Finite Group Databasebeta

← #28345950

Subgroups — #28345950

Order 13696.

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×#16812138C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic
3#124×D107C211yesmaximal, characteristic, complemented
4C4×#13133817C211yesmaximal, characteristic, complemented
5#20292852C211yesmaximal, characteristic, complemented
6#20292850C211yesmaximal, characteristic, complemented
7#20292848C211yesmaximal, characteristic, complemented
8#20292846C211yesmaximal, characteristic, complemented
9#20292844C211yesmaximal, characteristic, complemented
10#20292842C211yesmaximal, characteristic, complemented
11#20292840C211yesmaximal, characteristic, complemented
12#20292649C211yesmaximal, characteristic, complemented
13#20292634C211yesmaximal, characteristic, complemented
14#20292630C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812138—24no—
20#16812133—24no—
21C2×Dic428C2211yescharacteristic, complemented
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic
24C2×#13133806C2211yescharacteristic
25C2×#13133806C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic
27C2×#13133801C2211yescharacteristic
28C2×#13133801C2211yescharacteristic
29#16812098C2211yescharacteristic
30#16812097C2211yescharacteristic
31#16812096C2211yescharacteristic
32#16812095—21no—
33#16812095—21no—
34C4×Dic214C2211yescharacteristic, complemented
35#16812074C2211yescharacteristic, complemented
36D4×Dic107C2211yescharacteristic, complemented
37C4×C22⋊D107—21no—
38(C22⋊D107)⋊C4—21no—
39#16812067C2211yescharacteristic
40#16812065—21no—
41C22⋊C4×D107—21no—
42C22⋊C4×D107—21no—
43C2×C4×Dic107—21no—
44C42×D107C2211yescharacteristic
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812033C2211yescharacteristic, complemented
50#16812032C2211yescharacteristic, complemented
51#16812031C2211yescharacteristic
52#16812030C2211yescharacteristic
53#16812029C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic
61C23×D107—21no—
62#13133817—22no—
63#13133817D414yes—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—22no—
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72D4×D107—22no—
73D4×D107—22no—
74C22×Dic107—21no—
75C22×Dic107—21no—
76C2×C4×D107—21no—
77C2×C4×D107D411yescharacteristic
78C2×C4×D107D411yescharacteristic
79C2×C4×D107C2311yescharacteristic
80Dic428—22no—
81#13133808—22no—
82#13133807—22no—
83#13133806—22no—
84#13133806—22no—
85#13133803—22no—
86#13133801—22no—
87#13133801—22no—
88C2×C107⋊C8C2311yescharacteristic
89C2×C107⋊C8C2311yescharacteristic
90C4⋊Dic107C2311yescharacteristic
91Dic107⋊C4—21no—
92Dic107⋊C4—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107—21no—
96(C2×D107)⋊C4—21no—
97(C2×D107)⋊C4—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4—21no—
101C4×Dic107—21no—
102C4×Dic107—21no—
103C4×Dic107—21no—
104C4×Dic107D411yescharacteristic
105C4×Dic107D411yescharacteristic
106C4×Dic107C2311yescharacteristic
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107C2311yescharacteristic
109C2×D4×C107C2311yescharacteristic
110C22×C4×C107—21no—
111SD16×C107—22no—
112D8×C107—22no—
113C2×C8×C107C2311yescharacteristic
114C2×C8×C107C2311yescharacteristic
115C4⋊C4×C107C2311yescharacteristic
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C42×C107C2311yescharacteristic
120C23×C107—21no—
121C23×C107—21no—
122C22×D107—21no—
123C22×D107—42no—
124C22×D107—42no—
125C22×D107—42no—
126C22×D107C2×D411yescharacteristic
127Q8×C107—21no—
128Q8×C107—21no—
129D4×C107—21no—
130D4×C107—21no—
131D4×C107—21no—
132D4×C107C2×D412yes—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—42no—
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2411yescharacteristic
139C22⋊D107—24no—
140C22⋊D107—42no—
141C2×Dic107—21no—
142C2×Dic107—24no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—42no—
146C2×Dic107C2×D411yescharacteristic
147C2×Dic107C2×D411yescharacteristic
148C2×Dic107C2×D411yescharacteristic
149D428—21no—
150D428—21no—
151C4×D107—41no—
152C4×D107—41no—
153C4×D107—42no—
154C4×D107C2×D412yes—
155C4×D107#4112yes—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—42no—
160Dic214C2×D412yes—
161C8×C107—21no—
162C8×C107—21no—
163C107⋊C8—21no—
164C107⋊C8—21no—
165C22×C107—22no—
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—21no—
180C4×C107—21no—
181C4×C107—41no—
182C4×C107—41no—
183C4×C107C22×D411yescharacteristic
184C4×C107C2×#4111yescharacteristic
185Dic107—22no—
186Dic107—21no—
187Dic107—21no—
188Dic107—41no—
189Dic107—41no—
190Dic107#12112yes—
191C2×C107—42no—
192C2×C107—81no—
193C2×C107C2×#12111yescharacteristic
194C2×C107#57511yescharacteristic
