Finite Group Databasebeta

← #36365343

Subgroups — #36365343

Order 22144.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012306C211yesmaximal, characteristic, complemented
2#26012298C211yesmaximal, characteristic, complemented
3#26012294C211yesmaximal, characteristic, complemented
4#26011872C211yesmaximal, characteristic, complemented
5#26011764C211yesmaximal, characteristic, complemented
6#26011759C211yesmaximal, characteristic, complemented
7#26011754C211yesmaximal, characteristic, complemented
8#26011751C211yesmaximal, characteristic, complemented
9#26011669C211yesmaximal, characteristic, complemented
10#26011667C211yesmaximal, characteristic, complemented
11#26011664C211yesmaximal, characteristic, complemented
12#26011662C211yesmaximal, characteristic, complemented
13#26011608C211yesmaximal, characteristic, complemented
14#26011606C211yesmaximal, characteristic, complemented
15#577×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic
18#19149711—21no—
19#19149710—21no—
20#19149710C2211yescharacteristic, complemented
21#19149709—21no—
22#19149709C2211yescharacteristic
23#19149709C2211yescharacteristic, complemented
24#19149705—21no—
25#19149705C2211yescharacteristic
26#19149704—21no—
27#19149704C2211yescharacteristic
28#19149704C2211yescharacteristic, complemented
29#19149703—21no—
30#19149700—21no—
31#19149699—21no—
32#19149699—21no—
33#19149699C2211yescharacteristic
34C2×#14583816—21no—
35C2×#14583816C2211yescharacteristic, complemented
36C2×#14583815C2211yescharacteristic, complemented
37SD16×D173—21no—
38C2×#14583811—21no—
39C2×#14583811C2211yescharacteristic
40#19149686C2211yescharacteristic
41#19149686C2211yescharacteristic, complemented
42#19149685C2211yescharacteristic, complemented
43#19149668C2211yescharacteristic, complemented
44#19149666C2211yescharacteristic, complemented
45#19149615C2211yescharacteristic, complemented
46#19149614C2211yescharacteristic, complemented
47#19149606C2211yescharacteristic
48#19149606C2211yescharacteristic, complemented
49#19149606C2211yescharacteristic, complemented
50#19149605C2211yescharacteristic, complemented
51#19149598C2211yescharacteristic, complemented
52#19149597C2211yescharacteristic, complemented
53#19149596C2211yescharacteristic, complemented
54#19149595C2211yescharacteristic, complemented
55C8×Dic173C2211yescharacteristic, complemented
562−1+4×C173C2211yescharacteristic, complemented
57H(5,𝔽2)×C173C2211yescharacteristic
58#137×C173—21no—
59#137×C173C2211yescharacteristic
60AGL(1,ℤ/8)×C173—21no—
61AGL(1,ℤ/8)×C173C2211yescharacteristic, complemented
62#135×C173—21no—
63#135×C173C2211yescharacteristic
64C2×SD16×C173—21no—
65C2×SD16×C173C2211yescharacteristic, complemented
66#131×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583826—21no—
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583825—21no—
72#14583825—41no—
73C2×Dic346—21no—
74C2×Dic346C2311yescharacteristic
75Q8×D173—21no—
76C2×D692—21no—
77C2×D692C2311yescharacteristic
78D4×D173—21no—
79D4×D173—41no—
80#14583816—21no—
81#14583816—21no—
82#14583816—21no—
83#14583816—21no—
84#14583816—21no—
85#14583816D411yescharacteristic
86#14583815—21no—
87#14583815—21no—
88#14583814—21no—
89#14583814—21no—
90#14583814—21no—
91#14583814D411yescharacteristic
92#14583813—21no—
93#14583813—21no—
94#14583813—21no—
95#14583813D411yescharacteristic
96#14583811—21no—
97#14583811—21no—
98#14583811—21no—
99#14583811—21no—
100#14583811—21no—
101#14583811D411yescharacteristic
102#14583810—21no—
103#14583810—21no—
104#14583810C2311yescharacteristic
105#14583810C2311yescharacteristic
106#14583810C2311yescharacteristic
107#14583809—21no—
108C2×#13063436—21no—
109C2×#13063436—21no—
110C2×#13063436C2311yescharacteristic
111C8×D173—21no—
112C22⋊Dic173—41no—
113#14583803—41no—
114C4×Dic173C2311yescharacteristic
115#41×C173—21no—
116#41×C173—21no—
117#41×C173—21no—
