Finite Group Databasebeta

← #36365326

Subgroups — #36365326

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#26012303C211yesmaximal, characteristic, complemented
2#26012301C211yesmaximal, characteristic, complemented
3#26012295C211yesmaximal, characteristic, complemented
4#26012292C211yesmaximal, characteristic, complemented
5#26011879C211yesmaximal, characteristic, complemented
6#26011760C211yesmaximal, characteristic, complemented
7#26011755C211yesmaximal, characteristic, complemented
8#26011753C211yesmaximal, characteristic, complemented
9#26011671C211yesmaximal, characteristic, complemented
10#26011662C211yesmaximal, characteristic, complemented
11#26011661C211yesmaximal, characteristic, complemented
12#26011657C211yesmaximal, characteristic, complemented
13#26011612C211yesmaximal, characteristic, complemented
14#26011611C211yesmaximal, characteristic, complemented
15#472×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic, complemented
18#19149724C2211yescharacteristic
19#19149724C2211yescharacteristic, complemented
20#19149712—21no—
21#19149712C2211yescharacteristic, complemented
22#19149712C2211yescharacteristic, complemented
23#19149708—21no—
24#19149707—21no—
25#19149707C2211yescharacteristic
26#19149707C2211yescharacteristic, complemented
27#19149706—21no—
28#19149705C2211yescharacteristic
29#19149705C2211yescharacteristic, complemented
30#19149703—21no—
31#19149703—21no—
32#19149703—21no—
33#19149703—21no—
34#19149701—21no—
35#19149701C2211yescharacteristic
36#19149700—21no—
37#19149699C2211yescharacteristic
38#19149698—21no—
39#19149698—21no—
40C2×#14583815—21no—
41C2×#14583815C2211yescharacteristic, complemented
42C2×#14583814C2211yescharacteristic, complemented
43SD16×D173—21no—
44C2×#14583811C2211yescharacteristic, complemented
45D8×D173—21no—
46D8×D173—21no—
47#19149687C2211yescharacteristic, complemented
48#19149687C2211yescharacteristic, complemented
49#19149685C2211yescharacteristic, complemented
50#19149675C2211yescharacteristic, complemented
51#19149666C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149606C2211yescharacteristic, complemented
54#19149605C2211yescharacteristic, complemented
55#19149605C2211yescharacteristic, complemented
56#19149605C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149589C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic, complemented
62C2×D8×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic, complemented
64#108×C173C2211yescharacteristic, complemented
65#100×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583826—21no—
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583826C2311yescharacteristic
72#14583826C2311yescharacteristic
73#14583825—21no—
74#14583825—21no—
75#14583825—21no—
76#14583825—21no—
77#14583825—21no—
78#14583825—41no—
79C2×Dic346—21no—
80C2×Dic346C2311yescharacteristic
81Q8×D173—21no—
82C2×D692—21no—
83C2×D692C2311yescharacteristic
84C2×C22⋊D173—41no—
85C2×C22⋊D173—41no—
86C2×C22⋊D173—41no—
87D4×D173—21no—
88D4×D173—21no—
89D4×D173—21no—
90D4×D173—21no—
91D4×D173—21no—
92D4×D173—41no—
93Dic692—21no—
94Dic692—21no—
95Dic692D411yescharacteristic, complemented
96#14583816D411yescharacteristic, complemented
97#14583815—21no—
98#14583815—21no—
99#14583815—21no—
100#14583815—21no—
101#14583815—21no—
102#14583815—21no—
103#14583815D411yescharacteristic, complemented
104#14583815D411yescharacteristic, complemented
105#14583814—21no—
106#14583814—21no—
107#14583814—21no—
108#14583814—21no—
109#14583814D411yescharacteristic, complemented
110#14583813D411yescharacteristic, complemented
111D1384—21no—
112D1384—21no—
113D1384D411yescharacteristic, complemented
114#14583811—21no—
115#14583811—21no—
116#14583811—21no—
117#14583811—21no—
