Finite Group Databasebeta

← #36363289

Subgroups — #36363289

Order 22144.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012287C211yesmaximal, characteristic, complemented
2C2×#19149703C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D173C211yesmaximal, characteristic, complemented
4#26012256C211yesmaximal, characteristic, complemented
5#26011760C211yesmaximal, characteristic, complemented
6#26011759C211yesmaximal, characteristic, complemented
7#26011758C211yesmaximal, characteristic, complemented
8#26011755C211yesmaximal, characteristic, complemented
9#26011754C211yesmaximal, characteristic, complemented
10#26011753C211yesmaximal, characteristic, complemented
11#26011752C211yesmaximal, characteristic, complemented
12#26011538C211yesmaximal, characteristic, complemented
13#26011472C211yesmaximal, characteristic, complemented
14#100×D173C211yesmaximal, characteristic, complemented
15#472×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17#41×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149712C2211yescharacteristic, complemented
20#19149709C2211yescharacteristic, complemented
21#19149708—21no—
22#19149707C2211yescharacteristic
23#19149706—21no—
24#19149705C2211yescharacteristic
25#19149705C2211yescharacteristic, complemented
26#19149704C2211yescharacteristic
27#19149703—21no—
28#19149703—21no—
29#19149703—21no—
30#19149703—21no—
31#19149703—21no—
32#19149703—21no—
33#19149700—21no—
34#19149698—21no—
35C2×#14583815C2211yescharacteristic, complemented
36C2×#14583814C2211yescharacteristic, complemented
37SD16×D173—21no—
38C2×#14583811C2211yescharacteristic, complemented
39D8×D173—21no—
40#19149687—21no—
41#19149687C2211yescharacteristic, complemented
42#19149686C2211yescharacteristic, complemented
43#19149685—21no—
44#19149685C2211yescharacteristic, complemented
45C2×#14583809—21no—
46C2×#14583809C2211yescharacteristic
47M16×D173—21no—
48M16×D173C2211yescharacteristic
49M16×D173C2211yescharacteristic
50M16×D173C2211yescharacteristic
51#19149615C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149597C2211yescharacteristic, complemented
55#19149597C2211yescharacteristic, complemented
56#19149596C2211yescharacteristic, complemented
57#19149596C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149595C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C173C2211yescharacteristic
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
67C23×D173—21no—
68#14583827D411yescharacteristic
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583825—21no—
72#14583825—21no—
73#14583825—21no—
74#14583825—21no—
75#14583825—21no—
76#14583825D411yescharacteristic
77C2×Dic346C2311yescharacteristic
78Q8×D173D411yescharacteristic
79C2×D692C2311yescharacteristic
80C2×C22⋊D173—21no—
81C2×C22⋊D173—21no—
82D4×D173—21no—
83D4×D173—21no—
84D4×D173—21no—
85D4×D173—21no—
86D4×D173—21no—
87D4×D173D411yescharacteristic
88C22×Dic173—21no—
89C2×C4×D173—21no—
90C2×C4×D173C2311yescharacteristic
91Dic692—21no—
92#14583816—21no—
93#14583815—21no—
94#14583815—21no—
95#14583815—21no—
96#14583815—21no—
97#14583814—21no—
98#14583814—21no—
99#14583814—21no—
100#14583814—21no—
101#14583814—21no—
102#14583813—21no—
103D1384—21no—
104#14583811—21no—
105#14583811—21no—
106#14583811—21no—
107#14583811—21no—
108#14583811—21no—
109#14583810—21no—
110#14583810C2311yescharacteristic
111#14583810C2311yescharacteristic
112#14583810C2311yescharacteristic
113#14583809—21no—
114#14583809—21no—
115#14583809—21no—
116#14583809—21no—
117#14583809—21no—
118#14583809D411yescharacteristic
119#14583809D411yescharacteristic
120#14583809D411yescharacteristic
121#14583809D411yescharacteristic
122C2×#13063436—21no—
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126C8×D173—21no—
127C8×D173—21no—
128C8×D173—21no—
