Finite Group Databasebeta

← #28348223

Subgroups — #28348223

Order 13696.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293435C211yesmaximal, characteristic, complemented
2#20293430C211yesmaximal, characteristic, complemented
3#20293429C211yesmaximal, characteristic, complemented
4#20293013C211yesmaximal, characteristic, complemented
5#20292897C211yesmaximal, characteristic, complemented
6#20292895C211yesmaximal, characteristic, complemented
7#20292892C211yesmaximal, characteristic, complemented
8#20292887C211yesmaximal, characteristic, complemented
9#20292803C211yesmaximal, characteristic, complemented
10#20292800C211yesmaximal, characteristic, complemented
11#20292795C211yesmaximal, characteristic, complemented
12#20292793C211yesmaximal, characteristic, complemented
13#20292749C211yesmaximal, characteristic, complemented
14#20292747C211yesmaximal, characteristic, complemented
15#577×C107C211yesmaximal, characteristic, complemented
16#16812155C2211yescharacteristic
17#16812155C2211yescharacteristic, complemented
18#16812140—21no—
19#16812140C2211yescharacteristic, complemented
20#16812136—21no—
21#16812136—21no—
22#16812135—21no—
23#16812135C2211yescharacteristic
24#16812134—21no—
25#16812134—21no—
26#16812134C2211yescharacteristic
27#16812134C2211yescharacteristic, complemented
28#16812134C2211yescharacteristic, complemented
29#16812134C2211yescharacteristic, complemented
30#16812132—21no—
31#16812129—21no—
32#16812129—21no—
33#16812129C2211yescharacteristic
34C2×#13133805—21no—
35C2×#13133805C2211yescharacteristic, complemented
36C2×D856C2211yescharacteristic, complemented
37C2×#13133803—21no—
38C2×#13133803C2211yescharacteristic, complemented
39D8×D107—21no—
40#16812116C2211yescharacteristic, complemented
41#16812116C2211yescharacteristic, complemented
42#16812115C2211yescharacteristic, complemented
43#16812105C2211yescharacteristic, complemented
44#16812098C2211yescharacteristic, complemented
45#16812045C2211yescharacteristic, complemented
46#16812044C2211yescharacteristic, complemented
47#16812036C2211yescharacteristic, complemented
48#16812036C2211yescharacteristic, complemented
49#16812036C2211yescharacteristic, complemented
50#16812035C2211yescharacteristic, complemented
51#16812028C2211yescharacteristic, complemented
52#16812026C2211yescharacteristic, complemented
53#16812025C2211yescharacteristic, complemented
54#16812025C2211yescharacteristic, complemented
55#16812019C2211yescharacteristic, complemented
562−1+4×C107C2211yescharacteristic, complemented
57H(5,𝔽2)×C107C2211yescharacteristic, complemented
58#137×C107—21no—
59#137×C107C2211yescharacteristic, complemented
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#13133819—21no—
69#13133818—21no—
70#13133818—21no—
71#13133818C2311yescharacteristic
72C2×D428—21no—
73C2×D428—21no—
74C2×D428C2311yescharacteristic
75C2×D428C2311yescharacteristic
76C2×C22⋊D107—41no—
77D4×D107—21no—
78D4×D107—21no—
79D4×D107—41no—
80D4×D107—41no—
81#13133808—21no—
82#13133808D411yescharacteristic, complemented
83#13133806—21no—
