Finite Group Databasebeta

← #36365342

Subgroups — #36365342

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#26012301C211yesmaximal, characteristic, complemented
3#26012299C211yesmaximal, characteristic, complemented
4#26012295C211yesmaximal, characteristic, complemented
5#26011873C211yesmaximal, characteristic, complemented
6#26011765C211yesmaximal, characteristic, complemented
7#26011759C211yesmaximal, characteristic, complemented
8#26011757C211yesmaximal, characteristic, complemented
9#26011752C211yesmaximal, characteristic, complemented
10#26011670C211yesmaximal, characteristic, complemented
11#26011668C211yesmaximal, characteristic, complemented
12#26011665C211yesmaximal, characteristic, complemented
13#26011605C211yesmaximal, characteristic, complemented
14#26011605C211yesmaximal, characteristic, complemented
15#454×C173C211yesmaximal, characteristic, complemented
16#19149726C2211yescharacteristic, complemented
17#19149725C2211yescharacteristic
18#19149724C2211yescharacteristic
19#19149712C2211yescharacteristic
20#19149710—21no—
21#19149710—21no—
22#19149710C2211yescharacteristic
23#19149710C2211yescharacteristic, complemented
24#19149709—21no—
25#19149709C2211yescharacteristic, complemented
26#19149708—21no—
27#19149707C2211yescharacteristic
28#19149706—21no—
29#19149704—21no—
30#19149704C2211yescharacteristic, complemented
31#19149703—21no—
32#19149703—21no—
33#19149700—21no—
34#19149699—21no—
35#19149699—21no—
36#19149699C2211yescharacteristic
37#19149699C2211yescharacteristic
38#19149698—21no—
39C2×#14583816—21no—
40C2×#14583816C2211yescharacteristic, complemented
41C2×#14583814C2211yescharacteristic, complemented
42C2×#14583813—21no—
43C2×#14583813C2211yescharacteristic, complemented
44SD16×D173—21no—
45C2×#14583811C2211yescharacteristic
46D8×D173—21no—
47#19149687C2211yescharacteristic
48#19149687C2211yescharacteristic, complemented
49#19149686C2211yescharacteristic, complemented
50#19149686C2211yescharacteristic, complemented
51#19149667C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149607C2211yescharacteristic
55#19149607C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149596C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59C4×#13063436C2211yescharacteristic, complemented
602−1+4×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62AGL(1,ℤ/8)×C173C2211yescharacteristic, complemented
63#124×C173C2211yescharacteristic, complemented
64#104×C173C2211yescharacteristic, complemented
65#104×C173C2211yescharacteristic, complemented
66#101×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583826—21no—
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583826C2311yescharacteristic
72#14583825—21no—
73#14583825—21no—
74#14583825—21no—
75#14583825—41no—
76C2×Dic346—21no—
77C2×Dic346C2311yescharacteristic
78Q8×D173—21no—
79C2×D692—21no—
80C2×D692C2311yescharacteristic
81C2×C22⋊D173—41no—
82D4×D173—21no—
83D4×D173—21no—
84D4×D173—21no—
85D4×D173—41no—
86Dic692—21no—
87#14583816—21no—
88#14583816—21no—
89#14583816—21no—
90#14583816—21no—
91#14583816—21no—
92#14583816D411yescharacteristic
93#14583816D411yescharacteristic
94#14583815—21no—
95#14583815—21no—
96#14583814—21no—
97#14583814—21no—
98#14583814—21no—
99#14583814D411yescharacteristic
100#14583814D411yescharacteristic
101#14583813—21no—
102#14583813—21no—
103#14583813—21no—
104#14583813—21no—
105#14583813—21no—
106#14583813D411yescharacteristic
107#14583813D411yescharacteristic
108D1384—21no—
109#14583811—21no—
110#14583811—21no—
111#14583811—21no—
112#14583811D411yescharacteristic
113#14583811D411yescharacteristic
114#14583810—21no—
115#14583810—21no—
116#14583810C2311yescharacteristic
117#14583810C2311yescharacteristic
118#14583809—21no—
119#14583809—21no—