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—21no—
146C22×D173—41no—
147C22×D173—41no—
148C22×D173—82no—
149C22×D173#4111yescharacteristic
150Dic346—21no—
151Dic346—21no—
152Dic346—21no—
153Dic346C2×D411yescharacteristic
154D692—21no—
155D692—21no—
156D692—21no—
157D692C2×D411yescharacteristic
158C22⋊D173—21no—
159C22⋊D173—21no—
160C22⋊D173—21no—
161C22⋊D173—41no—
162C22⋊D173—41no—
163C22⋊D173—41no—
164C22⋊D173—41no—
165C2×Dic173—21no—
166C2×Dic173—21no—
167C2×Dic173—41no—
168C2×Dic173—41no—
169C2×Dic173#4111yescharacteristic
170C4×D173—21no—
171C4×D173—21no—
172C4×D173—21no—
173C4×D173C2×D411yescharacteristic
174C4×D173C2×D411yescharacteristic
175C4×D173C2×D411yescharacteristic
176C4×D173C2×D411yescharacteristic
177#13063436—21no—
178#13063436—21no—
179#13063436—21no—
180#13063436C2×D411yescharacteristic
181#13063436C2×D411yescharacteristic
182C23×C173—21no—
183C23×C173—21no—
184Q8×C173C2×D411yescharacteristic
185D4×C173—21no—
186D4×C173—21no—
187D4×C173—21no—
188D4×C173—21no—
189D4×C173—21no—
190D4×C173C2×D411yescharacteristic
191C2×C4×C173—21no—
192C2×C4×C173C2411yescharacteristic
193C8×C173—21no—
194C8×C173—21no—
195C8×C173—21no—
196C8×C173C2×D411yescharacteristic
197C8×C173C2×D411yescharacteristic
198C22×C173—21no—
199C22×C173—41no—
200C22×C173—41no—
201C22×C173—81no—
202C22×C173—81no—
203C22×C173C2×#4111yescharacteristic
204C2×D173—21no—
205C2×D173—41no—
206C2×D173—41no—
207C2×D173—81no—
208C2×D173—82no—
209C2×D173—82no—
210C2×D173—81no—
211C2×D173#12111yescharacteristic
212C2×D173#12111yescharacteristic
213C4×C173—21no—
214C4×C173C22×D411yescharacteristic
215C4×C173C22×D411yescharacteristic
216Dic173—21no—
217Dic173—41no—
218Dic173#12111yescharacteristic
219Dic173#12111yescharacteristic
220C2×C173—21no—
221C2×C173—41no—
222C2×C173—81no—
223C2×C173—81no—
224C2×C173C2×#12111yescharacteristic
225D173—21no—
226D173—21no—
227D173—41no—
228D173—81no—
229C173#306811yescharacteristic, complemented, Sylow
230#3068—1731nomaximal, Sylow
231#575—1731no—
232C2×AGL(1,ℤ/8)—1731no—
233C2×AGL(1,ℤ/8)—1731no—
234#568—1731no—
235#472—1731no—
236#472—1731no—
237#472—1731no—
238#472D17311yescharacteristic, complemented
239#471—1731no—
240#471—1731no—
241#471—1731no—
242#471—1731no—
243#430—1731no—
244#413—1731no—
245C2×#100—1731no—
246C2×#41—1731no—
247C2×#41—1731no—
248C22×D4—1731no—
249#137—3461no—
250#137—1731no—
251#137—1731no—
252AGL(1,ℤ/8)—3461no—
253AGL(1,ℤ/8)—3461no—
254AGL(1,ℤ/8)—3461no—
255AGL(1,ℤ/8)—3461no—
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—3461no—
259AGL(1,ℤ/8)—1731no—
260AGL(1,ℤ/8)—1731no—
261AGL(1,ℤ/8)—1731no—
262AGL(1,ℤ/8)—1731no—
263AGL(1,ℤ/8)C2×D17311yescharacteristic
264AGL(1,ℤ/8)C2×D17311yescharacteristic, complemented
265#135—3461no—
266#135—3461no—
267C2×SD16—3461no—
268C2×SD16—1731no—
269C2×SD16—1731no—
270C2×D8—3461no—
271C2×D8—1731no—
272C2×D8C2×D17311yescharacteristic, complemented
273#131—3461no—
274#131—3461no—
275#131—1731no—
276#131—1731no—
277#131—1731no—
278#131C2×D17311yescharacteristic, complemented
279C2×M16—3461no—
280C2×M16—3461no—
281C2×M16—1731no—
282C2×M16—1731no—
283C2×M16—1731no—
284C2×M16—1731no—
285#108—1731no—
286#108—1731no—
287#108—1731no—
288#108C2×D17311yescharacteristic, complemented
289#101—1731no—
290#101—1731no—
291#100—1731no—
292#100—1731no—
293#100—1731no—
294#100—1731no—
295#100C2×D17311yescharacteristic, complemented
296#100C2×D17311yescharacteristic, complemented
297C24—3461no—
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—3461no—
304#41—1731no—
305#41—1731no—
306#41—1731no—
307#41C22×D17311yescharacteristic
308C2×Q8—1731no—
309C2×Q8—1731no—
310C2×D4—3461no—
311C2×D4—3461no—
312C2×D4—3461no—
313C2×D4—3461no—
314C2×D4—3461no—
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—1731no—
