Finite Group Databasebeta

← #28348310

Subgroups — #28348310

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293330C211yesmaximal, characteristic, complemented
2#20293270C211yesmaximal, characteristic, complemented
3#20292961C211yesmaximal, characteristic, complemented
4#20292949C211yesmaximal, characteristic, complemented
5#20292924C211yesmaximal, characteristic, complemented
6#20292862C211yesmaximal, characteristic, complemented
7#20292859C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292775C211yesmaximal, characteristic, complemented
10#20292762C211yesmaximal, characteristic, complemented
11C4×D856C211yesmaximal, characteristic, complemented
12C8×D428C211yesmaximal, characteristic, complemented
13#20292622C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#488×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×#13133805C2211yescharacteristic, complemented
19C2×D856C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic, complemented
23#16812104—21no—
24#16812094—21no—
25#16812091—21no—
26#16812085C2211yescharacteristic
27#16812084—21no—
28#16812084C2211yescharacteristic
29#16812084C2211yescharacteristic, complemented
30#16812082—21no—
31#16812080C2211yescharacteristic
32#16812079—21no—
33C4×D428C2211yescharacteristic
34C4×D428C2211yescharacteristic
35C4×D428C2211yescharacteristic, complemented
36#16812071—21no—
37#16812067C2211yescharacteristic, complemented
38C4⋊C4×D107C2211yescharacteristic
39C22⋊C4×D107—21no—
40#16812043C2211yescharacteristic, complemented
41#16812042C2211yescharacteristic, complemented
42#16812040C2211yescharacteristic, complemented
43C4⋊(C107⋊C8)C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52#125×C107C2211yescharacteristic, complemented
53#124×C107C2211yescharacteristic
54#103×C107C2211yescharacteristic, complemented
55#103×C107C2211yescharacteristic, complemented
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133819—22no—
61#13133819—41no—
62C2×D428—21no—
63C2×D428—21no—
64C2×D428—21no—
65C2×D428D411yescharacteristic
66C2×D428C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70D4×D107—22no—
71D4×D107—41no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76C2×C4×D107C2311yescharacteristic
77C2×C4×D107C2311yescharacteristic
78#13133805—41no—
79D856—22no—
80#13133803—41no—
81C2×C107⋊C8C2311yescharacteristic
82C2×C107⋊C8C2311yescharacteristic
83C8×D107—41no—
84C8×D107—41no—
85C4⋊Dic107D411yescharacteristic
86C4⋊Dic107C2311yescharacteristic
87C4⋊Dic107C2311yescharacteristic
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90C22⋊Dic107—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)⋊C4—21no—
97(C2×D107)⋊C4D411yescharacteristic
98(C2×D107)⋊C4D411yescharacteristic
99C4×Dic107—21no—
100C2×Q8×C107C2311yescharacteristic
101C2×D4×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C22⋊C4×C107—21no—
109C22⋊C4×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C22×D107—21no—
113C22×D107—21no—
114C22×D107—42no—
115C22×D107—42no—
116C22×D107—81no—
117C22×D107C2×D411yescharacteristic
118C22×D107C2×D411yescharacteristic
119Q8×C107—21no—
120Q8×C107—21no—
121D4×C107—21no—
122D4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—42no—
130C22⋊D107—81no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135D428—22no—
136D428—21no—
137D428—21no—
138D428—21no—
139D428—42no—
140D428—42no—
141D428—81no—
142D428#4112yes—
143C4×D107—21no—
144C4×D107—21no—
145C4×D107—21no—
146C4×D107—21no—
147C4×D107—42no—
148C4×D107—81no—
149C4×D107—81no—
150C4×D107—81no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—41no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107C22×D411yescharacteristic
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—21no—
162C2×D107—22no—
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—84no—
171C2×D107#13512yes—
172C4×C107—21no—
173C4×C107—21no—
174C4×C107—41no—
175C4×C107—41no—
176C4×C107—41no—
177C4×C107C2×#4111yescharacteristic
178C4×C107H(5,𝔽2)11yescharacteristic
179Dic107—21no—
180Dic107—21no—
181Dic107—41no—
182Dic107—41no—
183C2×C107—81no—
184C2×C107#54611yescharacteristic
185C2×C107C2×#13511yescharacteristic
186C2×C107#57611yescharacteristic
187D107—22no—
188D107—41no—
189D107—42no—
190D107—81no—
191#3297—1071nomaximal, Sylow
192C107#329711yescharacteristic, complemented, Sylow
193#553—1071no—
194#546—1071no—
195#488D10711yescharacteristic, complemented
196#484—1071no—
197#482—1071no—
198#478—1071no—
199#466—1071no—
200#465—1071no—
201#463—1071no—
202#449—1071no—
203#447—1071no—
204C4×D8—1071no—
