Finite Group Databasebeta

← #28346184

Subgroups — #28346184

Order 13696.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293426C211yesmaximal, characteristic, complemented
2C2×#16812136C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D107C211yesmaximal, characteristic, complemented
4#20293392C211yesmaximal, characteristic, complemented
5#20292898C211yesmaximal, characteristic, complemented
6#20292897C211yesmaximal, characteristic, complemented
7#20292896C211yesmaximal, characteristic, complemented
8#20292895C211yesmaximal, characteristic, complemented
9#20292893C211yesmaximal, characteristic, complemented
10#20292892C211yesmaximal, characteristic, complemented
11#20292887C211yesmaximal, characteristic, complemented
12#20292674C211yesmaximal, characteristic, complemented
13#20292610C211yesmaximal, characteristic, complemented
14#100×D107C211yesmaximal, characteristic, complemented
15#471×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17#41×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812141—21no—
20#16812140C2211yescharacteristic, complemented
21#16812137C2211yescharacteristic
22#16812137C2211yescharacteristic, complemented
23#16812136—21no—
24#16812136—21no—
25#16812136—21no—
26#16812136—21no—
27#16812136—21no—
28#16812136—21no—
29#16812135C2211yescharacteristic
30#16812134C2211yescharacteristic
31#16812134C2211yescharacteristic, complemented
32#16812133—21no—
33#16812132—21no—
34#16812130—21no—
35C2×#13133805C2211yescharacteristic, complemented
36SD16×D107—21no—
37C2×D856C2211yescharacteristic, complemented
38C2×#13133803C2211yescharacteristic, complemented
39D8×D107—21no—
40#16812117—21no—
41#16812117C2211yescharacteristic, complemented
42#16812116C2211yescharacteristic, complemented
43#16812115—21no—
44#16812115C2211yescharacteristic, complemented
45C2×#13133801—21no—
46C2×#13133801C2211yescharacteristic
47M16×D107—21no—
48M16×D107C2211yescharacteristic
49M16×D107C2211yescharacteristic
50M16×D107C2211yescharacteristic
51#16812045C2211yescharacteristic, complemented
52#16812044C2211yescharacteristic, complemented
53#16812044C2211yescharacteristic, complemented
54#16812028C2211yescharacteristic, complemented
55#16812026C2211yescharacteristic, complemented
56#16812025C2211yescharacteristic, complemented
57#16812025C2211yescharacteristic, complemented
58#16812025C2211yescharacteristic, complemented
59#16812025C2211yescharacteristic, complemented
60#137×C107C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C107C2211yescharacteristic
62C2×SD16×C107C2211yescharacteristic, complemented
63#131×C107C2211yescharacteristic, complemented
64#108×C107C2211yescharacteristic, complemented
65#101×C107C2211yescharacteristic, complemented
66#100×C107C2211yescharacteristic, complemented
67C23×D107—21no—
68#13133819—21no—
69#13133819—21no—
70#13133819—21no—
71#13133819—21no—
72#13133819D411yescharacteristic
73#13133818—21no—
74#13133818C2311yescharacteristic
75#13133817—21no—
76#13133817D411yescharacteristic
77Q8×D107D411yescharacteristic
78C2×D428—21no—
79C2×D428C2311yescharacteristic
