Finite Group Databasebeta

← #28348241

Subgroups — #28348241

Order 13696.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293442C211yesmaximal, characteristic, complemented
2#20293434C211yesmaximal, characteristic, complemented
3#20293430C211yesmaximal, characteristic, complemented
4#20293008C211yesmaximal, characteristic, complemented
5#20292900C211yesmaximal, characteristic, complemented
6#20292895C211yesmaximal, characteristic, complemented
7#20292890C211yesmaximal, characteristic, complemented
8#20292887C211yesmaximal, characteristic, complemented
9#20292805C211yesmaximal, characteristic, complemented
10#20292803C211yesmaximal, characteristic, complemented
11#20292800C211yesmaximal, characteristic, complemented
12#20292798C211yesmaximal, characteristic, complemented
13#20292744C211yesmaximal, characteristic, complemented
14#20292742C211yesmaximal, characteristic, complemented
15#577×C107C211yesmaximal, characteristic, complemented
16#16812156C2211yescharacteristic, complemented
17#16812155C2211yescharacteristic
18#16812141—21no—
19#16812140—21no—
20#16812140C2211yescharacteristic, complemented
21#16812139—21no—
22#16812139C2211yescharacteristic
23#16812139C2211yescharacteristic, complemented
24#16812135—21no—
25#16812135C2211yescharacteristic
26#16812134—21no—
27#16812134C2211yescharacteristic
28#16812134C2211yescharacteristic, complemented
29#16812133—21no—
30#16812130—21no—
31#16812129—21no—
32#16812129—21no—
33#16812129C2211yescharacteristic
34C2×#13133808—21no—
35C2×#13133808C2211yescharacteristic, complemented
36C2×#13133807C2211yescharacteristic, complemented
37SD16×D107—21no—
38C2×#13133803—21no—
39C2×#13133803C2211yescharacteristic
40#16812116C2211yescharacteristic
41#16812116C2211yescharacteristic, complemented
42#16812115C2211yescharacteristic, complemented
43#16812098C2211yescharacteristic, complemented
44#16812096C2211yescharacteristic, complemented
45#16812045C2211yescharacteristic, complemented
46#16812044C2211yescharacteristic, complemented
47#16812036C2211yescharacteristic
48#16812036C2211yescharacteristic, complemented
49#16812036C2211yescharacteristic, complemented
50#16812035C2211yescharacteristic, complemented
51#16812028C2211yescharacteristic, complemented
52#16812027C2211yescharacteristic, complemented
53#16812026C2211yescharacteristic, complemented
54#16812025C2211yescharacteristic, complemented
55C8×Dic107C2211yescharacteristic, complemented
562−1+4×C107C2211yescharacteristic, complemented
57H(5,𝔽2)×C107C2211yescharacteristic
58#137×C107—21no—
59#137×C107C2211yescharacteristic
60AGL(1,ℤ/8)×C107—21no—
61AGL(1,ℤ/8)×C107C2211yescharacteristic, complemented
62#135×C107—21no—
63#135×C107C2211yescharacteristic
64C2×SD16×C107—21no—
65C2×SD16×C107C2211yescharacteristic, complemented
66#131×C107C2211yescharacteristic, complemented
67#13133819—21no—
68#13133818—21no—
69#13133818—21no—
70#13133818C2311yescharacteristic
71#13133817—21no—
72#13133817—41no—
73C2×Dic214—21no—
74C2×Dic214C2311yescharacteristic
75Q8×D107—21no—
76C2×D428—21no—
77C2×D428C2311yescharacteristic
78D4×D107—21no—
79D4×D107—41no—
80#13133808—21no—
81#13133808—21no—
82#13133808—21no—
83#13133808—21no—
84#13133808—21no—
85#13133808D411yescharacteristic
86#13133807—21no—
87#13133807—21no—
88#13133806—21no—
89#13133806—21no—
90#13133806—21no—
91#13133806D411yescharacteristic
92#13133805—21no—
93#13133805—21no—
94#13133805—21no—
95#13133805D411yescharacteristic
96#13133803—21no—
97#13133803—21no—
98#13133803—21no—
99#13133803—21no—
100#13133803—21no—
101#13133803D411yescharacteristic
102#13133802—21no—
103#13133802—21no—
104#13133802C2311yescharacteristic
105#13133802C2311yescharacteristic
106#13133802C2311yescharacteristic
107#13133801—21no—
108C2×C107⋊C8—21no—
109C2×C107⋊C8—21no—
110C2×C107⋊C8C2311yescharacteristic
111C8×D107—21no—
112C22⋊Dic107—41no—
113(C2×D107)⋊C4—41no—
114C4×Dic107C2311yescharacteristic
115#41×C107—21no—
116#41×C107—21no—
