Finite Group Databasebeta

← #28374795

Subgroups — #28374795

Order 13952.

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#20295838C211yesmaximal, characteristic, complemented
2#20295836C211yesmaximal, characteristic, complemented
3#20295830C211yesmaximal, characteristic, complemented
4#20295827C211yesmaximal, characteristic, complemented
5#20295414C211yesmaximal, characteristic, complemented
6#20295295C211yesmaximal, characteristic, complemented
7#20295290C211yesmaximal, characteristic, complemented
8#20295288C211yesmaximal, characteristic, complemented
9#20295206C211yesmaximal, characteristic, complemented
10#20295197C211yesmaximal, characteristic, complemented
11#20295196C211yesmaximal, characteristic, complemented
12#20295192C211yesmaximal, characteristic, complemented
13#20295147C211yesmaximal, characteristic, complemented
14#20295146C211yesmaximal, characteristic, complemented
15#472×C109C211yesmaximal, characteristic, complemented
16#16812746C2211yescharacteristic, complemented
17#16812745C2211yescharacteristic, complemented
18#16812744C2211yescharacteristic
19#16812744C2211yescharacteristic, complemented
20#16812732—21no—
21#16812732C2211yescharacteristic, complemented
22#16812732C2211yescharacteristic, complemented
23#16812728—21no—
24#16812727—21no—
25#16812727C2211yescharacteristic
26#16812727C2211yescharacteristic, complemented
27#16812726—21no—
28#16812725C2211yescharacteristic
29#16812725C2211yescharacteristic, complemented
30#16812723—21no—
31#16812723—21no—
32#16812723—21no—
33#16812723—21no—
34#16812721—21no—
35#16812721C2211yescharacteristic
36#16812720—21no—
37#16812719C2211yescharacteristic
38#16812718—21no—
39#16812718—21no—
40C2×#13182018—21no—
41C2×#13182018C2211yescharacteristic, complemented
42C2×#13182017C2211yescharacteristic, complemented
43SD16×D109—21no—
44C2×#13182014C2211yescharacteristic, complemented
45D8×D109—21no—
46D8×D109—21no—
47#16812707C2211yescharacteristic, complemented
48#16812707C2211yescharacteristic, complemented
49#16812705C2211yescharacteristic, complemented
50#16812695C2211yescharacteristic, complemented
51#16812686C2211yescharacteristic, complemented
52#16812634C2211yescharacteristic, complemented
53#16812626C2211yescharacteristic, complemented
54#16812625C2211yescharacteristic, complemented
55#16812625C2211yescharacteristic, complemented
56#16812625C2211yescharacteristic, complemented
57#16812617C2211yescharacteristic, complemented
58#16812615C2211yescharacteristic, complemented
59#16812609C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C109C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C109C2211yescharacteristic, complemented
62C2×D8×C109C2211yescharacteristic, complemented
63#131×C109C2211yescharacteristic, complemented
64#108×C109C2211yescharacteristic, complemented
65#100×C109C2211yescharacteristic, complemented
66#100×C109C2211yescharacteristic, complemented
67#13182030—21no—
68#13182029—21no—
69#13182029—21no—
70#13182029C2311yescharacteristic
71#13182029C2311yescharacteristic
72#13182029C2311yescharacteristic
73#13182028—21no—
74#13182028—21no—
75#13182028—21no—
76#13182028—21no—
77#13182028—21no—
78#13182028—41no—
79C2×Dic218—21no—
