Finite Group Databasebeta

← #36365579

Subgroups — #36365579

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
2#26012125C211yesmaximal, characteristic
3#26011854C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011806C211yesmaximal, characteristic, complemented
6#26011794C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011641C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011594C211yesmaximal, characteristic, complemented
12#26011560C211yesmaximal, characteristic, complemented
13#26011497C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic, complemented
21C2×C8×D173C2211yescharacteristic, complemented
22#19149675C2211yescharacteristic
23#19149672—21no—
24#19149670C2211yescharacteristic, complemented
25#19149669—21no—
26#19149661—21no—
27#19149655C2211yescharacteristic, complemented
28#19149654C2211yescharacteristic
29#19149652—21no—
30#19149649—21no—
31#19149647—21no—
32Q8×Dic173C2211yescharacteristic, complemented
33#19149641C2211yescharacteristic
34C4×C22⋊D173C2211yescharacteristic
35#19149638C2211yescharacteristic, complemented
36#19149637—21no—
37C4⋊C4×D173C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149610C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149601C2211yescharacteristic
42#19149601C2211yescharacteristic
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C2×M16×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173—21no—
61C2×D692—22no—
62C2×D692—21no—
63C2×D692—21no—
64C2×D692—41no—
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71D4×D173—22no—
72D4×D173—41no—
73D4×D173—44no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583813—22no—
78#14583813—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C8×D173—41no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103SD16×C173—41no—
104M16×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
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
121D692—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125D692—42no—
126D692#4112yes—
127C22⋊D173—22no—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C22⋊D173—81no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—42no—
139C4×D173—42no—
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—42no—
149C2×C4×C173—21no—
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—21no—
160C22×C173—21no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—22no—
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—41no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173C2×#4111yescharacteristic
181C4×C173H(5,𝔽2)11yescharacteristic
182Dic173—21no—
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187C2×C173—41no—
188C2×C173—81no—
189C2×C173#54611yescharacteristic
190C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C173#57711yescharacteristic
192D173—22no—
193D173—42no—
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#468D17311yescharacteristic, complemented
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×SD16C2×D17311yescharacteristic, complemented
218C2×M16C2×D17311yescharacteristic, complemented
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#121C2×D17311yescharacteristic
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#107C2×D17311yescharacteristic, complemented
240#106—1731no—
241C4⋊C8—1731no—
242#103C2×D17311yescharacteristic, complemented
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102C2×D17311yescharacteristic, complemented
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C24—3461no—
256C24—3461no—
257C2×Q8C22×D17311yescharacteristic
258C2×D4—21no—
259C2×D4—3462no—
260C2×D4—3461no—
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—6921no—
270C2×D4—6921no—
271C2×D4—6924no—
272C2×D4C22×D17311yescharacteristic
273C22×C4—3461no—
274C22×C4—3461no—
275C22×C4—1731no—
276C22×C4C22×D17311yescharacteristic
277SD16—41no—
278SD16—3462no—
279SD16—6921no—
280M16—41no—
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⋊C4C22×D17311yescharacteristic
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
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—3461no—
310C23—3461no—
311C23—13841no—
312C23—13841no—
313C23—1731no—
314C23—6922no—
315C23—6922no—
316C23—6922no—
317C23—6922no—
318C23D4×D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—3462no—
325D4—3461no—
326D4—3461no—
327D4—3461no—
328D4—13841no—
329D4—1732no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
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—41no—
358C22—41no—
359C22—41no—
360C22—3461no—
361C22—3461no—
362C22—3462no—
363C22—13844no—
364C22—13844no—
365C22—1732no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22C2×D4×D17311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3461no—
377C4—3461no—
378C4—3461no—
379C4—6921no—
380C4—6921no—
381C4#41×D17311yescharacteristic
382C4#1914972411yescharacteristic
383C2—41no—
384C2—81no—
385C2—3462no—
386C2—13841no—
387C2—6922no—
388C2#2601213011yescharacteristic
389C2AGL(1,ℤ/8)×D17311yescharacteristic
390C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#26012125C2
hypercenter#468D173
hypocenterC173#3310
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #26012175C2semidirect
3. #26011854C2semidirect
4. #26011812C2semidirect
5. #26011806C2semidirect
6. #26011794C2semidirect
7. #26011723C2semidirect
8. #26011689C2semidirect
9. #26011641C2semidirect
10. #26011640C2semidirect
11. #26011594C2semidirect
12. #26011560C2semidirect
13. #26011497C2semidirect
14. #102×D173C2semidirect
15. #468×C173C2semidirect
20. C2×#14583813C22semidirect
21. C2×C8×D173C22semidirect
24. #19149670C22semidirect
27. #19149655C22semidirect
32. Q8×Dic173C22semidirect
35. #19149638C22semidirect
37. C4⋊C4×D173C22semidirect
38. #19149612C22semidirect
39. #19149610C22semidirect
40. #19149608C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
45. #19149600C22semidirect
46. #19149600C22semidirect
48. #19149599C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
52. C2×SD16×C173C22semidirect
53. C2×M16×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
195. C173#3310semidirect
203. #468D173semidirect
217. C2×SD16C2×D173semidirect
218. C2×M16C2×D173semidirect
226. C22⋊Q8C2×D173semidirect
239. #107C2×D173semidirect
242. #103C2×D173semidirect
251. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121752
#260121252
#260118542
#260118122
#260118062
#260117942
#260117232
#260116892
#260116412
#260116402
#260115942
#260115602
#260114972
#102×D1732
#468×C1732
#3310173