80C2×D428C2311yescharacteristic
81C2×C22⋊D107—21no—
82D4×D107—21no—
83D4×D107—21no—
84D4×D107—21no—
85D4×D107—21no—
86D4×D107—21no—
87D4×D107D411yescharacteristic
88C2×C4×D107—21no—
89C2×C4×D107—21no—
90C2×C4×D107C2311yescharacteristic
91#13133808—21no—
92#13133807—21no—
93#13133807—21no—
94#13133806—21no—
95#13133805—21no—
96#13133805—21no—
97#13133805—21no—
98#13133805—21no—
99#13133805—21no—
100D856—21no—
101D856—21no—
102D856—21no—
103D856—21no—
104#13133803—21no—
105#13133803—21no—
106#13133803—21no—
107#13133803—21no—
108#13133803—21no—
109#13133802—21no—
110#13133802C2311yescharacteristic
111#13133802C2311yescharacteristic
112#13133802C2311yescharacteristic
113#13133801—21no—
114#13133801—21no—
115#13133801—21no—
116#13133801—21no—
117#13133801—21no—
118#13133801D411yescharacteristic
119#13133801D411yescharacteristic
120#13133801D411yescharacteristic
121#13133801D411yescharacteristic
122C2×C107⋊C8—21no—
123C2×C107⋊C8C2311yescharacteristic
124C8×D107—21no—
125C8×D107—21no—
126C8×D107—21no—
127C8×D107—21no—
128C8×D107—21no—
129#41×C107C2311yescharacteristic
130C2×Q8×C107C2311yescharacteristic
131C2×D4×C107C2311yescharacteristic
132Q16×C107—21no—
133SD16×C107—21no—
134SD16×C107—21no—
135SD16×C107—21no—
136SD16×C107—21no—
137D8×C107—21no—
138M16×C107—21no—
139M16×C107C2311yescharacteristic
140M16×C107C2311yescharacteristic
141M16×C107C2311yescharacteristic
142C2×C8×C107—21no—
143C2×C8×C107C2311yescharacteristic
144C23×C107—21no—
145C22×D107—21no—
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—41no—
149C22×D107—41no—
150C22×D107—82no—
151C22×D107#4111yescharacteristic
152Q8×C107—21no—
153Q8×C107—21no—
154Q8×C107C2×D411yescharacteristic
155D4×C107—21no—
156D4×C107—21no—
157D4×C107—21no—
158D4×C107C2×D411yescharacteristic
159C2×C4×C107—21no—
160C2×C4×C107—21no—
161C2×C4×C107C2411yescharacteristic
162C22⋊D107—21no—
163C22⋊D107—21no—
164C22⋊D107—21no—
165C22⋊D107—41no—
166C22⋊D107—41no—
167C2×Dic107—21no—
168C2×Dic107#4111yescharacteristic
169D428—21no—
170D428—21no—
171D428—21no—
172D428—21no—
173D428—21no—
174D428—41no—
175D428—41no—
176D428C2×D411yescharacteristic
177C4×D107—21no—
178C4×D107—21no—
179C4×D107—21no—
180C4×D107—41no—
181C4×D107—41no—
182C4×D107C2×D411yescharacteristic
183C4×D107C2×D411yescharacteristic
184C4×D107C2×D411yescharacteristic
185C4×D107C2×D411yescharacteristic
186Dic214—21no—
187Dic214C2×D411yescharacteristic
188C8×C107—21no—
189C8×C107—21no—
190C8×C107—21no—
191C8×C107C2×D411yescharacteristic
192C8×C107C2×D411yescharacteristic
193C107⋊C8—21no—
194C107⋊C8—21no—
195C107⋊C8—21no—
196C107⋊C8C2×D411yescharacteristic
197C107⋊C8C2×D411yescharacteristic
198C22×C107—21no—
199C22×C107—41no—
200C22×C107—81no—
201C22×C107C2×#4111yescharacteristic
202C2×D107—21no—
203C2×D107—41no—
204C2×D107—41no—
205C2×D107—41no—
206C2×D107—81no—
207C2×D107—82no—
208C2×D107—82no—
209C2×D107—81no—
210C2×D107—81no—
211C2×D107#12111yescharacteristic
212C2×D107#12111yescharacteristic
213C4×C107—21no—
214C4×C107—41no—
215C4×C107C22×D411yescharacteristic
216C4×C107C22×D411yescharacteristic
