Finite Group Databasebeta

← #36365581

Subgroups — #36365581

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#26012179C211yesmaximal, characteristic, complemented
2D4×D692C211yesmaximal, characteristic
3#26011857C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011806C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011692C211yesmaximal, characteristic, complemented
9#26011643C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011561C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C22×D692—21no—
17C2×D4×D173C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19C2×#14583815C2211yescharacteristic, complemented
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic
22C2×#14583809C2211yescharacteristic, complemented
23C2×C8×D173C2211yescharacteristic, complemented
24#19149676C2211yescharacteristic
25#19149674—21no—
26#19149670—21no—
27#19149664—21no—
28#19149659C2211yescharacteristic, complemented
29#19149659C2211yescharacteristic, complemented
30#19149654C2211yescharacteristic
31#19149654C2211yescharacteristic
32#19149653—21no—
33#19149649—21no—
34#19149649—21no—
35C4×Dic346C2211yescharacteristic, complemented
36C4×D692C2211yescharacteristic
37#19149637—21no—
38#19149637C2211yescharacteristic, complemented
39C4⋊C4×D173C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149602C2211yescharacteristic, complemented
42#19149601C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic, complemented
44#19149600C2211yescharacteristic
45#19149599C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49#19149591C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149590C2211yescharacteristic, complemented
52#125×C173C2211yescharacteristic, complemented
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56C4⋊C8×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C23×D173—21no—
60C23×D173—21no—
61C2×Dic346C2311yescharacteristic
62C2×D692—21no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—44no—
67C2×D692D411yescharacteristic
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72D4×D173—22no—
73D4×D173—22no—
74D4×D173—41no—
75D4×D173—41no—
76C2×C4×D173—21no—
77C2×C4×D173C2311yescharacteristic
78C2×C4×D173C2311yescharacteristic
79#14583815—41no—
80#14583815—41no—
81#14583814—22no—
82#14583809—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C8×D173—41no—
86C4⋊Dic173—21no—
87C4⋊Dic173—21no—
88C4⋊Dic173D411yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91#14583805—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98#14583803D411yescharacteristic
99C4×Dic173—21no—
100C4×Dic173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—42no—
117C22×D173—42no—
118C22×D173—81no—
119C22×D173—81no—
120C22×D173C2×D411yescharacteristic
121C22×D173C2×D411yescharacteristic
122Dic346—21no—
123Dic346—21no—
124D692—22no—
125D692—21no—
126D692—21no—
127D692—21no—
128D692—21no—
129D692—42no—
130D692—42no—
131D692—42no—
132D692—81no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—42no—
144#13063436—21no—
145#13063436—21no—
146C23×C173—21no—
147D4×C173—21no—
148D4×C173#4112yes—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2411yescharacteristic
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—22no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173C22×D411yescharacteristic
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—84no—
174C2×D173—84no—
175C2×D173AGL(1,ℤ/8)12yes—
176C4×C173—21no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173C2×#4111yescharacteristic
182C4×C173H(5,𝔽2)11yescharacteristic
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187C2×C173—42no—
188C2×C173#54611yescharacteristic
189C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C173#57711yescharacteristic
191D173—22no—
192D173—41no—
193D173—81no—
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#478D17311yescharacteristic, complemented
203#468—1731no—
204#465—1731no—
205#460—1731no—
206#450—1731no—
207#450—1731no—
208#440—1731no—
209#435—1731no—
210#418—1731no—
211C2×#102—1731no—
212C22×D4—3461no—
213C22×D4—1731no—
214C22×D4—1731no—
