,𝔽2)11yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178#3202—531nomaximal, Sylow
179C2×C53—41no—
180C2×C53#53411yescharacteristic
181C2×C53#57811yescharacteristic
182C2×C53#57811yescharacteristic
183D53—41no—
184D53—41no—
185#536—531no—
186#536—531no—
187#485—531no—
188#485—531no—
189#485—531no—
190#485—531no—
191#470—531no—
192#470—531no—
193#467—531no—
194#467—531no—
195#451—531no—
196#451—531no—
197#451—531no—
198#451D5311yescharacteristic, complemented
199#408—531no—
200C53#320211yescharacteristic, complemented, Sylow
201C2×#41—531no—
202C22×Q8—531no—
203C22×Q8C2×D5311yescharacteristic
204C2×Q16—531no—
205C2×Q16—531no—
206C2×Q16C2×D5311yescharacteristic
207C2×Q16C2×D5311yescharacteristic
208C2×M16—531no—
209C22×C8—531no—
210C4⋊Q8—1061no—
211C4⋊Q8—1061no—
212#124—1061no—
213#124—1061no—
214#123—1061no—
215#123—1061no—
216#123—1061no—
217#123—1061no—
218C22⋊Q8—531no—
219C22⋊Q8—531no—
220C22⋊Q8—531no—
221C22⋊Q8—531no—
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—531no—
225C22⋊Q8C2×D5311yescharacteristic
226#117—531no—
227#117—531no—
228#107—531no—
229#107—531no—
230#106—531no—
231#106—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#103C2×D5311yescharacteristic
239#103C2×D5311yescharacteristic
240C22⋊C8—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243C22⋊C8C2×D5311yescharacteristic
244#41—2121no—
245#41—2121no—
246C2×Q8—21no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—531no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8—2121no—
253C2×Q8—2121no—
254C2×Q8C22×D5311yescharacteristic
255C2×Q8C22×D5311yescharacteristic
256C2×Q8C22×D5311yescharacteristic
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—531no—
260C22×C4—21no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4C22×D5311yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—2121no—
268Q16—2121no—
269M16—2121no—
270C2×C8—531no—
271C2×C8—531no—
272C2×C8—2121no—
273C2×C8C22×D5311yescharacteristic
274C2×C8C22×D5311yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4C22×D5311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C42—1061no—
298C42—1061no—
299C23—531no—
300C23—531no—
301C23D4×D5311yescharacteristic
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—42no—
309Q8—1061no—
310Q8—1061no—
311Q8—2121no—
312Q8—2122no—
313D4—2121no—
314D4—2121no—
315D4—2121no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—531no—
324C2×C4—531no—
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—2122no—
334C2×C4D4×D5311yescharacteristic
335C2×C4C23×D5311yescharacteristic
336C8—21no—
337C8—21no—
338C8—1061no—
339C8—1061no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—1061no—
344C22—1061no—
345C22—1061no—
346C22—1061no—
347C22—1061no—
348C22—1061no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4—2121no—
361C4#1313326211yescharacteristic
362C4#1313326211yescharacteristic
363C2—41no—
364C2—2121no—
365C2—2121no—
366C2#1681118311yescharacteristic
367C2#1681143611yescharacteristic
368C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53#3202
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #16810965C2semidirect
4. #16810963C2semidirect
5. #16810962C2semidirect
6. #16810961C2semidirect
7. #16810880C2semidirect
8. #16810878C2semidirect
9. #16810863C2semidirect
10. #16810861C2semidirect
11. #16810779C2semidirect
12. #16810777C2semidirect
13. #16810777C2semidirect
15. C53×#451C2semidirect
198. #451D53semidirect
200. C53#3202semidirect