Finite Group Databasebeta

← #36365324

Subgroups — #36365324

Order 22144.

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#26012302C211yesmaximal, characteristic, complemented
2#26012301C211yesmaximal, characteristic, complemented
3#26012295C211yesmaximal, characteristic, complemented
4#26012294C211yesmaximal, characteristic, complemented
5#26011878C211yesmaximal, characteristic, complemented
6#26011760C211yesmaximal, characteristic, complemented
7#26011759C211yesmaximal, characteristic, complemented
8#26011753C211yesmaximal, characteristic, complemented
9#26011752C211yesmaximal, characteristic, complemented
10#26011670C211yesmaximal, characteristic, complemented
11#26011665C211yesmaximal, characteristic, complemented
12#26011658C211yesmaximal, characteristic, complemented
13#26011610C211yesmaximal, characteristic, complemented
14#26011610C211yesmaximal, characteristic, complemented
15#453×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic
18#19149724C2211yescharacteristic, complemented
19#19149712C2211yescharacteristic, complemented
20#19149709—21no—
21#19149709C2211yescharacteristic, complemented
22#19149708—21no—
23#19149707C2211yescharacteristic, complemented
24#19149706—21no—
25#19149705—21no—
26#19149705—21no—
27#19149705C2211yescharacteristic
28#19149705C2211yescharacteristic, complemented
29#19149704—21no—
30#19149704C2211yescharacteristic, complemented
31#19149703—21no—
32#19149703—21no—
33#19149700—21no—
34#19149699—21no—
35#19149699—21no—
36#19149699C2211yescharacteristic
37#19149699C2211yescharacteristic
38#19149698—21no—
39C2×#14583814—21no—
40C2×#14583814C2211yescharacteristic, complemented
41C2×#14583814C2211yescharacteristic, complemented
42SD16×D173—21no—
43C2×#14583811—21no—
44C2×#14583811C2211yescharacteristic, complemented
45C2×#14583811C2211yescharacteristic, complemented
46D8×D173—21no—
47#19149687C2211yescharacteristic, complemented
48#19149687C2211yescharacteristic, complemented
49#19149686C2211yescharacteristic, complemented
50#19149686C2211yescharacteristic, complemented
51#19149667C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149607C2211yescharacteristic, complemented
55#19149607C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149596C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149590C2211yescharacteristic, complemented
60H(5,𝔽2)×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62AGL(1,ℤ/8)×C173C2211yescharacteristic, complemented
63#127×C173C2211yescharacteristic, complemented
64#104×C173C2211yescharacteristic, complemented
65#104×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583826—21no—
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583826C2311yescharacteristic
72#14583825—21no—
73#14583825—21no—
74#14583825—21no—
75#14583825—41no—
76C2×Dic346—21no—
77C2×Dic346C2311yescharacteristic
78Q8×D173—21no—
79C2×D692—21no—
80C2×D692C2311yescharacteristic
81C2×C22⋊D173—41no—
82D4×D173—21no—
83D4×D173—21no—
84D4×D173—21no—
85D4×D173—41no—
86Dic692—21no—
87#14583816—21no—
88#14583816D411yescharacteristic, complemented
89#14583816D411yescharacteristic, complemented
90#14583815—21no—
91#14583815—21no—
92#14583814—21no—
93#14583814—21no—
94#14583814—21no—
95#14583814—21no—
96#14583814—21no—
97#14583814—21no—
98#14583814—21no—
99#14583814D411yescharacteristic, complemented
100#14583814D411yescharacteristic, complemented
101#14583813—21no—
102#14583813D411yescharacteristic, complemented
103#14583813D411yescharacteristic, complemented
104D1384—21no—
105#14583811—21no—
106#14583811—21no—
107#14583811—21no—
108#14583811—21no—
109#14583811—21no—
110#14583811—21no—
111#14583811—21no—
112#14583811D411yescharacteristic, complemented
113#14583811D411yescharacteristic, complemented
114#14583810—21no—
115#14583810—21no—
116#14583810C2311yescharacteristic
117#14583810C2311yescharacteristic
118#14583809—21no—
119#14583809—21no—
120C2×#13063436—21no—
121C2×#13063436—21no—
122C2×#13063436C2311yescharacteristic
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126#14583803—41no—
