Finite Group Databasebeta

← #20286916

Subgroups — #20286916

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#16811319C211yesmaximal, characteristic, complemented
2#16810977C211yesmaximal, characteristic, complemented
3#16810961C211yesmaximal, characteristic, complemented
4#16810934C211yesmaximal, characteristic, complemented
5#16810923C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810871C211yesmaximal, characteristic, complemented
8#16810835C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810757C211yesmaximal, characteristic, complemented
11C4×#11719922C211yesmaximal, characteristic, complemented
12#16810701C211yesmaximal, characteristic, complemented
13#16810628C211yesmaximal, characteristic, complemented
14#16810624C211yesmaximal, characteristic, complemented
15C53×#546C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic, complemented
17C2×#11719922C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C2×#11719917C2211yescharacteristic, complemented
21#13133212—21no—
22#13133209C2211yescharacteristic, complemented
23#13133199—21no—
24#13133191C2211yescharacteristic
25#13133191C2211yescharacteristic, complemented
26#13133189—21no—
27C4×D212C2211yescharacteristic, complemented
28#13133177—21no—
29#13133176C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31#13133150C2211yescharacteristic, complemented
32#13133150C2211yescharacteristic, complemented
33#13133148C2211yescharacteristic, complemented
34C4⋊#88322C2211yescharacteristic, complemented
35#13133141C2211yescharacteristic, complemented
36#13133141C2211yescharacteristic, complemented
37#13133141C2211yescharacteristic, complemented
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic, complemented
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic, complemented
47C4×#88322C2211yescharacteristic, complemented
48C2×#41×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50#124×C53C2211yescharacteristic
51#123×C53—21no—
52C22⋊Q8×C53C2211yescharacteristic
53#121×C53—21no—
54#121×C53C2211yescharacteristic
55#120×C53—21no—
56C4×D4×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic
58C2×C22⋊C4×C53—21no—
59C24×C53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—22no—
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—41no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C22×Dic53—21no—
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53D411yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53D411yescharacteristic
89C22⋊C4×C53D411yescharacteristic
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92#11719924—41no—
93#11719922—22no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106#11719917—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53C2×D411yescharacteristic
117C23×C53C2×D411yescharacteristic
118C22×D53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—41no—
125D4×C53—42no—
126D4×C53—42no—
127D4×C53—81no—
128D4×C53#4112yes—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53—81no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—42no—
143C22⋊D53—81no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—81no—
148D212—21no—
149D212—21no—
150C4×D53—81no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—22no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—84no—
168C22×C53C22×D411yescharacteristic
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53C2×#4111yescharacteristic
179C4×C53H(5,𝔽2)11yescharacteristic
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#3322—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—41no—
186C2×C53—42no—
187C2×C53—81no—
188C2×C53#54611yescharacteristic
189C2×C53#57511yescharacteristic
190C2×C53#57611yescharacteristic
191D53—81no—
192#553—531no—
193#546D5311yescharacteristic, complemented
194#488—531no—
195#485—531no—
196#480—531no—
197#478—531no—
198#469—531no—
199#468—531no—
200#463—531no—
201#449—531no—
202#447—531no—
203C4×D8—531no—
204#436—531no—
205#418—531no—
206#417—531no—
207C53#332211yescharacteristic, complemented, Sylow
208C2×#41C2×D5311yescharacteristic
209C22×D4C2×D5311yescharacteristic
210C2×SD16—531no—
211C2×D8—531no—
212C2×D8—531no—
213C2×M16—531no—
214C2×M16—531no—
215#126—1061no—
216#125—531no—
217#124C2×D5311yescharacteristic
218#123—21no—
219#123—1061no—
220C22⋊Q8C2×D5311yescharacteristic
221#121—21no—
222#121—531no—
223#121—531no—
224#121—1061no—
225#121C2×D5311yescharacteristic
226#120—21no—
227C4×D4—531no—
228C4×D4—1061no—
229C4×D4C2×D5311yescharacteristic
230C4×D4C2×D5311yescharacteristic
231#117—531no—
232C2×C4⋊C4—531no—
233C2×C22⋊C4—21no—
234#107—531no—
235#107—531no—
236#106—531no—
237C4⋊C8—531no—
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⋊C8—531no—
249C22⋊C8—531no—
250C4×C8—531no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—531no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4C22⋊D5311yescharacteristic
264C2×D4C22×D5311yescharacteristic
265C2×D4C22×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272SD16—2121no—
273D8—1062no—
274D8—2121no—
275M16—2121no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—531no—
280C2×C8—531no—
281C4⋊C4—21no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4C22⋊D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4C22⋊D5311yescharacteristic
300C22⋊C4C22⋊D5311yescharacteristic
301C42—1061no—
302C42C22×D5311yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—1061no—
309C23C2×C22⋊D5311yescharacteristic
310C23C2×C22⋊D5311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—1061no—
321D4—1061no—
322D4—4241no—
323D4—2122no—
324D4#1171994712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4C2×C22⋊D5311yescharacteristic
339C2×C4C2×C22⋊D5311yescharacteristic
340C2×C4C2×C22⋊D5311yescharacteristic
341C2×C4C2×C22⋊D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—1061no—
344C8—1061no—
345C8—1061no—
346C8—1061no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—84no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C22×C22⋊D5311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—2121no—
371C4—2121no—
372C4—2121no—
373C4C2×#1171994711yescharacteristic
374C4#1313326211yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—4241no—
380C2#1681126711yescharacteristic
381C2#1681142111yescharacteristic
382C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#546C2
socleC22×C53C22×D4
hypercenter#546D53
hypocenterC53#3322
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. #16810977C2semidirect
3. #16810961C2semidirect
4. #16810934C2semidirect
5. #16810923C2semidirect
6. #16810874C2semidirect
7. #16810871C2semidirect
8. #16810835C2semidirect
9. #16810768C2semidirect
10. #16810757C2semidirect
11. C4×#11719922C2semidirect
12. #16810701C2semidirect
13. #16810628C2semidirect
14. #16810624C2semidirect
15. C53×#546C2semidirect
16. C2×#11719924C22semidirect
20. C2×#11719917C22semidirect
22. #13133209C22semidirect
25. #13133191C22semidirect
27. C4×D212C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. #13133150C22semidirect
32. #13133150C22semidirect
33. #13133148C22semidirect
34. C4⋊#88322C22semidirect
35. #13133141C22semidirect
36. #13133141C22semidirect
37. #13133141C22semidirect
44. #13133137C22semidirect
46. C22⋊#88322C22semidirect
47. C4×#88322C22semidirect
193. #546D53semidirect
207. C53#3322semidirect