Finite Group Databasebeta

← #28346649

Subgroups — #28346649

Order 13696.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293249C211yesmaximal, characteristic, complemented
2#20293099C211yesmaximal, characteristic, complemented
3#20292966C211yesmaximal, characteristic, complemented
4#20292954C211yesmaximal, characteristic, complemented
5#20292915C211yesmaximal, characteristic, complemented
6#20292906C211yesmaximal, characteristic, complemented
7#20292845C211yesmaximal, characteristic, complemented
8#20292844C211yesmaximal, characteristic, complemented
9#20292776C211yesmaximal, characteristic, complemented
10#20292762C211yesmaximal, characteristic
11#20292740C211yesmaximal, characteristic, complemented
12C4×#13133805C211yesmaximal, characteristic, complemented
13C4×#13133803C211yesmaximal, characteristic, complemented
14#20292689C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18C2×D856C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20#16812108C2211yescharacteristic, complemented
21#16812103—21no—
22#16812098C2211yescharacteristic, complemented
23#16812097—21no—
24#16812094—21no—
25#16812087—21no—
26#16812085C2211yescharacteristic
27#16812084C2211yescharacteristic
28C4×D428C2211yescharacteristic
29C4×D428C2211yescharacteristic, complemented
30D4×Dic107C2211yescharacteristic
31#16812071C2211yescharacteristic
32C4×C22⋊D107—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067C2211yescharacteristic
35#16812066C2211yescharacteristic
36C2×(C2×D107)⋊C4—21no—
37C22⋊C4×D107—21no—
38#16812042C2211yescharacteristic, complemented
39#16812041C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic, complemented
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic, complemented
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C4×C107⋊C8C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic, complemented
54C4×D4×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C2×Dic214—21no—
61C2×D428—21no—
62C2×D428C2311yescharacteristic
63C2×D428C2311yescharacteristic
64C2×C22⋊D107—21no—
65D4×D107—22no—
66D4×D107—22no—
67C22×Dic107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107C2311yescharacteristic
72#13133805—22no—
73D856—22no—
74#13133803—22no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4D411yescharacteristic
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99SD16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107—21no—
110C22×D107—24no—
111C22×D107—21no—
112C22×D107—81no—
113C22×D107—81no—
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—21no—
118D4×C107#4112yes—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—81no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C22⋊D107—24no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—81no—
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133D428—21no—
134D428—21no—
135D428—21no—
136D428—81no—
137D428#4112yes—
138C4×D107—21no—
139C4×D107—21no—
140C4×D107—81no—
141C4×D107—81no—
142C4×D107—81no—
143Dic214—81no—
144C8×C107—21no—
145C8×C107—21no—
146C107⋊C8—21no—
147C107⋊C8—21no—
148C22×C107—22no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107C22×D411yescharacteristic
152C2×D107—21no—
153C2×D107—22no—
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—84no—
162C4×C107—21no—
163C4×C107—21no—
164C4×C107—41no—
165C4×C107—41no—
166C4×C107—41no—
167C4×C107C2×#4111yescharacteristic
168C4×C107C2×#4111yescharacteristic
169Dic107—21no—
170Dic107—21no—
171Dic107—21no—
172Dic107—41no—
173Dic107—41no—
174C2×C107—42no—
175C2×C107#52511yescharacteristic
176C2×C107#57611yescharacteristic
177C2×C107#57711yescharacteristic
178D107—41no—
179D107—42no—
180D107—81no—
181#3118—1071nomaximal, Sylow
182C107#311811yescharacteristic, complemented, Sylow
183#542—1071no—
184#518—1071no—
185#485—1071no—
186#482—1071no—
187#476—1071no—
188#474—1071no—
189#463—1071no—
190#463—1071no—
191#450—1071no—
192#447—1071no—
193#442—1071no—
194#440D10711yescharacteristic, complemented
195C4×SD16—1071no—
196C4×D8—1071no—
197#433—1071no—
198C22×D4—1071no—
199C2×SD16—1071no—
200C2×SD16C2×D10711yescharacteristic
201C2×D8—1071no—
202C2×D8—1071no—
203C4⋊Q8—1071no—