127#41×C173—21no—
128#41×C173—21no—
129#41×C173C2311yescharacteristic
130#41×C173C2311yescharacteristic
131C2×D4×C173—21no—
132C2×D4×C173—21no—
133C2×D4×C173—41no—
134C2×D4×C173—41no—
135C2×D4×C173C2311yescharacteristic
136C2×D4×C173C2311yescharacteristic
137SD16×C173—21no—
138SD16×C173—21no—
139D8×C173—21no—
140D8×C173—21no—
141M16×C173C2311yescharacteristic
142M16×C173C2311yescharacteristic
143C42×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146C22×D173—41no—
147Dic346—21no—
148Dic346—21no—
149Dic346—41no—
150Dic346C2×D411yescharacteristic
151Dic346C2×D411yescharacteristic
152D692—21no—
153D692—21no—
154D692—41no—
155D692C2×D411yescharacteristic
156D692C2×D411yescharacteristic
157C22⋊D173—21no—
158C22⋊D173—21no—
159C22⋊D173—41no—
160C22⋊D173—41no—
161C22⋊D173—41no—
162C22⋊D173—41no—
163C2×Dic173—21no—
164C2×Dic173—41no—
165C2×Dic173—41no—
166C4×D173—21no—
167C4×D173—21no—
168C4×D173—41no—
169C4×D173—41no—
170#13063436—21no—
171#13063436—21no—
172#13063436C2×D411yescharacteristic
173#13063436C2×D411yescharacteristic
174#13063436C2×D411yescharacteristic
175#13063436C2×D411yescharacteristic
176C23×C173—21no—
177C23×C173—21no—
178C23×C173—41no—
179Q8×C173C2×D411yescharacteristic
180Q8×C173C2×D411yescharacteristic
181D4×C173—21no—
182D4×C173—21no—
183D4×C173—21no—
184D4×C173—21no—
185D4×C173—21no—
186D4×C173—21no—
187D4×C173—41no—
188D4×C173—41no—
189D4×C173—82no—
190D4×C173C2×D411yescharacteristic
191D4×C173C2×D411yescharacteristic
192C2×C4×C173—21no—
193C2×C4×C173—21no—
194C2×C4×C173—21no—
195C2×C4×C173—41no—
196C2×C4×C173C2411yescharacteristic
197C8×C173—21no—
198C8×C173—21no—
199C22×C173—21no—
200C22×C173—21no—
201C22×C173—41no—
202C22×C173—41no—
203C22×C173—81no—
204C22×C173—81no—
205C22×C173—161no—
206C22×C173H(5,𝔽2)11yescharacteristic
207C2×D173—21no—
208C2×D173—21no—
209C2×D173—41no—
210C2×D173—81no—
211C2×D173—161no—
212C2×D173—161no—
213C4×C173—21no—
214C4×C173—21no—
215C4×C173—41no—
216C4×C173C22×D411yescharacteristic
217C4×C173C22×D411yescharacteristic
218Dic173—21no—
219Dic173—21no—
220Dic173—41no—
221C2×C173—21no—
222C2×C173—41no—
223C2×C173—41no—
224C2×C173—81no—
225C2×C173—81no—
226C2×C173D4211yescharacteristic
227D173—41no—
228D173—41no—
229D173—81no—
230C173#328811yescharacteristic, complemented, Sylow
231#3288—1731nomaximal, Sylow
232#577—1731no—
233#577—1731no—
234#576—1731no—
235#576—1731no—
236#496—1731no—
237#472—1731no—
238#472—1731no—
239#471—1731no—
240#471—1731no—
241#455—1731no—
242#454—1731no—
243#453—1731no—
244#453D17311yescharacteristic, complemented
245#444—1731no—
246#444—1731no—
2472−1+4—1731no—
248H(5,𝔽2)—1731no—
249H(5,𝔽2)—1731no—
250H(5,𝔽2)C2×D17311yescharacteristic, complemented
251#137—3461no—
252#137—3461no—
253#137—1731no—
254#137—1731no—
255AGL(1,ℤ/8)—3461no—
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—3461no—
259AGL(1,ℤ/8)—3461no—
260AGL(1,ℤ/8)—3461no—
261AGL(1,ℤ/8)—1731no—
262AGL(1,ℤ/8)—1731no—
263AGL(1,ℤ/8)—1731no—
264AGL(1,ℤ/8)—1731no—
265AGL(1,ℤ/8)C2×D17311yescharacteristic
266AGL(1,ℤ/8)C2×D17311yescharacteristic, complemented
267#135—3461no—
268#135—3461no—
269#135—3461no—
270#135—3461no—
271#135—1731no—
272#135—1731no—
273C2×SD16—3461no—
274C2×SD16—3461no—
275C2×SD16—1731no—
276C2×SD16—1731no—
277C2×D8—3461no—
278C2×D8—3461no—
279C2×D8—1731no—
280C2×D8—1731no—
281#131—1731no—
282#131—1731no—
283#131—1731no—
284#131—1731no—
285#127C2×D17311yescharacteristic, complemented
286#124—1731no—
287#108—1731no—
288#108—1731no—
289#104—1731no—
290#104—1731no—
291#104C2×D17311yescharacteristic, complemented
292#104C2×D17311yescharacteristic, complemented
293#101—1731no—
294#100—1731no—
295#100—1731no—
296#100C2×D17311yescharacteristic, complemented
297#97—1731no—
298#41—21no—
299#41—21no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—3461no—
304#41—3461no—
305#41—3461no—
306#41—1731no—
307#41—1731no—
308#41—6921no—
309#41C22×D17311yescharacteristic
310#41C22×D17311yescharacteristic
311C2×Q8—3461no—
312C2×Q8—3461no—
313C2×Q8—1731no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—3461no—
