Finite Group Databasebeta

← #471×D173 #36363258

Subgroups — #471×D173 #36363258

Order 22144.

468 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#137×D173C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D173C211yesmaximal, characteristic
3C2×SD16×D173C211yesmaximal, characteristic, complemented
4#131×D173C211yesmaximal, characteristic, complemented
5#26011757C211yesmaximal, characteristic, complemented
6#26011756C211yesmaximal, characteristic, complemented
7#26011755C211yesmaximal, characteristic, complemented
8#26011754C211yesmaximal, characteristic, complemented
9#26011753C211yesmaximal, characteristic, complemented
10#26011752C211yesmaximal, characteristic, complemented
11#26011751C211yesmaximal, characteristic, complemented
12#108×D173C211yesmaximal, characteristic, complemented
13#101×D173C211yesmaximal, characteristic, complemented
14#100×D173C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16#41×D173C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149712C2211yescharacteristic, complemented
20#19149711—21no—
21#19149710C2211yescharacteristic, complemented
22#19149709C2211yescharacteristic, complemented
23#19149708—21no—
24#19149707C2211yescharacteristic
25#19149706—21no—
26#19149705C2211yescharacteristic
27#19149704C2211yescharacteristic
28#19149703—21no—
29Q16×D173—21no—
30C2×#14583815C2211yescharacteristic, complemented
31C2×#14583814C2211yescharacteristic, complemented
32C2×#14583813C2211yescharacteristic, complemented
33SD16×D173—21no—
34SD16×D173—21no—
35SD16×D173—21no—
36SD16×D173—21no—
37SD16×D173—21no—
38SD16×D173—21no—
39D8×D173—21no—
40#19149687—21no—
41#19149687C2211yescharacteristic, complemented
42#19149686C2211yescharacteristic, complemented
43#19149685—21no—
44#19149685C2211yescharacteristic, complemented
45M16×D173—21no—
46M16×D173C2211yescharacteristic
47M16×D173C2211yescharacteristic
48M16×D173C2211yescharacteristic
49C2×C8×D173—21no—
50C2×C8×D173C2211yescharacteristic
51#19149615C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149598C2211yescharacteristic, complemented
55#19149597C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149596C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59#19149595C2211yescharacteristic, 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
67C23×D173—21no—
68#14583827D411yescharacteristic
69#14583826—21no—
70#14583826C2311yescharacteristic
71#14583825—21no—
72#14583825D411yescharacteristic
73C2×Dic346—21no—
74C2×Dic346C2311yescharacteristic
75Q8×D173—21no—
76Q8×D173—21no—
77Q8×D173—21no—
78Q8×D173—21no—
79Q8×D173D411yescharacteristic
80C2×D692C2311yescharacteristic
81C2×C22⋊D173—21no—
82D4×D173—21no—
83D4×D173—21no—
84D4×D173—21no—
85D4×D173—21no—
86D4×D173—21no—
87D4×D173D411yescharacteristic
88C2×C4×D173—21no—
89C2×C4×D173—21no—
90C2×C4×D173C2311yescharacteristic
91Dic692—21no—
92#14583816—21no—
93#14583816—21no—
94#14583815—21no—
95#14583815—21no—
96#14583815—21no—
97#14583815—21no—
98#14583814—21no—
99#14583814—21no—
100#14583814—21no—
101#14583814—21no—
102#14583813—21no—
103#14583813—21no—
104#14583813—21no—
105#14583813—21no—
106D1384—21no—
107#14583811—21no—
108#14583811—21no—
109#14583810—21no—
110#14583810C2311yescharacteristic
111#14583810C2311yescharacteristic
112#14583810C2311yescharacteristic
113#14583809—21no—
114#14583809—21no—
115#14583809—21no—
116#14583809—21no—
117#14583809—21no—
118C2×#13063436—21no—
119C2×#13063436C2311yescharacteristic
120C8×D173—21no—
121C8×D173—21no—
122C8×D173—21no—
123C8×D173—21no—
124C8×D173—21no—
125C8×D173D411yescharacteristic
126C8×D173D411yescharacteristic
