Finite Group Databasebeta

← #28348252

Subgroups — #28348252

Order 13696.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293438C211yesmaximal, characteristic, complemented
2#20293435C211yesmaximal, characteristic, complemented
3#20293434C211yesmaximal, characteristic, complemented
4#20292998C211yesmaximal, characteristic, complemented
5#20292892C211yesmaximal, characteristic, complemented
6#20292890C211yesmaximal, characteristic, complemented
7#20292888C211yesmaximal, characteristic, complemented
8#20292887C211yesmaximal, characteristic, complemented
9#20292810C211yesmaximal, characteristic, complemented
10#20292808C211yesmaximal, characteristic, complemented
11#20292795C211yesmaximal, characteristic, complemented
12#20292793C211yesmaximal, characteristic, complemented
13#20292744C211yesmaximal, characteristic, complemented
14#20292742C211yesmaximal, characteristic, complemented
15#577×C107C211yesmaximal, characteristic, complemented
16#16812156C2211yescharacteristic
17#16812155C2211yescharacteristic, complemented
18#16812141—21no—
19#16812140—21no—
20#16812140C2211yescharacteristic, complemented
21#16812139—21no—
22#16812139C2211yescharacteristic, complemented
23#16812139C2211yescharacteristic, complemented
24#16812135—21no—
25#16812135C2211yescharacteristic
26#16812134—21no—
27#16812134C2211yescharacteristic
28#16812134C2211yescharacteristic
29#16812133—21no—
30#16812130—21no—
31#16812129—21no—
32#16812129—21no—
33#16812129C2211yescharacteristic
34C2×#13133807C2211yescharacteristic, complemented
35C2×#13133806—21no—
36C2×#13133806C2211yescharacteristic, complemented
37C2×#13133805—21no—
38C2×#13133805C2211yescharacteristic, complemented
39SD16×D107—21no—
40#16812116C2211yescharacteristic, complemented
41#16812116C2211yescharacteristic, complemented
42#16812115C2211yescharacteristic, complemented
43#16812107C2211yescharacteristic, complemented
44#16812105C2211yescharacteristic
45#16812045C2211yescharacteristic, complemented
46#16812044C2211yescharacteristic, complemented
47#16812036C2211yescharacteristic, complemented
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
57H(5,𝔽2)×C107C2211yescharacteristic, complemented
58#137×C107—21no—
59#137×C107C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C107—21no—
61AGL(1,ℤ/8)×C107C2211yescharacteristic
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
78C2×C22⋊D107—41no—
79D4×D107—21no—
80D4×D107—41no—
81#13133808—21no—
82#13133808—21no—
83#13133808—21no—
84#13133808D411yescharacteristic, complemented
85#13133807—21no—
86#13133807—21no—
87#13133806—21no—
88#13133806—21no—
89#13133806—21no—
90#13133806—21no—
91#13133806—21no—
92#13133806D411yescharacteristic, complemented
93#13133805—21no—
94#13133805—21no—
95#13133805—21no—
96#13133805—21no—
97#13133805—21no—
98#13133805D411yescharacteristic, complemented
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—
113Dic107⋊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—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—
176C22⋊D107—82no—
177C2×Dic107—21no—
178C2×Dic107—21no—
179C2×Dic107—41no—
180D428—21no—
181D428—21no—
182D428—41no—
183D428C2×D411yescharacteristic
184C4×D107—21no—
185C4×D107—41no—
186C4×D107—41no—
187Dic214—21no—
188Dic214—21no—
189Dic214—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—81no—
209C2×D107—161no—
210C4×C107—21no—
211C4×C107—21no—
212C4×C107—21no—
213C4×C107—41no—
214C4×C107C22×D411yescharacteristic
215C4×C107C22×D411yescharacteristic
216Dic107—22no—
217Dic107—21no—
218Dic107—41no—
219C2×C107—21no—
220C2×C107—41no—
221C2×C107—41no—
222C2×C107—41no—
223C2×C107—81no—
224C2×C107D4211yescharacteristic
225D107—41no—
226D107—81no—
227#3291—1071nomaximal, Sylow
228C107#329111yescharacteristic, complemented, Sylow
229#577—1071no—
230#577—1071no—
231#577—1071no—
232#577D10711yescharacteristic, complemented
233#492—1071no—
234#471—1071no—
235#471—1071no—
236#471—1071no—
237#471—1071no—
238#456—1071no—
239#456—1071no—
240#453—1071no—
241#453—1071no—
242#443—1071no—
243#443—1071no—
2442−1+4—1071no—
2452−1+4C2×D10711yescharacteristic
246H(5,𝔽2)—1071no—
247H(5,𝔽2)C2×D10711yescharacteristic, complemented
248#137—21no—
249#137—1071no—
250#137—1071no—
251#137—1071no—
252#137—2141no—
253#137—2141no—
254#137—2141no—
255#137C2×D10711yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—1071no—
258AGL(1,ℤ/8)—1071no—
259AGL(1,ℤ/8)—1071no—
260AGL(1,ℤ/8)—2141no—
261AGL(1,ℤ/8)—2141no—
262AGL(1,ℤ/8)—2141no—
263AGL(1,ℤ/8)C2×D10711yescharacteristic
264#135—21no—
265#135—1071no—
266#135—2141no—
267#135—2141no—
268#135—2141no—
269#135C2×D10711yescharacteristic
270C2×SD16—21no—
271C2×SD16—1071no—
272C2×SD16—1071no—
273C2×SD16—1071no—
274C2×SD16—2141no—
275C2×SD16—2141no—
276C2×SD16—2141no—
