Finite Group Databasebeta

← #36365328

Subgroups — #36365328

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#26012300C211yesmaximal, characteristic, complemented
2#26012297C211yesmaximal, characteristic, complemented
3#26012296C211yesmaximal, characteristic, complemented
4#26012295C211yesmaximal, characteristic, complemented
5#26011877C211yesmaximal, characteristic, complemented
6#26011762C211yesmaximal, characteristic, complemented
7#26011760C211yesmaximal, characteristic, complemented
8#26011757C211yesmaximal, characteristic, complemented
9#26011667C211yesmaximal, characteristic, complemented
10#26011666C211yesmaximal, characteristic, complemented
11#26011661C211yesmaximal, characteristic, complemented
12#26011657C211yesmaximal, characteristic, complemented
13#26011612C211yesmaximal, characteristic, complemented
14#26011611C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16#19149727C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic
18#19149725C2211yescharacteristic, complemented
19#19149724C2211yescharacteristic, complemented
20#19149711—21no—
21#19149710C2211yescharacteristic, complemented
22#19149708—21no—
23#19149707—21no—
24#19149707—21no—
25#19149707C2211yescharacteristic
26#19149707C2211yescharacteristic, complemented
27#19149707C2211yescharacteristic, complemented
28#19149707C2211yescharacteristic, complemented
29#19149706—21no—
30#19149706—21no—
31#19149706—21no—
32#19149705C2211yescharacteristic, complemented
33#19149703—21no—
34#19149702—21no—
35#19149701—21no—
36#19149701C2211yescharacteristic
37#19149700—21no—
38#19149700—21no—
39#19149699C2211yescharacteristic
40C2×#14583813C2211yescharacteristic, complemented
41SD16×D173—21no—
42SD16×D173—21no—
43C2×D1384—21no—
44C2×D1384C2211yescharacteristic, complemented
45C2×#14583811C2211yescharacteristic, complemented
46D8×D173—21no—
47#19149687C2211yescharacteristic, complemented
48#19149687C2211yescharacteristic, complemented
49#19149685C2211yescharacteristic, complemented
50#19149675C2211yescharacteristic, complemented
51#19149668C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149606C2211yescharacteristic, complemented
54#19149605C2211yescharacteristic, complemented
55#19149605C2211yescharacteristic, complemented
56#19149605C2211yescharacteristic, complemented
57#19149595C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149589C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62C2×SD16×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic, complemented
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583827—21no—
71#14583826—21no—
72#14583826—21no—
73#14583826—41no—
74#14583826C2311yescharacteristic
75#14583826C2311yescharacteristic
76#14583826C2311yescharacteristic
77#14583825—21no—
78#14583825—21no—
79Q8×D173—21no—
80Q8×D173—21no—
81C2×D692—21no—
82C2×D692—21no—
83C2×D692C2311yescharacteristic
84C2×D692C2311yescharacteristic
85C2×C22⋊D173—41no—
86C2×C22⋊D173—41no—
87D4×D173—21no—
88D4×D173—21no—
89D4×D173—21no—
90D4×D173—21no—
91D4×D173—41no—
92D4×D173—41no—
93Dic692D411yescharacteristic, complemented
94#14583816—21no—
95#14583816D411yescharacteristic, complemented
96#14583815—21no—
97#14583815—21no—
98#14583815—21no—
99#14583815—21no—
100#14583815D411yescharacteristic, complemented
101#14583815D411yescharacteristic, complemented
102#14583814—21no—
103#14583814D411yescharacteristic, complemented
104#14583813—21no—
105#14583813—21no—
106#14583813—21no—
107#14583813D411yescharacteristic, complemented
108D1384—21no—
109D1384—21no—
110D1384—21no—
111D1384—21no—
112D1384—21no—
113D1384—21no—
114D1384D411yescharacteristic, complemented
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—
127#14583803—41no—
128C4×Dic173C2311yescharacteristic
129#41×C173C2311yescharacteristic
130C2×Q8×C173C2311yescharacteristic
131C2×D4×C173C2311yescharacteristic
132Q16×C173—21no—
133SD16×C173—21no—
134SD16×C173—21no—
135SD16×C173—21no—
136SD16×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—21no—
146C22×D173—41no—
147C22×D173—41no—
148C22×D173—41no—
149Dic346—41no—
150Dic346C2×D411yescharacteristic
151Dic346C2×D411yescharacteristic
152Dic346C2×D411yescharacteristic
153D692—21no—
154D692—21no—
155D692—21no—
156D692—21no—
157D692—41no—
158D692—41no—
159D692—41no—
160D692C2×D411yescharacteristic
161D692C2×D411yescharacteristic
162D692C2×D411yescharacteristic
163C22⋊D173—21no—
164C22⋊D173—21no—
165C22⋊D173—21no—
166C22⋊D173—41no—
167C22⋊D173—41no—
168C22⋊D173—41no—
169C22⋊D173—41no—
170C22⋊D173—82no—
171C2×Dic173—21no—
172C2×Dic173—41no—
173C4×D173—21no—
174C4×D173—21no—
175C4×D173—21no—
176C4×D173—41no—
177C4×D173—41no—
178C4×D173—41no—
179C4×D173—41no—
180#13063436—21no—
181#13063436C2×D411yescharacteristic
182#13063436C2×D411yescharacteristic