80C2×Dic218C2311yescharacteristic
81Q8×D109—21no—
82C2×D436—21no—
83C2×D436C2311yescharacteristic
84C2×C22⋊D109—41no—
85C2×C22⋊D109—41no—
86C2×C22⋊D109—41no—
87D4×D109—21no—
88D4×D109—21no—
89D4×D109—21no—
90D4×D109—21no—
91D4×D109—21no—
92D4×D109—41no—
93Dic436—21no—
94Dic436—21no—
95Dic436D411yescharacteristic, complemented
96#13182019D411yescharacteristic, complemented
97#13182018—21no—
98#13182018—21no—
99#13182018—21no—
100#13182018—21no—
101#13182018—21no—
102#13182018—21no—
103#13182018D411yescharacteristic, complemented
104#13182018D411yescharacteristic, complemented
105#13182017—21no—
106#13182017—21no—
107#13182017—21no—
108#13182017—21no—
109#13182017D411yescharacteristic, complemented
110#13182016D411yescharacteristic, complemented
111D872—21no—
112D872—21no—
113D872D411yescharacteristic, complemented
114#13182014—21no—
115#13182014—21no—
116#13182014—21no—
117#13182014—21no—
118#13182014D411yescharacteristic, complemented
119#13182013C2311yescharacteristic
120#13182012—21no—
121#13182012—21no—
122#13182012—21no—
123C2×#11724784C2311yescharacteristic
124C8×D109—21no—
125C8×D109—21no—
126C8×D109—21no—
127C22⋊Dic109—41no—
128C4×Dic109C2311yescharacteristic
129#41×C109C2311yescharacteristic
130C2×D4×C109C2311yescharacteristic
131C2×D4×C109C2311yescharacteristic
132SD16×C109—21no—
133SD16×C109—21no—
134D8×C109—21no—
135D8×C109—21no—
136D8×C109—21no—
137D8×C109—21no—
138M16×C109—21no—
139M16×C109C2311yescharacteristic
140M16×C109C2311yescharacteristic
141M16×C109C2311yescharacteristic
142C2×C8×C109—21no—
143C2×C8×C109C2311yescharacteristic
144C23×C109—21no—
145C23×C109—21no—
146C22×D109—21no—
147C22×D109—41no—
148C22×D109—41no—
149C22×D109—41no—
150Q8×C109C2×D411yescharacteristic
151D4×C109—21no—
152D4×C109—21no—
153D4×C109—21no—
154D4×C109—21no—
155D4×C109—21no—
156D4×C109C2×D411yescharacteristic
157C2×C4×C109—21no—
158C2×C4×C109C2411yescharacteristic
159C22⋊D109—21no—
160C22⋊D109—21no—
161C22⋊D109—21no—
162C22⋊D109—41no—
163C22⋊D109—41no—
164C22⋊D109—41no—
165C22⋊D109—41no—
166C22⋊D109—41no—
167C22⋊D109—41no—
168C22⋊D109—82no—
169C2×Dic109—21no—
170C2×Dic109—21no—
171C2×Dic109—41no—
172C2×Dic109—41no—
173C2×Dic109—41no—
174D436—21no—
175D436—21no—
176D436—41no—
177D436C2×D411yescharacteristic
178D436C2×D411yescharacteristic
179D436C2×D411yescharacteristic
180C4×D109—21no—
181C4×D109—21no—
182C4×D109—21no—
183C4×D109—41no—
184C4×D109—41no—
185Dic218—21no—
186Dic218—21no—
187Dic218—41no—
188Dic218C2×D411yescharacteristic
189Dic218C2×D411yescharacteristic
190Dic218C2×D411yescharacteristic
191C8×C109—21no—
192C8×C109—21no—
193C8×C109—21no—
194C8×C109C2×D411yescharacteristic
195C8×C109C2×D411yescharacteristic
196#11724784—21no—
197#11724784C2×D411yescharacteristic
198#11724784C2×D411yescharacteristic
199C22×C109—21no—
200C22×C109—41no—
201C22×C109—41no—
202C22×C109—81no—
203C22×C109—81no—
204C22×C109H(5,𝔽2)11yescharacteristic
205C2×D109—21no—
206C2×D109—21no—
207C2×D109—21no—
208C2×D109—41no—
209C2×D109—81no—
210C2×D109—161no—
211C2×D109—161no—
212C2×D109—161no—
213C4×C109—21no—
214C4×C109C22×D411yescharacteristic
215C4×C109C22×D411yescharacteristic
216Dic109—21no—
217Dic109—21no—
218Dic109—21no—
219Dic109—41no—
220Dic109—41no—