84#13133806D411yescharacteristic, complemented
85#13133805—21no—
86#13133805—21no—
87#13133805—21no—
88#13133805—21no—
89#13133805—21no—
90#13133805—21no—
91#13133805—21no—
92#13133805D411yescharacteristic, complemented
93D856—21no—
94D856—21no—
95#13133803—21no—
96#13133803—21no—
97#13133803—21no—
98#13133803—21no—
99#13133803—21no—
100#13133803—21no—
101#13133803—21no—
102#13133803D411yescharacteristic, complemented
103#13133802—21no—
104#13133802—21no—
105#13133802C2311yescharacteristic
106#13133802C2311yescharacteristic
107#13133802C2311yescharacteristic
108#13133801—21no—
109C2×C107⋊C8—21no—
110C2×C107⋊C8—21no—
111C2×C107⋊C8C2311yescharacteristic
112C8×D107—21no—
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, complemented
132SD16×C107—21no—
133SD16×C107—21no—
134SD16×C107—21no—
135SD16×C107—21no—
136SD16×C107D411yescharacteristic, complemented
137SD16×C107D411yescharacteristic, complemented
138D8×C107—21no—
139D8×C107D411yescharacteristic, complemented
140M16×C107—21no—
141M16×C107C2311yescharacteristic
142C2×C8×C107—21no—
143C2×C8×C107C2311yescharacteristic
144C23×C107—21no—
145C23×C107—41no—
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—41no—
149C22×D107—41no—
150Q8×C107—21no—
151Q8×C107—21no—
152Q8×C107—41no—
153Q8×C107C2×D411yescharacteristic
154Q8×C107C2×D411yescharacteristic
155Q8×C107C2×D411yescharacteristic
156D4×C107—21no—
157D4×C107—21no—
158D4×C107—21no—
159D4×C107—21no—
160D4×C107—21no—
161D4×C107—41no—
162D4×C107—41no—
163D4×C107—41no—
164D4×C107C2×D411yescharacteristic
165D4×C107C2×D411yescharacteristic
166D4×C107C2×D411yescharacteristic
167C2×C4×C107—21no—
168C2×C4×C107—21no—
169C2×C4×C107—21no—
170C2×C4×C107—21no—
171C2×C4×C107—41no—
172C2×C4×C107—41no—
173C2×C4×C107—41no—
174C2×C4×C107C2411yescharacteristic
175C22⋊D107—21no—
176C22⋊D107—41no—
177C22⋊D107—41no—
178C22⋊D107—82no—
179C2×Dic107—21no—
180D428—21no—
181D428—21no—
182D428—21no—
183D428—21no—
184D428—41no—
185D428—41no—
186D428C2×D411yescharacteristic
187C4×D107—21no—
188C4×D107—41no—
189C4×D107—41no—
190Dic214C2×D411yescharacteristic
191C8×C107—21no—
192C8×C107C2×D411yescharacteristic
193C8×C107C2×D411yescharacteristic
194C107⋊C8—21no—
195C107⋊C8—21no—
196C107⋊C8—21no—
197C107⋊C8C2×D411yescharacteristic
198C107⋊C8C2×D411yescharacteristic
199C22×C107—21no—
200C22×C107—21no—
201C22×C107—21no—
202C22×C107—41no—
203C22×C107—81no—
204C22×C107—161no—
205C22×C107H(5,𝔽2)11yescharacteristic
206C2×D107—21no—
207C2×D107—41no—
208C2×D107—41no—
209C2×D107—81no—
210C2×D107—81no—
211C2×D107—161no—
212C2×D107—161no—
213C4×C107—21no—
214C4×C107—21no—
215C4×C107—21no—
216C4×C107—41no—
217C4×C107C22×D411yescharacteristic
218C4×C107C22×D411yescharacteristic
219Dic107—21no—
220Dic107—41no—
221C2×C107—21no—
222C2×C107—41no—
223C2×C107—41no—
224C2×C107—41no—
225C2×C107—81no—
226C2×C107D4211yescharacteristic
227D107—41no—
228D107—81no—
229D107—81no—
230#3288—1071nomaximal, Sylow
231C107#328811yescharacteristic, complemented, Sylow
232#577—1071no—
233#577D10711yescharacteristic, complemented
