Finite Group Databasebeta

← #28345980

Subgroups — #28345980

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×#16812141C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293195C211yesmaximal, characteristic
4#20293103C211yesmaximal, characteristic, complemented
5#20292852C211yesmaximal, characteristic, complemented
6#20292851C211yesmaximal, characteristic, complemented
7#20292846C211yesmaximal, characteristic, complemented
8#20292828C211yesmaximal, characteristic, complemented
9#20292827C211yesmaximal, characteristic, complemented
10#20292826C211yesmaximal, characteristic, complemented
11#20292825C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic, complemented
14#20292622C211yesmaximal, characteristic, complemented
15#464×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812141—24no—
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133807C2211yescharacteristic
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic, complemented
26SD16×D107—24no—
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812106C2211yescharacteristic
30#16812105C2211yescharacteristic
31#16812098C2211yescharacteristic
32#16812082—21no—
33#16812082—21no—
34#16812079—21no—
35#16812079—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#16812033C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic
52#16812029C2211yescharacteristic
53#16812029C2211yescharacteristic, complemented
54C2×Q16×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic
57C4×Q8×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic
61C23×D107—21no—
62#13133819—22no—
63#13133819—22no—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66Q8×D107—22no—
67Q8×D107D414yescomplemented
68C2×D428—21no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76D4×D107—22no—
77D4×D107—22no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107C2311yescharacteristic
81#13133808—22no—
82#13133807—22no—
83#13133807—22no—
84#13133806—22no—
85#13133805—22no—
86#13133805—22no—
87#13133801—22no—
88C2×C107⋊C8C2311yescharacteristic
89C2×C107⋊C8C2311yescharacteristic
90C8×D107—22no—
91C4⋊Dic107—21no—
92C4⋊Dic107C2311yescharacteristic
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4D411yescharacteristic
96Dic107⋊C4D411yescharacteristic
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×Q8×C107C2311yescharacteristic
109C2×D4×C107C2311yescharacteristic
110Q16×C107—22no—
111SD16×C107—22no—
112C2×C8×C107C2311yescharacteristic
113C2×C8×C107C2311yescharacteristic
114C4⋊C4×C107—21no—
115C4⋊C4×C107C2311yescharacteristic
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C42×C107—21no—
119C42×C107C2311yescharacteristic
120C23×C107—21no—
121C22×D107—21no—
122C22×D107—21no—
123C22×D107—42no—
124C22×D107—81no—
125C22×D107—81no—
126C22×D107C2×D411yescharacteristic
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107C2×D412yes—
131D4×C107—21no—
132D4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2411yescharacteristic
139C22⋊D107—42no—
140C22⋊D107—81no—
141C22⋊D107—81no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107C2×D411yescharacteristic
145C2×Dic107C2×D411yescharacteristic
146C2×Dic107C2×D411yescharacteristic
147D428—21no—
148D428—21no—
149D428—21no—
150D428—21no—
151D428—42no—
152D428—81no—
153D428—81no—
154C4×D107—24no—
155C4×D107—42no—
156C4×D107C2×D412yes—
157C4×D107#4112yes—
158Dic214—21no—
159Dic214—24no—
160Dic214C2×D412yes—
161C8×C107—21no—
162C8×C107—21no—
163C107⋊C8—21no—
164C107⋊C8—21no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107C22×D411yescharacteristic
169C2×D107—21no—
170C2×D107—21no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—41no—
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×C107—41no—
184C4×C107C22×D411yescharacteristic
185C4×C107C2×#4111yescharacteristic
186Dic107—22no—
187Dic107—21no—
188Dic107—21no—
189Dic107—41no—
190Dic107#12112yes—
191C2×C107—81no—
192C2×C107C2×#12111yescharacteristic
