Finite Group Databasebeta

← #20284206

Subgroups — #20284206

Order 6784.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133249C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic
3#124×D53C211yesmaximal, characteristic, complemented
4C4×#11719949C211yesmaximal, characteristic, complemented
5#16810847C211yesmaximal, characteristic, complemented
6#16810845C211yesmaximal, characteristic, complemented
7#16810843C211yesmaximal, characteristic, complemented
8#16810841C211yesmaximal, characteristic, complemented
9#16810839C211yesmaximal, characteristic, complemented
10#16810837C211yesmaximal, characteristic, complemented
11#16810835C211yesmaximal, characteristic, complemented
12#16810645C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810626C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133249—24no—
20#13133244—24no—
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719913C2211yescharacteristic, complemented
23C2×#11719924C2211yescharacteristic
24C2×#11719924C2211yescharacteristic, complemented
25C2×D424C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×#11719912C2211yescharacteristic
29#13133206C2211yescharacteristic
30#13133205C2211yescharacteristic
31#13133204C2211yescharacteristic
32#13133203—21no—
33#13133203—21no—
34Q8×Dic53C2211yescharacteristic, complemented
35C4×D212—21no—
36C4×D212C2211yescharacteristic, complemented
37#13133179—21no—
38#13133179C2211yescharacteristic, complemented
39#13133175—21no—
40#13133175C2211yescharacteristic
41C22⋊C4×D53—21no—
42C22⋊C4×D53—21no—
43C42×D53—21no—
44C42×D53C2211yescharacteristic
45C4⋊#88322C2211yescharacteristic, complemented
46#13133146C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133141C2211yescharacteristic, complemented
50#13133140C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic
52#13133138C2211yescharacteristic
53#13133137C2211yescharacteristic
54C2×Q16×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic
57C4×Q8×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66#11719949—22no—
67#11719949D414yes—
68Q8×D53—22no—
69Q8×D53—22no—
70D4×D53—22no—
71D4×D53—22no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53D411yescharacteristic
80C2×C4×D53D411yescharacteristic
81C2×C4×D53C2311yescharacteristic
82C2×Dic106—21no—
83C2×Dic106C2311yescharacteristic
84Q16×C53—22no—
85SD16×C53—22no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53—21no—
89C4⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53—21no—
93C42×C53C2311yescharacteristic
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96#11719925—22no—
97#11719924—22no—
98#11719924—22no—
99#11719922—22no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53C2311yescharacteristic
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53D411yescharacteristic
112C4×Dic53D411yescharacteristic
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116D424—22no—
117#11719913—22no—
118#11719912—22no—
119#11719912—22no—
120C23×C53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—42no—
124C22×D53—42no—
125C22×D53—42no—
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53C2×D412yes—
131D4×C53—21no—
132D4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2411yescharacteristic
139C22⋊D53—42no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—24no—
146D212—21no—
147D212—21no—
148D212—21no—
149D212C2×D412yes—
150C4×D53—24no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53—42no—
156C4×D53C2×D412yes—
157C4×D53#4112yes—
158Dic106—21no—
159Dic106—21no—
160Dic106—42no—
161C8×C53—21no—
162C8×C53—21no—
163#88322—21no—
164#88322—21no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53C22×D411yescharacteristic
169C2×D53—21no—
170C2×D53—21no—
171C2×D53—22no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—84no—
178C2×D53#12112yes—
179C4×C53—22no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53C22×D411yescharacteristic
185C4×C53C2×#4111yescharacteristic
186Dic53—21no—
187Dic53—21no—
188Dic53—41no—
189Dic53—41no—
190Dic53#12112yes—
191#3040—531nomaximal, Sylow
192C2×C53—81no—
193C2×C53C2×#12111yescharacteristic
194C2×C53#57511yescharacteristic
195C2×C53#57511yescharacteristic
196D53—22no—
197D53—42no—
198D53—81no—
199C2×#137—531no—
200C2×AGL(1,ℤ/8)—531no—
201C2×#124—531no—
202C4×#41—531no—
203#464—531no—
204#464—531no—
205#464—531no—
