Finite Group Databasebeta

← #36365334

Subgroups — #36365334

Order 22144.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012306C211yesmaximal, characteristic, complemented
2#26012299C211yesmaximal, characteristic, complemented
3#26012298C211yesmaximal, characteristic, complemented
4#26011884C211yesmaximal, characteristic, complemented
5#26011764C211yesmaximal, characteristic, complemented
6#26011763C211yesmaximal, characteristic, complemented
7#26011756C211yesmaximal, characteristic, complemented
8#26011754C211yesmaximal, characteristic, complemented
9#26011674C211yesmaximal, characteristic, complemented
10#26011672C211yesmaximal, characteristic, complemented
11#26011667C211yesmaximal, characteristic, complemented
12#26011664C211yesmaximal, characteristic, complemented
13#26011613C211yesmaximal, characteristic, complemented
14#26011611C211yesmaximal, characteristic, complemented
15#578×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic
17#19149725C2211yescharacteristic
18#19149711—21no—
19#19149710—21no—
20#19149710—21no—
21#19149710C2211yescharacteristic
22#19149710C2211yescharacteristic
23#19149709—21no—
24#19149709C2211yescharacteristic
25#19149709C2211yescharacteristic, complemented
26#19149704—21no—
27#19149704C2211yescharacteristic
28#19149704C2211yescharacteristic
29#19149703—21no—
30#19149700—21no—
31#19149699—21no—
32#19149699—21no—
33#19149699C2211yescharacteristic
34C2×#14583816—21no—
35C2×#14583816C2211yescharacteristic, complemented
36C2×#14583815C2211yescharacteristic
37C2×#14583813—21no—
38C2×#14583813C2211yescharacteristic
39SD16×D173—21no—
40#19149686C2211yescharacteristic
41#19149686C2211yescharacteristic, complemented
42#19149685C2211yescharacteristic
43#19149677C2211yescharacteristic, complemented
44#19149668C2211yescharacteristic
45#19149615C2211yescharacteristic, complemented
46#19149614C2211yescharacteristic, complemented
47#19149606C2211yescharacteristic
48#19149606C2211yescharacteristic, complemented
49#19149606C2211yescharacteristic, complemented
50#19149605C2211yescharacteristic
51#19149598C2211yescharacteristic, complemented
52#19149598C2211yescharacteristic, complemented
53#19149597C2211yescharacteristic, complemented
54#19149595C2211yescharacteristic
55#19149589C2211yescharacteristic, complemented
562−1+4×C173C2211yescharacteristic
572−1+4×C173C2211yescharacteristic, complemented
58#137×C173—21no—
59#137×C173—21no—
60#137×C173C2211yescharacteristic
61#137×C173C2211yescharacteristic, complemented
62#135×C173—21no—
63#135×C173C2211yescharacteristic
64C2×Q16×C173—21no—
65C2×Q16×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#14583816—21no—
86#14583816D411yescharacteristic
87#14583815—21no—
88#14583815—21no—
89#14583814—21no—
90#14583814—21no—
91#14583814D411yescharacteristic
92#14583813—21no—
93#14583813—21no—
94#14583813—21no—
95#14583813—21no—
96#14583813—21no—
97#14583813—21no—
98#14583813D411yescharacteristic
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—
112#14583805—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×C173—41no—
121#41×C173C2311yescharacteristic
122#41×C173C2311yescharacteristic
123#41×C173C2311yescharacteristic
124C2×Q8×C173—21no—
125C2×Q8×C173—21no—
126C2×Q8×C173—21no—
127C2×Q8×C173—21no—
128C2×Q8×C173C2311yescharacteristic
129C2×Q8×C173C2311yescharacteristic
130Q16×C173—21no—
131Q16×C173—21no—
132Q16×C173—21no—
133Q16×C173—21no—
134Q16×C173D411yescharacteristic
135SD16×C173—21no—
136SD16×C173—21no—
137SD16×C173D411yescharacteristic
138SD16×C173D411yescharacteristic
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
168Q8×C173—21no—
169Q8×C173—21no—
170Q8×C173—21no—
171Q8×C173—21no—
172Q8×C173—41no—
173Q8×C173—41no—
174Q8×C173C2×D411yescharacteristic
175Q8×C173C2×D411yescharacteristic
176Q8×C173C2×D411yescharacteristic
177D4×C173—21no—
178D4×C173—21no—
179D4×C173—21no—
180D4×C173—41no—
181D4×C173—41no—
182D4×C173C2×D411yescharacteristic
183D4×C173C2×D411yescharacteristic
184D4×C173C2×D411yescharacteristic
185C2×C4×C173—21no—
186C2×C4×C173—21no—
187C2×C4×C173—21no—
188C2×C4×C173—21no—
189C2×C4×C173—21no—
190C2×C4×C173—41no—
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×C173H(5,𝔽2)11yescharacteristic
202C2×D173—21no—
203C2×D173—41no—
204C2×D173—81no—
205C2×D173—161no—
206C4×C173—21no—
207C4×C173—21no—
208C4×C173—21no—
209C4×C173—41no—
210C4×C173—41no—
211C4×C173C22×D411yescharacteristic
212C4×C173C22×D411yescharacteristic
213Dic173—21no—
214Dic173—41no—
215Dic173—41no—
216C2×C173—21no—
217C2×C173—41no—
218C2×C173—41no—
219C2×C173—41no—
220C2×C173D4211yescharacteristic
221D173—41no—
222D173—81no—
223C173#328911yescharacteristic, complemented, Sylow
224#3289—1731nomaximal, Sylow
225#578—1731no—
226#578D17311yescharacteristic, complemented
227#577—1731no—
228#577—1731no—
229#497—1731no—
