Finite Group Databasebeta

← #20286670

Subgroups — #20286670

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811269C211yesmaximal, characteristic, complemented
2#16810985C211yesmaximal, characteristic, complemented
3#16810953C211yesmaximal, characteristic, complemented
4#16810941C211yesmaximal, characteristic, complemented
5#16810910C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810838C211yesmaximal, characteristic, complemented
9#16810766C211yesmaximal, characteristic, complemented
10#16810756C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810622C211yesmaximal, characteristic, complemented
14C2×#13133137C211yesmaximal, characteristic, complemented
15C53×#547C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×#11719923C2211yescharacteristic, complemented
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719917C2211yescharacteristic, complemented
22C22×#88322C2211yescharacteristic
23#13133205C2211yescharacteristic
24#13133202—21no—
25#13133196C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133190—21no—
28#13133185—21no—
29C4×D212C2211yescharacteristic
30C22⋊C4×D53—21no—
31#13133148C2211yescharacteristic, complemented
32#13133148C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133140C2211yescharacteristic, complemented
36#13133140C2211yescharacteristic, complemented
37#13133140C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic, complemented
39#13133137C2211yescharacteristic
40#13133137C2211yescharacteristic
41#13133137C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47#13133128C2211yescharacteristic, complemented
48C2×#41×C53—21no—
49C4⋊Q8×C53C2211yescharacteristic, complemented
50#123×C53—21no—
51#123×C53—21no—
52C22⋊Q8×C53—21no—
53#121×C53C2211yescharacteristic
54#121×C53C2211yescharacteristic, complemented
55C4×D4×C53C2211yescharacteristic
56C4×D4×C53C2211yescharacteristic, complemented
57C2×C4⋊C4×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60#41×C53—44no—
61C2×Q8×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53D411yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73#11719945—22no—
74#11719945—41no—
75C2×D212—22no—
76C2×D212—21no—
77C2×D212—41no—
78C2×D212C2311yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×Dic106C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53D411yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53D411yescharacteristic
94C22⋊C4×C53D411yescharacteristic
95C42×C53C2311yescharacteristic
96C22⋊Dic53—21no—
97#11719923—41no—
98#11719922—22no—
99#11719922—41no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104C4⋊Dic53C2311yescharacteristic
105#11719919—21no—
106#11719917—41no—
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—81no—
120Q8×C53—42no—
121D4×C53—22no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C22⋊D53—42no—
142C22⋊D53—81no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145D212—21no—
146D212—21no—
147D212—21no—
148D212—42no—
149D212#4112yescomplemented
150C4×D53—42no—
151C4×D53—81no—
152Dic106—21no—
153Dic106—21no—
154#88322—21no—
155#88322—21no—
156#88322—21no—
157#88322—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53AGL(1,ℤ/8)12yes—
167C22×C53C22×D411yescharacteristic
168C2×D53—22no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—84no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53C2×#4111yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—41no—
185Dic53—41no—
186#3306—531nomaximal, Sylow
187C2×C53—22no—
188C2×C53—41no—
189C2×C53—81no—
190C2×C53#54611yescharacteristic
191C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C53#57711yescharacteristic
193D53—42no—
194D53—81no—
195#547D5311yescharacteristic, complemented
196#546—531no—
197#489—531no—
198#483—531no—
199#481—531no—
200#476—531no—
201#469—531no—
202#465—531no—
203#463—531no—
204#449—531no—
205#447—531no—
206#442—531no—
207#435—531no—
208#417—531no—
209C2×#102—531no—
210C53#330611yescharacteristic, complemented, Sylow