183C23×C173—21no—
184Q8×C173—21no—
185Q8×C173—21no—
186Q8×C173C2×D411yescharacteristic
187D4×C173—21no—
188D4×C173—21no—
189D4×C173—21no—
190D4×C173C2×D411yescharacteristic
191C2×C4×C173—21no—
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—81no—
202C22×C173H(5,𝔽2)11yescharacteristic
203C2×D173—21no—
204C2×D173—21no—
205C2×D173—21no—
206C2×D173—41no—
207C2×D173—41no—
208C2×D173—81no—
209C2×D173—81no—
210C2×D173—161no—
211C2×D173—161no—
212C2×D173—161no—
213C4×C173—21no—
214C4×C173—41no—
215C4×C173C22×D411yescharacteristic
216C4×C173C22×D411yescharacteristic
217Dic173—21no—
218Dic173—21no—
219Dic173—21no—
220Dic173—41no—
221C2×C173—21no—
222C2×C173—41no—
223C2×C173—81no—
224C2×C173D4211yescharacteristic
225D173—41no—
226D173—41no—
227D173—41no—
228D173—81no—
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#471D17311yescharacteristic, complemented
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#137C2×D17311yescharacteristic, complemented
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)—1731no—
266AGL(1,ℤ/8)C2×D17311yescharacteristic
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×SD16C2×D17311yescharacteristic, complemented
277C2×D8—3461no—
278C2×D8—3461no—
279C2×D8—1731no—
280C2×D8—1731no—
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#101C2×D17311yescharacteristic, complemented
294#100—1731no—
295#100—1731no—
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×Q8C22×D17311yescharacteristic
314C2×D4—3461no—
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—3461no—
318C2×D4—3461no—
319C2×D4—3461no—
320C2×D4—1731no—
321C2×D4—1731no—
322C2×D4—6921no—
323C2×D4—6921no—
324C2×D4—6921no—
325C2×D4—6921no—
326C2×D4C22×D17311yescharacteristic
327Q16—21no—
328Q16—3461no—
329Q16—1731no—
330Q16—1731no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
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—3461no—
349D8—3461no—
350D8—3461no—
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—3461no—
377C23—3461no—
378C23—6921no—
379C23—6921no—
380C23—6921no—
381Q8—21no—
382Q8—21no—
383Q8—1731no—
384Q8—1731no—
385Q8—1731no—
386Q8—6921no—
387Q8D4×D17311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—3461no—
392D4—3461no—
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—21no—
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—81no—
433C22—3461no—
434C22—3461no—
435C22—3461no—
436C22—13841no—
437C22—13841no—
438C22—6921no—
439C22—6921no—
440C22—27681no—
441C22—27681no—
442C22—27681no—
443C22#1914972511yescharacteristic
444C4—21no—
445C4—41no—
446C4—3461no—
447C4—3461no—
448C4—3461no—
449C4—6921no—
450C4C2×D4×D17311yescharacteristic
451C4C2×D4×D17311yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—13841no—
456C2—13841no—
457C2—6921no—
458C2—6921no—
459C2—6921no—
460C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173D42
hypercenter#471D173
hypocenterC173#3288
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #26012300C2semidirect
2. #26012297C2semidirect
3. #26012296C2semidirect
4. #26012295C2semidirect
5. #26011877C2semidirect
6. #26011762C2semidirect
7. #26011760C2semidirect
8. #26011757C2semidirect
9. #26011667C2semidirect
10. #26011666C2semidirect
11. #26011661C2semidirect
12. #26011657C2semidirect
13. #26011612C2semidirect
14. #26011611C2semidirect
15. #471×C173C2semidirect
16. #19149727C22semidirect
18. #19149725C22semidirect
19. #19149724C22semidirect
21. #19149710C22semidirect
26. #19149707C22semidirect
27. #19149707C22semidirect
28. #19149707C22semidirect
32. #19149705C22semidirect
40. C2×#14583813C22semidirect
44. C2×D1384C22semidirect
45. C2×#14583811C22semidirect
47. #19149687C22semidirect
48. #19149687C22semidirect
49. #19149685C22semidirect
50. #19149675C22semidirect
51. #19149668C22semidirect
52. #19149614C22semidirect
53. #19149606C22semidirect
54. #19149605C22semidirect
55. #19149605C22semidirect
56. #19149605C22semidirect
57. #19149595C22semidirect
58. #19149595C22semidirect
59. #19149589C22semidirect
60. #137×C173C22semidirect
62. C2×SD16×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #100×C173C22semidirect
93. Dic692D4semidirect
95. #14583816D4semidirect
100. #14583815D4semidirect
101. #14583815D4semidirect
103. #14583814D4semidirect
107. #14583813D4semidirect
114. D1384D4semidirect
118. #14583811D4semidirect
230. C173#3288semidirect
240. #471D173semidirect
254. #137C2×D173semidirect
276. C2×SD16C2×D173semidirect
284. #131C2×D173semidirect
288. #108C2×D173semidirect
293. #101C2×D173semidirect
296. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123002
#260122972
#260122962
#260122952
#260118772
#260117622
#260117602
#260117572
#260116672
#260116662
#260116612
#260116572
#260116122
#260116112
#471×C1732
#3288173