230#473—1731no—
231#473—1731no—
232#471—1731no—
233#471—1731no—
234#456—1731no—
235#456—1731no—
236#455—1731no—
237#454—1731no—
238#444—1731no—
239#444—1731no—
2402−1+4—1731no—
2412−1+4C2×D17311yescharacteristic
2422−1+4C2×D17311yescharacteristic, complemented
243H(5,𝔽2)—1731no—
244#137—21no—
245#137—21no—
246#137—3461no—
247#137—3461no—
248#137—3461no—
249#137—3461no—
250#137—1731no—
251#137—1731no—
252#137—1731no—
253#137—1731no—
254#137C2×D17311yescharacteristic
255#137C2×D17311yescharacteristic, complemented
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—1731no—
259AGL(1,ℤ/8)—1731no—
260#135—21no—
261#135—3461no—
262#135—3461no—
263#135—3461no—
264#135—1731no—
265#135C2×D17311yescharacteristic
266C2×Q16—21no—
267C2×Q16—3461no—
268C2×Q16—1731no—
269C2×Q16C2×D17311yescharacteristic, complemented
270C2×SD16—3461no—
271C2×SD16—3461no—
272C2×SD16—1731no—
273C2×SD16—1731no—
274#131—1731no—
275#131—1731no—
276#131—1731no—
277#131C2×D17311yescharacteristic, complemented
278C4⋊Q8—1731no—
279#124—1731no—
280#108—1731no—
281#108—1731no—
282#104—1731no—
283#104—1731no—
284#104—1731no—
285#104—1731no—
286#101—1731no—
287#101—1731no—
288#101—1731no—
289#100—1731no—
290#97—1731no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295#41—41no—
296#41—41no—
297#41—3461no—
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—1731no—
302#41—6921no—
303#41C22×D17311yescharacteristic
304#41C22×D17311yescharacteristic
305#41C22×D17311yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—3461no—
311C2×Q8—3461no—
312C2×Q8—1731no—
313C2×Q8C22×D17311yescharacteristic
314C2×Q8C22×D17311yescharacteristic
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—1731no—
318C2×D4—6921no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16C22⋊D17311yescharacteristic
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—
345SD16C22⋊D17311yescharacteristic
346SD16C22⋊D17311yescharacteristic
347D8—3461no—
348D8—3461no—
349D8—1731no—
350D8C22⋊D17311yescharacteristic
351M16—21no—
352M16—3461no—
353M16—3461no—
354M16—3461no—
355M16—1731no—
356M16—1731no—
357M16—1731no—
358M16C22×D17311yescharacteristic
359C2×C8—21no—
360C2×C8—3461no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8—1731no—
364C2×C8C22×D17311yescharacteristic
365C4⋊C4—6921no—
366C22⋊C4—6921no—
367C42—1731no—
368C23—3461no—
369C23—6921no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—41no—
376Q8—3461no—
377Q8—3461no—
378Q8—1731no—
379Q8—6921no—
380Q8D4×D17311yescharacteristic
381Q8C2×C22⋊D17311yescharacteristic
382Q8C2×C22⋊D17311yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388D4—3461no—
389D4—3461no—
390D4—3461no—
391D4—1731no—
392D4—6921no—
393D4—6921no—
394D4—6921no—
395D4D4×D17311yescharacteristic
396D4C2×C22⋊D17311yescharacteristic
397D4C2×C22⋊D17311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—6921no—
411C2×C4—6921no—
412C2×C4—6921no—
413C2×C4C23×D17311yescharacteristic
414C8—21no—
415C8—3461no—
416C8—3461no—
417C8—3461no—
418C8—1731no—
419C8—1731no—
420C8C2×C22⋊D17311yescharacteristic
421C8C2×C22⋊D17311yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—3461no—
426C22—13841no—
427C22—6921no—
428C22—27681no—
429C22#1914972411yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3461no—
436C4—6921no—
437C4—6921no—
438C4C2×D4×D17311yescharacteristic
439C4C22×C22⋊D17311yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2—13841no—
445C2—6921no—
446C2D4×C22⋊D17311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#578×C173C2
socleC2×C173D42
hypercenter#578D173
hypocenterC173#3289
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #26012306C2semidirect
2. #26012299C2semidirect
3. #26012298C2semidirect
4. #26011884C2semidirect
5. #26011764C2semidirect
6. #26011763C2semidirect
7. #26011756C2semidirect
8. #26011754C2semidirect
9. #26011674C2semidirect
10. #26011672C2semidirect
11. #26011667C2semidirect
12. #26011664C2semidirect
13. #26011613C2semidirect
14. #26011611C2semidirect
15. #578×C173C2semidirect
25. #19149709C22semidirect
35. C2×#14583816C22semidirect
41. #19149686C22semidirect
43. #19149677C22semidirect
45. #19149615C22semidirect
46. #19149614C22semidirect
48. #19149606C22semidirect
49. #19149606C22semidirect
51. #19149598C22semidirect
52. #19149598C22semidirect
53. #19149597C22semidirect
55. #19149589C22semidirect
57. 2−1+4×C173C22semidirect
61. #137×C173C22semidirect
65. C2×Q16×C173C22semidirect
66. #131×C173C22semidirect
223. C173#3289semidirect
226. #578D173semidirect
242. 2−1+4C2×D173semidirect
255. #137C2×D173semidirect
269. C2×Q16C2×D173semidirect
277. #131C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123062
#260122992
#260122982
#260118842
#260117642
#260117632
#260117562
#260117542
#260116742
#260116722
#260116672
#260116642
#260116132
#260116112
#578×C1732
#3289173