118#41×C173—21no—
119#41×C173—41no—
120#41×C173C2311yescharacteristic
121#41×C173C2311yescharacteristic
122#41×C173C2311yescharacteristic
123C2×Q8×C173—21no—
124C2×Q8×C173—21no—
125C2×Q8×C173C2311yescharacteristic
126C2×D4×C173—21no—
127C2×D4×C173—21no—
128C2×D4×C173—41no—
129C2×D4×C173C2311yescharacteristic
130Q16×C173—21no—
131Q16×C173D411yescharacteristic
132SD16×C173—21no—
133SD16×C173—21no—
134SD16×C173—21no—
135SD16×C173—21no—
136SD16×C173D411yescharacteristic
137SD16×C173D411yescharacteristic
138D8×C173—21no—
139D8×C173D411yescharacteristic
140M16×C173—21no—
141M16×C173C2311yescharacteristic
142C2×C8×C173—21no—
143C2×C8×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146Dic346—21no—
147Dic346—21no—
148Dic346—41no—
149Dic346C2×D411yescharacteristic
150D692—21no—
151D692—21no—
152D692—41no—
153D692C2×D411yescharacteristic
154C22⋊D173—21no—
155C22⋊D173—41no—
156C22⋊D173—41no—
157C2×Dic173—21no—
158C2×Dic173—21no—
159C2×Dic173—41no—
160C4×D173—21no—
161C4×D173—41no—
162C4×D173—41no—
163#13063436—21no—
164#13063436—21no—
165#13063436—21no—
166#13063436C2×D411yescharacteristic
167#13063436C2×D411yescharacteristic
168C23×C173—21no—
169C23×C173—41no—
170Q8×C173—21no—
171Q8×C173—21no—
172Q8×C173—41no—
173Q8×C173C2×D411yescharacteristic
174Q8×C173C2×D411yescharacteristic
175Q8×C173C2×D411yescharacteristic
176D4×C173—21no—
177D4×C173—21no—
178D4×C173—21no—
179D4×C173—21no—
180D4×C173—21no—
181D4×C173—41no—
182D4×C173—41no—
183D4×C173—41no—
184D4×C173C2×D411yescharacteristic
185D4×C173C2×D411yescharacteristic
186D4×C173C2×D411yescharacteristic
187C2×C4×C173—21no—
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—41no—
192C2×C4×C173—41no—
193C2×C4×C173—41no—
194C2×C4×C173C2411yescharacteristic
195C8×C173—21no—
196C8×C173C2×D411yescharacteristic
197C8×C173C2×D411yescharacteristic
198C22×C173—21no—
199C22×C173—21no—
200C22×C173—21no—
201C22×C173—41no—
202C22×C173—81no—
203C22×C173—161no—
204C22×C173H(5,𝔽2)11yescharacteristic
205C2×D173—21no—
206C2×D173—41no—
207C2×D173—81no—
208C2×D173—161no—
209C4×C173—21no—
210C4×C173—21no—
211C4×C173—21no—
212C4×C173—41no—
213C4×C173C22×D411yescharacteristic
214C4×C173C22×D411yescharacteristic
215Dic173—21no—
216Dic173—41no—
217Dic173—41no—
218C2×C173—21no—
219C2×C173—41no—
220C2×C173—41no—
221C2×C173—41no—
222C2×C173—81no—
223C2×C173D4211yescharacteristic
224D173—41no—
225D173—81no—
226C173#329011yescharacteristic, complemented, Sylow
227#3290—1731nomaximal, Sylow
228#578—1731no—
229#577—1731no—
230#577D17311yescharacteristic, complemented
231#576—1731no—
232#495—1731no—
233#473—1731no—
234#472—1731no—
235#471—1731no—
236#471—1731no—
237#455—1731no—
238#455—1731no—
239#454—1731no—
240#454—1731no—
241#443—1731no—
242#443—1731no—
2432−1+4—1731no—
2442−1+4C2×D17311yescharacteristic, complemented
245H(5,𝔽2)—1731no—
246H(5,𝔽2)C2×D17311yescharacteristic
247#137—21no—
248#137—3461no—
249#137—3461no—
250#137—3461no—
251#137—1731no—
252#137—1731no—
253#137—1731no—
254#137C2×D17311yescharacteristic
255AGL(1,ℤ/8)—21no—
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)C2×D17311yescharacteristic, complemented
263#135—21no—
264#135—3461no—
265#135—3461no—
266#135—3461no—
267#135—1731no—
268#135C2×D17311yescharacteristic
269C2×Q16—3461no—
270C2×Q16—1731no—
271C2×SD16—21no—
272C2×SD16—3461no—
273C2×SD16—1731no—
274C2×SD16C2×D17311yescharacteristic, complemented
275C2×D8—3461no—
276C2×D8—1731no—
277#131—1731no—
278#131—1731no—
279#131—1731no—
280#131C2×D17311yescharacteristic, complemented
281#124—1731no—
282#124—1731no—
283#108—1731no—
284#108—1731no—
285#104—1731no—
286#104—1731no—
287#104—1731no—
288#104—1731no—
289#101—1731no—
290#101—1731no—
291#100—1731no—
292#100—1731no—
293C4×C8—1731no—
294#41—21no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—41no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—1731no—
304#41—6921no—
