Finite Group Databasebeta

← #36365346

Subgroups — #36365346

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#26012307C211yesmaximal, characteristic, complemented
2#26012303C211yesmaximal, characteristic, complemented
3#26012297C211yesmaximal, characteristic, complemented
4#26012295C211yesmaximal, characteristic, complemented
5#26011872C211yesmaximal, characteristic, complemented
6#26011765C211yesmaximal, characteristic, complemented
7#26011760C211yesmaximal, characteristic, complemented
8#26011755C211yesmaximal, characteristic, complemented
9#26011671C211yesmaximal, characteristic, complemented
10#26011667C211yesmaximal, characteristic, complemented
11#26011666C211yesmaximal, characteristic, complemented
12#26011662C211yesmaximal, characteristic, complemented
13#26011607C211yesmaximal, characteristic, complemented
14#26011606C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16#19149727C2211yescharacteristic, complemented
17#19149726C2211yescharacteristic, complemented
18#19149725C2211yescharacteristic
19#19149724C2211yescharacteristic
20#19149712—21no—
21#19149712C2211yescharacteristic
22#19149712C2211yescharacteristic, complemented
23#19149711—21no—
24#19149710C2211yescharacteristic
25#19149708—21no—
26#19149708—21no—
27#19149707—21no—
28#19149707C2211yescharacteristic
29#19149707C2211yescharacteristic, complemented
30#19149706—21no—
31#19149706—21no—
32#19149705C2211yescharacteristic, complemented
33#19149703—21no—
34#19149701—21no—
35#19149701C2211yescharacteristic
36#19149700—21no—
37#19149700—21no—
38#19149700—21no—
39#19149699C2211yescharacteristic
40C2×#14583816C2211yescharacteristic, complemented
41Q16×D173—21no—
42C2×#14583815—21no—
43C2×#14583815C2211yescharacteristic, complemented
44SD16×D173—21no—
45C2×#14583811C2211yescharacteristic
46D8×D173—21no—
47#19149687C2211yescharacteristic
48#19149687C2211yescharacteristic, complemented
49#19149685C2211yescharacteristic, complemented
50#19149668C2211yescharacteristic, complemented
51#19149666C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149606C2211yescharacteristic, complemented
54#19149605C2211yescharacteristic
55#19149605C2211yescharacteristic, complemented
56#19149605C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59C8×Dic173C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62C2×SD16×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic, complemented
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583826—21no—
71#14583826—21no—
72#14583826—41no—
73#14583826C2311yescharacteristic
74#14583826C2311yescharacteristic
75#14583826C2311yescharacteristic
76#14583825—21no—
77#14583825—21no—
78#14583825—21no—
79#14583825—41no—
80C2×Dic346—21no—
81C2×Dic346C2311yescharacteristic
82Q8×D173—21no—
83Q8×D173—21no—
84Q8×D173—21no—
85C2×D692—21no—
86C2×D692C2311yescharacteristic
87C2×C22⋊D173—41no—
88D4×D173—21no—
89D4×D173—21no—
90D4×D173—21no—
91D4×D173—41no—
92Dic692—21no—
93Dic692—21no—
94Dic692D411yescharacteristic
95#14583816—21no—
96#14583816—21no—
97#14583816—21no—
98#14583816D411yescharacteristic
99#14583815—21no—
100#14583815—21no—
101#14583815—21no—
102#14583815—21no—
103#14583815—21no—
104#14583815—21no—
105#14583815D411yescharacteristic
106#14583815D411yescharacteristic
107#14583814—21no—
108#14583814D411yescharacteristic
109#14583813—21no—
110#14583813D411yescharacteristic
111D1384—21no—
112D1384—21no—
113D1384D411yescharacteristic
114#14583811—21no—
115#14583811—21no—
116#14583811—21no—
117#14583811D411yescharacteristic
118#14583810C2311yescharacteristic
119#14583809—21no—
120#14583809—21no—
121#14583809—21no—
122C2×#13063436C2311yescharacteristic
123C8×D173—21no—
124C8×D173—21no—
125C8×D173—21no—
126C22⋊Dic173—41no—
127#14583803—41no—
128C4×Dic173C2311yescharacteristic
129#41×C173C2311yescharacteristic
130C2×Q8×C173C2311yescharacteristic
131C2×D4×C173C2311yescharacteristic
132Q16×C173—21no—
133SD16×C173—21no—
134SD16×C173—21no—
135SD16×C173—21no—
136SD16×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—41no—
146C22×D173—41no—
147Dic346—21no—
148Dic346—21no—
149Dic346—41no—
150Dic346—41no—
151Dic346C2×D411yescharacteristic
152Dic346C2×D411yescharacteristic
153Dic346C2×D411yescharacteristic
154D692—21no—
155D692—21no—
156D692—41no—
157D692—41no—
158D692C2×D411yescharacteristic
159D692C2×D411yescharacteristic
160D692C2×D411yescharacteristic
161C22⋊D173—21no—
162C22⋊D173—21no—
163C22⋊D173—21no—
164C22⋊D173—41no—
165C22⋊D173—41no—
166C22⋊D173—41no—
167C22⋊D173—41no—
168C2×Dic173—21no—
169C2×Dic173—21no—
170C2×Dic173—41no—
171C2×Dic173—41no—
172C4×D173—21no—
173C4×D173—21no—
174C4×D173—21no—
175C4×D173—41no—
176C4×D173—41no—
177C4×D173—41no—
178C4×D173—41no—
179#13063436—21no—
180#13063436C2×D411yescharacteristic
181#13063436C2×D411yescharacteristic
182C23×C173—21no—
183Q8×C173—21no—
184Q8×C173—21no—
185Q8×C173C2×D411yescharacteristic
186D4×C173—21no—
187D4×C173—21no—
188D4×C173—21no—
189D4×C173C2×D411yescharacteristic
