Finite Group Databasebeta

← #28345977

Subgroups — #28345977

Order 13696.

490 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×#16812138C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293190C211yesmaximal, characteristic
4#20293103C211yesmaximal, characteristic, complemented
5#20292850C211yesmaximal, characteristic, complemented
6#20292849C211yesmaximal, characteristic, complemented
7#20292848C211yesmaximal, characteristic, complemented
8#20292847C211yesmaximal, characteristic, complemented
9#20292826C211yesmaximal, characteristic, complemented
10#20292825C211yesmaximal, characteristic, complemented
11#20292824C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic, complemented
14#20292620C211yesmaximal, characteristic, complemented
15#460×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812138—24no—
20C2×Dic428C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133807C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133806C2211yescharacteristic, complemented
25C2×#13133805C2211yescharacteristic
26SD16×D107—24no—
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812105C2211yescharacteristic
30#16812097C2211yescharacteristic
31#16812096C2211yescharacteristic
32#16812081—21no—
33#16812081—21no—
34#16812077—21no—
35#16812077—21no—
36Q8×Dic107C2211yescharacteristic, complemented
37C4×Dic214—21no—
38C4×Dic214C2211yescharacteristic, complemented
39#16812074—21no—
40#16812074C2211yescharacteristic, complemented
41#16812067—21no—
42#16812067C2211yescharacteristic
43#16812066—21no—
44#16812066C2211yescharacteristic
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic
51#16812030C2211yescharacteristic
52#16812030C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55C2×SD16×C107C2211yescharacteristic, complemented
56#127×C107C2211yescharacteristic
57C4×Q8×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—22no—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107D414yescomplemented
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74D4×D107—22no—
75D4×D107—22no—
76C22×Dic107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79Dic428—22no—
80#13133808—22no—
81#13133807—22no—
82#13133806—22no—
83#13133806—22no—
84#13133805—22no—
85#13133801—22no—
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C8×D107—22no—
89C4⋊Dic107—21no—
90C4⋊Dic107C2311yescharacteristic
91Dic107⋊C4—21no—
92Dic107⋊C4—21no—
93Dic107⋊C4D411yescharacteristic
94Dic107⋊C4D411yescharacteristic
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97(C2×D107)⋊C4—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4D411yescharacteristic
101(C2×D107)⋊C4D411yescharacteristic
102C4×Dic107—21no—
103C4×Dic107—21no—
104C4×Dic107—21no—
105C4×Dic107—21no—
106C4×Dic107C2311yescharacteristic
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107—21no—
109C2×D4×C107—21no—
110C2×D4×C107C2311yescharacteristic
111C2×D4×C107C2311yescharacteristic
112SD16×C107—22no—
113SD16×C107—22no—
114C2×C8×C107C2311yescharacteristic
115C2×C8×C107C2311yescharacteristic
116C4⋊C4×C107—21no—
117C4⋊C4×C107C2311yescharacteristic
118C42×C107—21no—
119C42×C107C2311yescharacteristic
120C23×C107—21no—
121C23×C107—21no—
122C22×D107—21no—
123C22×D107—42no—
124C22×D107—81no—
125C22×D107—81no—
126C22×D107C2×D411yescharacteristic
127Q8×C107—21no—
128Q8×C107C2×D412yes—
129D4×C107—21no—
130D4×C107—21no—
131D4×C107—21no—
132D4×C107—21no—
133D4×C107—81no—
134D4×C107—81no—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2411yescharacteristic
140C22⋊D107—42no—
141C22⋊D107—42no—
142C22⋊D107—81no—
143C22⋊D107—81no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—42no—
148C2×Dic107C2×D411yescharacteristic
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151D428—21no—
152D428—21no—
153C4×D107—24no—
154C4×D107C2×D412yes—
155C4×D107#4112yes—
156Dic214—21no—
157Dic214—24no—
158Dic214—21no—
159Dic214—21no—
160Dic214C2×D412yes—
161C8×C107—21no—
162C8×C107—21no—
163C107⋊C8—21no—
164C107⋊C8—21no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107—41no—
169C22×C107—41no—
170C22×C107—41no—
171C22×C107C22×D411yescharacteristic
172C2×D107—21no—
173C2×D107—21no—
174C2×D107—41no—
175C2×D107—41no—
176C2×D107—41no—
177C2×D107—84no—
178C2×D107#12112yes—
179C4×C107—22no—
180C4×C107—21no—
181C4×C107—21no—
182C4×C107—41no—
183C4×C107C22×D411yescharacteristic
184C4×C107C2×#4111yescharacteristic
185Dic107—22no—
186Dic107—21no—
187Dic107—21no—
188Dic107—41no—
189Dic107—41no—
190Dic107#12112yes—
191C2×C107—81no—
192C2×C107—81no—
193C2×C107C2×#12111yescharacteristic
194C2×C107#57711yescharacteristic
195C2×C107#57711yescharacteristic
196D107—22no—
197D107—81no—
198#3043—1071nomaximal, Sylow
199C107#304311yescharacteristic, complemented, Sylow
200C2×#137—1071no—
201C22×SD16—1071no—
202#535—1071no—
203#519—1071no—