120C2×#13063436—21no—
121C2×#13063436—21no—
122C2×#13063436C2311yescharacteristic
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126#14583803—41no—
127#41×C173—21no—
128#41×C173—21no—
129#41×C173—41no—
130#41×C173C2311yescharacteristic
131#41×C173C2311yescharacteristic
132C2×Q8×C173—21no—
133C2×Q8×C173—21no—
134C2×Q8×C173C2311yescharacteristic
135C2×D4×C173C2311yescharacteristic
136SD16×C173—21no—
137SD16×C173—21no—
138D8×C173—21no—
139D8×C173—21no—
140M16×C173C2311yescharacteristic
141M16×C173C2311yescharacteristic
142C22⋊C4×C173—41no—
143C42×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146C22×D173—41no—
147Dic346—21no—
148Dic346—21no—
149Dic346—41no—
150Dic346C2×D411yescharacteristic
151Dic346C2×D411yescharacteristic
152D692—21no—
153D692—21no—
154D692—41no—
155D692C2×D411yescharacteristic
156D692C2×D411yescharacteristic
157C22⋊D173—21no—
158C22⋊D173—21no—
159C22⋊D173—41no—
160C22⋊D173—41no—
161C22⋊D173—41no—
162C22⋊D173—41no—
163C2×Dic173—21no—
164C2×Dic173—41no—
165C2×Dic173—41no—
166C4×D173—21no—
167C4×D173—21no—
168C4×D173—41no—
169C4×D173—41no—
170#13063436—21no—
171#13063436—21no—
172#13063436C2×D411yescharacteristic
173#13063436C2×D411yescharacteristic
174#13063436C2×D411yescharacteristic
175#13063436C2×D411yescharacteristic
176C23×C173—21no—
177Q8×C173—21no—
178Q8×C173—21no—
179Q8×C173—41no—
180Q8×C173C2×D411yescharacteristic
181Q8×C173C2×D411yescharacteristic
182D4×C173—21no—
183D4×C173—21no—
184D4×C173—21no—
185D4×C173—21no—
186D4×C173—41no—
187D4×C173C2×D411yescharacteristic
188D4×C173C2×D411yescharacteristic
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173—41no—
194C2×C4×C173—41no—
195C2×C4×C173C2411yescharacteristic
196C8×C173—21no—
197C8×C173—21no—
198C22×C173—21no—
199C22×C173—21no—
200C22×C173—41no—
201C22×C173—81no—
202C22×C173H(5,𝔽2)11yescharacteristic
203C2×D173—21no—
204C2×D173—21no—
205C2×D173—41no—
206C2×D173—81no—
207C2×D173—161no—
208C2×D173—161no—
209C4×C173—21no—
210C4×C173—21no—
211C4×C173—41no—
212C4×C173—41no—
213C4×C173C22×D411yescharacteristic
214C4×C173C22×D411yescharacteristic
215Dic173—21no—
216Dic173—21no—
217Dic173—41no—
218C2×C173—21no—
219C2×C173—41no—
220C2×C173—41no—
221C2×C173—81no—
222C2×C173D4211yescharacteristic
223D173—41no—
224D173—41no—
225D173—81no—
226C173#329011yescharacteristic, complemented, Sylow
227#3290—1731nomaximal, Sylow
228#578—1731no—
229#577—1731no—
230#577—1731no—
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#454D17311yescharacteristic, complemented
241#443—1731no—
242#443—1731no—
2432−1+4—1731no—
2442−1+4C2×D17311yescharacteristic, complemented
245H(5,𝔽2)—1731no—
246H(5,𝔽2)—1731no—
247#137—3461no—
248#137—3461no—
249#137—3461no—
250#137—3461no—
251#137—1731no—
252#137—1731no—
253#137—1731no—
254#137—1731no—
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)C2×D17311yescharacteristic
262AGL(1,ℤ/8)C2×D17311yescharacteristic, complemented
263#135—3461no—
264#135—3461no—
265#135—3461no—
266#135—3461no—
267#135—1731no—
268#135—1731no—
269C2×Q16—3461no—
270C2×Q16—1731no—
271C2×SD16—3461no—
272C2×SD16—3461no—
273C2×SD16—1731no—
274C2×SD16—1731no—
275C2×D8—3461no—
276C2×D8—1731no—
277#131—1731no—
278#131—1731no—
279#131—1731no—
280#131—1731no—
281#124—1731no—
282#124C2×D17311yescharacteristic, complemented
283#108—1731no—
284#108—1731no—
285#104—1731no—
286#104—1731no—
287#104C2×D17311yescharacteristic, complemented
288#104C2×D17311yescharacteristic, complemented
289#101—1731no—
290#101C2×D17311yescharacteristic, complemented
291#100—1731no—
292#100—1731no—
293C4×C8—1731no—
294#41—21no—
295#41—21no—
296#41—41no—
297#41—3461no—
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—1731no—
304#41—1731no—
305#41—6921no—