193C2×C107#57711yescharacteristic
194C2×C107#57711yescharacteristic
195D107—22no—
196D107—81no—
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#464D10711yescharacteristic, complemented
208#460—1071no—
209#460—1071no—
210#460—1071no—
211#460—1071no—
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×Q16C2×D10711yescharacteristic, complemented
221C2×SD16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—1071no—
225C2×SD16—1071no—
226C2×SD16—2144no—
227C2×SD16C2×D10711yescharacteristic
228C2×M16—1071no—
229C22×C8—1071no—
230#127—1071no—
231#127—1071no—
232#124—1071no—
233#124C2×D10711yescharacteristic
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#103C2×D10711yescharacteristic, complemented
254#102—1071no—
255#102—1071no—
256#102—1071no—
257#102—1071no—
258#102—1071no—
259#102C2×D10711yescharacteristic
260C24—2141no—
261#41—2142no—
262#41—2142no—
263C2×Q8—1074no—
264C2×Q8—1071no—
265C2×Q8—2141no—
266C2×Q8—2142no—
267C2×Q8C22×D10711yescharacteristic
268C2×Q8C22×D10711yescharacteristic
269C2×D4—1071no—
270C2×D4—1071no—
271C2×D4—2141no—
272C2×D4—2142no—
273C2×D4—2142no—
274C2×D4—2141no—
275C2×D4—2141no—
276C2×D4—2141no—
277C2×D4—2141no—
278C2×D4—2141no—
279C2×D4C22×D10711yescharacteristic
280C22×C4—1071no—
281C22×C4—2141no—
282C22×C4—2141no—
283Q16—22no—
284Q16—2142no—
285SD16—22no—
286SD16—2142no—
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—21no—
306C22⋊C4—21no—
307C22⋊C4—1071no—
308C22⋊C4—1071no—
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—8561no—
321C23—8561no—
322C23—1071no—
323C23—4282no—
324C23—2141no—
325C23—2141no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—1072no—
330Q8—2144no—
331Q8—2141no—
332Q8D4×D10712yes—
333D4—21no—
334D4—21no—
335D4—8561no—
336D4—8561no—
337D4—8561no—
338D4—8561no—
339D4—4282no—
340D4—4282no—
341D4—2141no—
342D4—2141no—
343D4—2141no—
344D4—2141no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—1071no—
349C2×C4—1072no—
350C2×C4—1072no—
351C2×C4—1071no—
352C2×C4—1071no—
353C2×C4—4282no—
354C2×C4—2144no—
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—8564no—
368C22—1072no—
369C22—4281no—
370C22—4281no—
371C22—4281no—
372C22—4281no—
373C22—4281no—
374C22—4281no—
375C22—2141no—
376C22—2141no—
377C22C2×D4×D10711yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—1072no—
384C4—4281no—
385C4—2141no—
386C4—2141no—
387C4—2142no—
388C4C2×D4×D10711yescharacteristic
389C4#41×D10711yescharacteristic
390C2—81no—
391C2—8561no—
392C2—8561no—
393C2—2142no—
394C2#121×D10711yescharacteristic
395C2#2029343411yescharacteristic
396C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#464×C107C2
socleC22×C107C22×D4
prime radical#20293195C2
hypercenter#464D107
hypocenterC107#3043
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. C2×SD16×D107C2semidirect
4. #20293103C2semidirect
5. #20292852C2semidirect
6. #20292851C2semidirect
7. #20292846C2semidirect
8. #20292828C2semidirect
9. #20292827C2semidirect
10. #20292826C2semidirect
11. #20292825C2semidirect
12. #20292652C2semidirect
13. #20292630C2semidirect
14. #20292622C2semidirect
15. #464×C107C2semidirect
20. C2×#13133808C22semidirect
22. C2×#13133807C22semidirect
25. C2×#13133805C22semidirect
36. Q8×Dic107C22semidirect
38. C4×Dic214C22semidirect
40. #16812074C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812033C22semidirect
50. #16812031C22semidirect
53. #16812029C22semidirect
54. C2×Q16×C107C22semidirect
57. C4×Q8×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
67. Q8×D107D4semidirect
199. C107#3043semidirect
207. #464D107semidirect
220. C2×Q16C2×D107semidirect
243. C4×Q8C2×D107semidirect
251. C4⋊C8C2×D107semidirect
253. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
C2×SD16×D1072
#202931952
#202931032
#202928522
#202928512
#202928462
#202928282
#202928272
#202928262
#202928252
#202926522
#202926302
#202926222
#464×C1072
#3043107