204#464—1071no—
205#464—1071no—
206#464—1071no—
207#464—1071no—
208#460—1071no—
209#460—1071no—
210#460—1071no—
211#460D10711yescharacteristic, complemented
212#424—1071no—
213#418—1071no—
214#416—1071no—
215C2×#41—1071no—
216C22×Q8—1071no—
217C22×D4—1071no—
218#137—2144no—
219C2×Q16—1071no—
220C2×Q16—1071no—
221C2×SD16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—1071no—
225C2×SD16—2144no—
226C2×SD16C2×D10711yescharacteristic
227C2×SD16C2×D10711yescharacteristic, complemented
228C2×M16—1071no—
229C22×C8—1071no—
230#127—1071no—
231#127C2×D10711yescharacteristic
232#124—1071no—
233#124—1071no—
234#121—2141no—
235#121—2141no—
236#120—2141no—
237#120—2141no—
238C4×Q8—1071no—
239C4×Q8—1071no—
240C4×Q8—1071no—
241C4×Q8—2141no—
242C4×Q8—2141no—
243C4×Q8C2×D10711yescharacteristic, complemented
244#117—1071no—
245#117—1071no—
246#117—2141no—
247#117—2141no—
248C4⋊C8—1071no—
249C4⋊C8—1071no—
250C4⋊C8—1071no—
251C4⋊C8C2×D10711yescharacteristic, complemented
252#103—1071no—
253#103—1071no—
254#102—1071no—
255#102—1071no—
256#102—1071no—
257#102—1071no—
258#102C2×D10711yescharacteristic
259#102C2×D10711yescharacteristic, complemented
260C24—2141no—
261#41—2142no—
262#41—2142no—
263C2×Q8—1074no—
264C2×Q8—1071no—
265C2×Q8—1071no—
266C2×Q8—2141no—
267C2×Q8—2142no—
268C2×Q8C22×D10711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—1071no—
272C2×D4—2142no—
273C2×D4—2142no—
274C2×D4—2141no—
275C2×D4—2141no—
276C2×D4—2141no—
277C2×D4—2141no—
278C2×D4C22×D10711yescharacteristic
279C2×D4C22×D10711yescharacteristic
280C22×C4—1071no—
281C22×C4—2141no—
282C22×C4—2141no—
283Q16—2142no—
284Q16—2142no—
285SD16—22no—
286SD16—22no—
287SD16—2142no—
288SD16—2142no—
289SD16—2142no—
290SD16—2142no—
291M16—2142no—
292C2×C8—1071no—
293C2×C8—1071no—
294C2×C8—2142no—
295C2×C8C22×D10711yescharacteristic
296C2×C8C22×D10711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—1071no—
299C4⋊C4—1071no—
300C4⋊C4—1071no—
301C4⋊C4—2141no—
302C4⋊C4—2141no—
303C4⋊C4—2141no—
304C4⋊C4C22×D10711yescharacteristic
305C22⋊C4—1071no—
306C22⋊C4—1071no—
307C22⋊C4—2141no—
308C22⋊C4—2141no—
309C22⋊C4—2141no—
310C22⋊C4—2141no—
311C22⋊C4—2141no—
312C42—21no—
313C42—1071no—
314C42—2141no—
315C42—2141no—
316C42—2141no—
317C42—2141no—
318C42C22×D10711yescharacteristic
319C23—21no—
320C23—21no—
321C23—8561no—
322C23—8561no—
323C23—1071no—
324C23—4282no—
325C23—2141no—
326Q8—21no—
327Q8—1072no—
328Q8—2144no—
329Q8—2141no—
330Q8—2141no—
331Q8—2141no—
332Q8D4×D10712yes—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—81no—
338D4—81no—
339D4—8561no—
340D4—8561no—
341D4—4282no—
342D4—4282no—
343D4—2141no—
344D4—2141no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—1071no—
348C2×C4—1072no—
349C2×C4—1072no—
350C2×C4—1071no—
351C2×C4—1071no—
352C2×C4—4282no—
353C2×C4—2144no—
354C2×C4—2141no—
355C2×C4—2141no—
356C2×C4—2141no—
357C2×C4D4×D10711yescharacteristic
358C2×C4D4×D10711yescharacteristic
359C2×C4C23×D10711yescharacteristic
360C8—21no—
361C8—21no—
362C8—2141no—
363C8—2141no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—8564no—
371C22—1072no—
372C22—4281no—
373C22—4281no—
374C22—4281no—
375C22—2141no—
376C22—2141no—
377C22C2×D4×D10711yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—1072no—
383C4—4281no—
384C4—4281no—
385C4—2141no—
386C4—2141no—
387C4—2142no—
388C4C2×D4×D10711yescharacteristic
389C4#41×D10711yescharacteristic
390C2—81no—
391C2—81no—
392C2—8561no—
393C2—2142no—
394C2#121×D10711yescharacteristic
395C2#2029343311yescharacteristic
396C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293190C2
hypercenter#460D107
hypocenterC107#3043
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. C2×SD16×D107C2semidirect
4. #20293103C2semidirect
5. #20292850C2semidirect
6. #20292849C2semidirect
7. #20292848C2semidirect
8. #20292847C2semidirect
9. #20292826C2semidirect
10. #20292825C2semidirect
11. #20292824C2semidirect
12. #20292652C2semidirect
13. #20292630C2semidirect
14. #20292620C2semidirect
15. #460×C107C2semidirect
20. C2×Dic428C22semidirect
21. C2×#13133808C22semidirect
24. C2×#13133806C22semidirect
36. Q8×Dic107C22semidirect
38. C4×Dic214C22semidirect
40. #16812074C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812032C22semidirect
52. #16812030C22semidirect
55. C2×SD16×C107C22semidirect
57. C4×Q8×C107C22semidirect
58. C4⋊C8×C107C22semidirect
60. #102×C107C22semidirect
68. Q8×D107D4semidirect
199. C107#3043semidirect
211. #460D107semidirect
227. C2×SD16C2×D107semidirect
243. C4×Q8C2×D107semidirect
251. C4⋊C8C2×D107semidirect
259. #102C2×D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168121382
C2×SD16×D1072
#202931902
#202931032
#202928502
#202928492
#202928482
#202928472
#202928262
#202928252
#202928242
#202926522
#202926302
#202926202
#460×C1072
#3043107