Finite Group Databasebeta

← #20284662

Subgroups — #20284662

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D53C211yesmaximal, characteristic
2#117×D53C211yesmaximal, characteristic
3#16810965C211yesmaximal, characteristic, complemented
4#16810964C211yesmaximal, characteristic, complemented
5#16810961C211yesmaximal, characteristic, complemented
6#16810960C211yesmaximal, characteristic, complemented
7#16810957C211yesmaximal, characteristic, complemented
8#16810955C211yesmaximal, characteristic, complemented
9#16810952C211yesmaximal, characteristic, complemented
10#16810664C211yesmaximal, characteristic, complemented
11#16810662C211yesmaximal, characteristic, complemented
12#16810630C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810585C211yesmaximal, characteristic
15C53×#485C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×#11719912C2211yescharacteristic
20#13133197C2211yescharacteristic
21#13133196C2211yescharacteristic
22#13133195—21no—
23#13133194—21no—
24#13133193C2211yescharacteristic
25#13133176C2211yescharacteristic
26#13133175C2211yescharacteristic
27#13133175C2211yescharacteristic
28#13133174—21no—
29#13133173—21no—
30C4⋊C4×D53—21no—
31C4⋊C4×D53C2211yescharacteristic
32C4⋊C4×D53C2211yescharacteristic
33C4⋊C4×D53C2211yescharacteristic
34C22⋊C4×D53—21no—
35C22⋊C4×D53—21no—
36C42×D53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133149C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133141C2211yescharacteristic, complemented
46#13133141C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic
59C23×D53D411yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×Dic53D411yescharacteristic
63Q8×D53—22no—
64Q8×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—21no—
71C2×C4×D53D411yescharacteristic
72C2×C4×D53D411yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×C4×D53C2311yescharacteristic
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88#11719921—21no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107#11719912—22no—
108C23×C53C2×D411yescharacteristic
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—41no—
112C22×D53C2×D411yescharacteristic
113C22×D53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×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×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131C4×D53—22no—
132C4×D53—21no—
133C4×D53—21no—
134C4×D53—22no—
135C4×D53—42no—
136C4×D53—42no—
137C4×D53—42no—
138C4×D53—42no—
139C4×D53#4112yes—
140C4×D53#4112yes—
141Dic106—21no—
142Dic106—21no—
143Dic106—42no—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53C22×D411yescharacteristic
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—44no—
157C2×D53#12312yes—
158C2×D53#13712yes—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C2×#4111yescharacteristic
165C4×C53C2×#4111yescharacteristic
166Dic53—21no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171Dic53#12312yes—
172#3095—531nomaximal, Sylow
173C2×C53—41no—
174C2×C53C2×#12311yescharacteristic
175C2×C53C2×#13711yescharacteristic
176C2×C53#57511yescharacteristic
177D53—22no—
178D53—22no—
179C2×C22⋊Q8—531no—
180C2×#117—531no—
181#485—531no—
182#485—531no—
183#485—531no—
184#485D5311yescharacteristic, complemented
185#484—531no—
186#484—531no—
187#483—531no—
188#483—531no—
189#428—531no—
190#428—531no—
191#419—531no—
192#419—531no—
193#407—531no—
194C53#309511yescharacteristic, complemented, Sylow
195C22×Q8—531no—
196C23×C4—531no—
197C2×M16—531no—
198C2×M16—531no—
199C22⋊Q8—531no—
200C22⋊Q8—531no—
201C22⋊Q8—531no—
202C22⋊Q8—1061no—
203C22⋊Q8—1061no—
204C22⋊Q8C2×D5311yescharacteristic
205#117—531no—
206#117—531no—
207#117—531no—
208#117—1061no—
209#117—1061no—
210#117C2×D5311yescharacteristic
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C42—1061no—
218#107—531no—
219#107—531no—
220#107—531no—
221#107C2×D5311yescharacteristic, complemented
222#106—531no—
223#106—531no—
224#106—531no—
225#106C2×D5311yescharacteristic, complemented
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103C2×D5311yescharacteristic, complemented
233#103C2×D5311yescharacteristic, complemented
234C22⋊C8—531no—
235C22⋊C8—531no—
236C22⋊C8—531no—
237C22⋊C8C2×D5311yescharacteristic
238C24—531no—
239C2×Q8—531no—
240C2×Q8—1061no—
241C2×Q8—1062no—
242C2×Q8—1062no—
243C2×Q8C22×D5311yescharacteristic
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4C22×D5311yescharacteristic
256M16—1062no—
257M16—1062no—
258C2×C8—531no—
259C2×C8—531no—
260C2×C8C22×D5311yescharacteristic
261C2×C8C22×D5311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—531no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4C22×D5311yescharacteristic
273C4⋊C4C22×D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C42—21no—
284C42—1061no—
285C42—1061no—
286C42—1061no—
287C23—531no—
288C23—531no—
289C23—1062no—
290C23—1062no—
291C23—2121no—
292C23D4×D5311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1061no—
296Q8—1061no—
297Q8—2122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—531no—
305C2×C4—532no—
306C2×C4—532no—
307C2×C4—531no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—1062no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4D4×D5311yescharacteristic
322C2×C4C23×D5311yescharacteristic
323C8—21no—
324C8—21no—
325C8—1061no—
326C8—1061no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—532no—
331C22—532no—
332C22—1061no—
333C22—1061no—
334C22—1061no—
335C22—1061no—
336C22—2124no—
337C22C2×D4×D5311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—532no—
344C4—1061no—
345C4—2121no—
346C4—2121no—
347C4—2121no—
348C4—2121no—
349C4#41×D5311yescharacteristic
350C4#41×D5311yescharacteristic
351C2—41no—
352C2—1062no—
353C2—1062no—
354C2#123×D5311yescharacteristic
355C2#137×D5311yescharacteristic
356C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#485C2
socleC22×C53C22×D4
hypercenter#485D53
hypocenterC53#3095
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #16810965C2semidirect
4. #16810964C2semidirect
5. #16810961C2semidirect
6. #16810960C2semidirect
7. #16810957C2semidirect
8. #16810955C2semidirect
9. #16810952C2semidirect
10. #16810664C2semidirect
11. #16810662C2semidirect
12. #16810630C2semidirect
13. #16810630C2semidirect
15. C53×#485C2semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133150C22semidirect
40. #13133149C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133141C22semidirect
46. #13133141C22semidirect
47. #13133140C22semidirect
48. #13133140C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
56. #103×C53C22semidirect
57. #103×C53C22semidirect
184. #485D53semidirect
194. C53#3095semidirect
221. #107C2×D53semidirect
225. #106C2×D53semidirect
232. #103C2×D53semidirect
233. #103C2×D53semidirect