Finite Group Databasebeta

← #20286873

Subgroups — #20286873

Order 6784.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811338C211yesmaximal, characteristic, complemented
2#16811269C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810965C211yesmaximal, characteristic, complemented
5#16810948C211yesmaximal, characteristic, complemented
6#16810904C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810859C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810770C211yesmaximal, characteristic, complemented
11#16810759C211yesmaximal, characteristic, complemented
12#16810736C211yesmaximal, characteristic, complemented
13#16810629C211yesmaximal, characteristic, complemented
14#16810620C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719913C2211yescharacteristic, complemented
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21#13133217C2211yescharacteristic, complemented
22#13133205C2211yescharacteristic
23#13133203—21no—
24#13133203—21no—
25#13133202—21no—
26#13133196C2211yescharacteristic
27#13133191C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic, complemented
30#13133190—21no—
31#13133188—21no—
32#13133185—21no—
33C4×D212C2211yescharacteristic
34C4×D212C2211yescharacteristic, complemented
35#13133176C2211yescharacteristic
36#13133176C2211yescharacteristic, complemented
37#13133175—21no—
38C22⋊C4×D53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133151C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic, complemented
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic, complemented
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic, complemented
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic, complemented
58#97×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53D411yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67#11719945—22no—
68#11719945—41no—
69C2×D212—22no—
70C2×D212—21no—
71C2×D212—41no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×Dic106—21no—
79C2×Dic106C2311yescharacteristic
80M16×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53C2311yescharacteristic
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108D424—22no—
109D424—41no—
110#11719913—41no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—81no—
119D4×C53—41no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—42no—
130C22⋊D53—81no—
131C22⋊D53—81no—
132C22⋊D53—81no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143D212—42no—
144D212#4112yes—
145C4×D53—42no—
146C4×D53—42no—
147C4×D53—81no—
148Dic106—21no—
149Dic106—21no—
150Dic106—81no—
151C8×C53—21no—
152C8×C53—21no—
153C8×C53—21no—
154C8×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53C22×D411yescharacteristic
162C2×D53—22no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—41no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3319—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—41no—
186C2×C53#54611yescharacteristic
187C2×C53#57511yescharacteristic
188C2×C53#57611yescharacteristic
189D53—42no—
190D53—81no—
191D53—81no—
192#555—531no—
193#546—531no—
194#489—531no—
195#485—531no—
196#482—531no—
197#474—531no—
198#468—531no—
199#466—531no—
200#463—531no—
201#449—531no—
202#447—531no—
203#442—531no—
204#435D5311yescharacteristic, complemented
205#418—531no—
206#416—531no—
207C53#331911yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C22×D4—531no—
210C2×SD16—531no—
211C2×D8—531no—
212C2×D8—531no—
213C2×M16C2×D5311yescharacteristic
214C22×C8C2×D5311yescharacteristic, complemented
215C4⋊Q8—531no—
216#124—531no—
217#124—1061no—
218#124—1061no—
219#123—1061no—
220C22⋊Q8—531no—
221#121—531no—
222#121—531no—
223#121—531no—
224#121—1061no—
225#121—1061no—
226#120—1061no—
227C4×D4—531no—
228C4×D4—531no—
229C4×D4C2×D5311yescharacteristic
230#117—531no—
231#117—531no—
232#117—1061no—
233C2×C22⋊C4—1061no—
234#107—531no—
235#107—531no—
236#106—531no—
237C4⋊C8C2×D5311yescharacteristic, complemented
238#103—531no—
239#103—531no—
240#103—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248C22⋊C8C2×D5311yescharacteristic
249C22⋊C8C2×D5311yescharacteristic, complemented
250#97C2×D5311yescharacteristic, complemented
251C24—1061no—
252#41—1062no—
253#41—2121no—
254C2×Q8—531no—
255C2×Q8—1061no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1062no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—2121no—
266C2×D4D21211yescharacteristic
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272SD16—2121no—
273D8—1062no—
274D8—2121no—
275M16—41no—
276C2×C8—41no—
277C2×C8C22×D5311yescharacteristic
278C2×C8C22×D5311yescharacteristic
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4D21211yescharacteristic
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—
297C22⋊C4—1061no—
298C22⋊C4D21211yescharacteristic
299C22⋊C4D21211yescharacteristic
300C42—1061no—
301C42—1061no—
302C42C22×D5311yescharacteristic
303C23—1061no—
304C23—1061no—
305C23—1061no—
306C23—4241no—
307C23—2122no—
308C23—2122no—
309C23C2×D21211yescharacteristic
310C23C2×D21211yescharacteristic
311Q8—1061no—
312Q8—1061no—
313Q8—4241no—
314D4—41no—
315D4—532no—
316D4—1061no—
317D4—1061no—
318D4—1061no—
319D4—1061no—
320D4—1061no—
321D4—4241no—
322D4—4241no—
323D4—4241no—
324D4—2122no—
325D4—2122no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—4241no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4C2×D21211yescharacteristic
339C2×C4C2×D21211yescharacteristic
340C2×C4C2×D21211yescharacteristic
341C2×C4C2×D21211yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—1062no—
354C22—4244no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C22×D21211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—2121no—
369C4—2121no—
370C4—2121no—
371C4—2121no—
372C4—2121no—
373C4#41×D5311yescharacteristic
374C4#1313326211yescharacteristic
375C2—41no—
376C2—41no—
377C2—4241no—
378C2—4241no—
379C2—2122no—
380C2#1681126811yescharacteristic
381C2#1681142311yescharacteristic
382C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3319
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811338C2semidirect
2. #16811269C2semidirect
3. #16810983C2semidirect
4. #16810965C2semidirect
5. #16810948C2semidirect
6. #16810904C2semidirect
7. #16810869C2semidirect
8. #16810859C2semidirect
9. #16810839C2semidirect
10. #16810770C2semidirect
11. #16810759C2semidirect
12. #16810736C2semidirect
13. #16810629C2semidirect
14. #16810620C2semidirect
15. C53×#435C2semidirect
18. C2×#11719913C22semidirect
21. #13133217C22semidirect
29. #13133191C22semidirect
34. C4×D212C22semidirect
36. #13133176C22semidirect
39. #13133151C22semidirect
40. #13133151C22semidirect
41. #13133149C22semidirect
42. #13133142C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
51. #13133139C22semidirect
53. C22×C8×C53C22semidirect
55. C4⋊C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. #97×C53C22semidirect
204. #435D53semidirect
207. C53#3319semidirect
214. C22×C8C2×D53semidirect
237. C4⋊C8C2×D53semidirect
249. C22⋊C8C2×D53semidirect
250. #97C2×D53semidirect