211C2×#41—21no—
212C2×#41—531no—
213C22×D4—531no—
214C2×SD16—531no—
215C2×D8—531no—
216C2×D8—531no—
217C2×M16—531no—
218C22×C8—531no—
219C4⋊Q8C2×D5311yescharacteristic, complemented
220#124—531no—
221#123—21no—
222#123—21no—
223#123—1061no—
224C22⋊Q8—21no—
225C22⋊Q8—531no—
226#121—531no—
227#121—1061no—
228#121C2×D5311yescharacteristic
229#121C2×D5311yescharacteristic, complemented
230#120—1061no—
231C4×D4—531no—
232C4×D4C2×D5311yescharacteristic
233C4×D4C2×D5311yescharacteristic, complemented
234C2×C4⋊C4C2×D5311yescharacteristic
235C2×C4⋊C4C2×D5311yescharacteristic, complemented
236C2×C22⋊C4—1061no—
237#106—531no—
238#106—531no—
239#106—531no—
240C4⋊C8—531no—
241#103—531no—
242#103—531no—
243#103—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251C22⋊C8—531no—
252C22⋊C8—531no—
253#97—531no—
254C24—1061no—
255#41—44no—
256#41—1062no—
257#41—2121no—
258C2×Q8—21no—
259C2×Q8—531no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—1062no—
265C2×D4—1061no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—2121no—
269C2×D4C22⋊D5311yescharacteristic
270C2×D4C22×D5311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4C22×D5311yescharacteristic
277C22×C4C22×D5311yescharacteristic
278SD16—2121no—
279D8—1062no—
280D8—2121no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—2121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—531no—
291C4⋊C4—1061no—
292C4⋊C4C22⋊D5311yescharacteristic
293C4⋊C4C22×D5311yescharacteristic
294C4⋊C4C22×D5311yescharacteristic
295C4⋊C4C22×D5311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4C22⋊D5311yescharacteristic
305C22⋊C4C22⋊D5311yescharacteristic
306C42C22×D5311yescharacteristic
307C23—21no—
308C23—1061no—
309C23—1061no—
310C23—4241no—
311C23—2122no—
312C23—2122no—
313C23C2×C22⋊D5311yescharacteristic
314C23C2×C22⋊D5311yescharacteristic
315Q8—42no—
316Q8—1061no—
317Q8—1061no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—532no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—4241no—
328D4—2122no—
329D4—2122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4C2×C22⋊D5311yescharacteristic
345C2×C4C2×C22⋊D5311yescharacteristic
346C2×C4C2×C22⋊D5311yescharacteristic
347C2×C4C2×C22⋊D5311yescharacteristic
348C2×C4C23×D5311yescharacteristic
349C8—1061no—
350C8—1061no—
351C8—1061no—
352C8—1061no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—1062no—
362C22—4244no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22#1313324312yes—
369C22C22×C22⋊D5311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—2121no—
378C4—2121no—
379C4#41×D5311yescharacteristic
380C4#1313326311yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—4241no—
385C2—2122no—
386C2#1681126111yescharacteristic
387C2C2×#1313324311yescharacteristic
388C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#547C2
socleC22×C53C22×D4
hypercenter#547D53
hypocenterC53#3306
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #16811269C2semidirect
2. #16810985C2semidirect
3. #16810953C2semidirect
4. #16810941C2semidirect
5. #16810910C2semidirect
6. #16810874C2semidirect
7. #16810850C2semidirect
8. #16810838C2semidirect
9. #16810766C2semidirect
10. #16810756C2semidirect
11. #16810733C2semidirect
12. #16810695C2semidirect
13. #16810622C2semidirect
14. C2×#13133137C2semidirect
15. C53×#547C2semidirect
18. C2×#11719923C22semidirect
21. C2×#11719917C22semidirect
31. #13133148C22semidirect
32. #13133148C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133140C22semidirect
36. #13133140C22semidirect
37. #13133140C22semidirect
38. #13133138C22semidirect
46. C22⋊#88322C22semidirect
47. #13133128C22semidirect
49. C4⋊Q8×C53C22semidirect
54. #121×C53C22semidirect
56. C4×D4×C53C22semidirect
58. C2×C4⋊C4×C53C22semidirect
149. D212#41semidirect
195. #547D53semidirect
210. C53#3306semidirect
219. C4⋊Q8C2×D53semidirect
229. #121C2×D53semidirect
233. C4×D4C2×D53semidirect
235. C2×C4⋊C4C2×D53semidirect