195C2×C107#57511yescharacteristic
196D107—22no—
197D107—81no—
198#3040—1071nomaximal, Sylow
199C107#304011yescharacteristic, complemented, Sylow
200C2×#137—1071no—
201C2×AGL(1,ℤ/8)—1071no—
202C2×#124—1071no—
203C4×#41—1071no—
204#464—1071no—
205#464—1071no—
206#464—1071no—
207#464—1071no—
208#463—1071no—
209#463—1071no—
210#463—1071no—
211#463D10711yescharacteristic, complemented
212#423—1071no—
213#419—1071no—
214#418—1071no—
215C2×#41—1071no—
216C22×Q8—1071no—
217C22×D4—1071no—
218#137—2144no—
219AGL(1,ℤ/8)—2144no—
220C2×Q16—1071no—
221C2×Q16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—1071no—
225C2×SD16C2×D10711yescharacteristic, complemented
226C2×D8—1071no—
227C2×D8C2×D10711yescharacteristic
228C2×M16—1071no—
229C2×M16—1071no—
230#124—1071no—
231#124—1071no—
232#124—1071no—
233#124—2141no—
234#124—2141no—
235#124C2×D10711yescharacteristic
236C4×Q8—1071no—
237C4×Q8—1071no—
238C4×D4—1071no—
239C4×D4—2141no—
240C4×D4—2141no—
241C4×D4C2×D10711yescharacteristic, complemented
242#117—1071no—
243#117—2141no—
244C2×C22⋊C4—2141no—
245C2×C22⋊C4—2141no—
246C2×C42—1071no—
247C2×C42—2141no—
248C4⋊C8—1071no—
249C4⋊C8—1071no—
250C4⋊C8—1071no—
251C4⋊C8C2×D10711yescharacteristic, complemented
252#103—1071no—
253#103—1071no—
254#103—1071no—
255#103C2×D10711yescharacteristic, complemented
256#102—1071no—
257#102—1071no—
258#102—1071no—
259#102C2×D10711yescharacteristic
260C24—2141no—
261#41—1074no—
262#41—2142no—
263C2×Q8—1071no—
264C2×Q8—1071no—
265C2×Q8—2141no—
266C2×Q8—2142no—
267C2×Q8—2142no—
268C2×Q8C22×D10711yescharacteristic
269C2×D4—1071no—
270C2×D4—2142no—
271C2×D4—2142no—
272C2×D4—2141no—
273C2×D4—2141no—
274C2×D4C22×D10711yescharacteristic
275C2×D4C22×D10711yescharacteristic
276C22×C4—21no—
277C22×C4—1071no—
278C22×C4—1071no—
279C22×C4—1071no—
280C22×C4—2141no—
281C22×C4—2141no—
282C22×C4—2141no—
283Q16—2142no—
284Q16—2142no—
285SD16—22no—
286SD16—2142no—
287SD16—2142no—
288SD16—2142no—
289D8—22no—
290D8—2142no—
291M16—2142no—
292M16—2142no—
293C2×C8—1071no—
294C2×C8—1071no—
295C2×C8C22×D10711yescharacteristic
296C2×C8C22×D10711yescharacteristic
297C4⋊C4—1071no—
298C4⋊C4—2141no—
299C4⋊C4—2141no—
300C4⋊C4C22×D10711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—2141no—
305C22⋊C4—2141no—
306C22⋊C4—2141no—
307C22⋊C4—2141no—
308C22⋊C4—2141no—
309C22⋊C4—2141no—
310C22⋊C4—2141no—
311C22⋊C4—2141no—
312C42—1071no—
313C42—1071no—
314C42—1071no—
315C42—2141no—
316C42—2141no—
317C42—2141no—
318C42C22×D10711yescharacteristic
319C23—21no—
320C23—21no—
321C23—1071no—
322C23—4282no—
323C23—4282no—
324C23—4282no—
325C23—2141no—
326Q8—21no—
327Q8—21no—
328Q8—1072no—
329Q8—4282no—
330Q8—2141no—
331Q8—2141no—
332Q8—2141no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—4282no—
337D4—2144no—
338D4—2141no—
339D4—2141no—
340D4D4×D10712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—1071no—
345C2×C4—1072no—
346C2×C4—1072no—
347C2×C4—1071no—
348C2×C4—1071no—
349C2×C4—4281no—
350C2×C4—4281no—
351C2×C4—4282no—
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—22no—
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—41no—
381C4—41no—
382C4—1072no—
383C4—4281no—
384C4—4281no—
385C4—2141no—
386C4—2141no—
387C4—2142no—
388C4C2×D4×D10711yescharacteristic
389C4#41×D10711yescharacteristic
390C2—42no—
391C2—81no—
392C2—8561no—
393C2—2142no—
394C2#121×D10711yescharacteristic
395C2#2029342311yescharacteristic
396C2#2029342411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812133C2
hypercenter#463D107
hypocenterC107#3040
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
3. #124×D107C2semidirect
4. C4×#13133817C2semidirect
5. #20292852C2semidirect
6. #20292850C2semidirect
7. #20292848C2semidirect
8. #20292846C2semidirect
9. #20292844C2semidirect
10. #20292842C2semidirect
11. #20292840C2semidirect
12. #20292649C2semidirect
13. #20292634C2semidirect
14. #20292630C2semidirect
15. #463×C107C2semidirect
21. C2×Dic428C22semidirect
22. C2×#13133808C22semidirect
25. C2×#13133806C22semidirect
34. C4×Dic214C22semidirect
35. #16812074C22semidirect
36. D4×Dic107C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812033C22semidirect
50. #16812032C22semidirect
54. C2×SD16×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
199. C107#3040semidirect
211. #463D107semidirect
225. C2×SD16C2×D107semidirect
241. C4×D4C2×D107semidirect
251. C4⋊C8C2×D107semidirect
255. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
C2×#168121332
#124×D1072
C4×#131338172
#202928522
#202928502
#202928482
#202928462
#202928442
#202928422
#202928402
#202926492
#202926342
#202926302
#463×C1072
#3040107