118#14583811D411yescharacteristic, complemented
119#14583810C2311yescharacteristic
120#14583809—21no—
121#14583809—21no—
122#14583809—21no—
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126C8×D173—21no—
127C22⋊Dic173—41no—
128C4×Dic173C2311yescharacteristic
129#41×C173C2311yescharacteristic
130C2×D4×C173C2311yescharacteristic
131C2×D4×C173C2311yescharacteristic
132SD16×C173—21no—
133SD16×C173—21no—
134D8×C173—21no—
135D8×C173—21no—
136D8×C173—21no—
137D8×C173—21no—
138M16×C173—21no—
139M16×C173C2311yescharacteristic
140M16×C173C2311yescharacteristic
141M16×C173C2311yescharacteristic
142C2×C8×C173—21no—
143C2×C8×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146C22×D173—41no—
147C22×D173—41no—
148Dic346—21no—
149Dic346—21no—
150Dic346—41no—
151Dic346C2×D411yescharacteristic
152Dic346C2×D411yescharacteristic
153Dic346C2×D411yescharacteristic
154D692—21no—
155D692—21no—
156D692—41no—
157D692C2×D411yescharacteristic
158D692C2×D411yescharacteristic
159D692C2×D411yescharacteristic
160C22⋊D173—21no—
161C22⋊D173—21no—
162C22⋊D173—21no—
163C22⋊D173—41no—
164C22⋊D173—41no—
165C22⋊D173—41no—
166C22⋊D173—41no—
167C22⋊D173—41no—
168C22⋊D173—41no—
169C22⋊D173—82no—
170C2×Dic173—21no—
171C2×Dic173—21no—
172C2×Dic173—41no—
173C2×Dic173—41no—
174C2×Dic173—41no—
175C4×D173—21no—
176C4×D173—21no—
177C4×D173—21no—
178C4×D173—41no—
179C4×D173—41no—
180#13063436—21no—
181#13063436C2×D411yescharacteristic
182#13063436C2×D411yescharacteristic
183C23×C173—21no—
184C23×C173—21no—
185Q8×C173C2×D411yescharacteristic
186D4×C173—21no—
187D4×C173—21no—
188D4×C173—21no—
189D4×C173—21no—
190D4×C173—21no—
191D4×C173C2×D411yescharacteristic
192C2×C4×C173—21no—
193C2×C4×C173C2411yescharacteristic
194C8×C173—21no—
195C8×C173—21no—
196C8×C173—21no—
197C8×C173C2×D411yescharacteristic
198C8×C173C2×D411yescharacteristic
199C22×C173—21no—
200C22×C173—41no—
201C22×C173—41no—
202C22×C173—81no—
203C22×C173—81no—
204C22×C173H(5,𝔽2)11yescharacteristic
205C2×D173—21no—
206C2×D173—21no—
207C2×D173—21no—
208C2×D173—41no—
209C2×D173—81no—
210C2×D173—161no—
211C2×D173—161no—
212C2×D173—161no—
213C4×C173—21no—
214C4×C173C22×D411yescharacteristic
215C4×C173C22×D411yescharacteristic
216Dic173—21no—
217Dic173—21no—
218Dic173—21no—
219Dic173—41no—
220Dic173—41no—
221C2×C173—21no—
222C2×C173—41no—
223C2×C173—81no—
224C2×C173—81no—
225C2×C173D4211yescharacteristic
226D173—41no—
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#472D17311yescharacteristic, complemented
239#471—1731no—
240#471—1731no—
241#455—1731no—
242#454—1731no—
243#453—1731no—
244#453—1731no—
245#444—1731no—
246#444—1731no—
2472−1+4—1731no—
248H(5,𝔽2)—1731no—
249H(5,𝔽2)—1731no—
250H(5,𝔽2)—1731no—
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, complemented
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×D8C2×D17311yescharacteristic, complemented
281#131—1731no—
282#131—1731no—
283#131—1731no—
284#131C2×D17311yescharacteristic, complemented
285#127—1731no—
286#124—1731no—
287#108—1731no—
288#108C2×D17311yescharacteristic, complemented
289#104—1731no—
290#104—1731no—
291#104—1731no—
292#104—1731no—
293#101—1731no—
294#100—1731no—
295#100C2×D17311yescharacteristic, complemented
296#100C2×D17311yescharacteristic, complemented
297#97—1731no—
298#41—3461no—
299#41—3461no—
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—1731no—
309#41—6921no—
310#41C22×D17311yescharacteristic
311C2×Q8—3461no—
312C2×Q8—3461no—
313C2×Q8—1731no—
314C2×D4—3461no—
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—3461no—
318C2×D4—3461no—
319C2×D4—3461no—
320C2×D4—1731no—
321C2×D4—6921no—
322C2×D4—6921no—
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—21no—
350D8—21no—
351D8—3461no—