204#126—2141no—
205#124—1071no—
206#124—2141no—
207#123—2141no—
208C22⋊Q8—1071no—
209C22⋊Q8—2141no—
210#121—1071no—
211C4×Q8C2×D10711yescharacteristic, complemented
212C4×D4—1071no—
213C4×D4—1071no—
214C4×D4—1071no—
215C4×D4—1071no—
216C4×D4—2141no—
217C4×D4—2141no—
218C4×D4C2×D10711yescharacteristic
219#117—1071no—
220#117—1071no—
221C2×C22⋊C4—2141no—
222C2×C22⋊C4—2141no—
223#107—1071no—
224#107C2×D10711yescharacteristic, complemented
225#106—1071no—
226#106—1071no—
227C4⋊C8—1071no—
228C4⋊C8—1071no—
229#103—1071no—
230#103C2×D10711yescharacteristic, complemented
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
235#102—1071no—
236#102C2×D10711yescharacteristic
237C22⋊C8—1071no—
238C22⋊C8—1071no—
239#97C2×D10711yescharacteristic, complemented
240C4×C8—1071no—
241C24—2141no—
242C2×Q8—2141no—
243C2×Q8C22×D10711yescharacteristic
244C2×D4—1071no—
245C2×D4—1071no—
246C2×D4—2141no—
247C2×D4—2141no—
248C2×D4—2142no—
249C2×D4—2142no—
250C2×D4C22×D10711yescharacteristic
251C22×C4—21no—
252C22×C4—1071no—
253C22×C4—2141no—
254C22×C4—2141no—
255C22×C4—2141no—
256C22×C4—2141no—
257SD16—22no—
258SD16—2142no—
259D8—2142no—
260D8—2142no—
261C2×C8—1071no—
262C2×C8—1071no—
263C2×C8C22×D10711yescharacteristic
264C2×C8C22×D10711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—1071no—
269C4⋊C4—1071no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4C22×D10711yescharacteristic
273C4⋊C4C22×D10711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—1071no—
276C22⋊C4—1071no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—21no—
287C42—1071no—
288C42—2141no—
289C42C22×D10711yescharacteristic
290C23—21no—
291C23—8561no—
292C23—8561no—
293C23—1071no—
294C23—2144no—
295C23—2141no—
296C23—2141no—
297Q8—21no—
298Q8—21no—
299Q8—8561no—
300D4—21no—
301D4—8561no—
302D4—1072no—
303D4—2144no—
304D4—2141no—
305D4—2141no—
306D4—2141no—
307D4#1313381812yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—8561no—
313C2×C4—8561no—
314C2×C4—8561no—
315C2×C4—8561no—
316C2×C4—1071no—
317C2×C4—1071no—
318C2×C4—1071no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4D4×D10711yescharacteristic
324C2×C4D4×D10711yescharacteristic
325C2×C4C23×D10711yescharacteristic
326C8—21no—
327C8—21no—
328C8—2141no—
329C8—2141no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—8564no—
334C22—4281no—
335C22—4281no—
336C22—4281no—
337C22—4281no—
338C22—4281no—
339C22—2141no—
340C22—2142no—
341C22—2141no—
342C22—2141no—
343C22C2×D4×D10711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—4281no—
350C4—4281no—
351C4—2141no—
352C4—2141no—
353C4—2141no—
354C4#41×D10711yescharacteristic
355C4C2×#1313381811yescharacteristic
356C2—42no—
357C2—8561no—
358C2—4281no—
359C2—4282no—
360C2#2029313611yescharacteristic
361C2#2029343111yescharacteristic
362C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radical#20292762C2
hypercenter#440D107
hypocenterC107#3118
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20293249C2semidirect
2. #20293099C2semidirect
3. #20292966C2semidirect
4. #20292954C2semidirect
5. #20292915C2semidirect
6. #20292906C2semidirect
7. #20292845C2semidirect
8. #20292844C2semidirect
9. #20292776C2semidirect
11. #20292740C2semidirect
12. C4×#13133805C2semidirect
13. C4×#13133803C2semidirect
14. #20292689C2semidirect
15. #440×C107C2semidirect
20. #16812108C22semidirect
22. #16812098C22semidirect
29. C4×D428C22semidirect
38. #16812042C22semidirect
39. #16812041C22semidirect
40. #16812040C22semidirect
41. Dic107⋊C8C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812033C22semidirect
45. #16812031C22semidirect
48. #16812029C22semidirect
51. C4×C107⋊C8C22semidirect
53. C4×Q8×C107C22semidirect
55. #107×C107C22semidirect
56. #103×C107C22semidirect
58. #97×C107C22semidirect
182. C107#3118semidirect
194. #440D107semidirect
211. C4×Q8C2×D107semidirect
224. #107C2×D107semidirect
230. #103C2×D107semidirect
239. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932492
#202930992
#202929662
#202929542
#202929152
#202929062
#202928452
#202928442
#202927762
#202927622
#202927402
C4×#131338052
C4×#131338032
#202926892
#440×C1072
#3118107