Finite Group Databasebeta

← #28348423

Subgroups — #28348423

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#20293278C211yesmaximal, characteristic, complemented
2#20293265C211yesmaximal, characteristic, complemented
3#20292988C211yesmaximal, characteristic, complemented
4#20292956C211yesmaximal, characteristic, complemented
5#20292945C211yesmaximal, characteristic, complemented
6#20292915C211yesmaximal, characteristic, complemented
7#20292880C211yesmaximal, characteristic, complemented
8#20292859C211yesmaximal, characteristic, complemented
9#20292845C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292736C211yesmaximal, characteristic, complemented
13#20292696C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#417×C107C211yesmaximal, characteristic, complemented
16C2×#13133818—21no—
17C2×D4×D107C2211yescharacteristic
18C2×#13133805C2211yescharacteristic, complemented
19C2×D856C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22#16812108C2211yescharacteristic, complemented
23#16812098C2211yescharacteristic
24#16812094—21no—
25#16812092—21no—
26#16812092—21no—
27#16812087—21no—
28#16812085C2211yescharacteristic
29#16812084C2211yescharacteristic
30#16812083C2211yescharacteristic
31#16812083C2211yescharacteristic, complemented
32#16812081—21no—
33#16812077—21no—
34D4×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic, complemented
36C4×C22⋊D107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38C2×(C2×D107)⋊C4—21no—
39#16812041C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic, complemented
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic
51#16812019C2211yescharacteristic, complemented
52C2×#41×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic, complemented
54C2×C4⋊C4×C107C2211yescharacteristic, complemented
55#103×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C23×D107—21no—
60#13133818—44no—
61C2×Dic214—21no—
62C2×D428—21no—
63C2×D428C2311yescharacteristic
64C2×D428C2311yescharacteristic
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107D411yescharacteristic
69D4×D107—22no—
70D4×D107—41no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75#13133805—41no—
76D856—41no—
77#13133803—22no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—41no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107—21no—
88C22⋊Dic107D411yescharacteristic
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107C2311yescharacteristic
98#41×C107—22no—
99#41×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×D4×C107—21no—
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107—21no—
105C22×C4×C107C2311yescharacteristic
106M16×C107—41no—
107C2×C8×C107C2311yescharacteristic
108C2×C8×C107C2311yescharacteristic
109C4⋊C4×C107—21no—
110C4⋊C4×C107C2311yescharacteristic
111C4⋊C4×C107C2311yescharacteristic
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—42no—
124D4×C107#4112yescomplemented
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—42no—
131C2×C4×C107—42no—
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2411yescharacteristic
134C22⋊D107—22no—
135C22⋊D107—42no—
136C22⋊D107—42no—
137C22⋊D107—81no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—81no—
141C2×Dic107C2×D411yescharacteristic
142C2×Dic107C2×D411yescharacteristic
143C2×Dic107C2×D411yescharacteristic
144D428—21no—
145D428—21no—
146D428—21no—
147D428—21no—
148D428—42no—
149C4×D107—21no—
150C4×D107—21no—
151C4×D107—42no—
152C4×D107—42no—
153Dic214—42no—
154C8×C107—21no—
155C8×C107—21no—
156C107⋊C8—21no—
157C107⋊C8—21no—
158C22×C107—22no—
159C22×C107—21no—
160C22×C107—21no—
161C22×C107—21no—
162C22×C107—41no—
163C22×C107—41no—
164C22×C107C22×D411yescharacteristic
165C2×D107—21no—
166C2×D107—21no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—84no—
174C2×D107AGL(1,ℤ/8)12yes—
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—42no—
187C2×C107—41no—
188C2×C107#54611yescharacteristic
189C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C107#57711yescharacteristic
191D107—22no—
192D107—81no—
