Finite Group Databasebeta

← #36365354

Subgroups — #36365354

Order 22144.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012302C211yesmaximal, characteristic, complemented
2#26012299C211yesmaximal, characteristic, complemented
3#26012298C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011756C211yesmaximal, characteristic, complemented
6#26011754C211yesmaximal, characteristic, complemented
7#26011752C211yesmaximal, characteristic, complemented
8#26011751C211yesmaximal, characteristic, complemented
9#26011674C211yesmaximal, characteristic, complemented
10#26011672C211yesmaximal, characteristic, complemented
11#26011659C211yesmaximal, characteristic, complemented
12#26011657C211yesmaximal, characteristic, complemented
13#26011608C211yesmaximal, characteristic, complemented
14#26011606C211yesmaximal, characteristic, complemented
15#577×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic
17#19149725C2211yescharacteristic, complemented
18#19149711—21no—
19#19149710—21no—
20#19149710C2211yescharacteristic, complemented
21#19149709—21no—
22#19149709C2211yescharacteristic, complemented
23#19149709C2211yescharacteristic, complemented
24#19149705—21no—
25#19149705C2211yescharacteristic
26#19149704—21no—
27#19149704C2211yescharacteristic
28#19149704C2211yescharacteristic
29#19149703—21no—
30#19149700—21no—
31#19149699—21no—
32#19149699—21no—
33#19149699C2211yescharacteristic
34C2×#14583815C2211yescharacteristic, complemented
35C2×#14583814—21no—
36C2×#14583814C2211yescharacteristic, complemented
37C2×#14583813—21no—
38C2×#14583813C2211yescharacteristic, complemented
39SD16×D173—21no—
40#19149686C2211yescharacteristic, complemented
41#19149686C2211yescharacteristic, complemented
42#19149685C2211yescharacteristic, complemented
43#19149677C2211yescharacteristic, complemented
44#19149675C2211yescharacteristic
45#19149615C2211yescharacteristic, complemented
46#19149614C2211yescharacteristic, complemented
47#19149606C2211yescharacteristic, complemented
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
57H(5,𝔽2)×C173C2211yescharacteristic, complemented
58#137×C173—21no—
59#137×C173C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C173—21no—
61AGL(1,ℤ/8)×C173C2211yescharacteristic
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
78C2×C22⋊D173—41no—
79D4×D173—21no—
80D4×D173—41no—
81#14583816—21no—
82#14583816—21no—
83#14583816—21no—
84#14583816D411yescharacteristic, complemented
85#14583815—21no—
86#14583815—21no—
87#14583814—21no—
88#14583814—21no—
89#14583814—21no—
90#14583814—21no—
91#14583814—21no—
92#14583814D411yescharacteristic, complemented
93#14583813—21no—
94#14583813—21no—
95#14583813—21no—
96#14583813—21no—
97#14583813—21no—
98#14583813D411yescharacteristic, complemented
99#14583811—21no—
100#14583811—21no—
101#14583811—21no—
102#14583811D411yescharacteristic, complemented
103#14583810—21no—
104#14583810—21no—
105#14583810C2311yescharacteristic
106#14583810C2311yescharacteristic
107#14583810C2311yescharacteristic
108#14583809—21no—
109C2×#13063436—21no—
110C2×#13063436—21no—
111C2×#13063436C2311yescharacteristic
112C8×D173—21no—
113#14583805—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, complemented
132SD16×C173—21no—
133SD16×C173—21no—
134SD16×C173—21no—
135SD16×C173—21no—
136SD16×C173D411yescharacteristic, complemented
137SD16×C173D411yescharacteristic, complemented
138D8×C173—21no—
139D8×C173D411yescharacteristic, complemented
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—
157C22⋊D173—82no—
158C2×Dic173—21no—
159C2×Dic173—21no—
160C2×Dic173—41no—
161C4×D173—21no—