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—3461no—
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—3461no—
318C2×D4—1731no—
319C2×D4—6921no—
320C2×D4—6921no—
321C2×D4C22×D17311yescharacteristic
322Q16—3461no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—1731no—
329Q16—1731no—
330SD16—21no—
331SD16—21no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—1731no—
343SD16—1731no—
344SD16—1731no—
345SD16—1731no—
346D8—21no—
347D8—21no—
348D8—3461no—
349D8—3461no—
350D8—3461no—
351D8—3461no—
352D8—1731no—
353D8—1731no—
354M16—3461no—
355M16—3461no—
356M16—3461no—
357M16—3461no—
358M16—1731no—
359M16—1731no—
360M16C22×D17311yescharacteristic
361M16C22×D17311yescharacteristic
362C2×C8—3461no—
363C2×C8—3461no—
364C2×C8—3461no—
365C2×C8—3461no—
366C2×C8—1731no—
367C2×C8—1731no—
368C22⋊C4—41no—
369C22⋊C4—6921no—
370C42C22×D17311yescharacteristic
371C23—21no—
372C23—3461no—
373C23—6921no—
374C23—6921no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—3461no—
379Q8—3461no—
380Q8—1731no—
381Q8—1731no—
382Q8—6921no—
383Q8D4×D17311yescharacteristic
384Q8D4×D17311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—3461no—
391D4—3461no—
392D4—3461no—
393D4—3461no—
394D4—1731no—
395D4—1731no—
396D4—6921no—
397D4—6921no—
398D4—6921no—
399D4—6921no—
400D4—6921no—
401D4D4×D17311yescharacteristic
402D4D4×D17311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—3461no—
410C2×C4—3461no—
411C2×C4—3461no—
412C2×C4—6921no—
413C2×C4—6921no—
414C2×C4—6921no—
415C2×C4—6921no—
416C2×C4C23×D17311yescharacteristic
417C8—21no—
418C8—21no—
419C8—3461no—
420C8—3461no—
421C8—1731no—
422C8—1731no—
423C8—1731no—
424C8—1731no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—81no—
429C22—3461no—
430C22—3461no—
431C22—13841no—
432C22—6921no—
433C22—27681no—
434C22—27681no—
435C22#1914972411yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—3461no—
441C4—3461no—
442C4—6921no—
443C4C2×D4×D17311yescharacteristic
444C4C2×D4×D17311yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—13841no—
450C2—6921no—
451C2—6921no—
452C2#2601212311yescharacteristic
subgroupG/H
centerC2#26012123
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#454×C173C2
socleC2×C173D42
hypercenter#454D173
hypocenterC173#3290
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26012306C2semidirect
2. #26012301C2semidirect
3. #26012299C2semidirect
4. #26012295C2semidirect
5. #26011873C2semidirect
6. #26011765C2semidirect
7. #26011759C2semidirect
8. #26011757C2semidirect
9. #26011752C2semidirect
10. #26011670C2semidirect
11. #26011668C2semidirect
12. #26011665C2semidirect
13. #26011605C2semidirect
14. #26011605C2semidirect
15. #454×C173C2semidirect
16. #19149726C22semidirect
23. #19149710C22semidirect
25. #19149709C22semidirect
30. #19149704C22semidirect
40. C2×#14583816C22semidirect
41. C2×#14583814C22semidirect
43. C2×#14583813C22semidirect
48. #19149687C22semidirect
49. #19149686C22semidirect
50. #19149686C22semidirect
51. #19149667C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
55. #19149607C22semidirect
56. #19149597C22semidirect
57. #19149596C22semidirect
58. #19149595C22semidirect
59. C4×#13063436C22semidirect
60. 2−1+4×C173C22semidirect
62. AGL(1,ℤ/8)×C173C22semidirect
63. #124×C173C22semidirect
64. #104×C173C22semidirect
65. #104×C173C22semidirect
66. #101×C173C22semidirect
226. C173#3290semidirect
240. #454D173semidirect
244. 2−1+4C2×D173semidirect
262. AGL(1,ℤ/8)C2×D173semidirect
282. #124C2×D173semidirect
287. #104C2×D173semidirect
288. #104C2×D173semidirect
290. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123062
#260123012
#260122992
#260122952
#260118732
#260117652
#260117592
#260117572
#260117522
#260116702
#260116682
#260116652
#260116052
#454×C1732
#3290173