Finite Group Databasebeta

← #20284225

Subgroups — #20284225

Order 6784.

494 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×#13133241C211yesmaximal, characteristic
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3#16811193C211yesmaximal, characteristic, complemented
4#16811095C211yesmaximal, characteristic, complemented
5#16810837C211yesmaximal, characteristic, complemented
6#16810836C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810829C211yesmaximal, characteristic, complemented
9#16810828C211yesmaximal, characteristic, complemented
10#16810827C211yesmaximal, characteristic, complemented
11#16810826C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810616C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133241—24no—
20C2×#11719913C2211yescharacteristic
21C2×#11719913C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719924C2211yescharacteristic
25SD16×D53—24no—
26C2×#11719922C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133217C2211yescharacteristic, complemented
30#13133215C2211yescharacteristic, complemented
31#13133204C2211yescharacteristic, complemented
32#13133199—21no—
33#13133199—21no—
34#13133194—21no—
35#13133194—21no—
36C4×D212C2211yescharacteristic
37D4×Dic53C2211yescharacteristic
38#13133179C2211yescharacteristic
39C4×C22⋊D53—21no—
40#13133177—21no—
41#13133175C2211yescharacteristic
42#13133174C2211yescharacteristic
43C2×#11719921—21no—
44C22⋊C4×D53—21no—
45C4⋊#88322C2211yescharacteristic, complemented
46#13133146C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133142C2211yescharacteristic, complemented
50#13133141C2211yescharacteristic, complemented
51#13133140C2211yescharacteristic, complemented
52#13133140C2211yescharacteristic, complemented
53#13133138C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C2×D8×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic, complemented
57C4×D4×C53C2211yescharacteristic
58C4⋊C8×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70Q8×D53—22no—
71Q8×D53—22no—
72#11719947—22no—
73#11719947—22no—
74D4×D53—22no—
75D4×D53D414yes—
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106C2311yescharacteristic
83C2×Dic106C2311yescharacteristic
84SD16×C53—22no—
85D8×C53—22no—
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96#11719924—22no—
97#11719923—22no—
98#11719923—22no—
99#11719922—22no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921D411yescharacteristic
104#11719921D411yescharacteristic
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110#11719919D411yescharacteristic
111#11719919D411yescharacteristic
112C4×Dic53—21no—
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116#11719913—22no—
117#11719913—22no—
118#11719912—22no—
119C8×D53—22no—
120C23×C53—21no—
121C23×C53—21no—
122C22×D53—21no—
123C22×D53—24no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53C2×D411yescharacteristic
127Q8×C53—21no—
128Q8×C53—21no—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53C2×D412yes—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—81no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2411yescharacteristic
139C22⋊D53—24no—
140C22⋊D53—42no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—42no—
145C2×Dic53—81no—
146C2×Dic53C2×D411yescharacteristic
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149D212—21no—
150D212C2×D412yes—
151C4×D53—42no—
152C4×D53—81no—
153C4×D53—81no—
154C4×D53C2×D412yes—
155C4×D53#4112yes—
156Dic106—21no—
157Dic106—21no—
158Dic106—21no—
159Dic106—21no—
160Dic106—42no—
161C8×C53—21no—
162C8×C53—21no—
163#88322—21no—
164#88322—21no—
165C22×C53—22no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53C22×D411yescharacteristic
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—22no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—84no—
178C2×D53#12112yes—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53C22×D411yescharacteristic
184C4×C53C2×#4111yescharacteristic
185Dic53—21no—
186Dic53—21no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190Dic53#12112yes—
191#3042—531nomaximal, Sylow
192C2×C53—42no—
193C2×C53—81no—
194C2×C53C2×#12111yescharacteristic
195C2×C53#57711yescharacteristic
196C2×C53#57711yescharacteristic
197D53—22no—
198D53—42no—
199C2×AGL(1,ℤ/8)—531no—
200C22×SD16—531no—
201#536—531no—
202#518—531no—
203#463—531no—