234#576—1071no—
235#576—1071no—
236#496—1071no—
237#472—1071no—
238#472—1071no—
239#471—1071no—
240#471—1071no—
241#455—1071no—
242#454—1071no—
243#453—1071no—
244#453—1071no—
245#444—1071no—
246#444—1071no—
2472−1+4C2×D10711yescharacteristic, complemented
248H(5,𝔽2)—1071no—
249H(5,𝔽2)—1071no—
250H(5,𝔽2)C2×D10711yescharacteristic, complemented
251#137—21no—
252#137—1071no—
253#137—2141no—
254#137C2×D10711yescharacteristic, complemented
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—1071no—
257AGL(1,ℤ/8)—1071no—
258AGL(1,ℤ/8)—1071no—
259AGL(1,ℤ/8)—1071no—
260AGL(1,ℤ/8)—1071no—
261AGL(1,ℤ/8)—2141no—
262AGL(1,ℤ/8)—2141no—
263AGL(1,ℤ/8)—2141no—
264AGL(1,ℤ/8)—2141no—
265AGL(1,ℤ/8)—2141no—
266AGL(1,ℤ/8)C2×D10711yescharacteristic, complemented
267#135—21no—
268#135—1071no—
269#135—2141no—
270#135—2141no—
271#135—2141no—
272#135C2×D10711yescharacteristic
273C2×SD16—21no—
274C2×SD16—1071no—
275C2×SD16—2141no—
276C2×SD16C2×D10711yescharacteristic, complemented
277C2×D8—1071no—
278C2×D8—1071no—
279C2×D8—2141no—
280C2×D8—2141no—
281#131—1071no—
282#131—1071no—
283#131—1071no—
284#131C2×D10711yescharacteristic, complemented
285#127—1071no—
286#124—1071no—
287#108—1071no—
288#108—1071no—
289#104—1071no—
290#104—1071no—
291#104—1071no—
292#104—1071no—
293#101—1071no—
294#100—1071no—
295#100—1071no—
296#100—1071no—
297#97—1071no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—41no—
303#41—1071no—
304#41—2141no—
305#41—2141no—
306#41—2141no—
307#41—2141no—
308#41C22×D10711yescharacteristic
309#41C22×D10711yescharacteristic
310#41C22×D10711yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C22×D10711yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—1071no—
318C2×D4—1071no—
319C2×D4—4281no—
320C2×D4—4281no—
321C2×D4—4281no—
322C2×D4—2141no—
323C2×D4—2141no—
324C2×D4—2141no—
325C2×D4—2141no—
326C2×D4C22×D10711yescharacteristic
327Q16—21no—
328Q16—1071no—
329Q16—2141no—
330Q16C22⋊D10711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—1071no—
336SD16—1071no—
337SD16—2141no—
338SD16—2141no—
339SD16—2141no—
340SD16—2141no—
341SD16—2141no—
342SD16—2141no—
343SD16—2141no—
344SD16—2141no—
345SD16C22⋊D10711yescharacteristic, complemented
346SD16C22⋊D10711yescharacteristic, complemented
347D8—21no—
348D8—1071no—
349D8—2141no—
350D8—2141no—
351D8—2141no—
352D8—2141no—
353D8—2141no—
354D8—2141no—
355D8—2141no—
356D8—2141no—
357D8—2141no—
358D8C22⋊D10711yescharacteristic, complemented
359M16—21no—
360M16—1071no—
361M16—1071no—
362M16—1071no—
363M16—2141no—
364M16—2141no—
365M16—2141no—
366M16C22×D10711yescharacteristic
367C2×C8—21no—
368C2×C8—1071no—
369C2×C8—2141no—
370C2×C8—2141no—
371C2×C8—2141no—
372C2×C8C22×D10711yescharacteristic
373C22⋊C4—4281no—
374C42—1071no—
375C23—21no—
376C23—41no—
377C23—4281no—
378C23—4281no—
379C23—2141no—
380C23—2141no—
381Q8—21no—
382Q8—21no—
383Q8—41no—
384Q8—1071no—
385Q8D4×D10711yescharacteristic
386Q8C2×C22⋊D10711yescharacteristic
387Q8C2×C22⋊D10711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—8562no—
397D4—1071no—
398D4—4281no—