221C2×C109—21no—
222C2×C109—41no—
223C2×C109—81no—
224C2×C109—81no—
225C2×C109D4211yescharacteristic
226D109—41no—
227D109—41no—
228D109—41no—
229D109—81no—
230#3288—1091nomaximal, Sylow
231C109#328811yescharacteristic, complemented, Sylow
232#577—1091no—
233#577—1091no—
234#576—1091no—
235#576—1091no—
236#496—1091no—
237#472—1091no—
238#472D10911yescharacteristic, complemented
239#471—1091no—
240#471—1091no—
241#455—1091no—
242#454—1091no—
243#453—1091no—
244#453—1091no—
245#444—1091no—
246#444—1091no—
2472−1+4—1091no—
248H(5,𝔽2)—1091no—
249H(5,𝔽2)—1091no—
250H(5,𝔽2)—1091no—
251#137—1091no—
252#137—1091no—
253#137—2181no—
254#137—2181no—
255AGL(1,ℤ/8)—1091no—
256AGL(1,ℤ/8)—1091no—
257AGL(1,ℤ/8)—1091no—
258AGL(1,ℤ/8)—1091no—
259AGL(1,ℤ/8)—2181no—
260AGL(1,ℤ/8)—2181no—
261AGL(1,ℤ/8)—2181no—
262AGL(1,ℤ/8)—2181no—
263AGL(1,ℤ/8)—2181no—
264AGL(1,ℤ/8)—2181no—
265AGL(1,ℤ/8)C2×D10911yescharacteristic, complemented
266AGL(1,ℤ/8)C2×D10911yescharacteristic, complemented
267#135—1091no—
268#135—1091no—
269#135—2181no—
270#135—2181no—
271#135—2181no—
272#135—2181no—
273C2×SD16—1091no—
274C2×SD16—1091no—
275C2×SD16—2181no—
276C2×SD16—2181no—
277C2×D8—1091no—
278C2×D8—2181no—
279C2×D8—2181no—
280C2×D8C2×D10911yescharacteristic, complemented
281#131—1091no—
282#131—1091no—
283#131—1091no—
284#131C2×D10911yescharacteristic, complemented
285#127—1091no—
286#124—1091no—
287#108—1091no—
288#108C2×D10911yescharacteristic, complemented
289#104—1091no—
290#104—1091no—
291#104—1091no—
292#104—1091no—
293#101—1091no—
294#100—1091no—
295#100C2×D10911yescharacteristic, complemented
296#100C2×D10911yescharacteristic, complemented
297#97—1091no—
298#41—1091no—
299#41—1091no—
300#41—1091no—
301#41—4361no—
302#41—2181no—
303#41—2181no—
304#41—2181no—
305#41—2181no—
306#41—2181no—
307#41—2181no—
308#41—2181no—
309#41—2181no—
310#41C22×D10911yescharacteristic
311C2×Q8—1091no—
312C2×Q8—2181no—
313C2×Q8—2181no—
314C2×D4—1091no—
315C2×D4—4361no—
316C2×D4—4361no—
317C2×D4—4361no—
318C2×D4—4361no—
319C2×D4—2181no—
320C2×D4—2181no—
321C2×D4—2181no—
322C2×D4—2181no—
323C2×D4—2181no—
324C2×D4—2181no—
325C2×D4C22×D10911yescharacteristic
326C2×D4C22×D10911yescharacteristic
327Q16—1091no—
328Q16—1091no—
329Q16—2181no—
330Q16—2181no—
331SD16—21no—
332SD16—21no—
333SD16—1091no—
334SD16—1091no—
335SD16—1091no—
336SD16—1091no—
337SD16—2181no—
338SD16—2181no—
339SD16—2181no—
340SD16—2181no—
341SD16—2181no—
342SD16—2181no—
343SD16—2181no—
344SD16—2181no—
345SD16—2181no—
346SD16—2181no—
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—1091no—
352D8—1091no—
353D8—2181no—
354D8—2181no—
355D8—2181no—
356D8—2181no—
357D8—2181no—
358D8—2181no—
359M16—21no—
360M16—1091no—
361M16—2181no—
362M16—2181no—
363M16—2181no—
364M16C22×D10911yescharacteristic
365M16C22×D10911yescharacteristic
366M16C22×D10911yescharacteristic
367C2×C8—21no—
368C2×C8—1091no—
369C2×C8—2181no—
370C2×C8—2181no—
371C2×C8—2181no—
372C2×C8C22×D10911yescharacteristic
373C22⋊C4—4361no—
374C42—1091no—
375C23—21no—
376C23—21no—
377C23—4361no—
378C23—4361no—
379C23—4361no—
380C23—2181no—
381Q8—1091no—
382Q8—1091no—
383Q8—1091no—
384Q8—4361no—
385Q8—2181no—
386Q8—2181no—