204#463—531no—
205#463—531no—
206#463D5311yescharacteristic, complemented
207#461—531no—
208#461—531no—
209#461—531no—
210#461—531no—
211#424—531no—
212#419—531no—
213#416—531no—
214C53#304211yescharacteristic, complemented, Sylow
215C2×#41—531no—
216C22×Q8—531no—
217C22×D4—531no—
218AGL(1,ℤ/8)—1064no—
219C2×SD16—531no—
220C2×SD16—531no—
221C2×SD16—531no—
222C2×SD16—531no—
223C2×SD16—531no—
224C2×SD16—1064no—
225C2×SD16C2×D5311yescharacteristic
226C2×D8—531no—
227C2×D8C2×D5311yescharacteristic
228C2×M16—531no—
229C22×C8—531no—
230C4⋊Q8—531no—
231C4⋊Q8—531no—
232#124—531no—
233#124C2×D5311yescharacteristic, complemented
234#123—1061no—
235#123—1061no—
236C22⋊Q8—1061no—
237C22⋊Q8—1061no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—531no—
241C4×D4—1061no—
242C4×D4—1061no—
243C4×D4C2×D5311yescharacteristic
244#117—531no—
245#117—531no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C4⋊C8—531no—
249C4⋊C8—531no—
250C4⋊C8—531no—
251C4⋊C8C2×D5311yescharacteristic, complemented
252#103—531no—
253#103—531no—
254#103—531no—
255#103—531no—
256#103—531no—
257#103C2×D5311yescharacteristic, complemented
258#102—531no—
259#102C2×D5311yescharacteristic, complemented
260C24—1061no—
261#41—1062no—
262#41—1062no—
263C2×Q8—531no—
264C2×Q8—531no—
265C2×Q8—1061no—
266C2×Q8—1062no—
267C2×Q8—1062no—
268C2×Q8C22×D5311yescharacteristic
269C2×D4—534no—
270C2×D4—531no—
271C2×D4—1062no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4C22×D5311yescharacteristic
275C2×D4C22×D5311yescharacteristic
276C22×C4—21no—
277C22×C4—531no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283SD16—22no—
284SD16—1062no—
285SD16—1062no—
286SD16—1062no—
287SD16—1062no—
288SD16—1062no—
289D8—22no—
290D8—1062no—
291M16—1062no—
292C2×C8—531no—
293C2×C8—531no—
294C2×C8—1062no—
295C2×C8C22×D5311yescharacteristic
296C2×C8C22×D5311yescharacteristic
297C4⋊C4—531no—
298C4⋊C4—531no—
299C4⋊C4—531no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4C22×D5311yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—531no—
309C22⋊C4—531no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C42—531no—
317C42—1061no—
318C42C22×D5311yescharacteristic
319C23—21no—
320C23—21no—
321C23—531no—
322C23—1064no—
323C23—1061no—
324C23—4241no—
325C23—4241no—
326Q8—21no—
327Q8—21no—
328Q8—1061no—
329Q8—1061no—
330Q8—1061no—
331Q8—1061no—
332Q8—2122no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—532no—
337D4—1064no—
338D4—1061no—
339D4—2122no—
340D4D4×D5312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—81no—
344C2×C4—531no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—531no—
348C2×C4—531no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4D4×D5311yescharacteristic
358C2×C4D4×D5311yescharacteristic
359C2×C4C23×D5311yescharacteristic
360C8—21no—
361C8—21no—
362C8—1061no—
363C8—1061no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—532no—
371C22—1061no—
372C22—1061no—
373C22—1062no—
374C22—4244no—
375C22—2121no—
376C22—2121no—
377C22C2×D4×D5311yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—532no—
383C4—1061no—
384C4—1061no—
385C4—2121no—
386C4—2121no—
387C4—2121no—
388C4C2×D4×D5311yescharacteristic
389C4#41×D5311yescharacteristic
390C2—42no—
391C2—81no—
392C2—1062no—
393C2—2122no—
394C2#121×D5311yescharacteristic
395C2#1681143011yescharacteristic
396C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3042
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D53C2semidirect
3. #16811193C2semidirect
4. #16811095C2semidirect
5. #16810837C2semidirect
6. #16810836C2semidirect
7. #16810835C2semidirect
8. #16810829C2semidirect
9. #16810828C2semidirect
10. #16810827C2semidirect
11. #16810826C2semidirect
12. #16810648C2semidirect
13. #16810630C2semidirect
14. #16810616C2semidirect
15. C53×#463C2semidirect
29. #13133217C22semidirect
30. #13133215C22semidirect
31. #13133204C22semidirect
45. C4⋊#88322C22semidirect
46. #13133146C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133142C22semidirect
50. #13133141C22semidirect
51. #13133140C22semidirect
52. #13133140C22semidirect
53. #13133138C22semidirect
56. #124×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
206. #463D53semidirect
214. C53#3042semidirect
233. #124C2×D53semidirect
251. C4⋊C8C2×D53semidirect
257. #103C2×D53semidirect
259. #102C2×D53semidirect