319C2×D4—3461no—
320C2×D4—3461no—
321C2×D4—3461no—
322C2×D4—1731no—
323C2×D4—6921no—
324C2×D4—6921no—
325C2×D4C22×D17311yescharacteristic
326C2×D4C22×D17311yescharacteristic
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16—1731no—
331SD16—21no—
332SD16—21no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—1731no—
344SD16—1731no—
345SD16—1731no—
346SD16—1731no—
347D8—21no—
348D8—21no—
349D8—3461no—
350D8—3461no—
351D8—3461no—
352D8—3461no—
353D8—3461no—
354D8—3461no—
355D8—3461no—
356D8—3461no—
357D8—1731no—
358D8—1731no—
359M16—3461no—
360M16—3461no—
361M16—3461no—
362M16—3461no—
363M16—1731no—
364M16—1731no—
365M16C22×D17311yescharacteristic
366M16C22×D17311yescharacteristic
367C2×C8—3461no—
368C2×C8—3461no—
369C2×C8—3461no—
370C2×C8—3461no—
371C2×C8—1731no—
372C2×C8—1731no—
373C22⋊C4—6921no—
374C42C22×D17311yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—3461no—
379C23—6921no—
380C23—6921no—
381Q8—3461no—
382Q8—3461no—
383Q8—1731no—
384Q8—1731no—
385Q8—6921no—
386Q8D4×D17311yescharacteristic
387Q8D4×D17311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—3461no—
398D4—3461no—
399D4—3461no—
400D4—3461no—
401D4—1731no—
402D4—1731no—
403D4—6921no—
404D4—6921no—
405D4—6921no—
406D4—6921no—
407D4—6921no—
408D4D4×D17311yescharacteristic
409D4D4×D17311yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—3461no—
415C2×C4—3461no—
416C2×C4—3461no—
417C2×C4—6921no—
418C2×C4—6921no—
419C2×C4—6921no—
420C2×C4—6921no—
421C2×C4C23×D17311yescharacteristic
422C8—21no—
423C8—21no—
424C8—3461no—
425C8—3461no—
426C8—1731no—
427C8—1731no—
428C8—1731no—
429C8—1731no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—3461no—
438C22—3461no—
439C22—13841no—
440C22—6921no—
441C22—27681no—
442C22—27681no—
443C22#1914972411yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3461no—
448C4—3461no—
449C4—6921no—
450C4C2×D4×D17311yescharacteristic
451C4C2×D4×D17311yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—13841no—
458C2—6921no—
459C2—6921no—
460C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#453×C173C2
socleC2×C173D42
hypercenter#453D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #26012302C2semidirect
2. #26012301C2semidirect
3. #26012295C2semidirect
4. #26012294C2semidirect
5. #26011878C2semidirect
6. #26011760C2semidirect
7. #26011759C2semidirect
8. #26011753C2semidirect
9. #26011752C2semidirect
10. #26011670C2semidirect
11. #26011665C2semidirect
12. #26011658C2semidirect
13. #26011610C2semidirect
14. #26011610C2semidirect
15. #453×C173C2semidirect
16. #19149726C22semidirect
18. #19149724C22semidirect
19. #19149712C22semidirect
21. #19149709C22semidirect
23. #19149707C22semidirect
28. #19149705C22semidirect
30. #19149704C22semidirect
40. C2×#14583814C22semidirect
41. C2×#14583814C22semidirect
44. C2×#14583811C22semidirect
45. C2×#14583811C22semidirect
47. #19149687C22semidirect
48. #19149687C22semidirect
49. #19149686C22semidirect
50. #19149686C22semidirect
51. #19149667C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
54. #19149607C22semidirect
55. #19149607C22semidirect
56. #19149597C22semidirect
57. #19149596C22semidirect
58. #19149595C22semidirect
59. #19149590C22semidirect
60. H(5,𝔽2)×C173C22semidirect
62. AGL(1,ℤ/8)×C173C22semidirect
63. #127×C173C22semidirect
64. #104×C173C22semidirect
65. #104×C173C22semidirect
66. #100×C173C22semidirect
88. #14583816D4semidirect
89. #14583816D4semidirect
99. #14583814D4semidirect
100. #14583814D4semidirect
102. #14583813D4semidirect
103. #14583813D4semidirect
112. #14583811D4semidirect
113. #14583811D4semidirect
230. C173#3288semidirect
244. #453D173semidirect
250. H(5,𝔽2)C2×D173semidirect
266. AGL(1,ℤ/8)C2×D173semidirect
285. #127C2×D173semidirect
291. #104C2×D173semidirect
292. #104C2×D173semidirect
296. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123022
#260123012
#260122952
#260122942
#260118782
#260117602
#260117592
#260117532
#260117522
#260116702
#260116652
#260116582
#260116102
#453×C1732
#3288173