205C8×D4—1071no—
206#416—1071no—
207C2×#102—1071no—
208C2×#41—1071no—
209C22×D4—1071no—
210C2×SD16—1071no—
211C2×D8—1071no—
212C2×D8—1071no—
213C22×C8—1071no—
214C22×C8—1071no—
215#126—2141no—
216#125C2×D10711yescharacteristic, complemented
217#124C2×D10711yescharacteristic
218#123—2141no—
219#123—2141no—
220C22⋊Q8—1071no—
221#121—1071no—
222#121—1071no—
223#121—1071no—
224#121—2141no—
225#121—2141no—
226#120—2141no—
227C4×D4—1071no—
228C4×D4—1071no—
229C4×D4—1071no—
230C4×D4—2141no—
231#117—1071no—
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—2141no—
234#107—1071no—
235#107—1071no—
236#106—1071no—
237C4⋊C8—1071no—
238#103—1071no—
239#103C2×D10711yescharacteristic, complemented
240#103C2×D10711yescharacteristic, complemented
241#102—1071no—
242#102—1071no—
243#102—1071no—
244#102—1071no—
245#102—1071no—
246#102C2×D10711yescharacteristic
247#102C2×D10711yescharacteristic
248C22⋊C8—1071no—
249C22⋊C8—1071no—
250C4×C8C2×D10711yescharacteristic, complemented
251C24—2141no—
252#41—4281no—
253#41—2142no—
254C2×Q8C22×D10711yescharacteristic
255C2×D4—1071no—
256C2×D4—1071no—
257C2×D4—1071no—
258C2×D4—4281no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2142no—
264C2×D4—2141no—
265C2×D4C22×D10711yescharacteristic
266C22×C4—1071no—
267C22×C4—1071no—
268C22×C4—2141no—
269C22×C4—2141no—
270C22×C4—2141no—
271C22×C4—2141no—
272SD16—4281no—
273D8—4281no—
274D8—2142no—
275C2×C8—1071no—
276C2×C8—1071no—
277C2×C8—4281no—
278C2×C8—4281no—
279C2×C8C22×D10711yescharacteristic
280C2×C8C22×D10711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—1071no—
284C4⋊C4—1071no—
285C4⋊C4—1071no—
286C4⋊C4—2141no—
287C4⋊C4—2141no—
288C4⋊C4C22×D10711yescharacteristic
289C4⋊C4C22×D10711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1071no—
293C22⋊C4—1071no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4—2141no—
300C22⋊C4—2141no—
301C42—2141no—
302C42C22×D10711yescharacteristic
303C23—21no—
304C23—8561no—
305C23—1071no—
306C23—1071no—
307C23—4282no—
308C23—4282no—
309C23—2141no—
310C23—2141no—
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—8561no—
316D4—8561no—
317D4—1072no—
318D4—4282no—
319D4—4282no—
320D4—4282no—
321D4—2142no—
322D4—2141no—
323D4—2141no—
324D4—2141no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—8561no—
329C2×C4—8561no—
330C2×C4—8561no—
331C2×C4—1071no—
332C2×C4—1071no—
333C2×C4—4282no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4D4×D10711yescharacteristic
341C2×C4D4×D10711yescharacteristic
342C2×C4C23×D10711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2141no—
346C8—2141no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—8564no—
351C22—1072no—
352C22—4281no—
353C22—4281no—
354C22—4281no—
355C22—4281no—
356C22—4281no—
357C22—2141no—
358C22—2141no—
359C22—2142no—
360C22—2141no—
361C22—2141no—
362C22—2141no—
363C22C2×D4×D10711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—4281no—
370C4—4281no—
371C4—2141no—
372C4—2141no—
373C4#41×D10711yescharacteristic
374C4#1681215511yescharacteristic
375C2—81no—
376C2—8561no—
377C2—4282no—
378C2—4281no—
379C2—2142no—
380C2#2029327011yescharacteristic
381C2#135×D10711yescharacteristic
382C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#488×C107C2
socleC22×C107C22×D4
hypercenter#488D107
hypocenterC107#3297
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20293330C2semidirect
2. #20293270C2semidirect
3. #20292961C2semidirect
4. #20292949C2semidirect
5. #20292924C2semidirect
6. #20292862C2semidirect
7. #20292859C2semidirect
8. #20292839C2semidirect
9. #20292775C2semidirect
10. #20292762C2semidirect
11. C4×D856C2semidirect
12. C8×D428C2semidirect
13. #20292622C2semidirect
14. #102×D107C2semidirect
15. #488×C107C2semidirect
18. C2×#13133805C22semidirect
22. C2×C8×D107C22semidirect
29. #16812084C22semidirect
35. C4×D428C22semidirect
37. #16812067C22semidirect
40. #16812043C22semidirect
41. #16812042C22semidirect
42. #16812040C22semidirect
43. C4⋊(C107⋊C8)C22semidirect
44. #16812033C22semidirect
46. #16812031C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. #125×C107C22semidirect
54. #103×C107C22semidirect
55. #103×C107C22semidirect
58. C4×C8×C107C22semidirect
192. C107#3297semidirect
195. #488D107semidirect
216. #125C2×D107semidirect
239. #103C2×D107semidirect
240. #103C2×D107semidirect
250. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933302
#202932702
#202929612
#202929492
#202929242
#202928622
#202928592
#202928392
#202927752
#202927622
C4×D8562
C8×D4282
#202926222
#102×D1072
#488×C1072
#3297107