399D4—4281no—
400D4—4281no—
401D4—4281no—
402D4—2141no—
403D4—2141no—
404D4—2141no—
405D4—2141no—
406D4—2141no—
407D4D4×D10711yescharacteristic
408D4C2×C22⋊D10711yescharacteristic
409D4C2×C22⋊D10711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—4281no—
418C2×C4—4281no—
419C2×C4—2141no—
420C2×C4—2141no—
421C2×C4C23×D10711yescharacteristic
422C8—21no—
423C8—1071no—
424C8—1071no—
425C8—2141no—
426C8—2141no—
427C8—2141no—
428C8C2×C22⋊D10711yescharacteristic
429C8C2×C22⋊D10711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—81no—
435C22—161no—
436C22—8561no—
437C22—8561no—
438C22—4281no—
439C22—4281no—
440C22—17121no—
441C22—17121no—
442C22—2141no—
443C22#1681215411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—4281no—
449C4—2141no—
450C4C2×D4×D10711yescharacteristic
451C4C22×C22⋊D10711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—8561no—
458C2—8561no—
459C2—4281no—
460C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#577×C107C2
socleC2×C107D42
hypercenter#577D107
hypocenterC107#3288
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. #20293435C2semidirect
2. #20293430C2semidirect
3. #20293429C2semidirect
4. #20293013C2semidirect
5. #20292897C2semidirect
6. #20292895C2semidirect
7. #20292892C2semidirect
8. #20292887C2semidirect
9. #20292803C2semidirect
10. #20292800C2semidirect
11. #20292795C2semidirect
12. #20292793C2semidirect
13. #20292749C2semidirect
14. #20292747C2semidirect
15. #577×C107C2semidirect
17. #16812155C22semidirect
19. #16812140C22semidirect
27. #16812134C22semidirect
28. #16812134C22semidirect
29. #16812134C22semidirect
35. C2×#13133805C22semidirect
36. C2×D856C22semidirect
38. C2×#13133803C22semidirect
40. #16812116C22semidirect
41. #16812116C22semidirect
42. #16812115C22semidirect
43. #16812105C22semidirect
44. #16812098C22semidirect
45. #16812045C22semidirect
46. #16812044C22semidirect
47. #16812036C22semidirect
48. #16812036C22semidirect
49. #16812036C22semidirect
50. #16812035C22semidirect
51. #16812028C22semidirect
52. #16812026C22semidirect
53. #16812025C22semidirect
54. #16812025C22semidirect
55. #16812019C22semidirect
56. 2−1+4×C107C22semidirect
57. H(5,𝔽2)×C107C22semidirect
59. #137×C107C22semidirect
61. AGL(1,ℤ/8)×C107C22semidirect
65. C2×SD16×C107C22semidirect
66. #131×C107C22semidirect
82. #13133808D4semidirect
84. #13133806D4semidirect
92. #13133805D4semidirect
102. #13133803D4semidirect
131. Q16×C107D4semidirect
136. SD16×C107D4semidirect
137. SD16×C107D4semidirect
139. D8×C107D4semidirect
231. C107#3288semidirect
233. #577D107semidirect
247. 2−1+4C2×D107semidirect
250. H(5,𝔽2)C2×D107semidirect
254. #137C2×D107semidirect
266. AGL(1,ℤ/8)C2×D107semidirect
276. C2×SD16C2×D107semidirect
284. #131C2×D107semidirect
330. Q16C22⋊D107semidirect
345. SD16C22⋊D107semidirect
346. SD16C22⋊D107semidirect
358. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934352
#202934302
#202934292
#202930132
#202928972
#202928952
#202928922
#202928872
#202928032
#202928002
#202927952
#202927932
#202927492
#202927472
#577×C1072
#3288107