Finite Group Databasebeta

← #28346379

Subgroups — #28346379

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293140C211yesmaximal, characteristic
2#117×D107C211yesmaximal, characteristic, complemented
3#20292965C211yesmaximal, characteristic, complemented
4#20292961C211yesmaximal, characteristic, complemented
5#20292956C211yesmaximal, characteristic, complemented
6#20292952C211yesmaximal, characteristic, complemented
7#20292951C211yesmaximal, characteristic, complemented
8#20292947C211yesmaximal, characteristic, complemented
9#20292942C211yesmaximal, characteristic, complemented
10#20292664C211yesmaximal, characteristic, complemented
11#20292662C211yesmaximal, characteristic, complemented
12#20292622C211yesmaximal, characteristic, complemented
13#20292622C211yesmaximal, characteristic, complemented
14C22⋊C8×D107C211yesmaximal, characteristic, complemented
15#485×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18C2×C8×D107C2211yescharacteristic
19C2×C8×D107C2211yescharacteristic
20#16812094—21no—
21#16812085C2211yescharacteristic
22#16812084C2211yescharacteristic
23#16812083C2211yescharacteristic
24#16812079—21no—
25#16812071—21no—
26(C22⋊D107)⋊C4—21no—
27#16812068C2211yescharacteristic
28#16812067C2211yescharacteristic
29#16812067C2211yescharacteristic
30#16812067C2211yescharacteristic
31#16812066—21no—
32#16812065—21no—
33C4⋊C4×D107C2211yescharacteristic
34C4⋊C4×D107C2211yescharacteristic
35C22⋊C4×D107—21no—
36C42×D107—21no—
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42#16812040C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic, complemented
47#16812029C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic, complemented
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic
52C22⋊Q8×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C23×D107D411yescharacteristic
60#13133819—22no—
61#13133819—22no—
62C2×D428—21no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×C22⋊D107—21no—
66C22×Dic107D411yescharacteristic
67C2×C4×D107—22no—
68C2×C4×D107—22no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107D411yescharacteristic
73C2×C4×D107D411yescharacteristic
74C2×C4×D107C2311yescharacteristic
75C2×C4×D107C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C8×D107—22no—
79C8×D107—22no—
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86C22⋊Dic107—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)⋊C4—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107—21no—
97C2×Q8×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107—21no—
108C23×C107C2×D411yescharacteristic
109C22×D107—22no—
110C22×D107—22no—
111C22×D107—21no—
112C22×D107—41no—
113C22×D107C2×D411yescharacteristic
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107—42no—
126C2×Dic107—22no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132D428—21no—
133D428—21no—
134D428—42no—
135D428—42no—
136C4×D107—22no—
137C4×D107—21no—
138C4×D107—21no—
139C4×D107—22no—
140C4×D107—42no—
141C4×D107—42no—
142C4×D107—42no—
143C4×D107#4112yes—
144C4×D107#4112yes—
145C8×C107—21no—
146C8×C107—21no—
147C107⋊C8—21no—
148C107⋊C8—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107C22×D411yescharacteristic
153C2×D107—21no—
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—21no—
157C2×D107—44no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107#12312yes—
162C2×D107#13512yes—
163C4×C107—21no—
164C4×C107—41no—
165C4×C107—41no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107C2×#4111yescharacteristic
169C4×C107C2×#4111yescharacteristic
170Dic107—21no—
171Dic107—41no—
172Dic107—41no—
173Dic107—41no—
174Dic107#12312yes—
175C2×C107—41no—
176C2×C107C2×#12311yescharacteristic
177C2×C107C2×#13511yescharacteristic
178C2×C107#57511yescharacteristic
179D107—22no—
180D107—22no—
181D107—81no—
182#3091—1071nomaximal, Sylow
183C107#309111yescharacteristic, complemented, Sylow
184#525—1071no—
185C2×#117—1071no—
186#485—1071no—
187#485D10711yescharacteristic, complemented
188#484—1071no—
189#483—1071no—
190#482—1071no—
191#482—1071no—
192#481—1071no—
193#480—1071no—
194#427—1071no—
195#427—1071no—
196#416—1071no—
197#416—1071no—