206#464D5311yescharacteristic, complemented
207#463—531no—
208#463—531no—
209#463—531no—
210#463—531no—
211#423—531no—
212#419—531no—
213#418—531no—
214C53#304011yescharacteristic, complemented, Sylow
215C2×#41—531no—
216C22×Q8—531no—
217C22×D4—531no—
218#137—1064no—
219AGL(1,ℤ/8)—1064no—
220C2×Q16—531no—
221C2×Q16C2×D5311yescharacteristic, complemented
222C2×SD16—531no—
223C2×SD16—531no—
224C2×SD16—531no—
225C2×SD16C2×D5311yescharacteristic
226C2×D8—531no—
227C2×D8—531no—
228C2×M16—531no—
229C2×M16—531no—
230#124—531no—
231#124—531no—
232#124—531no—
233#124—1061no—
234#124—1061no—
235#124C2×D5311yescharacteristic
236C4×Q8—531no—
237C4×Q8C2×D5311yescharacteristic, complemented
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—1061no—
241C4×D4—1061no—
242#117—531no—
243#117—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42—531no—
247C2×C42—1061no—
248C4⋊C8—531no—
249C4⋊C8—531no—
250C4⋊C8—531no—
251C4⋊C8C2×D5311yescharacteristic, complemented
252#103—531no—
253#103—531no—
254#103—531no—
255#103C2×D5311yescharacteristic, complemented
256#102—531no—
257#102—531no—
258#102—531no—
259#102C2×D5311yescharacteristic
260C24—1061no—
261#41—534no—
262#41—1062no—
263C2×Q8—531no—
264C2×Q8—1061no—
265C2×Q8—1062no—
266C2×Q8—1062no—
267C2×Q8C22×D5311yescharacteristic
268C2×Q8C22×D5311yescharacteristic
269C2×D4—531no—
270C2×D4—531no—
271C2×D4—1061no—
272C2×D4—1062no—
273C2×D4—1062no—
274C2×D4—1061no—
275C2×D4C22×D5311yescharacteristic
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283Q16—22no—
284Q16—1062no—
285SD16—22no—
286SD16—1062no—
287SD16—1062no—
288SD16—1062no—
289D8—1062no—
290D8—1062no—
291M16—1062no—
292M16—1062no—
293C2×C8—531no—
294C2×C8—531no—
295C2×C8C22×D5311yescharacteristic
296C2×C8C22×D5311yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—531no—
299C4⋊C4—1061no—
300C4⋊C4C22×D5311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4—1061no—
307C22⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C42—21no—
313C42—531no—
314C42—531no—
315C42—531no—
316C42—1061no—
317C42—1061no—
318C42C22×D5311yescharacteristic
319C23—21no—
320C23—531no—
321C23—1061no—
322C23—1061no—
323C23—2122no—
324C23—2122no—
325C23—2122no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—1061no—
330Q8—1061no—
331Q8—2122no—
332Q8D4×D5312yes—
333D4—21no—
334D4—21no—
335D4—532no—
336D4—1064no—
337D4—1061no—
338D4—1061no—
339D4—1061no—
340D4—2122no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—531no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—531no—
348C2×C4—531no—
349C2×C4—1064no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4D4×D5311yescharacteristic
358C2×C4D4×D5311yescharacteristic
359C2×C4C23×D5311yescharacteristic
360C8—21no—
361C8—21no—
362C8—1061no—
363C8—1061no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—532no—
368C22—1061no—
369C22—1061no—
370C22—1062no—
371C22—4244no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22—2121no—
376C22—2121no—
377C22C2×D4×D5311yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—532no—
384C4—1061no—
385C4—1061no—
386C4—2121no—
387C4—2121no—
388C4C2×D4×D5311yescharacteristic
389C4#41×D5311yescharacteristic
390C2—81no—
391C2—1062no—
392C2—4241no—
393C2—2122no—
394C2#121×D5311yescharacteristic
395C2#1681142011yescharacteristic
396C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3040
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#13133249C2semidirect
3. #124×D53C2semidirect
4. C4×#11719949C2semidirect
5. #16810847C2semidirect
6. #16810845C2semidirect
7. #16810843C2semidirect
8. #16810841C2semidirect
9. #16810839C2semidirect
10. #16810837C2semidirect
11. #16810835C2semidirect
12. #16810645C2semidirect
13. #16810630C2semidirect
14. #16810626C2semidirect
15. C53×#464C2semidirect
21. C2×#11719925C22semidirect
22. C2×#11719913C22semidirect
24. C2×#11719924C22semidirect
34. Q8×Dic53C22semidirect
36. C4×D212C22semidirect
38. #13133179C22semidirect
45. C4⋊#88322C22semidirect
46. #13133146C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133141C22semidirect
50. #13133140C22semidirect
54. C2×Q16×C53C22semidirect
57. C4×Q8×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #103×C53C22semidirect
206. #464D53semidirect
214. C53#3040semidirect
221. C2×Q16C2×D53semidirect
237. C4×Q8C2×D53semidirect
251. C4⋊C8C2×D53semidirect
255. #103C2×D53semidirect