217Dic107—21no—
218Dic107#12111yescharacteristic
219Dic107#12111yescharacteristic
220C2×C107—21no—
221C2×C107—41no—
222C2×C107—81no—
223C2×C107C2×#12111yescharacteristic
224D107—21no—
225D107—21no—
226D107—41no—
227D107—81no—
228D107—81no—
229#3068—1071nomaximal, Sylow
230C107#306811yescharacteristic, complemented, Sylow
231#575—1071no—
232C2×AGL(1,ℤ/8)—1071no—
233C2×AGL(1,ℤ/8)—1071no—
234#568—1071no—
235#472—1071no—
236#472—1071no—
237#472—1071no—
238#472—1071no—
239#471—1071no—
240#471—1071no—
241#471—1071no—
242#471D10711yescharacteristic, complemented
243#430—1071no—
244#413—1071no—
245C2×#100—1071no—
246C2×#41—1071no—
247C2×#41—1071no—
248C22×D4—1071no—
249#137—1071no—
250#137—2141no—
251#137C2×D10711yescharacteristic, complemented
252AGL(1,ℤ/8)—1071no—
253AGL(1,ℤ/8)—1071no—
254AGL(1,ℤ/8)—1071no—
255AGL(1,ℤ/8)—1071no—
256AGL(1,ℤ/8)—1071no—
257AGL(1,ℤ/8)—2141no—
258AGL(1,ℤ/8)—2141no—
259AGL(1,ℤ/8)—2141no—
260AGL(1,ℤ/8)—2141no—
261AGL(1,ℤ/8)—2141no—
262AGL(1,ℤ/8)—2141no—
263AGL(1,ℤ/8)—2141no—
264AGL(1,ℤ/8)C2×D10711yescharacteristic
265#135—2141no—
266#135—2141no—
267C2×SD16—1071no—
268C2×SD16—2141no—
269C2×SD16C2×D10711yescharacteristic, complemented
270C2×D8—1071no—
271C2×D8—1071no—
272C2×D8—2141no—
273#131—1071no—
274#131—1071no—
275#131—1071no—
276#131—2141no—
277#131—2141no—
278#131C2×D10711yescharacteristic, complemented
279C2×M16—1071no—
280C2×M16—1071no—
281C2×M16—1071no—
282C2×M16—1071no—
283C2×M16—2141no—
284C2×M16—2141no—
285#108—1071no—
286#108—1071no—
287#108—1071no—
288#108C2×D10711yescharacteristic, complemented
289#101—1071no—
290#101C2×D10711yescharacteristic, complemented
291#100—1071no—
292#100—1071no—
293#100—1071no—
294#100—1071no—
295#100—1071no—
296#100C2×D10711yescharacteristic, complemented
297C24—2141no—
298#41—1071no—
299#41—1071no—
300#41—1071no—
301#41—2141no—
302#41—2141no—
303#41—2141no—
304#41—2141no—
305#41—2141no—
306#41—2141no—
307#41C22×D10711yescharacteristic
308C2×Q8—1071no—
309C2×Q8C22×D10711yescharacteristic
310C2×D4—1071no—
311C2×D4—1071no—
312C2×D4—1071no—
313C2×D4—2141no—
314C2×D4—2141no—
315C2×D4—2141no—
316C2×D4—2141no—
317C2×D4—2141no—
318C2×D4—2141no—
319C2×D4—2141no—
320C2×D4C22×D10711yescharacteristic
321C22×C4—1071no—
322C22×C4—2141no—
323C22×C4—2141no—
324Q16—21no—
325Q16—2141no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—2141no—
331SD16—2141no—
332SD16—2141no—
333SD16—2141no—
334SD16—2141no—
335SD16—2141no—
336SD16—2141no—
337SD16—2141no—
338D8—21no—
339D8—2141no—
340D8—2141no—
341D8—2141no—
342D8—2141no—
343D8—2141no—
344D8—2141no—
345D8—2141no—
346D8—2141no—
347D8—2141no—
348M16—21no—
349M16—1071no—
350M16—1071no—
351M16—1071no—
352M16—1071no—
353M16—1071no—
354M16—1071no—
355M16—1071no—
356M16—2141no—
357M16—2141no—
358M16—2141no—
359M16—2141no—
360M16—2141no—
361M16—2141no—
362M16C22×D10711yescharacteristic
363M16C22×D10711yescharacteristic
364M16C22×D10711yescharacteristic
365C2×C8—21no—
366C2×C8—1071no—
367C2×C8—2141no—
368C2×C8—2141no—
369C2×C8—2141no—
370C2×C8—2141no—
371C2×C8—2141no—
372C2×C8—2141no—
373C2×C8C22×D10711yescharacteristic