277C2×SD16C2×D10711yescharacteristic, complemented
278#131—1071no—
279#131—1071no—
280#131—1071no—
281#131C2×D10711yescharacteristic, complemented
282C4⋊Q8—1071no—
283#127—1071no—
284#108—1071no—
285#108—1071no—
286#104—1071no—
287#104—1071no—
288#104—1071no—
289#104—1071no—
290#101—1071no—
291#101—1071no—
292#100—1071no—
293#100—1071no—
294C4×C8—1071no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—1071no—
301#41—4281no—
302#41—2141no—
303#41—2141no—
304#41—2141no—
305#41—2141no—
306#41C22×D10711yescharacteristic
307#41C22×D10711yescharacteristic
308#41C22×D10711yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—1071no—
312C2×Q8—2141no—
313C2×Q8—2141no—
314C2×Q8C22×D10711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—1071no—
319C2×D4—4281no—
320C2×D4—4281no—
321C2×D4—2141no—
322C2×D4—2141no—
323C2×D4C22×D10711yescharacteristic
324Q16—21no—
325Q16—1071no—
326Q16—2141no—
327Q16—2141no—
328Q16—2141no—
329Q16C22⋊D10711yescharacteristic, complemented
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—
344SD16—2141no—
345SD16—2141no—
346SD16—2141no—
347SD16—2141no—
348SD16C22⋊D10711yescharacteristic, complemented
349SD16C22⋊D10711yescharacteristic, complemented
350D8—21no—
351D8—1071no—
352D8—2141no—
353D8—2141no—
354D8—2141no—
355D8C22⋊D10711yescharacteristic, complemented
356M16—21no—
357M16—1071no—
358M16—1071no—
359M16—1071no—
360M16—2141no—
361M16—2141no—
362M16—2141no—
363M16C22×D10711yescharacteristic
364C2×C8—21no—
365C2×C8—1071no—
366C2×C8—2141no—
367C2×C8—2141no—
368C2×C8—2141no—
369C2×C8C22×D10711yescharacteristic
370C4⋊C4—4281no—
371C42—1071no—
372C23—21no—
373C23—41no—
374C23—4281no—
375C23—2141no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—1071no—
380Q8—4281no—
381Q8—2141no—
382Q8—2141no—
383Q8D4×D10711yescharacteristic
384Q8C2×C22⋊D10711yescharacteristic
385Q8C2×C22⋊D10711yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—8562no—
395D4—1071no—
396D4—4281no—
397D4—4281no—
398D4—4281no—
399D4—2141no—
400D4—2141no—
401D4—2141no—
402D4D4×D10711yescharacteristic
403D4C2×C22⋊D10711yescharacteristic
404D4C2×C22⋊D10711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—4281no—
413C2×C4—4281no—
414C2×C4—4281no—
415C2×C4—2141no—
416C2×C4—2141no—
417C2×C4—2141no—
418C2×C4C23×D10711yescharacteristic
419C8—21no—
420C8—1071no—
421C8—1071no—
422C8—2141no—
423C8—2141no—
424C8—2141no—
425C8C2×C22⋊D10711yescharacteristic
426C8C2×C22⋊D10711yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—8561no—
434C22—4281no—
435C22—17121no—
436C22—2141no—
437C22#1681215411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—4281no—
443C4—2142no—
444C4—2141no—
445C4C2×D4×D10711yescharacteristic
446C4C22×C22⋊D10711yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—8561no—
453C2—4281no—
454C2D4×C22⋊D10711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#577×C107C2
socleC2×C107D42
hypercenter#577D107
hypocenterC107#3291
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. #20293438C2semidirect
2. #20293435C2semidirect
3. #20293434C2semidirect
4. #20292998C2semidirect
5. #20292892C2semidirect
6. #20292890C2semidirect
7. #20292888C2semidirect
8. #20292887C2semidirect
9. #20292810C2semidirect
10. #20292808C2semidirect
11. #20292795C2semidirect
12. #20292793C2semidirect
13. #20292744C2semidirect
14. #20292742C2semidirect
15. #577×C107C2semidirect
17. #16812155C22semidirect
20. #16812140C22semidirect
22. #16812139C22semidirect
23. #16812139C22semidirect
34. C2×#13133807C22semidirect
36. C2×#13133806C22semidirect
38. C2×#13133805C22semidirect
40. #16812116C22semidirect
41. #16812116C22semidirect
42. #16812115C22semidirect
43. #16812107C22semidirect
45. #16812045C22semidirect
46. #16812044C22semidirect
47. #16812036C22semidirect
48. #16812036C22semidirect
49. #16812036C22semidirect
50. #16812035C22semidirect
51. #16812028C22semidirect
52. #16812027C22semidirect
53. #16812026C22semidirect
54. #16812025C22semidirect
55. C8×Dic107C22semidirect
57. H(5,𝔽2)×C107C22semidirect
59. #137×C107C22semidirect
65. C2×SD16×C107C22semidirect
66. #131×C107C22semidirect
84. #13133808D4semidirect
92. #13133806D4semidirect
98. #13133805D4semidirect
102. #13133803D4semidirect
131. Q16×C107D4semidirect
136. SD16×C107D4semidirect
137. SD16×C107D4semidirect
139. D8×C107D4semidirect
228. C107#3291semidirect
232. #577D107semidirect
247. H(5,𝔽2)C2×D107semidirect
255. #137C2×D107semidirect
277. C2×SD16C2×D107semidirect
281. #131C2×D107semidirect
329. Q16C22⋊D107semidirect
348. SD16C22⋊D107semidirect
349. SD16C22⋊D107semidirect
355. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202934382
#202934352
#202934342
#202929982
#202928922
#202928902
#202928882
#202928872
#202928102
#202928082
#202927952
#202927932
#202927442
#202927422
#577×C1072
#3291107