190C2×C4×C173—21no—
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—81no—
201C22×C173H(5,𝔽2)11yescharacteristic
202C2×D173—21no—
203C2×D173—21no—
204C2×D173—21no—
205C2×D173—41no—
206C2×D173—81no—
207C2×D173—161no—
208C2×D173—161no—
209C4×C173—21no—
210C4×C173—41no—
211C4×C173C22×D411yescharacteristic
212C4×C173C22×D411yescharacteristic
213Dic173—21no—
214Dic173—21no—
215Dic173—21no—
216Dic173—41no—
217Dic173—41no—
218C2×C173—21no—
219C2×C173—41no—
220C2×C173—81no—
221C2×C173D4211yescharacteristic
222D173—41no—
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#471D17311yescharacteristic, complemented
237#455—1731no—
238#455—1731no—
239#454—1731no—
240#454—1731no—
241#443—1731no—
242#443—1731no—
2432−1+4—1731no—
2442−1+4—1731no—
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#137C2×D17311yescharacteristic, complemented
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)C2×D17311yescharacteristic
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×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#108C2×D17311yescharacteristic, complemented
285#104—1731no—
286#104—1731no—
287#104—1731no—
288#104—1731no—
289#101—1731no—
290#101C2×D17311yescharacteristic, complemented
291#100—1731no—
292#100C2×D17311yescharacteristic, complemented
293C4×C8—1731no—
294#41—3461no—
295#41—3461no—
296#41—3461no—
297#41—3461no—
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—1731no—
303#41—1731no—
304#41—1731no—
305#41—6921no—
306#41—6921no—
307#41C22×D17311yescharacteristic
308C2×Q8—3461no—
309C2×Q8—3461no—
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—21no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—1731no—
329Q16—1731no—
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—
344SD16—1731no—
345SD16—1731no—
346D8—21no—
347D8—3461no—
348D8—3461no—
349D8—3461no—
350D8—3461no—
351D8—3461no—
352D8—1731no—
353D8—1731no—
354M16—21no—
355M16—3461no—
356M16—3461no—
357M16—3461no—
358M16—1731no—
359M16C22×D17311yescharacteristic
360M16C22×D17311yescharacteristic
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—3461no—
373C23—6921no—
374C23—6921no—
375Q8—21no—
376Q8—21no—
377Q8—3461no—
378Q8—3461no—
379Q8—1731no—
380Q8—1731no—
381Q8—1731no—
382Q8—6921no—
383Q8—6921no—
384Q8D4×D17311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—3461no—
389D4—3461no—
390D4—3461no—
391D4—3461no—
392D4—3461no—
393D4—1731no—
394D4—1731no—
395D4—1731no—
396D4—6921no—
397D4—6921no—
398D4—6921no—
399D4—6921no—
400D4—6921no—
401D4—6921no—
402D4D4×D17311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—3461no—
406C2×C4—3461no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—6921no—
411C2×C4—6921no—
412C2×C4—6921no—
413C2×C4—6921no—
414C2×C4—6921no—
415C2×C4—6921no—
416C2×C4C23×D17311yescharacteristic
417C8—21no—
418C8—21no—
419C8—21no—
420C8—3461no—
421C8—1731no—
422C8—1731no—
423C8D4×D17311yescharacteristic
424C8D4×D17311yescharacteristic
425C22—21no—
426C22—41no—
427C22—81no—
428C22—3461no—
429C22—3461no—
430C22—3461no—
431C22—13841no—
432C22—6921no—
433C22—27681no—
434C22—27681no—
435C22#1914972511yescharacteristic
436C4—21no—
437C4—41no—
438C4—3461no—
439C4—3461no—
440C4—3461no—
441C4—6921no—
442C4—6921no—
443C4C2×D4×D17311yescharacteristic
444C4C2×D4×D17311yescharacteristic
445C2—21no—
446C2—41no—
447C2—81no—
448C2—13841no—
449C2—6921no—
450C2—6921no—
451C2—6921no—
452C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173D42
hypercenter#471D173
hypocenterC173#3290
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #26012307C2semidirect
2. #26012303C2semidirect
3. #26012297C2semidirect
4. #26012295C2semidirect
5. #26011872C2semidirect
6. #26011765C2semidirect
7. #26011760C2semidirect
8. #26011755C2semidirect
9. #26011671C2semidirect
10. #26011667C2semidirect
11. #26011666C2semidirect
12. #26011662C2semidirect
13. #26011607C2semidirect
14. #26011606C2semidirect
15. #471×C173C2semidirect
16. #19149727C22semidirect
17. #19149726C22semidirect
22. #19149712C22semidirect
29. #19149707C22semidirect
32. #19149705C22semidirect
40. C2×#14583816C22semidirect
43. C2×#14583815C22semidirect
48. #19149687C22semidirect
49. #19149685C22semidirect
50. #19149668C22semidirect
51. #19149666C22semidirect
52. #19149614C22semidirect
53. #19149606C22semidirect
55. #19149605C22semidirect
56. #19149605C22semidirect
57. #19149597C22semidirect
58. #19149595C22semidirect
59. C8×Dic173C22semidirect
60. #137×C173C22semidirect
62. C2×SD16×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #100×C173C22semidirect
226. C173#3290semidirect
236. #471D173semidirect
254. #137C2×D173semidirect
274. C2×SD16C2×D173semidirect
280. #131C2×D173semidirect
284. #108C2×D173semidirect
290. #101C2×D173semidirect
292. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123072
#260123032
#260122972
#260122952
#260118722
#260117652
#260117602
#260117552
#260116712
#260116672
#260116662
#260116622
#260116072
#260116062
#471×C1732
#3290173