Finite Group Databasebeta

← #28345971

Subgroups — #28345971

Order 13696.

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×#16812133C211yesmaximal, characteristic
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293197C211yesmaximal, characteristic, complemented
4#20293099C211yesmaximal, characteristic, complemented
5#20292841C211yesmaximal, characteristic, complemented
6#20292840C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292833C211yesmaximal, characteristic, complemented
9#20292832C211yesmaximal, characteristic, complemented
10#20292831C211yesmaximal, characteristic, complemented
11#20292830C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292634C211yesmaximal, characteristic, complemented
14#20292620C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812133—24no—
20C2×#13133807C2211yescharacteristic
21C2×#13133807C2211yescharacteristic
22C2×#13133806C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25SD16×D107—24no—
26C2×#13133803C2211yescharacteristic
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812109C2211yescharacteristic, complemented
30#16812107C2211yescharacteristic, complemented
31#16812096C2211yescharacteristic, complemented
32#16812091—21no—
33#16812091—21no—
34#16812086—21no—
35#16812086—21no—
36C4×D428C2211yescharacteristic
37D4×Dic107C2211yescharacteristic
38#16812071C2211yescharacteristic
39C4×C22⋊D107—21no—
40(C22⋊D107)⋊C4—21no—
41#16812067C2211yescharacteristic
42#16812066C2211yescharacteristic
43C2×(C2×D107)⋊C4—21no—
44C22⋊C4×D107—21no—
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812034C2211yescharacteristic, complemented
50#16812033C2211yescharacteristic, complemented
51#16812032C2211yescharacteristic, complemented
52#16812032C2211yescharacteristic, complemented
53#16812030C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—22no—
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×D107D414yes—
74C22×Dic107—21no—
75C22×Dic107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107—21no—
79C2×C4×D107C2311yescharacteristic
80#13133807—22no—
81#13133807—22no—
82#13133806—22no—
83#13133806—22no—
84#13133805—22no—
85#13133803—22no—
86#13133801—22no—
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C8×D107—22no—
90C4⋊Dic107—21no—
91C4⋊Dic107—21no—
92C4⋊Dic107C2311yescharacteristic
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4D411yescharacteristic
96Dic107⋊C4D411yescharacteristic
97C22⋊Dic107—21no—
98C22⋊Dic107—21no—
99C22⋊Dic107—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4D411yescharacteristic
104(C2×D107)⋊C4D411yescharacteristic
105C4×Dic107—21no—
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—24no—
124C22×D107—81no—
125C22×D107—81no—
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—81no—
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2411yescharacteristic
139C22⋊D107—24no—
140C22⋊D107—42no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—42no—
145C2×Dic107—81no—
146C2×Dic107C2×D411yescharacteristic
147C2×Dic107C2×D411yescharacteristic
148C2×Dic107C2×D411yescharacteristic
149D428—21no—
150D428C2×D412yes—
151C4×D107—42no—
152C4×D107—81no—
153C4×D107—81no—
154C4×D107C2×D412yes—
155C4×D107#4112yes—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—21no—
160Dic214—42no—
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—22no—
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—21no—
186Dic107—21no—
187Dic107—41no—
188Dic107—41no—
189Dic107—41no—
190Dic107#12112yes—
191C2×C107—42no—
192C2×C107—81no—
193C2×C107C2×#12111yescharacteristic
194C2×C107#57711yescharacteristic
195C2×C107#57711yescharacteristic
196D107—22no—
197D107—42no—
198#3042—1071nomaximal, Sylow
199C107#304211yescharacteristic, complemented, Sylow
200C2×AGL(1,ℤ/8)—1071no—
201C22×SD16—1071no—
202#536—1071no—
203#518—1071no—
204#463—1071no—
205#463—1071no—
206#463—1071no—
207#463D10711yescharacteristic, complemented
208#461—1071no—
209#461—1071no—
210#461—1071no—
211#461—1071no—
212#424—1071no—
213#419—1071no—
214#416—1071no—
215C2×#41—1071no—
216C22×Q8—1071no—
217C22×D4—1071no—
218AGL(1,ℤ/8)—2144no—
219C2×SD16—1071no—
220C2×SD16—1071no—
