Finite Group Databasebeta

← #36365577

Subgroups — #36365577

Order 22144.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012175C211yesmaximal, characteristic, complemented
2D4×C22⋊D173C211yesmaximal, characteristic
3#26011854C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011793C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011717C211yesmaximal, characteristic, complemented
9#26011689C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011496C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173—21no—
17C2×D4×D173C2211yescharacteristic
18C2×#14583813C2211yescharacteristic, complemented
19C2×#14583810C2211yescharacteristic, complemented
20C2×C8×D173C2211yescharacteristic, complemented
21#19149675C2211yescharacteristic
22#19149672—21no—
23#19149670C2211yescharacteristic, complemented
24#19149669—21no—
25#19149661—21no—
26#19149655C2211yescharacteristic, complemented
27#19149653C2211yescharacteristic
28#19149651—21no—
29#19149650C2211yescharacteristic
30#19149648—21no—
31#19149647—21no—
32Q8×Dic173C2211yescharacteristic, complemented
33D4×Dic173C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35#19149638C2211yescharacteristic, complemented
36#19149637—21no—
37C4⋊C4×D173C2211yescharacteristic, complemented
38#19149613C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49#19149599C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic, complemented
55C22⋊Q8×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—44no—
67C2×C22⋊D173D411yescharacteristic
68D4×D173—22no—
69D4×D173—41no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583813—41no—
74#14583810—41no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—41no—
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C24×C173—21no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—22no—
100C2×D4×C173—21no—
101C2×D4×C173—41no—
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105SD16×C173—22no—
106SD16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117D692—21no—
118D692—21no—
119C22⋊D173—22no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C22⋊D173—42no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173—21no—
139C23×C173—42no—
140C23×C173—42no—
141C23×C173—81no—
142C23×C173C2×D411yescharacteristic
143Q8×C173—21no—
144Q8×C173—21no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—42no—
149D4×C173#4112yes—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—22no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—84no—
168C22×C173C22×D411yescharacteristic
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—84no—
175C2×D173AGL(1,ℤ/8)12yes—
176C4×C173—21no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173C2×#4111yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—21no—
183Dic173—21no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187C2×C173—42no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173#54611yescharacteristic
191C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C173#57711yescharacteristic
193D173—22no—
194D173—81no—
195C173#331011yescharacteristic, complemented, Sylow
196#3310—1731nomaximal, Sylow
197#552—1731no—
198D42—1731no—
199#490—1731no—
200#482—1731no—
201#480—1731no—
202#478—1731no—
203#468—1731no—
204#465—1731no—
205#460—1731no—
206#450—1731no—
207#450D17311yescharacteristic, complemented
208#440—1731no—
209#435—1731no—
210#418—1731no—
211C2×#102—1731no—
212C22×D4—3461no—
213C22×D4—1731no—
214C22×D4C2×D17311yescharacteristic
215C2×SD16—1731no—
216C2×SD16C2×D17311yescharacteristic
217C2×SD16C2×D17311yescharacteristic, complemented
218C2×M16—1731no—
219C22×C8—1731no—
220#127—1731no—
221#126—3461no—
222#125—3461no—
223#125—1731no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8C2×D17311yescharacteristic, complemented
227#121—3461no—
228#121—1731no—
229#121—1731no—
230#120—3461no—
231#120—3461no—
232C4×Q8—1731no—
233C4×D4—1731no—
234C4×D4—1731no—
235#117—3461no—
236#117—1731no—
237C2×C4⋊C4—1731no—
238#107—1731no—
239#107—1731no—
240#106—1731no—
241C4⋊C8—1731no—
242#103—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102C2×D17311yescharacteristic
251#102C2×D17311yescharacteristic, complemented
252C22⋊C8—1731no—
253C22⋊C8C2×D17311yescharacteristic, complemented
254#97—1731no—
255C24—21no—
256C24—3461no—
257C2×Q8C22×D17311yescharacteristic
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—3462no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6921no—
270C2×D4—6924no—
271C2×D4C22×D17311yescharacteristic
272C2×D4C22×D17311yescharacteristic
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—1731no—
276C22×C4C22×D17311yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—6921no—
280M16—6921no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—6921no—
284C2×C8C22×D17311yescharacteristic
285C2×C8C22×D17311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—1731no—
304C22⋊C4—1731no—
305C42—3461no—
306C42—3461no—
307C42—1731no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—3461no—
314C23—13841no—
315C23—1731no—
316C23—6922no—
317C23—6922no—
318C23D4×D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—3462no—
326D4—3461no—
327D4—3461no—
328D4—13841no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4#1458382512yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—1731no—
344C2×C4—1731no—
345C2×C4—1731no—
346C2×C4—6922no—
347C2×C4—6922no—
348C2×C4D4×D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3461no—
353C8—3461no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—3461no—
365C22—3461no—
366C22—13844no—
367C22—1732no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22C2×D4×D17311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3461no—
376C4—3461no—
377C4—3461no—
378C4—6921no—
379C4—6921no—
380C4—6921no—
381C4C2×#1458382511yescharacteristic
382C4#1914972411yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—3462no—
387C2—13841no—
388C2#2601212811yescharacteristic
389C2AGL(1,ℤ/8)×D17311yescharacteristic
390C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalD4×C22⋊D173C2
hypercenter#450D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012175C2semidirect
3. #26011854C2semidirect
4. #26011816C2semidirect
5. #26011803C2semidirect
6. #26011793C2semidirect
7. #26011739C2semidirect
8. #26011717C2semidirect
9. #26011689C2semidirect
10. #26011640C2semidirect
11. #26011592C2semidirect
12. #26011565C2semidirect
13. #26011496C2semidirect
14. #102×D173C2semidirect
15. #450×C173C2semidirect
18. C2×#14583813C22semidirect
19. C2×#14583810C22semidirect
20. C2×C8×D173C22semidirect
23. #19149670C22semidirect
26. #19149655C22semidirect
32. Q8×Dic173C22semidirect
35. #19149638C22semidirect
37. C4⋊C4×D173C22semidirect
38. #19149613C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149608C22semidirect
42. #19149603C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
49. #19149599C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
54. C2×SD16×C173C22semidirect
55. C22⋊Q8×C173C22semidirect
57. #102×C173C22semidirect
58. C22⋊C8×C173C22semidirect
195. C173#3310semidirect
207. #450D173semidirect
217. C2×SD16C2×D173semidirect
226. C22⋊Q8C2×D173semidirect
251. #102C2×D173semidirect
253. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121752
D4×C22⋊D1732
#260118542
#260118162
#260118032
#260117932
#260117392
#260117172
#260116892
#260116402
#260115922
#260115652
#260114962
#102×D1732
#450×C1732
#3310173