Finite Group Databasebeta

← #28348428

Subgroups — #28348428

Order 13696.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293277C211yesmaximal, characteristic, complemented
2#20293268C211yesmaximal, characteristic, complemented
3#20292988C211yesmaximal, characteristic, complemented
4#20292955C211yesmaximal, characteristic, complemented
5#20292947C211yesmaximal, characteristic, complemented
6#20292916C211yesmaximal, characteristic, complemented
7#20292879C211yesmaximal, characteristic, complemented
8#20292844C211yesmaximal, characteristic, complemented
9#20292771C211yesmaximal, characteristic, complemented
10#20292760C211yesmaximal, characteristic, complemented
11#20292736C211yesmaximal, characteristic, complemented
12#20292698C211yesmaximal, characteristic, complemented
13#20292627C211yesmaximal, characteristic, complemented
14C2×#16812031C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817—21no—
18C22×D428C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C2×#13133801C2211yescharacteristic, complemented
22#16812108C2211yescharacteristic, complemented
23#16812098C2211yescharacteristic
24#16812094—21no—
25#16812091—21no—
26#16812090—21no—
27#16812086—21no—
28#16812084—21no—
29#16812084C2211yescharacteristic
30#16812080C2211yescharacteristic, complemented
31#16812079—21no—
32D4×Dic107C2211yescharacteristic, complemented
33#16812071C2211yescharacteristic
34C4×C22⋊D107C2211yescharacteristic
35C2×C4⋊Dic107C2211yescharacteristic
36C4⋊C4×D107C2211yescharacteristic, complemented
37C22⋊C4×D107—21no—
38#16812040C2211yescharacteristic, complemented
39#16812040C2211yescharacteristic, complemented
40Dic107⋊C8C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812031C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51#16812019C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic, complemented
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133819—22no—
61#13133819—41no—
62#13133817—44no—
63C2×Dic214—21no—
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—41no—
69C2×D428C2311yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107D411yescharacteristic
73C22×Dic107—21no—
74C22×Dic107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79#13133803—22no—
80#13133803—41no—
81#13133801—41no—
82C2×C107⋊C8C2311yescharacteristic
83C2×C107⋊C8C2311yescharacteristic
84C4⋊Dic107—21no—
85C4⋊Dic107C2311yescharacteristic
86C4⋊Dic107C2311yescharacteristic
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92C22⋊Dic107D411yescharacteristic
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97(C2×D107)⋊C4D411yescharacteristic
98C4×Dic107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103SD16×C107—41no—
104C2×C8×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C22⋊C4×C107—21no—
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115C22×D107—21no—
116C22×D107—42no—
117C22×D107—42no—
118C22×D107—81no—
119C22×D107C2×D411yescharacteristic
120Q8×C107—21no—
121Q8×C107—21no—
122D4×C107—21no—
123D4×C107—21no—
124D4×C107—42no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2411yescharacteristic
132C22⋊D107—22no—
133C22⋊D107—42no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—42no—
137C2×Dic107—42no—
138C2×Dic107C2×D411yescharacteristic
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141D428—21no—
142D428—21no—
143D428—21no—
144D428—42no—
145D428—42no—
146D428—81no—
147D428#4112yescomplemented
148C4×D107—21no—
149C4×D107—21no—
150C4×D107—42no—
151C4×D107—42no—
152C4×D107—81no—
153Dic214—42no—
154C8×C107—21no—
155C8×C107—21no—
156C107⋊C8—21no—
157C107⋊C8—21no—
158C22×C107—21no—
159C22×C107—21no—
160C22×C107—41no—
161C22×C107—41no—
162C22×C107—41no—
163C22×C107AGL(1,ℤ/8)12yes—
164C22×C107C22×D411yescharacteristic
165C2×D107—21no—
166C2×D107—22no—
167C2×D107—21no—
168C2×D107—21no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—41no—
174C2×D107—84no—
175C4×C107—21no—
176C4×C107—41no—
177C4×C107—41no—
178C4×C107—41no—
179C4×C107C2×#4111yescharacteristic
180C4×C107H(5,𝔽2)11yescharacteristic
181Dic107—21no—
182Dic107—21no—
183Dic107—21no—
184Dic107—41no—
185Dic107—41no—
186C2×C107—22no—
187C2×C107—81no—
188C2×C107#54611yescharacteristic
189C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C107#57711yescharacteristic
191D107—41no—
192D107—42no—
193D107—81no—
194#3306—1071nomaximal, Sylow
195C107#330611yescharacteristic, complemented, Sylow
196#547—1071no—
197#546—1071no—
198#489—1071no—
199#483—1071no—
200#481—1071no—
201#476—1071no—
202#469—1071no—
203#465D10711yescharacteristic, complemented
204#463—1071no—
205#449—1071no—
206#447—1071no—
207#442—1071no—
208#435—1071no—
209#417—1071no—