127C8×D173D411yescharacteristic
128C8×D173D411yescharacteristic
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—21no—
146C22×D173—41no—
147C22×D173—41no—
148C22×D173—82no—
149C22×D173#4111yescharacteristic
150Dic346—21no—
151Dic346—21no—
152Dic346—21no—
153Dic346—41no—
154Dic346—41no—
155Dic346C2×D411yescharacteristic
156D692—21no—
157D692—21no—
158D692—21no—
159D692C2×D411yescharacteristic
160C22⋊D173—21no—
161C22⋊D173—21no—
162C22⋊D173—21no—
163C22⋊D173—41no—
164C22⋊D173—41no—
165C2×Dic173—21no—
166C2×Dic173—21no—
167C2×Dic173#4111yescharacteristic
168C4×D173—21no—
169C4×D173—21no—
170C4×D173—21no—
171C4×D173—41no—
172C4×D173—41no—
173C4×D173C2×D411yescharacteristic
174C4×D173C2×D411yescharacteristic
175C4×D173C2×D411yescharacteristic
176C4×D173C2×D411yescharacteristic
177#13063436—21no—
178#13063436—21no—
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×C173C2×#4111yescharacteristic
202C2×D173—22no—
203C2×D173—21no—
204C2×D173—42no—
205C2×D173—41no—
206C2×D173—82no—
207C2×D173—82no—
208C2×D173—81no—
209C2×D173#12111yescharacteristic
210C2×D173#12111yescharacteristic
211C4×C173—21no—
212C4×C173—41no—
213C4×C173C22×D411yescharacteristic
214C4×C173C22×D411yescharacteristic
215Dic173—21no—
216Dic173—41no—
217Dic173#12111yescharacteristic
218Dic173#12111yescharacteristic
219C2×C173—21no—
220C2×C173—41no—
221C2×C173—81no—
222C2×C173C2×#12111yescharacteristic
223D173—21no—
224D173—41no—
225D173—81no—
226D173#47112yesdirect product factor
227C173C2×#47111yescharacteristic, complemented, Sylow
228C2×#471—1731nomaximal, Sylow
229C2×#137—1731no—
230C2×AGL(1,ℤ/8)—1731no—
231C22×SD16—1731no—
232C2×#131—1731no—
233#471—1731no—
234#471—1731no—
235#471—1731no—
236#471—1731no—
237#471—1731no—
238#471—1731no—
239#471—1731no—
240#471D17311yescharacteristic, direct product factor
241C2×#108—1731no—
242C2×#101—1731no—
243C2×#100—1731no—
244C2×#41—1731no—
245C22×Q8—1731no—
246C22×D4—1731no—
247#137—3461no—
248#137—3461no—
249#137—1731no—
250#137—1731no—
251#137—1731no—
252#137C2×D17311yescharacteristic, complemented
253AGL(1,ℤ/8)—3461no—
254AGL(1,ℤ/8)—3461no—
255AGL(1,ℤ/8)—1731no—
256AGL(1,ℤ/8)—1731no—
257AGL(1,ℤ/8)—1731no—
258AGL(1,ℤ/8)C2×D17311yescharacteristic
259C2×Q16—3461no—
260C2×SD16—3461no—
261C2×SD16—3461no—
262C2×SD16—3461no—
263C2×SD16—3461no—
264C2×SD16—3461no—
265C2×SD16—3461no—
266C2×SD16—1731no—
267C2×SD16—1731no—
268C2×SD16—1731no—
269C2×SD16C2×D17311yescharacteristic, complemented
270C2×D8—3461no—
271#131—3461no—
272#131—3461no—
273#131—1731no—
274#131—1731no—
275#131—1731no—
276#131C2×D17311yescharacteristic, complemented
277C2×M16—3461no—
278C2×M16—1731no—
279C2×M16—1731no—
280C2×M16—1731no—
281C22×C8—3461no—
282C22×C8—1731no—
283#108—1731no—
284#108—1731no—
285#108—1731no—
286#108C2×D17311yescharacteristic, complemented
287#101—1731no—
288#101—1731no—
289#101—1731no—
290#101C2×D17311yescharacteristic, complemented
291#100—1731no—
292#100—1731no—
293#100—1731no—
294#100C2×D17311yescharacteristic, complemented
295C24—3461no—
296#41—3461no—
297#41—3461no—
298#41—1731no—
299#41—1731no—
300#41—1731no—
301#41C22×D17311yescharacteristic
302C2×Q8—3461no—
303C2×Q8—3461no—
304C2×Q8—3461no—
305C2×Q8—3461no—
306C2×Q8—3461no—
307C2×Q8—1731no—
308C2×Q8—1731no—
309C2×Q8C22×D17311yescharacteristic
310C2×D4—3461no—
311C2×D4—3461no—
312C2×D4—3461no—
313C2×D4—3461no—
314C2×D4—3461no—
315C2×D4—3461no—
316C2×D4—1731no—
317C2×D4—1731no—
318C2×D4C22×D17311yescharacteristic
319C22×C4—3461no—
320C22×C4—3461no—
321C22×C4—1731no—
322Q16—21no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—3461no—