352D8—3461no—
353D8—3461no—
354D8—3461no—
355D8—3461no—
356D8—3461no—
357D8—1731no—
358D8—1731no—
359M16—21no—
360M16—3461no—
361M16—3461no—
362M16—3461no—
363M16—1731no—
364M16C22×D17311yescharacteristic
365M16C22×D17311yescharacteristic
366M16C22×D17311yescharacteristic
367C2×C8—21no—
368C2×C8—3461no—
369C2×C8—3461no—
370C2×C8—3461no—
371C2×C8—1731no—
372C2×C8C22×D17311yescharacteristic
373C22⋊C4—6921no—
374C42—1731no—
375C23—21no—
376C23—21no—
377C23—3461no—
378C23—6921no—
379C23—6921no—
380C23—6921no—
381Q8—3461no—
382Q8—3461no—
383Q8—1731no—
384Q8—1731no—
385Q8—1731no—
386Q8—6921no—
387Q8D4×D17311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—3461no—
394D4—3461no—
395D4—3461no—
396D4—3461no—
397D4—3461no—
398D4—13842no—
399D4—1731no—
400D4—1731no—
401D4—1731no—
402D4—6921no—
403D4—6921no—
404D4—6921no—
405D4—6921no—
406D4—6921no—
407D4—6921no—
408D4—6921no—
409D4D4×D17311yescharacteristic
410C2×C4—21no—
411C2×C4—3461no—
412C2×C4—3461no—
413C2×C4—3461no—
414C2×C4—3461no—
415C2×C4—3461no—
416C2×C4—6921no—
417C2×C4—6921no—
418C2×C4—6921no—
419C2×C4—6921no—
420C2×C4—6921no—
421C2×C4C23×D17311yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—3461no—
426C8—1731no—
427C8—1731no—
428C8D4×D17311yescharacteristic
429C8D4×D17311yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—3461no—
436C22—3461no—
437C22—3461no—
438C22—13841no—
439C22—6921no—
440C22—27681no—
441C22—27681no—
442C22—27681no—
443C22#1914972511yescharacteristic
444C4—21no—
445C4—3461no—
446C4—3461no—
447C4—3461no—
448C4—6921no—
449C4—6921no—
450C4C2×D4×D17311yescharacteristic
451C4C2×D4×D17311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—13841no—
457C2—6921no—
458C2—6921no—
459C2—6921no—
460C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#472×C173C2
socleC2×C173D42
hypercenter#472D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #26012303C2semidirect
2. #26012301C2semidirect
3. #26012295C2semidirect
4. #26012292C2semidirect
5. #26011879C2semidirect
6. #26011760C2semidirect
7. #26011755C2semidirect
8. #26011753C2semidirect
9. #26011671C2semidirect
10. #26011662C2semidirect
11. #26011661C2semidirect
12. #26011657C2semidirect
13. #26011612C2semidirect
14. #26011611C2semidirect
15. #472×C173C2semidirect
16. #19149726C22semidirect
17. #19149725C22semidirect
19. #19149724C22semidirect
21. #19149712C22semidirect
22. #19149712C22semidirect
26. #19149707C22semidirect
29. #19149705C22semidirect
41. C2×#14583815C22semidirect
42. C2×#14583814C22semidirect
44. C2×#14583811C22semidirect
47. #19149687C22semidirect
48. #19149687C22semidirect
49. #19149685C22semidirect
50. #19149675C22semidirect
51. #19149666C22semidirect
52. #19149614C22semidirect
53. #19149606C22semidirect
54. #19149605C22semidirect
55. #19149605C22semidirect
56. #19149605C22semidirect
57. #19149597C22semidirect
58. #19149595C22semidirect
59. #19149589C22semidirect
60. AGL(1,ℤ/8)×C173C22semidirect
61. AGL(1,ℤ/8)×C173C22semidirect
62. C2×D8×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #100×C173C22semidirect
66. #100×C173C22semidirect
95. Dic692D4semidirect
96. #14583816D4semidirect
103. #14583815D4semidirect
104. #14583815D4semidirect
109. #14583814D4semidirect
110. #14583813D4semidirect
113. D1384D4semidirect
118. #14583811D4semidirect
230. C173#3288semidirect
238. #472D173semidirect
265. AGL(1,ℤ/8)C2×D173semidirect
266. AGL(1,ℤ/8)C2×D173semidirect
280. C2×D8C2×D173semidirect
284. #131C2×D173semidirect
288. #108C2×D173semidirect
295. #100C2×D173semidirect
296. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123032
#260123012
#260122952
#260122922
#260118792
#260117602
#260117552
#260117532
#260116712
#260116622
#260116612
#260116572
#260116122
#260116112
#472×C1732
#3288173