117#41×C107—21no—
118#41×C107—21no—
119#41×C107—41no—
120#41×C107C2311yescharacteristic
121#41×C107C2311yescharacteristic
122#41×C107C2311yescharacteristic
123C2×Q8×C107—21no—
124C2×Q8×C107—21no—
125C2×Q8×C107C2311yescharacteristic
126C2×D4×C107—21no—
127C2×D4×C107—21no—
128C2×D4×C107—41no—
129C2×D4×C107C2311yescharacteristic
130Q16×C107—21no—
131Q16×C107D411yescharacteristic
132SD16×C107—21no—
133SD16×C107—21no—
134SD16×C107—21no—
135SD16×C107—21no—
136SD16×C107D411yescharacteristic
137SD16×C107D411yescharacteristic
138D8×C107—21no—
139D8×C107D411yescharacteristic
140M16×C107—21no—
141M16×C107C2311yescharacteristic
142C2×C8×C107—21no—
143C2×C8×C107C2311yescharacteristic
144C23×C107—21no—
145C23×C107—41no—
146C22×D107—21no—
147C22×D107—41no—
148Q8×C107—21no—
149Q8×C107—21no—
150Q8×C107—41no—
151Q8×C107C2×D411yescharacteristic
152Q8×C107C2×D411yescharacteristic
153Q8×C107C2×D411yescharacteristic
154D4×C107—21no—
155D4×C107—21no—
156D4×C107—21no—
157D4×C107—21no—
158D4×C107—21no—
159D4×C107—41no—
160D4×C107—41no—
161D4×C107—41no—
162D4×C107C2×D411yescharacteristic
163D4×C107C2×D411yescharacteristic
164D4×C107C2×D411yescharacteristic
165C2×C4×C107—21no—
166C2×C4×C107—21no—
167C2×C4×C107—21no—
168C2×C4×C107—21no—
169C2×C4×C107—41no—
170C2×C4×C107—41no—
171C2×C4×C107—41no—
172C2×C4×C107C2411yescharacteristic
173C22⋊D107—21no—
174C22⋊D107—41no—
175C22⋊D107—41no—
176C2×Dic107—21no—
177C2×Dic107—21no—
178C2×Dic107—41no—
179D428—21no—
180D428—21no—
181D428—41no—
182D428C2×D411yescharacteristic
183C4×D107—21no—
184C4×D107—41no—
185C4×D107—41no—
186Dic214—21no—
187Dic214—21no—
188Dic214—41no—
189Dic214C2×D411yescharacteristic
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—21no—
200C22×C107—21no—
201C22×C107—41no—
202C22×C107—81no—
203C22×C107—161no—
204C22×C107H(5,𝔽2)11yescharacteristic
205C2×D107—21no—
206C2×D107—41no—
207C2×D107—81no—
208C2×D107—161no—
209C4×C107—21no—
210C4×C107—21no—
211C4×C107—21no—
212C4×C107—41no—
213C4×C107C22×D411yescharacteristic
214C4×C107C22×D411yescharacteristic
215Dic107—21no—
216Dic107—41no—
217Dic107—41no—
218C2×C107—21no—
219C2×C107—41no—
220C2×C107—41no—
221C2×C107—41no—
222C2×C107—81no—
223C2×C107D4211yescharacteristic
224D107—41no—
225D107—81no—
226#3290—1071nomaximal, Sylow
227C107#329011yescharacteristic, complemented, Sylow
228#578—1071no—
229#577—1071no—
230#577D10711yescharacteristic, complemented
231#576—1071no—
232#495—1071no—
233#473—1071no—
234#472—1071no—
235#471—1071no—
236#471—1071no—
237#455—1071no—
238#455—1071no—
239#454—1071no—
240#454—1071no—
241#443—1071no—
242#443—1071no—
2432−1+4—1071no—
2442−1+4C2×D10711yescharacteristic, complemented
245H(5,𝔽2)—1071no—
246H(5,𝔽2)C2×D10711yescharacteristic
247#137—21no—
248#137—1071no—
249#137—1071no—
250#137—1071no—
251#137—2141no—
252#137—2141no—
253#137—2141no—
254#137C2×D10711yescharacteristic
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—1071no—
257AGL(1,ℤ/8)—1071no—
258AGL(1,ℤ/8)—1071no—
259AGL(1,ℤ/8)—2141no—
260AGL(1,ℤ/8)—2141no—
261AGL(1,ℤ/8)—2141no—
262AGL(1,ℤ/8)C2×D10711yescharacteristic, complemented
263#135—21no—
264#135—1071no—
265#135—2141no—
266#135—2141no—
267#135—2141no—
268#135C2×D10711yescharacteristic
269C2×Q16—1071no—
270C2×Q16—2141no—
271C2×SD16—21no—
272C2×SD16—1071no—
273C2×SD16—2141no—
274C2×SD16C2×D10711yescharacteristic, complemented
275C2×D8—1071no—
276C2×D8—2141no—
277#131—1071no—
278#131—1071no—
279#131—1071no—
280#131C2×D10711yescharacteristic, complemented
281#124—1071no—
282#124—1071no—
283#108—1071no—
284#108—1071no—
285#104—1071no—
286#104—1071no—
287#104—1071no—
288#104—1071no—
289#101—1071no—
290#101—1071no—
291#100—1071no—
292#100—1071no—
293C4×C8—1071no—
294#41—21no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—41no—
299#41—1071no—
300#41—4281no—
301#41—2141no—
302#41—2141no—
303#41—2141no—
304#41—2141no—
305#41C22×D10711yescharacteristic
306#41C22×D10711yescharacteristic