374C23—21no—
375C23—8562no—
376C23—1071no—
377C23—4281no—
378C23—4281no—
379C23—2141no—
380C23—2141no—
381C23—2141no—
382Q8—21no—
383Q8—21no—
384Q8—1071no—
385Q8—2141no—
386Q8D4×D10711yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—1071no—
391D4—4281no—
392D4—4281no—
393D4—4281no—
394D4—4281no—
395D4—2141no—
396D4—2141no—
397D4—2141no—
398D4—2141no—
399D4—2141no—
400D4—2141no—
401D4—2141no—
402D4—2141no—
403D4D4×D10711yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—1071no—
407C2×C4—1071no—
408C2×C4—1071no—
409C2×C4—1071no—
410C2×C4—1071no—
411C2×C4—4281no—
412C2×C4—4281no—
413C2×C4—2141no—
414C2×C4—2141no—
415C2×C4—2141no—
416C2×C4—2141no—
417C2×C4C23×D10711yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—1071no—
422C8—1071no—
423C8—2141no—
424C8—2141no—
425C8—2141no—
426C8D4×D10711yescharacteristic
427C8D4×D10711yescharacteristic
428C22—21no—
429C22—41no—
430C22—81no—
431C22—8561no—
432C22—8561no—
433C22—8561no—
434C22—8562no—
435C22—8562no—
436C22—1071no—
437C22—1071no—
438C22—4281no—
439C22—4281no—
440C22—4281no—
441C22—2141no—
442C22#41×D10711yescharacteristic
443C4—21no—
444C4—41no—
445C4—1071no—
446C4—1071no—
447C4—2141no—
448C4C2×D4×D10711yescharacteristic
449C4C2×D4×D10711yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—8561no—
454C2—8561no—
455C2—4281no—
456C2—2141no—
457C2—2141no—
458C2#121×D10711yescharacteristic
subgroupG/H
centerC2#121×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#471×C107C2
socleC2×C107C2×#121
hypercenter#471D107
hypocenterC107#3068
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #20293426C2semidirect
2. C2×#16812136C2semidirect
3. AGL(1,ℤ/8)×D107C2semidirect
4. #20293392C2semidirect
5. #20292898C2semidirect
6. #20292897C2semidirect
7. #20292896C2semidirect
8. #20292895C2semidirect
9. #20292893C2semidirect
10. #20292892C2semidirect
11. #20292887C2semidirect
12. #20292674C2semidirect
13. #20292610C2semidirect
14. #100×D107C2semidirect
15. #471×C107C2semidirect
20. #16812140C22semidirect
22. #16812137C22semidirect
31. #16812134C22semidirect
35. C2×#13133805C22semidirect
37. C2×D856C22semidirect
38. C2×#13133803C22semidirect
41. #16812117C22semidirect
42. #16812116C22semidirect
44. #16812115C22semidirect
51. #16812045C22semidirect
52. #16812044C22semidirect
53. #16812044C22semidirect
54. #16812028C22semidirect
55. #16812026C22semidirect
56. #16812025C22semidirect
57. #16812025C22semidirect
58. #16812025C22semidirect
59. #16812025C22semidirect
60. #137×C107C22semidirect
62. C2×SD16×C107C22semidirect
63. #131×C107C22semidirect
64. #108×C107C22semidirect
65. #101×C107C22semidirect
66. #100×C107C22semidirect
230. C107#3068semidirect
242. #471D107semidirect
251. #137C2×D107semidirect
269. C2×SD16C2×D107semidirect
278. #131C2×D107semidirect
288. #108C2×D107semidirect
290. #101C2×D107semidirect
296. #100C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934262
C2×#168121362
AGL(1,ℤ/8)×D1072
#202933922
#202928982
#202928972
#202928962
#202928952
#202928932
#202928922
#202928872
#202926742
#202926102
#100×D1072
#471×C1072
#3068107