331SD16—3461no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342D8—21no—
343D8—3461no—
344D8—3461no—
345D8—3461no—
346M16—21no—
347M16—3461no—
348M16—3461no—
349M16—3461no—
350M16—3461no—
351M16—3461no—
352M16—3461no—
353M16—1731no—
354M16—1731no—
355M16—1731no—
356M16C22×D17311yescharacteristic
357M16C22×D17311yescharacteristic
358M16C22×D17311yescharacteristic
359C2×C8—21no—
360C2×C8—3461no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8—3461no—
364C2×C8—3461no—
365C2×C8—3461no—
366C2×C8—1731no—
367C2×C8—1731no—
368C2×C8—1731no—
369C2×C8—1731no—
370C2×C8—1731no—
371C2×C8C22×D17311yescharacteristic
372C23—21no—
373C23—3461no—
374C23—3461no—
375C23—13842no—
376C23—1731no—
377C23—6921no—
378C23—6921no—
379Q8—21no—
380Q8—21no—
381Q8—3461no—
382Q8—3461no—
383Q8—3461no—
384Q8—1731no—
385Q8—6921no—
386Q8—6921no—
387Q8D4×D17311yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—3461no—
392D4—3461no—
393D4—3461no—
394D4—3461no—
395D4—3461no—
396D4—3461no—
397D4—1731no—
398D4—6921no—
399D4—6921no—
400D4D4×D17311yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—3461no—
404C2×C4—3461no—
405C2×C4—3461no—
406C2×C4—3461no—
407C2×C4—3461no—
408C2×C4—1731no—
409C2×C4—1731no—
410C2×C4—1731no—
411C2×C4—1731no—
412C2×C4—1731no—
413C2×C4—6921no—
414C2×C4—6921no—
415C2×C4C23×D17311yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—3461no—
420C8—3461no—
421C8—3461no—
422C8—1731no—
423C8—1731no—
424C8D4×D17311yescharacteristic
425C8D4×D17311yescharacteristic
426C22—21no—
427C22—41no—
428C22—81no—
429C22—3462no—
430C22—3461no—
431C22—13841no—
432C22—13842no—
433C22—13842no—
434C22—1731no—
435C22—1731no—
436C22—6922no—
437C22—6921no—
438C22#41×D17311yescharacteristic
439C4—21no—
440C4—41no—
441C4—3461no—
442C4—1731no—
443C4—1731no—
444C4—6921no—
445C4C2×D4×D17311yescharacteristic
446C4C2×D4×D17311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—3461no—
451C2—13841no—
452C2—1732no—
453C2—6921no—
454C2#121×D17311yescharacteristic
subgroupG/H
centerC2#121×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173C2×#121
prime radicalAGL(1,ℤ/8)×D173C2
hypercenter#471D173
hypocenterC173C2×#471
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. #137×D173C2semidirect
3. C2×SD16×D173C2semidirect
4. #131×D173C2semidirect
5. #26011757C2semidirect
6. #26011756C2semidirect
7. #26011755C2semidirect
8. #26011754C2semidirect
9. #26011753C2semidirect
10. #26011752C2semidirect
11. #26011751C2semidirect
12. #108×D173C2semidirect
13. #101×D173C2semidirect
14. #100×D173C2semidirect
15. #471×C173C2semidirect
19. #19149712C22semidirect
21. #19149710C22semidirect
22. #19149709C22semidirect
30. C2×#14583815C22semidirect
31. C2×#14583814C22semidirect
32. C2×#14583813C22semidirect
41. #19149687C22semidirect
42. #19149686C22semidirect
44. #19149685C22semidirect
51. #19149615C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
54. #19149598C22semidirect
55. #19149597C22semidirect
56. #19149597C22semidirect
57. #19149596C22semidirect
58. #19149595C22semidirect
59. #19149595C22semidirect
60. #137×C173C22semidirect
62. C2×SD16×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #100×C173C22semidirect
226. D173#471direct (N × H)
227. C173C2×#471semidirect
240. #471D173direct (N × H)
252. #137C2×D173semidirect
269. C2×SD16C2×D173semidirect
276. #131C2×D173semidirect
286. #108C2×D173semidirect
290. #101C2×D173semidirect
294. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#137×D1732
AGL(1,ℤ/8)×D1732
C2×SD16×D1732
#131×D1732
#260117572
#260117562
#260117552
#260117542
#260117532
#260117522
#260117512
#108×D1732
#101×D1732
#100×D1732
#471×C1732
C2×#471173