Finite Group Databasebeta

← #20286919

Subgroups — #20286919

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#16811321C211yesmaximal, characteristic, complemented
2#16811270C211yesmaximal, characteristic, complemented
3#16810976C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810926C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810870C211yesmaximal, characteristic, complemented
8#16810837C211yesmaximal, characteristic, complemented
9#16810770C211yesmaximal, characteristic, complemented
10#16810758C211yesmaximal, characteristic, complemented
11#16810705C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810626C211yesmaximal, characteristic, complemented
14#16810622C211yesmaximal, characteristic, complemented
15C53×#480C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719913C2211yescharacteristic, complemented
19C2×D424C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719917C2211yescharacteristic, complemented
22C2×#11719912C2211yescharacteristic
23#13133212—21no—
24#13133208C2211yescharacteristic, complemented
25#13133204C2211yescharacteristic
26#13133202—21no—
27#13133199—21no—
28#13133197C2211yescharacteristic
29#13133192C2211yescharacteristic
30#13133191—21no—
31#13133191C2211yescharacteristic, complemented
32#13133188—21no—
33#13133185—21no—
34#13133179C2211yescharacteristic
35#13133179C2211yescharacteristic, complemented
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38#13133175C2211yescharacteristic
39C22⋊C4×D53—21no—
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133140C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#107×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53D411yescharacteristic
68D4×D53—22no—
69D4×D53—41no—
70#11719945—22no—
71#11719945—41no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×Dic106C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53D411yescharacteristic
90#11719922—41no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921D411yescharacteristic
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919D411yescharacteristic
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105#11719917—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108D424—22no—
109#11719913—41no—
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—81no—
118C22×D53C2×D411yescharacteristic
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—81no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—81no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—41no—
131C22⋊D53—42no—
132C22⋊D53—42no—
133C22⋊D53—42no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141D212—21no—
142D212—81no—
143D212#4112yes—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—42no—
147C4×D53—81no—
148C4×D53—81no—
149Dic106—21no—
150Dic106—21no—
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—22no—
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53C2×#4111yescharacteristic
177C4×C53H(5,𝔽2)11yescharacteristic
178Dic53—21no—
179Dic53—21no—
180Dic53—21no—
181Dic53—41no—
182Dic53—41no—
183#3322—531nomaximal, Sylow
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#54611yescharacteristic
187C2×C53#57511yescharacteristic
188C2×C53#57611yescharacteristic
189D53—41no—
190D53—42no—
191D53—81no—
192#553—531no—
193#546—531no—
194#488—531no—
195#485—531no—
196#480D5311yescharacteristic, complemented
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×#41—531no—
209C22×D4—531no—
210C2×SD16—531no—
211C2×D8—531no—
212C2×D8—531no—
213C2×M16—531no—
214C2×M16—531no—
215#126—1061no—
216#125—531no—
217#124—531no—
218#123—1061no—
219#123—1061no—
220C22⋊Q8—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224#121—1061no—
225#121C2×D5311yescharacteristic
226#120—1061no—
227C4×D4—531no—
228C4×D4—531no—
229C4×D4—531no—
230C4×D4—1061no—
231#117—531no—
232C2×C4⋊C4C2×D5311yescharacteristic, complemented
233C2×C22⋊C4—1061no—
234#107C2×D5311yescharacteristic, complemented
235#107C2×D5311yescharacteristic, complemented
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#102C2×D5311yescharacteristic
247#102C2×D5311yescharacteristic
248C22⋊C8—531no—
249C22⋊C8C2×D5311yescharacteristic, complemented
250C4×C8—531no—
251C24—1061no—
252#41—1062no—
253#41—2121no—
254C2×Q8—531no—
255C2×D4—21no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1062no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—2121no—
265C2×D4C22×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—531no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272SD16—2121no—
273D8—1062no—
274D8—2121no—
275M16—2121no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—531no—
292C22⋊C4—531no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C42—531no—
302C42—1061no—
303C23—21no—
304C23—531no—
305C23—1061no—
306C23—1061no—
307C23—4241no—
308C23—2122no—
309C23—2122no—
310C23D4×D5311yescharacteristic
311Q8—1061no—
312Q8—1061no—
313D4—21no—
314D4—21no—
315D4—81no—
316D4—532no—
317D4—1061no—
318D4—1061no—
319D4—1061no—
320D4—4241no—
321D4—2121no—
322D4—2122no—
323D4—2122no—
324D4—2122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—81no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—2122no—
341C2×C4D4×D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1061no—
346C8—1061no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—1061no—
354C22—1062no—
355C22—1061no—
356C22—1061no—
357C22—4244no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22—2121no—
362C22—2121no—
363C22C2×D4×D5311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—1061no—
369C4—1061no—
370C4—1061no—
371C4—2121no—
372C4—2121no—
373C4#41×D5311yescharacteristic
374C4#1313326311yescharacteristic
375C2—41no—
376C2—81no—
377C2—4241no—
378C2—2121no—
379C2—2122no—
380C2#1681126511yescharacteristic
381C2#1681142211yescharacteristic
382C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#480C2
socleC22×C53C22×D4
hypercenter#480D53
hypocenterC53#3322
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #16811321C2semidirect
2. #16811270C2semidirect
3. #16810976C2semidirect
4. #16810966C2semidirect
5. #16810926C2semidirect
6. #16810874C2semidirect
7. #16810870C2semidirect
8. #16810837C2semidirect
9. #16810770C2semidirect
10. #16810758C2semidirect
11. #16810705C2semidirect
12. #16810698C2semidirect
13. #16810626C2semidirect
14. #16810622C2semidirect
15. C53×#480C2semidirect
18. C2×#11719913C22semidirect
21. C2×#11719917C22semidirect
24. #13133208C22semidirect
31. #13133191C22semidirect
35. #13133179C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133142C22semidirect
43. #13133140C22semidirect
44. #13133140C22semidirect
49. #13133137C22semidirect
51. C8×Dic53C22semidirect
53. C2×C4⋊C4×C53C22semidirect
54. #107×C53C22semidirect
55. #107×C53C22semidirect
58. C22⋊C8×C53C22semidirect
196. #480D53semidirect
207. C53#3322semidirect
232. C2×C4⋊C4C2×D53semidirect
234. #107C2×D53semidirect
235. #107C2×D53semidirect
249. C22⋊C8C2×D53semidirect