307#41C22×D10711yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—1071no—
311C2×Q8—2141no—
312C2×Q8—2141no—
313C2×Q8C22×D10711yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—1071no—
318C2×D4—4281no—
319C2×D4—2141no—
320C2×D4—2141no—
321C2×D4C22×D10711yescharacteristic
322Q16—21no—
323Q16—1071no—
324Q16—2141no—
325Q16—2141no—
326Q16—2141no—
327Q16—2141no—
328Q16—2141no—
329Q16C22⋊D10711yescharacteristic
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—1071no—
335SD16—1071no—
336SD16—2141no—
337SD16—2141no—
338SD16—2141no—
339SD16—2141no—
340SD16—2141no—
341SD16—2141no—
342SD16—2141no—
343SD16—2141no—
344SD16C22⋊D10711yescharacteristic
345SD16C22⋊D10711yescharacteristic
346D8—21no—
347D8—1071no—
348D8—2141no—
349D8—2141no—
350D8—2141no—
351D8—2141no—
352D8—2141no—
353D8C22⋊D10711yescharacteristic
354M16—21no—
355M16—1071no—
356M16—1071no—
357M16—1071no—
358M16—2141no—
359M16—2141no—
360M16—2141no—
361M16C22×D10711yescharacteristic
362C2×C8—21no—
363C2×C8—1071no—
364C2×C8—2141no—
365C2×C8—2141no—
366C2×C8—2141no—
367C2×C8C22×D10711yescharacteristic
368C22⋊C4—4281no—
369C22⋊C4—4281no—
370C42—1071no—
371C23—21no—
372C23—41no—
373C23—4281no—
374C23—2141no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—1071no—
379Q8—4281no—
380Q8—2141no—
381Q8—2141no—
382Q8D4×D10711yescharacteristic
383Q8C2×C22⋊D10711yescharacteristic
384Q8C2×C22⋊D10711yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—1071no—
394D4—4281no—
395D4—4281no—
396D4—4281no—
397D4—2141no—
398D4—2141no—
399D4—2141no—
400D4D4×D10711yescharacteristic
401D4C2×C22⋊D10711yescharacteristic
402D4C2×C22⋊D10711yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—4281no—
411C2×C4—4281no—
412C2×C4—4281no—
413C2×C4—2141no—
414C2×C4—2141no—
415C2×C4—2141no—
416C2×C4C23×D10711yescharacteristic
417C8—21no—
418C8—1071no—
419C8—1071no—
420C8—2141no—
421C8—2141no—
422C8—2141no—
423C8C2×C22⋊D10711yescharacteristic
424C8C2×C22⋊D10711yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—81no—
430C22—161no—
431C22—8561no—
432C22—4281no—
433C22—17121no—
434C22—2141no—
435C22#1681215411yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—4281no—
441C4—4281no—
442C4—2141no—
443C4C2×D4×D10711yescharacteristic
444C4C22×C22⋊D10711yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—81no—
450C2—8561no—
451C2—4281no—
452C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#577×C107C2
socleC2×C107D42
hypercenter#577D107
hypocenterC107#3290
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #20293442C2semidirect
2. #20293434C2semidirect
3. #20293430C2semidirect
4. #20293008C2semidirect
5. #20292900C2semidirect
6. #20292895C2semidirect
7. #20292890C2semidirect
8. #20292887C2semidirect
9. #20292805C2semidirect
10. #20292803C2semidirect
11. #20292800C2semidirect
12. #20292798C2semidirect
13. #20292744C2semidirect
14. #20292742C2semidirect
15. #577×C107C2semidirect
16. #16812156C22semidirect
20. #16812140C22semidirect
23. #16812139C22semidirect
28. #16812134C22semidirect
35. C2×#13133808C22semidirect
36. C2×#13133807C22semidirect
41. #16812116C22semidirect
42. #16812115C22semidirect
43. #16812098C22semidirect
44. #16812096C22semidirect
45. #16812045C22semidirect
46. #16812044C22semidirect
48. #16812036C22semidirect
49. #16812036C22semidirect
50. #16812035C22semidirect
51. #16812028C22semidirect
52. #16812027C22semidirect
53. #16812026C22semidirect
54. #16812025C22semidirect
55. C8×Dic107C22semidirect
56. 2−1+4×C107C22semidirect
61. AGL(1,ℤ/8)×C107C22semidirect
65. C2×SD16×C107C22semidirect
66. #131×C107C22semidirect
227. C107#3290semidirect
230. #577D107semidirect
244. 2−1+4C2×D107semidirect
262. AGL(1,ℤ/8)C2×D107semidirect
274. C2×SD16C2×D107semidirect
280. #131C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934422
#202934342
#202934302
#202930082
#202929002
#202928952
#202928902
#202928872
#202928052
#202928032
#202928002
#202927982
#202927442
#202927422
#577×C1072
#3290107