221C2×SD16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—2144no—
225C2×SD16C2×D10711yescharacteristic
226C2×D8—1071no—
227C2×D8C2×D10711yescharacteristic
228C2×M16—1071no—
229C22×C8—1071no—
230C4⋊Q8—1071no—
231C4⋊Q8—1071no—
232#124—1071no—
233#124C2×D10711yescharacteristic, complemented
234#123—2141no—
235#123—2141no—
236C22⋊Q8—2141no—
237C22⋊Q8—2141no—
238C4×D4—1071no—
239C4×D4—1071no—
240C4×D4—1071no—
241C4×D4—2141no—
242C4×D4—2141no—
243C4×D4C2×D10711yescharacteristic
244#117—1071no—
245#117—1071no—
246C2×C22⋊C4—2141no—
247C2×C22⋊C4—2141no—
248C4⋊C8—1071no—
249C4⋊C8—1071no—
250C4⋊C8—1071no—
251C4⋊C8C2×D10711yescharacteristic, complemented
252#103—1071no—
253#103—1071no—
254#103—1071no—
255#103—1071no—
256#103—1071no—
257#103C2×D10711yescharacteristic, complemented
258#102—1071no—
259#102C2×D10711yescharacteristic, complemented
260C24—2141no—
261#41—2142no—
262#41—2142no—
263C2×Q8—1071no—
264C2×Q8—1071no—
265C2×Q8—2141no—
266C2×Q8—2142no—
267C2×Q8—2142no—
268C2×Q8C22×D10711yescharacteristic
269C2×D4—1074no—
270C2×D4—1071no—
271C2×D4—2142no—
272C2×D4—2141no—
273C2×D4—2141no—
274C2×D4C22×D10711yescharacteristic
275C2×D4C22×D10711yescharacteristic
276C22×C4—21no—
277C22×C4—1071no—
278C22×C4—2141no—
279C22×C4—2141no—
280C22×C4—2141no—
281C22×C4—2141no—
282C22×C4—2141no—
283SD16—22no—
284SD16—2142no—
285SD16—2142no—
286SD16—2142no—
287SD16—2142no—
288SD16—2142no—
289D8—22no—
290D8—2142no—
291M16—2142no—
292C2×C8—1071no—
293C2×C8—1071no—
294C2×C8—2142no—
295C2×C8C22×D10711yescharacteristic
296C2×C8C22×D10711yescharacteristic
297C4⋊C4—1071no—
298C4⋊C4—1071no—
299C4⋊C4—1071no—
300C4⋊C4—2141no—
301C4⋊C4—2141no—
302C4⋊C4—2141no—
303C4⋊C4—2141no—
304C4⋊C4C22×D10711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—1071no—
309C22⋊C4—1071no—
310C22⋊C4—2141no—
311C22⋊C4—2141no—
312C22⋊C4—2141no—
313C22⋊C4—2141no—
314C22⋊C4—2141no—
315C22⋊C4—2141no—
316C42—1071no—
317C42—2141no—
318C42C22×D10711yescharacteristic
319C23—21no—
320C23—21no—
321C23—8561no—
322C23—8561no—
323C23—1071no—
324C23—2144no—
325C23—2141no—
326Q8—21no—
327Q8—21no—
328Q8—4282no—
329Q8—2141no—
330Q8—2141no—
331Q8—2141no—
332Q8—2141no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—1072no—
337D4—4282no—
338D4—2144no—
339D4—2141no—
340D4D4×D10712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—81no—
344C2×C4—8561no—
345C2×C4—8561no—
346C2×C4—8561no—
347C2×C4—1071no—
348C2×C4—1072no—
349C2×C4—1072no—
350C2×C4—1071no—
351C2×C4—1071no—
352C2×C4—4282no—
353C2×C4—4282no—
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—2142no—
375C22—2141no—
376C22—2141no—
377C22C2×D4×D10711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—1072no—
383C4—4281no—
384C4—4281no—
385C4—4281no—
386C4—2141no—
387C4—2141no—
388C4C2×D4×D10711yescharacteristic
389C4#41×D10711yescharacteristic
390C2—42no—
391C2—81no—
392C2—4282no—
393C2—2142no—
394C2#121×D10711yescharacteristic
395C2#2029343411yescharacteristic
396C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812133C2
hypercenter#463D107
hypocenterC107#3042
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
2. C2×SD16×D107C2semidirect
3. #20293197C2semidirect
4. #20293099C2semidirect
5. #20292841C2semidirect
6. #20292840C2semidirect
7. #20292839C2semidirect
8. #20292833C2semidirect
9. #20292832C2semidirect
10. #20292831C2semidirect
11. #20292830C2semidirect
12. #20292652C2semidirect
13. #20292634C2semidirect
14. #20292620C2semidirect
15. #463×C107C2semidirect
29. #16812109C22semidirect
30. #16812107C22semidirect
31. #16812096C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812034C22semidirect
50. #16812033C22semidirect
51. #16812032C22semidirect
52. #16812032C22semidirect
53. #16812030C22semidirect
56. #124×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
199. C107#3042semidirect
207. #463D107semidirect
233. #124C2×D107semidirect
251. C4⋊C8C2×D107semidirect
257. #103C2×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×#168121332
C2×SD16×D1072
#202931972
#202930992
#202928412
#202928402
#202928392
#202928332
#202928322
#202928312
#202928302
#202926522
#202926342
#202926202
#463×C1072
#3042107