318C2×D4—1731no—
319C2×D4C22×D17311yescharacteristic
320C2×D4C22×D17311yescharacteristic
321C22×C4—3461no—
322C22×C4—3461no—
323C22×C4—1731no—
324Q16—3461no—
325Q16—3461no—
326SD16—21no—
327SD16—21no—
328SD16—3461no—
329SD16—3461no—
330SD16—3461no—
331SD16—3461no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—3461no—
343D8—3461no—
344D8—3461no—
345D8—3461no—
346D8—3461no—
347D8—3461no—
348M16—21no—
349M16—3461no—
350M16—3461no—
351M16—3461no—
352M16—3461no—
353M16—3461no—
354M16—3461no—
355M16—1731no—
356M16—1731no—
357M16—1731no—
358M16—1731no—
359M16—1731no—
360M16—1731no—
361M16—1731no—
362M16C22×D17311yescharacteristic
363M16C22×D17311yescharacteristic
364M16C22×D17311yescharacteristic
365C2×C8—21no—
366C2×C8—3461no—
367C2×C8—3461no—
368C2×C8—3461no—
369C2×C8—3461no—
370C2×C8—3461no—
371C2×C8—3461no—
372C2×C8—1731no—
373C2×C8C22×D17311yescharacteristic
374C23—21no—
375C23—21no—
376C23—3461no—
377C23—3461no—
378C23—13842no—
379C23—1731no—
380C23—6921no—
381C23—6921no—
382Q8—3461no—
383Q8—3461no—
384Q8—3461no—
385Q8—1731no—
386Q8D4×D17311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—3461no—
393D4—3461no—
394D4—3461no—
395D4—3461no—
396D4—3461no—
397D4—3461no—
398D4—1731no—
399D4—6921no—
400D4—6921no—
401D4—6921no—
402D4—6921no—
403D4D4×D17311yescharacteristic
404C2×C4—21no—
405C2×C4—3461no—
406C2×C4—3461no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—1731no—
411C2×C4—1731no—
412C2×C4—1731no—
413C2×C4—1731no—
414C2×C4—1731no—
415C2×C4—6921no—
416C2×C4—6921no—
417C2×C4C23×D17311yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—3461no—
422C8—3461no—
423C8—3461no—
424C8—1731no—
425C8—1731no—
426C8D4×D17311yescharacteristic
427C8D4×D17311yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—3461no—
434C22—13841no—
435C22—13841no—
436C22—13842no—
437C22—13842no—
438C22—1731no—
439C22—1731no—
440C22—6921no—
441C22—6921no—
442C22#41×D17311yescharacteristic
443C4—21no—
444C4—3461no—
445C4—1731no—
446C4—1731no—
447C4—6921no—
448C4C2×D4×D17311yescharacteristic
449C4C2×D4×D17311yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—3461no—
455C2—3461no—
456C2—13841no—
457C2—6921no—
458C2#121×D17311yescharacteristic
subgroupG/H
centerC2#121×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#472×C173C2
socleC2×C173C2×#121
hypercenter#472D173
hypocenterC173#3068
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. #26012287C2semidirect
2. C2×#19149703C2semidirect
3. AGL(1,ℤ/8)×D173C2semidirect
4. #26012256C2semidirect
5. #26011760C2semidirect
6. #26011759C2semidirect
7. #26011758C2semidirect
8. #26011755C2semidirect
9. #26011754C2semidirect
10. #26011753C2semidirect
11. #26011752C2semidirect
12. #26011538C2semidirect
13. #26011472C2semidirect
14. #100×D173C2semidirect
15. #472×C173C2semidirect
19. #19149712C22semidirect
20. #19149709C22semidirect
25. #19149705C22semidirect
35. C2×#14583815C22semidirect
36. C2×#14583814C22semidirect
38. C2×#14583811C22semidirect
41. #19149687C22semidirect
42. #19149686C22semidirect
44. #19149685C22semidirect
51. #19149615C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
54. #19149597C22semidirect
55. #19149597C22semidirect
56. #19149596C22semidirect
57. #19149596C22semidirect
58. #19149595C22semidirect
59. #19149595C22semidirect
61. AGL(1,ℤ/8)×C173C22semidirect
62. C2×D8×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #100×C173C22semidirect
66. #100×C173C22semidirect
229. C173#3068semidirect
238. #472D173semidirect
264. AGL(1,ℤ/8)C2×D173semidirect
272. C2×D8C2×D173semidirect
278. #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
#260122872
C2×#191497032
AGL(1,ℤ/8)×D1732
#260122562
#260117602
#260117592
#260117582
#260117552
#260117542
#260117532
#260117522
#260115382
#260114722
#100×D1732
#472×C1732
#3068173