198C2×C22⋊C8—1071no—
199C2×#41—1071no—
200C23×C4—1071no—
201C22×C8—1071no—
202C22×C8—1071no—
203#123—2141no—
204C22⋊Q8—1071no—
205C22⋊Q8C2×D10711yescharacteristic
206#121—1071no—
207#121—1071no—
208#120—2141no—
209C4×D4—2141no—
210C4×D4—2141no—
211#117—1071no—
212#117—1071no—
213#117—1071no—
214#117—1071no—
215#117—2141no—
216#117—2141no—
217#117C2×D10711yescharacteristic
218C2×C4⋊C4—1071no—
219C2×C4⋊C4—1071no—
220C2×C22⋊C4—2141no—
221C2×C42—2141no—
222#107—1071no—
223#107—1071no—
224#107—1071no—
225#107C2×D10711yescharacteristic, complemented
226#106—1071no—
227#106—1071no—
228#106—1071no—
229#106C2×D10711yescharacteristic, complemented
230#103—1071no—
231#103—1071no—
232#103C2×D10711yescharacteristic, complemented
233#103C2×D10711yescharacteristic, complemented
234#102—1071no—
235#102—1071no—
236#102—1071no—
237#102—1071no—
238C22⋊C8—1071no—
239C22⋊C8—1071no—
240C22⋊C8—1071no—
241C22⋊C8C2×D10711yescharacteristic
242C24—1071no—
243#41—2142no—
244#41—2142no—
245C2×Q8C22×D10711yescharacteristic
246C2×D4—1071no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4—2141no—
250C22×C4—1071no—
251C22×C4—1071no—
252C22×C4—1071no—
253C22×C4—1071no—
254C22×C4—1071no—
255C22×C4—2141no—
256C22×C4—2142no—
257C22×C4—2142no—
258C22×C4—2141no—
259C22×C4—2141no—
260C22×C4C22×D10711yescharacteristic
261C2×C8—1071no—
262C2×C8—1071no—
263C2×C8—2142no—
264C2×C8—2142no—
265C2×C8C22×D10711yescharacteristic
266C2×C8C22×D10711yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—1071no—
269C4⋊C4—1071no—
270C4⋊C4—1071no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4C22×D10711yescharacteristic
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—21no—
287C42—2141no—
288C42—2141no—
289C42—2141no—
290C42—2141no—
291C23—1071no—
292C23—1071no—
293C23—4281no—
294C23—2142no—
295C23—2142no—
296C23—2141no—
297C23D4×D10711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—4282no—
301D4—4282no—
302D4—4282no—
303D4—2141no—
304D4—2141no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1071no—
312C2×C4—1072no—
313C2×C4—1072no—
314C2×C4—1071no—
315C2×C4—4282no—
316C2×C4—4282no—
317C2×C4—4282no—
318C2×C4—2142no—
319C2×C4—2142no—
320C2×C4—2142no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4D4×D10711yescharacteristic
327C2×C4C23×D10711yescharacteristic
328C8—21no—
329C8—21no—
330C8—2141no—
331C8—2141no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—1072no—
336C22—1072no—
337C22—4284no—
338C22—4281no—
339C22—4281no—
340C22—4281no—
341C22—2141no—
342C22—2141no—
343C22—2141no—
344C22—2141no—
345C22C2×D4×D10711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—1072no—
352C4—4281no—
353C4—4281no—
354C4—4281no—
355C4—2141no—
356C4#41×D10711yescharacteristic
357C4#41×D10711yescharacteristic
358C2—41no—
359C2—8561no—
360C2—2142no—
361C2—2142no—
362C2#123×D10711yescharacteristic
363C2#135×D10711yescharacteristic
364C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#485×C107C2
socleC22×C107C22×D4
prime radical#20293140C2
hypercenter#485D107
hypocenterC107#3091
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #117×D107C2semidirect
3. #20292965C2semidirect
4. #20292961C2semidirect
5. #20292956C2semidirect
6. #20292952C2semidirect
7. #20292951C2semidirect
8. #20292947C2semidirect
9. #20292942C2semidirect
10. #20292664C2semidirect
11. #20292662C2semidirect
12. #20292622C2semidirect
13. #20292622C2semidirect
14. C22⋊C8×D107C2semidirect
15. #485×C107C2semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. #16812040C22semidirect
42. #16812040C22semidirect
43. #16812033C22semidirect
44. #16812033C22semidirect
45. #16812031C22semidirect
46. #16812031C22semidirect
47. #16812029C22semidirect
48. #16812029C22semidirect
54. #107×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
57. #103×C107C22semidirect
183. C107#3091semidirect
187. #485D107semidirect
225. #107C2×D107semidirect
229. #106C2×D107semidirect
232. #103C2×D107semidirect
233. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931402
#117×D1072
#202929652
#202929612
#202929562
#202929522
#202929512
#202929472
#202929422
#202926642
#202926622
#202926222
C22⋊C8×D1072
#485×C1072
#3091107