Finite Group Databasebeta

← #14539880

Subgroups — #14539880

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763430C211yesmaximal, characteristic, complemented
2#11763378C211yesmaximal, characteristic, complemented
3#11763198C211yesmaximal, characteristic, complemented
4#11763193C211yesmaximal, characteristic, complemented
5#11763169C211yesmaximal, characteristic, complemented
6#11763156C211yesmaximal, characteristic, complemented
7#11763032C211yesmaximal, characteristic, complemented
8#11763025C211yesmaximal, characteristic, complemented
9#11763010C211yesmaximal, characteristic, complemented
10#11763005C211yesmaximal, characteristic, complemented
11#11762973C211yesmaximal, characteristic, complemented
12#11762962C211yesmaximal, characteristic, complemented
13#11762956C211yesmaximal, characteristic, complemented
14#11762945C211yesmaximal, characteristic, complemented
15C17×#433C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C17×#117C2211yescharacteristic
19#10587616C2211yescharacteristic
20#10587610C2211yescharacteristic
21C2×Dic68C2211yescharacteristic, complemented
22C2×D136C2211yescharacteristic
23C2×D136C2211yescharacteristic
24C2×D136C2211yescharacteristic, complemented
25C2×#63219C2211yescharacteristic
26C2×#63219C2211yescharacteristic
27C2×#63219C2211yescharacteristic, complemented
28C2×#63219C2211yescharacteristic, complemented
29#10587584—21no—
30#10587582—21no—
31#10587581—21no—
32#10587581—21no—
33#10587571—21no—
34#10587569—21no—
35#10587568—21no—
36#10587568—21no—
37#10587561C2211yescharacteristic
38#10587561C2211yescharacteristic
39#10587561C2211yescharacteristic, complemented
40#10587560C2211yescharacteristic
41#10587560C2211yescharacteristic
42C4×D68C2211yescharacteristic, complemented
43#10587555C2211yescharacteristic, complemented
44C4×Dic34C2211yescharacteristic, complemented
45C17×C4⋊C8C2211yescharacteristic, complemented
46C17×C4⋊C8C2211yescharacteristic, complemented
47C17×C22⋊C8C2211yescharacteristic
48C17×C22⋊C8C2211yescharacteristic
49C17×#97C2211yescharacteristic, complemented
50C4×C8×C17C2211yescharacteristic, complemented
51#10587492C2211yescharacteristic
52#10587492C2211yescharacteristic
53#10587492C2211yescharacteristic
54#10587492C2211yescharacteristic
55#10587492C2211yescharacteristic, complemented
56#10587492C2211yescharacteristic, complemented
57#10587487C2211yescharacteristic, complemented
58#10587487C2211yescharacteristic, complemented
59C23×D17—21no—
60C22×C4×C17C2311yescharacteristic
61C2×C22⋊D17—21no—
62C2×C22⋊D17—21no—
63#63252—41no—
64#63252—41no—
65C2×D68—21no—
66C2×D68—21no—
67C2×D68—21no—
68C2×D68—21no—
69C2×D68—21no—
70C2×D68—41no—
71C2×D68—41no—
72C2×D68C2311yescharacteristic
73C2×D68C2311yescharacteristic
74C2×D68C2311yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×Dic34—21no—
79C2×Dic34C2311yescharacteristic
80C2×Dic34C2311yescharacteristic
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C4⋊C4×C17D411yescharacteristic
86C4⋊C4×C17D411yescharacteristic
87C22⋊C4×C17D411yescharacteristic
88C22⋊C4×C17D411yescharacteristic
89C42×C17C2311yescharacteristic
90C42×C17C2311yescharacteristic
91C22⋊Dic17—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104#63225—21no—
105C4×Dic17—21no—
106Dic68—41no—
107D136—41no—
108D136—41no—
109D136—41no—
110#63219—41no—
111#63219—41no—
112#63219—41no—
113#63219—41no—
114C23×C17C2×D411yescharacteristic
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—42no—
120C22×D17—81no—
121C22×D17—81no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C22⋊D17—42no—
131C22⋊D17—81no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136D68—21no—
137D68—21no—
138D68—21no—
139D68—21no—
140D68—21no—
141D68—21no—
142D68—21no—
143D68—21no—
144D68—42no—
145D68—81no—
146D68—81no—
147D68—81no—
148D68—81no—
149C4×D17—42no—
150C4×D17—81no—
151Dic34—21no—
152Dic34—21no—
153Dic34—21no—
154Dic34—21no—
155Dic34—81no—
156C8×C17—21no—
157C8×C17—21no—
158C8×C17—21no—
159C8×C17—21no—
160#3224—171nomaximal, Sylow
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17C22×D411yescharacteristic
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—84no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17H(5,𝔽2)11yescharacteristic
184C4×C17H(5,𝔽2)11yescharacteristic
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189#542—171no—
190#535—171no—
191#496—171no—
192#495—171no—
193#489—171no—
194#486—171no—
195#464—171no—
196#463—171no—
197#460—171no—
198#459—171no—
199#452—171no—
200#450—171no—
201#449—171no—
202#447—171no—
203#433D1711yescharacteristic, complemented