210C2×#102—1071no—
211C2×#41—1071no—
212C2×#41—2141no—
213C22×D4—1071no—
214C2×SD16C2×D10711yescharacteristic, complemented
215C2×D8—1071no—
216C2×D8—1071no—
217C2×M16—1071no—
218C22×C8C2×D10711yescharacteristic
219C4⋊Q8—1071no—
220#124—1071no—
221#123—2141no—
222#123—2141no—
223#123—2141no—
224C22⋊Q8—2141no—
225C22⋊Q8C2×D10711yescharacteristic
226#121—1071no—
227#121—1071no—
228#121—2141no—
229#121C2×D10711yescharacteristic
230#120—2141no—
231C4×D4—1071no—
232C4×D4—1071no—
233C4×D4—1071no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4—1071no—
236C2×C22⋊C4—2141no—
237#106—1071no—
238#106—1071no—
239#106C2×D10711yescharacteristic, complemented
240C4⋊C8—1071no—
241#103—1071no—
242#103—1071no—
243#103C2×D10711yescharacteristic, complemented
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102—1071no—
250#102C2×D10711yescharacteristic, complemented
251C22⋊C8—1071no—
252C22⋊C8—1071no—
253#97—1071no—
254C24—2141no—
255#41—4284no—
256#41—4281no—
257#41—2142no—
258C2×Q8—2141no—
259C2×Q8C22×D10711yescharacteristic
260C2×D4—21no—
261C2×D4—1071no—
262C2×D4—1071no—
263C2×D4—1071no—
264C2×D4—4281no—
265C2×D4—2141no—
266C2×D4—2142no—
267C2×D4—2141no—
268C2×D4—2141no—
269C2×D4—2141no—
270C2×D4C22×D10711yescharacteristic
271C22×C4—1071no—
272C22×C4—2141no—
273C22×C4—2141no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4—2141no—
277C22×C4C22×D10711yescharacteristic
278SD16—41no—
279D8—4281no—
280D8—2142no—
281M16—4281no—
282C2×C8—41no—
283C2×C8—1071no—
284C2×C8—1071no—
285C2×C8C22×D10711yescharacteristic
286C2×C8C22×D10711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—1071no—
289C4⋊C4—1071no—
290C4⋊C4—1071no—
291C4⋊C4—2141no—
292C4⋊C4—2141no—
293C4⋊C4—2141no—
294C4⋊C4C22×D10711yescharacteristic
295C4⋊C4C22×D10711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—1071no—
299C22⋊C4—1071no—
300C22⋊C4—2141no—
301C22⋊C4—2141no—
302C22⋊C4—2141no—
303C22⋊C4—2141no—
304C22⋊C4—2141no—
305C22⋊C4—2141no—
306C42—1071no—
307C23—21no—
308C23—8561no—
309C23—1071no—
310C23—4282no—
311C23—4282no—
312C23—2141no—
313C23—2141no—
314C23D4×D10711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—4282no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—8561no—
322D4—1072no—
323D4—4282no—
324D4—4282no—
325D4—4282no—
326D4—2142no—
327D4—2141no—
328D4—2141no—
329D4—2141no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—8561no—
336C2×C4—1071no—
337C2×C4—1071no—
338C2×C4—1071no—
339C2×C4—4282no—
340C2×C4—4282no—
341C2×C4—4282no—
342C2×C4—4282no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4—2141no—
347C2×C4D4×D10711yescharacteristic
348C2×C4C23×D10711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2141no—
352C8—2141no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—8564no—
359C22—4281no—
360C22—4281no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—2141no—
365C22—2141no—
366C22—2142no—
367C22—2141no—
368C22#1681213612yes—
369C22C2×D4×D10711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—4281no—
375C4—4281no—
376C4—2141no—
377C4—2141no—
378C4—2141no—
379C4#41×D10711yescharacteristic
380C4#1681215411yescharacteristic
381C2—22no—
382C2—81no—
383C2—8561no—
384C2—4282no—
385C2—4281no—
386C2#2029326611yescharacteristic
387C2C2×#1681213611yescharacteristic
388C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
hypercenter#465D107
hypocenterC107#3306
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20293277C2semidirect
2. #20293268C2semidirect
3. #20292988C2semidirect
4. #20292955C2semidirect
5. #20292947C2semidirect
6. #20292916C2semidirect
7. #20292879C2semidirect
8. #20292844C2semidirect
9. #20292771C2semidirect
10. #20292760C2semidirect
11. #20292736C2semidirect
12. #20292698C2semidirect
13. #20292627C2semidirect
14. C2×#16812031C2semidirect
15. #465×C107C2semidirect
21. C2×#13133801C22semidirect
22. #16812108C22semidirect
30. #16812080C22semidirect
32. D4×Dic107C22semidirect
36. C4⋊C4×D107C22semidirect
38. #16812040C22semidirect
39. #16812040C22semidirect
40. Dic107⋊C8C22semidirect
41. #16812033C22semidirect
42. #16812033C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. #16812019C22semidirect
52. C2×SD16×C107C22semidirect
56. #106×C107C22semidirect
57. #103×C107C22semidirect
58. #102×C107C22semidirect
147. D428#41semidirect
195. C107#3306semidirect
203. #465D107semidirect
214. C2×SD16C2×D107semidirect
239. #106C2×D107semidirect
243. #103C2×D107semidirect
250. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932772
#202932682
#202929882
#202929552
#202929472
#202929162
#202928792
#202928442
#202927712
#202927602
#202927362
#202926982
#202926272
C2×#168120312
#465×C1072
#3306107