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#465—1071no—
204#463—1071no—
205#449—1071no—
206#447—1071no—
207#442—1071no—
208#435—1071no—
209#417D10711yescharacteristic, complemented
210C2×#102—1071no—
211C2×#41—2141no—
212C2×#41C2×D10711yescharacteristic
213C22×D4—1071no—
214C2×SD16—1071no—
215C2×D8—1071no—
216C2×D8—1071no—
217C2×M16C2×D10711yescharacteristic, complemented
218C22×C8—1071no—
219C4⋊Q8—1071no—
220#124—1071no—
221#123—2141no—
222#123—2141no—
223#123—2141no—
224C22⋊Q8—1071no—
225C22⋊Q8—2141no—
226#121—1071no—
227#121—1071no—
228#121—1071no—
229#121—2141no—
230#120—2141no—
231C4×D4—1071no—
232C4×D4—1071no—
233C4×D4—1071no—
234C2×C4⋊C4—1071no—
235C2×C4⋊C4C2×D10711yescharacteristic, complemented
236C2×C22⋊C4—2141no—
237#106—1071no—
238#106—1071no—
239#106—1071no—
240C4⋊C8—1071no—
241#103—1071no—
242#103C2×D10711yescharacteristic, complemented
243#103C2×D10711yescharacteristic, complemented
244#102—1071no—
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102C2×D10711yescharacteristic
250#102C2×D10711yescharacteristic
251C22⋊C8—1071no—
252C22⋊C8—1071no—
253#97—1071no—
254C24—2141no—
255#41—22no—
256#41—41no—
257#41—4284no—
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—2141no—
267C2×D4—2141no—
268C2×D4—2141no—
269C2×D4—2142no—
270C2×D4C22×D10711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—1071no—
274C22×C4—2141no—
275C22×C4—2141no—
276C22×C4—2141no—
277C22×C4C22×D10711yescharacteristic
278SD16—4281no—
279D8—4281no—
280D8—2142no—
281M16—41no—
282C2×C8—1071no—
283C2×C8—1071no—
284C2×C8—4281no—
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—1071no—
297C22⋊C4—1071no—
298C22⋊C4—2141no—
299C22⋊C4—2141no—
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—42no—
320D4—8561no—
321D4—4282no—
322D4—4282no—
323D4—4282no—
324D4—2142no—
325D4—2141no—
326D4—2141no—
327D4—2141no—
328D4—2141no—
329D4#1313381812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—8561no—
338C2×C4—1071no—
339C2×C4—1071no—
340C2×C4—1071no—
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—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—8564no—
360C22—1072no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—4281no—
365C22—4281no—
366C22—4281no—
367C22—2141no—
368C22—2141no—
369C22C2×D4×D10711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—4281no—
375C4—4281no—
376C4—2141no—
377C4—2141no—
378C4—2141no—
379C4C2×#1313381811yescharacteristic
380C4#1681215411yescharacteristic
381C2—41no—
382C2—42no—
383C2—8561no—
384C2—8561no—
385C2—2142no—
386C2#2029326711yescharacteristic
387C2AGL(1,ℤ/8)×D10711yescharacteristic
388C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
hypercenter#417D107
hypocenterC107#3306
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #20293278C2semidirect
2. #20293265C2semidirect
3. #20292988C2semidirect
4. #20292956C2semidirect
5. #20292945C2semidirect
6. #20292915C2semidirect
7. #20292880C2semidirect
8. #20292859C2semidirect
9. #20292845C2semidirect
10. #20292772C2semidirect
11. #20292762C2semidirect
12. #20292736C2semidirect
13. #20292696C2semidirect
14. #102×D107C2semidirect
15. #417×C107C2semidirect
18. C2×#13133805C22semidirect
22. #16812108C22semidirect
31. #16812083C22semidirect
35. #16812071C22semidirect
39. #16812041C22semidirect
40. #16812040C22semidirect
41. #16812040C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812033C22semidirect
45. #16812031C22semidirect
49. C22⋊(C107⋊C8)C22semidirect
51. #16812019C22semidirect
53. C2×M16×C107C22semidirect
54. C2×C4⋊C4×C107C22semidirect
55. #103×C107C22semidirect
56. #103×C107C22semidirect
124. D4×C107#41semidirect
195. C107#3306semidirect
209. #417D107semidirect
217. C2×M16C2×D107semidirect
235. C2×C4⋊C4C2×D107semidirect
242. #103C2×D107semidirect
243. #103C2×D107semidirect
329. D4#13133818semidirect

Maximal subgroups

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

subgroupindex
#202932782
#202932652
#202929882
#202929562
#202929452
#202929152
#202928802
#202928592
#202928452
#202927722
#202927622
#202927362
#202926962
#102×D1072
#417×C1072
#3306107