204C2×C17—41no—
205C2×C17#53511yescharacteristic
206C2×C17#57611yescharacteristic
207C2×C17#57711yescharacteristic
208D17—81no—
209D17—81no—
210D17—81no—
211D17—81no—
212C2×#41—171no—
213C22×D4—171no—
214C2×Q16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×D8—171no—
221C2×D8—171no—
222C4⋊Q8—171no—
223#127—171no—
224#127—171no—
225#126—341no—
226#124—171no—
227#124—171no—
228#124—171no—
229#124—341no—
230#123—341no—
231C22⋊Q8—171no—
232C22⋊Q8—341no—
233#121—171no—
234#121—341no—
235#121—341no—
236#120—341no—
237#120—341no—
238C4×Q8—171no—
239C4×D4—171no—
240#117C2×D1711yescharacteristic
241C4⋊C8C2×D1711yescharacteristic, complemented
242C4⋊C8C2×D1711yescharacteristic, complemented
243#103—171no—
244#103—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251C22⋊C8C2×D1711yescharacteristic
252C22⋊C8C2×D1711yescharacteristic
253#97C2×D1711yescharacteristic, complemented
254C4×C8C2×D1711yescharacteristic, complemented
255C17#322411yescharacteristic, complemented, Sylow
256C24—341no—
257#41—681no—
258#41—681no—
259C2×Q8—171no—
260C2×Q8—171no—
261C2×Q8—341no—
262C2×D4—171no—
263C2×D4—171no—
264C2×D4—171no—
265C2×D4—171no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—681no—
274C2×D4—681no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4C22×D1711yescharacteristic
278Q16—681no—
279SD16—681no—
280SD16—681no—
281SD16—681no—
282SD16—681no—
283D8—681no—
284D8—681no—
285D8—681no—
286C2×C8C22×D1711yescharacteristic
287C2×C8C22×D1711yescharacteristic
288C2×C8C22×D1711yescharacteristic
289C2×C8C22×D1711yescharacteristic
290C4⋊C4—171no—
291C4⋊C4—171no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4D6811yescharacteristic
296C4⋊C4D6811yescharacteristic
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4D6811yescharacteristic
307C22⋊C4D6811yescharacteristic
308C42—341no—
309C42C22×D1711yescharacteristic
310C42C22×D1711yescharacteristic
311C23—341no—
312C23—341no—
313C23—341no—
314C23—341no—
315C23—682no—
316C23—1361no—
317C23—1361no—
318C23C2×D6811yescharacteristic
319Q8—341no—
320Q8—341no—
321Q8—341no—
322Q8—341no—
323Q8—1361no—
324D4—341no—
325D4—341no—
326D4—341no—
327D4—341no—
328D4—341no—
329D4—341no—
330D4—341no—
331D4—341no—
332D4—682no—
333D4—682no—
334D4—1361no—
335D4—1361no—
336D4—1361no—
337D4—1361no—
338D4—1361no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—682no—
346C2×C4—1361no—
347C2×C4C2×D6811yescharacteristic
348C2×C4C2×D6811yescharacteristic
349C2×C4C2×D6811yescharacteristic
350C2×C4C2×D6811yescharacteristic
351C2×C4C2×D6811yescharacteristic
352C2×C4C23×D1711yescharacteristic
353C8—21no—
354C8—21no—
355C8—21no—
356C8—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—681no—
361C22—681no—
362C22—681no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22—681no—
371C22—681no—
372C22—1364no—
373C22C22×D6811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—681no—
380C4—681no—
381C4—681no—
382C4—681no—
383C4#1058769611yescharacteristic
384C4#1058769611yescharacteristic
385C2—41no—
386C2—1361no—
387C2—1361no—
388C2—1361no—
389C2—1361no—
390C2#1176337811yescharacteristic
391C2#1176361411yescharacteristic
392C2#1176362111yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#433C2
socleC22×C17C22×D4
hypercenter#433D17
hypocenterC17#3224
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #11763430C2semidirect
2. #11763378C2semidirect
3. #11763198C2semidirect
4. #11763193C2semidirect
5. #11763169C2semidirect
6. #11763156C2semidirect
7. #11763032C2semidirect
8. #11763025C2semidirect
9. #11763010C2semidirect
10. #11763005C2semidirect
11. #11762973C2semidirect
12. #11762962C2semidirect
13. #11762956C2semidirect
14. #11762945C2semidirect
15. C17×#433C2semidirect
21. C2×Dic68C22semidirect
24. C2×D136C22semidirect
27. C2×#63219C22semidirect
28. C2×#63219C22semidirect
39. #10587561C22semidirect
42. C4×D68C22semidirect
43. #10587555C22semidirect
44. C4×Dic34C22semidirect
45. C17×C4⋊C8C22semidirect
46. C17×C4⋊C8C22semidirect
49. C17×#97C22semidirect
50. C4×C8×C17C22semidirect
55. #10587492C22semidirect
56. #10587492C22semidirect
57. #10587487C22semidirect
58. #10587487C22semidirect
203. #433D17semidirect
241. C4⋊C8C2×D17semidirect
242. C4⋊C8C2×D17semidirect
253. #97C2×D17semidirect
254. C4×C8C2×D17semidirect
255. C17#3224semidirect

Maximal subgroups

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

subgroupindex
#117634302
#117633782
#117631982
#117631932
#117631692
#117631562
#117630322
#117630252
#117630102
#117630052
#117629732
#117629622
#117629562
#117629452
C17×#4332
#322417