305#41C22×D17311yescharacteristic
306#41C22×D17311yescharacteristic
307#41C22×D17311yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—3461no—
311C2×Q8—3461no—
312C2×Q8—1731no—
313C2×Q8C22×D17311yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—3461no—
318C2×D4—3461no—
319C2×D4—1731no—
320C2×D4—6921no—
321C2×D4C22×D17311yescharacteristic
322Q16—21no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—1731no—
329Q16C22⋊D17311yescharacteristic
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—1731no—
343SD16—1731no—
344SD16C22⋊D17311yescharacteristic
345SD16C22⋊D17311yescharacteristic
346D8—21no—
347D8—3461no—
348D8—3461no—
349D8—3461no—
350D8—3461no—
351D8—3461no—
352D8—1731no—
353D8C22⋊D17311yescharacteristic
354M16—21no—
355M16—3461no—
356M16—3461no—
357M16—3461no—
358M16—1731no—
359M16—1731no—
360M16—1731no—
361M16C22×D17311yescharacteristic
362C2×C8—21no—
363C2×C8—3461no—
364C2×C8—3461no—
365C2×C8—3461no—
366C2×C8—1731no—
367C2×C8C22×D17311yescharacteristic
368C22⋊C4—6921no—
369C22⋊C4—6921no—
370C42—1731no—
371C23—21no—
372C23—41no—
373C23—3461no—
374C23—6921no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—3461no—
379Q8—3461no—
380Q8—1731no—
381Q8—6921no—
382Q8D4×D17311yescharacteristic
383Q8C2×C22⋊D17311yescharacteristic
384Q8C2×C22⋊D17311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—3461no—
394D4—3461no—
395D4—3461no—
396D4—1731no—
397D4—6921no—
398D4—6921no—
399D4—6921no—
400D4D4×D17311yescharacteristic
401D4C2×C22⋊D17311yescharacteristic
402D4C2×C22⋊D17311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—3461no—
411C2×C4—3461no—
412C2×C4—3461no—
413C2×C4—6921no—
414C2×C4—6921no—
415C2×C4—6921no—
416C2×C4C23×D17311yescharacteristic
417C8—21no—
418C8—3461no—
419C8—3461no—
420C8—3461no—
421C8—1731no—
422C8—1731no—
423C8C2×C22⋊D17311yescharacteristic
424C8C2×C22⋊D17311yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—81no—
430C22—161no—
431C22—3461no—
432C22—13841no—
433C22—6921no—
434C22—27681no—
435C22#1914972411yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—3461no—
441C4—6921no—
442C4—6921no—
443C4C2×D4×D17311yescharacteristic
444C4C22×C22⋊D17311yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—81no—
450C2—13841no—
451C2—6921no—
452C2D4×C22⋊D17311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#577×C173C2
socleC2×C173D42
hypercenter#577D173
hypocenterC173#3290
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #26012306C2semidirect
2. #26012298C2semidirect
3. #26012294C2semidirect
4. #26011872C2semidirect
5. #26011764C2semidirect
6. #26011759C2semidirect
7. #26011754C2semidirect
8. #26011751C2semidirect
9. #26011669C2semidirect
10. #26011667C2semidirect
11. #26011664C2semidirect
12. #26011662C2semidirect
13. #26011608C2semidirect
14. #26011606C2semidirect
15. #577×C173C2semidirect
16. #19149726C22semidirect
20. #19149710C22semidirect
23. #19149709C22semidirect
28. #19149704C22semidirect
35. C2×#14583816C22semidirect
36. C2×#14583815C22semidirect
41. #19149686C22semidirect
42. #19149685C22semidirect
43. #19149668C22semidirect
44. #19149666C22semidirect
45. #19149615C22semidirect
46. #19149614C22semidirect
48. #19149606C22semidirect
49. #19149606C22semidirect
50. #19149605C22semidirect
51. #19149598C22semidirect
52. #19149597C22semidirect
53. #19149596C22semidirect
54. #19149595C22semidirect
55. C8×Dic173C22semidirect
56. 2−1+4×C173C22semidirect
61. AGL(1,ℤ/8)×C173C22semidirect
65. C2×SD16×C173C22semidirect
66. #131×C173C22semidirect
226. C173#3290semidirect
230. #577D173semidirect
244. 2−1+4C2×D173semidirect
262. AGL(1,ℤ/8)C2×D173semidirect
274. C2×SD16C2×D173semidirect
280. #131C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123062
#260122982
#260122942
#260118722
#260117642
#260117592
#260117542
#260117512
#260116692
#260116672
#260116642
#260116622
#260116082
#260116062
#577×C1732
#3290173