162C4×D173—41no—
163C4×D173—41no—
164#13063436—21no—
165#13063436—21no—
166#13063436—21no—
167#13063436C2×D411yescharacteristic
168#13063436C2×D411yescharacteristic
169C23×C173—21no—
170C23×C173—41no—
171Q8×C173—21no—
172Q8×C173—21no—
173Q8×C173—41no—
174Q8×C173C2×D411yescharacteristic
175Q8×C173C2×D411yescharacteristic
176Q8×C173C2×D411yescharacteristic
177D4×C173—21no—
178D4×C173—21no—
179D4×C173—21no—
180D4×C173—21no—
181D4×C173—21no—
182D4×C173—41no—
183D4×C173—41no—
184D4×C173—41no—
185D4×C173C2×D411yescharacteristic
186D4×C173C2×D411yescharacteristic
187D4×C173C2×D411yescharacteristic
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—41no—
193C2×C4×C173—41no—
194C2×C4×C173—41no—
195C2×C4×C173C2411yescharacteristic
196C8×C173—21no—
197C8×C173C2×D411yescharacteristic
198C8×C173C2×D411yescharacteristic
199C22×C173—21no—
200C22×C173—21no—
201C22×C173—21no—
202C22×C173—41no—
203C22×C173—81no—
204C22×C173—161no—
205C22×C173H(5,𝔽2)11yescharacteristic
206C2×D173—21no—
207C2×D173—41no—
208C2×D173—81no—
209C2×D173—161no—
210C4×C173—21no—
211C4×C173—21no—
212C4×C173—21no—
213C4×C173—41no—
214C4×C173C22×D411yescharacteristic
215C4×C173C22×D411yescharacteristic
216Dic173—22no—
217Dic173—21no—
218Dic173—41no—
219C2×C173—21no—
220C2×C173—41no—
221C2×C173—41no—
222C2×C173—41no—
223C2×C173—81no—
224C2×C173D4211yescharacteristic
225D173—41no—
226D173—81no—
227C173#329111yescharacteristic, complemented, Sylow
228#3291—1731nomaximal, Sylow
229#577—1731no—
230#577—1731no—
231#577—1731no—
232#577D17311yescharacteristic, complemented
233#492—1731no—
234#471—1731no—
235#471—1731no—
236#471—1731no—
237#471—1731no—
238#456—1731no—
239#456—1731no—
240#453—1731no—
241#453—1731no—
242#443—1731no—
243#443—1731no—
2442−1+4—1731no—
2452−1+4C2×D17311yescharacteristic
246H(5,𝔽2)—1731no—
247H(5,𝔽2)C2×D17311yescharacteristic, complemented
248#137—21no—
249#137—3461no—
250#137—3461no—
251#137—3461no—
252#137—1731no—
253#137—1731no—
254#137—1731no—
255#137C2×D17311yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—3461no—
259AGL(1,ℤ/8)—3461no—
260AGL(1,ℤ/8)—1731no—
261AGL(1,ℤ/8)—1731no—
262AGL(1,ℤ/8)—1731no—
263AGL(1,ℤ/8)C2×D17311yescharacteristic
264#135—21no—
265#135—3461no—
266#135—3461no—
267#135—3461no—
268#135—1731no—
269#135C2×D17311yescharacteristic
270C2×SD16—21no—
271C2×SD16—3461no—
272C2×SD16—3461no—
273C2×SD16—3461no—
274C2×SD16—1731no—
275C2×SD16—1731no—
276C2×SD16—1731no—
277C2×SD16C2×D17311yescharacteristic, complemented
278#131—1731no—
279#131—1731no—
280#131—1731no—
281#131C2×D17311yescharacteristic, complemented
282C4⋊Q8—1731no—
283#127—1731no—
284#108—1731no—
285#108—1731no—
286#104—1731no—
287#104—1731no—
288#104—1731no—
289#104—1731no—
290#101—1731no—
291#101—1731no—
292#100—1731no—
293#100—1731no—
294C4×C8—1731no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—3461no—
304#41—1731no—
305#41—6921no—
306#41C22×D17311yescharacteristic
307#41C22×D17311yescharacteristic
308#41C22×D17311yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—3461no—
312C2×Q8—3461no—
313C2×Q8—1731no—
314C2×Q8C22×D17311yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—3461no—
319C2×D4—3461no—
320C2×D4—1731no—
321C2×D4—6921no—
322C2×D4—6921no—
323C2×D4C22×D17311yescharacteristic
324Q16—21no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—1731no—
329Q16C22⋊D17311yescharacteristic, complemented
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—3461no—
343SD16—3461no—
344SD16—3461no—
345SD16—3461no—
346SD16—1731no—
347SD16—1731no—
348SD16C22⋊D17311yescharacteristic, complemented
349SD16C22⋊D17311yescharacteristic, complemented