215C2×SD16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×M16—1731no—
219C22×C8—1731no—
220#127—1731no—
221#126—3461no—
222#125—3461no—
223#125C2×D17311yescharacteristic, complemented
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227#121—3461no—
228#121—1731no—
229#121—1731no—
230#120—3461no—
231#120—3461no—
232C4×Q8—1731no—
233C4×D4—1731no—
234C4×D4C2×D17311yescharacteristic
235#117—3461no—
236#117—1731no—
237C2×C4⋊C4—1731no—
238#107—1731no—
239#107C2×D17311yescharacteristic, complemented
240#106C2×D17311yescharacteristic, complemented
241C4⋊C8C2×D17311yescharacteristic, complemented
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
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C24—3461no—
256C24—3461no—
257C2×Q8—1731no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3462no—
261C2×D4—3462no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—1731no—
267C2×D4—1731no—
268C2×D4—1731no—
269C2×D4—6924no—
270C2×D4—6921no—
271C2×D4—6921no—
272C2×D4C22×D17311yescharacteristic
273C22×C4—21no—
274C22×C4—3461no—
275C22×C4—1731no—
276C22×C4—1731no—
277SD16—3462no—
278SD16—6921no—
279SD16—6921no—
280M16—6921no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—6921no—
284C2×C8C22×D17311yescharacteristic
285C2×C8C22×D17311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4C22×D17311yescharacteristic
295C4⋊C4C22×D17311yescharacteristic
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—
307C42C22×D17311yescharacteristic
308C23—21no—
309C23—3461no—
310C23—3461no—
311C23—13841no—
312C23—13841no—
313C23—1731no—
314C23—1731no—
315C23—6922no—
316C23—6922no—
317C23—6922no—
318C23—6922no—
319Q8—3461no—
320Q8—3461no—
321D4—21no—
322D4—3462no—
323D4—3461no—
324D4—3461no—
325D4—3461no—
326D4—3461no—
327D4—13841no—
328D4—6922no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4#1458382712yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—1731no—
345C2×C4—1731no—
346C2×C4—6922no—
347C2×C4D4×D17311yescharacteristic
348C2×C4D4×D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3461no—
353C8—3461no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—3461no—
358C22—3461no—
359C22—3461no—
360C22—3461no—
361C22—3461no—
362C22—13844no—
363C22—13844no—
364C22—1732no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22C2×D4×D17311yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3461no—
378C4—3461no—
379C4—6921no—
380C4—6921no—
381C4C2×#1458382711yescharacteristic
382C4#1914972511yescharacteristic
383C2—42no—
384C2—3462no—
385C2—13841no—
386C2—13841no—
387C2—6921no—
388C2#2601213211yescharacteristic
389C2AGL(1,ℤ/8)×D17311yescharacteristic
390C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
prime radicalD4×D692C2
hypercenter#478D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012179C2semidirect
3. #26011857C2semidirect
4. #26011817C2semidirect
5. #26011806C2semidirect
6. #26011738C2semidirect
7. #26011721C2semidirect
8. #26011692C2semidirect
9. #26011643C2semidirect
10. #26011643C2semidirect
11. #26011591C2semidirect
12. #26011561C2semidirect
13. #26011494C2semidirect
14. #102×D173C2semidirect
15. #478×C173C2semidirect
19. C2×#14583815C22semidirect
20. C2×#14583815C22semidirect
22. C2×#14583809C22semidirect
23. C2×C8×D173C22semidirect
28. #19149659C22semidirect
29. #19149659C22semidirect
35. C4×Dic346C22semidirect
38. #19149637C22semidirect
39. C4⋊C4×D173C22semidirect
40. #19149612C22semidirect
41. #19149602C22semidirect
42. #19149601C22semidirect
43. #19149601C22semidirect
45. #19149599C22semidirect
46. #19149599C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
49. #19149591C22semidirect
50. #19149591C22semidirect
51. #19149590C22semidirect
52. #125×C173C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. C4⋊C8×C173C22semidirect
195. C173#3310semidirect
202. #478D173semidirect
223. #125C2×D173semidirect
239. #107C2×D173semidirect
240. #106C2×D173semidirect
241. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121792
D4×D6922
#260118572
#260118172
#260118062
#260117382
#260117212
#260116922
#260116432
#260115912
#260115612
#260114942
#102×D1732
#478×C1732
#3310173