387Q8D4×D10911yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—8722no—
394D4—1091no—
395D4—1091no—
396D4—1091no—
397D4—4361no—
398D4—4361no—
399D4—4361no—
400D4—4361no—
401D4—4361no—
402D4—4361no—
403D4—4361no—
404D4—2181no—
405D4—2181no—
406D4—2181no—
407D4—2181no—
408D4—2181no—
409D4D4×D10911yescharacteristic
410C2×C4—21no—
411C2×C4—4361no—
412C2×C4—4361no—
413C2×C4—4361no—
414C2×C4—4361no—
415C2×C4—4361no—
416C2×C4—2181no—
417C2×C4—2181no—
418C2×C4—2181no—
419C2×C4—2181no—
420C2×C4—2181no—
421C2×C4C23×D10911yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—1091no—
426C8—1091no—
427C8—2181no—
428C8D4×D10911yescharacteristic
429C8D4×D10911yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—8721no—
436C22—4361no—
437C22—17441no—
438C22—17441no—
439C22—17441no—
440C22—2181no—
441C22—2181no—
442C22—2181no—
443C22#1681274511yescharacteristic
444C4—21no—
445C4—4361no—
446C4—4361no—
447C4—2181no—
448C4—2181no—
449C4—2181no—
450C4C2×D4×D10911yescharacteristic
451C4C2×D4×D10911yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—8721no—
457C2—4361no—
458C2—4361no—
459C2—4361no—
460C2#2029566011yescharacteristic
subgroupG/H
centerC2#20295660
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#472×C109C2
socleC2×C109D42
hypercenter#472D109
hypocenterC109#3288
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20295838C2semidirect
2. #20295836C2semidirect
3. #20295830C2semidirect
4. #20295827C2semidirect
5. #20295414C2semidirect
6. #20295295C2semidirect
7. #20295290C2semidirect
8. #20295288C2semidirect
9. #20295206C2semidirect
10. #20295197C2semidirect
11. #20295196C2semidirect
12. #20295192C2semidirect
13. #20295147C2semidirect
14. #20295146C2semidirect
15. #472×C109C2semidirect
16. #16812746C22semidirect
17. #16812745C22semidirect
19. #16812744C22semidirect
21. #16812732C22semidirect
22. #16812732C22semidirect
26. #16812727C22semidirect
29. #16812725C22semidirect
41. C2×#13182018C22semidirect
42. C2×#13182017C22semidirect
44. C2×#13182014C22semidirect
47. #16812707C22semidirect
48. #16812707C22semidirect
49. #16812705C22semidirect
50. #16812695C22semidirect
51. #16812686C22semidirect
52. #16812634C22semidirect
53. #16812626C22semidirect
54. #16812625C22semidirect
55. #16812625C22semidirect
56. #16812625C22semidirect
57. #16812617C22semidirect
58. #16812615C22semidirect
59. #16812609C22semidirect
60. AGL(1,ℤ/8)×C109C22semidirect
61. AGL(1,ℤ/8)×C109C22semidirect
62. C2×D8×C109C22semidirect
63. #131×C109C22semidirect
64. #108×C109C22semidirect
65. #100×C109C22semidirect
66. #100×C109C22semidirect
95. Dic436D4semidirect
96. #13182019D4semidirect
103. #13182018D4semidirect
104. #13182018D4semidirect
109. #13182017D4semidirect
110. #13182016D4semidirect
113. D872D4semidirect
118. #13182014D4semidirect
231. C109#3288semidirect
238. #472D109semidirect
265. AGL(1,ℤ/8)C2×D109semidirect
266. AGL(1,ℤ/8)C2×D109semidirect
280. C2×D8C2×D109semidirect
284. #131C2×D109semidirect
288. #108C2×D109semidirect
295. #100C2×D109semidirect
296. #100C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202958382
#202958362
#202958302
#202958272
#202954142
#202952952
#202952902
#202952882
#202952062
#202951972
#202951962
#202951922
#202951472
#202951462
#472×C1092
#3288109