350D8—21no—
351D8—3461no—
352D8—3461no—
353D8—3461no—
354D8—1731no—
355D8C22⋊D17311yescharacteristic, complemented
356M16—21no—
357M16—3461no—
358M16—3461no—
359M16—3461no—
360M16—1731no—
361M16—1731no—
362M16—1731no—
363M16C22×D17311yescharacteristic
364C2×C8—21no—
365C2×C8—3461no—
366C2×C8—3461no—
367C2×C8—3461no—
368C2×C8—1731no—
369C2×C8C22×D17311yescharacteristic
370C4⋊C4—6921no—
371C42—1731no—
372C23—21no—
373C23—41no—
374C23—3461no—
375C23—6921no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—3461no—
380Q8—3461no—
381Q8—1731no—
382Q8—6921no—
383Q8D4×D17311yescharacteristic
384Q8C2×C22⋊D17311yescharacteristic
385Q8C2×C22⋊D17311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—3461no—
395D4—3461no—
396D4—3461no—
397D4—13842no—
398D4—1731no—
399D4—6921no—
400D4—6921no—
401D4—6921no—
402D4D4×D17311yescharacteristic
403D4C2×C22⋊D17311yescharacteristic
404D4C2×C22⋊D17311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—3461no—
413C2×C4—3461no—
414C2×C4—3461no—
415C2×C4—6921no—
416C2×C4—6921no—
417C2×C4—6921no—
418C2×C4C23×D17311yescharacteristic
419C8—21no—
420C8—3461no—
421C8—3461no—
422C8—3461no—
423C8—1731no—
424C8—1731no—
425C8C2×C22⋊D17311yescharacteristic
426C8C2×C22⋊D17311yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—3461no—
434C22—13841no—
435C22—6921no—
436C22—27681no—
437C22#1914972411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—3462no—
443C4—3461no—
444C4—6921no—
445C4C2×D4×D17311yescharacteristic
446C4C22×C22⋊D17311yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—13841no—
453C2—6921no—
454C2D4×C22⋊D17311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#577×C173C2
socleC2×C173D42
hypercenter#577D173
hypocenterC173#3291
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. #26012302C2semidirect
2. #26012299C2semidirect
3. #26012298C2semidirect
4. #26011862C2semidirect
5. #26011756C2semidirect
6. #26011754C2semidirect
7. #26011752C2semidirect
8. #26011751C2semidirect
9. #26011674C2semidirect
10. #26011672C2semidirect
11. #26011659C2semidirect
12. #26011657C2semidirect
13. #26011608C2semidirect
14. #26011606C2semidirect
15. #577×C173C2semidirect
17. #19149725C22semidirect
20. #19149710C22semidirect
22. #19149709C22semidirect
23. #19149709C22semidirect
34. C2×#14583815C22semidirect
36. C2×#14583814C22semidirect
38. C2×#14583813C22semidirect
40. #19149686C22semidirect
41. #19149686C22semidirect
42. #19149685C22semidirect
43. #19149677C22semidirect
45. #19149615C22semidirect
46. #19149614C22semidirect
47. #19149606C22semidirect
48. #19149606C22semidirect
49. #19149606C22semidirect
50. #19149605C22semidirect
51. #19149598C22semidirect
52. #19149597C22semidirect
53. #19149596C22semidirect
54. #19149595C22semidirect
55. C8×Dic173C22semidirect
57. H(5,𝔽2)×C173C22semidirect
59. #137×C173C22semidirect
65. C2×SD16×C173C22semidirect
66. #131×C173C22semidirect
84. #14583816D4semidirect
92. #14583814D4semidirect
98. #14583813D4semidirect
102. #14583811D4semidirect
131. Q16×C173D4semidirect
136. SD16×C173D4semidirect
137. SD16×C173D4semidirect
139. D8×C173D4semidirect
227. C173#3291semidirect
232. #577D173semidirect
247. H(5,𝔽2)C2×D173semidirect
255. #137C2×D173semidirect
277. C2×SD16C2×D173semidirect
281. #131C2×D173semidirect
329. Q16C22⋊D173semidirect
348. SD16C22⋊D173semidirect
349. SD16C22⋊D173semidirect
355. D8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260123022
#260122992
#260122982
#260118622
#260117562
#260117542
#260117522
#260117512
#260116742
#260116722
#260116592
#260116572
#260116082
#260116062
#577×C1732
#3291173