Finite Group Databasebeta

← #14539621

Subgroups — #14539621

Order 2176.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763320C211yesmaximal, characteristic, complemented
2#11763150C211yesmaximal, characteristic, complemented
3#11763147C211yesmaximal, characteristic, complemented
4#11763133C211yesmaximal, characteristic, complemented
5#11763130C211yesmaximal, characteristic, complemented
6#11763065C211yesmaximal, characteristic, complemented
7#11763060C211yesmaximal, characteristic, complemented
8#11763057C211yesmaximal, characteristic, complemented
9#11763053C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762969C211yesmaximal, characteristic, complemented
12#11762954C211yesmaximal, characteristic, complemented
13#11762952C211yesmaximal, characteristic, complemented
14#11762773C211yesmaximal, characteristic, complemented
15C17×#525C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C2×#63252C2211yescharacteristic
19C17×#123—21no—
20C17×C22⋊Q8C2211yescharacteristic
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic
24C17×#120—21no—
25C4×D4×C17—21no—
26C4×D4×C17—21no—
27C17×#117C2211yescharacteristic
28#10587639—21no—
29C2×#63231C2211yescharacteristic, complemented
30C2×#63230C2211yescharacteristic, complemented
31C2×#63229C2211yescharacteristic, complemented
32C2×#63228C2211yescharacteristic, complemented
33#10587616C2211yescharacteristic
34#10587616C2211yescharacteristic
35#10587615—21no—
36C4×C22⋊D17—21no—
37C4×C22⋊D17—21no—
38#10587612C2211yescharacteristic
39#10587610C2211yescharacteristic
40#10587610C2211yescharacteristic
41C2×#63223C2211yescharacteristic
42C2×#63223C2211yescharacteristic
43#10587506C2211yescharacteristic, complemented
44#10587506C2211yescharacteristic, complemented
45#10587503C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic, complemented
47C22⋊#3622C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic
49C22⋊#3622C2211yescharacteristic
50C22⋊#3622C2211yescharacteristic
51#10587481C2211yescharacteristic, complemented
52#10587481C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic, complemented
55#10587479C2211yescharacteristic, complemented
56#10587479C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C24×C17D411yescharacteristic
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17C2311yescharacteristic
67C22×C4×C17—21no—
68C22×C4×C17—41no—
69C22×C4×C17—41no—
70C22×C4×C17D411yescharacteristic
71C22×C4×C17D411yescharacteristic
72C22×C4×C17D411yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C2×C22⋊D17—21no—
77C2×C22⋊D17—21no—
78C2×C22⋊D17—21no—
79#63252—41no—
80#63252—41no—
81C2×D68C2311yescharacteristic
82C2×C4×D17—21no—
83C2×Dic34C2311yescharacteristic
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C42×C17—21no—
94C22⋊Dic17—21no—
95C22⋊Dic17—21no—
96C22⋊Dic17—21no—
97#63231—41no—
98#63230—41no—
99#63229—41no—
100#63228—41no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105#63225—21no—
106#63225—21no—
107C4×Dic17—21no—
108#63223—41no—
109#63223—41no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17C2×D411yescharacteristic
119C23×C17C2×D411yescharacteristic
120C23×C17C2×D411yescharacteristic
121C22×D17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—42no—
127D4×C17—42no—
128D4×C17—42no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—42no—
142C2×C4×C17C2×D411yescharacteristic
143C2×C4×C17C2×D411yescharacteristic
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2411yescharacteristic
146C22⋊D17—42no—
147C22⋊D17—42no—
148C22⋊D17—42no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152D68—21no—
153D68—21no—
154C4×D17—42no—
155Dic34—21no—
156Dic34—21no—
157#3622—21no—
158#3622—21no—
159#3622—21no—
160#3622—21no—
161#3197—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—44no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17C22×D411yescharacteristic
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17H(5,𝔽2)11yescharacteristic
186C4×C17H(5,𝔽2)11yescharacteristic
187Dic17—41no—
188Dic17—41no—
189Dic17—41no—
190#525—171no—
191#525D1711yescharacteristic, complemented
192#485—171no—
193#485—171no—
194#482—171no—
195#482—171no—
196#470—171no—
197#469—171no—
198#468—171no—
199#468—171no—
200#452—171no—
201#452—171no—
202#449—171no—
203#449—171no—
204#407—171no—
205C2×C17—41no—
206C2×C17—41no—
207C2×C17—41no—
208C2×C17—81no—
209C2×C17#53411yescharacteristic
210C2×C17#57511yescharacteristic
211C2×C17#57511yescharacteristic
212D17—81no—
213C2×#41—171no—
214C2×#41C2×D1711yescharacteristic
215C23×C4C2×D1711yescharacteristic
216C2×Q16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×M16—171no—
221C2×M16—171no—
222#123—21no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226C22⋊Q8C2×D1711yescharacteristic
227C22⋊Q8C2×D1711yescharacteristic
228#121—171no—
229#121—171no—
230#121C2×D1711yescharacteristic
231#121C2×D1711yescharacteristic
232#120—21no—
233#120—341no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—341no—
237C4×D4—341no—
238#117—171no—
239#117C2×D1711yescharacteristic
240#107—171no—
241#107—171no—
242#106—171no—
243#106—171no—
244#103—171no—
245#103—171no—
246#103—171no—
247#103—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251#102—171no—
252C22⋊C8—171no—
253C22⋊C8—171no—
254C22⋊C8—171no—
255C22⋊C8—171no—
256C17#319711yescharacteristic, complemented, Sylow
257C24C22⋊D1711yescharacteristic
258#41—41no—
259#41—41no—
260#41—681no—
261#41—681no—
262C2×Q8—171no—
263C2×Q8C22×D1711yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—171no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4C22×D1711yescharacteristic
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—341no—
276C22×C4C22⋊D1711yescharacteristic
277C22×C4C22⋊D1711yescharacteristic
278C22×C4C22⋊D1711yescharacteristic
279C22×C4C22×D1711yescharacteristic
280C22×C4C22×D1711yescharacteristic
281C22×C4C22×D1711yescharacteristic
282Q16—681no—
283SD16—681no—
284SD16—681no—
285D8—681no—
286M16—681no—
287M16—681no—
288C2×C8—171no—
289C2×C8—171no—
290C2×C8—171no—
291C2×C8—171no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—171no—
295C4⋊C4—171no—
296C4⋊C4—341no—
297C4⋊C4—341no—
298C4⋊C4C22×D1711yescharacteristic
299C4⋊C4C22×D1711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C42—21no—
311C42—341no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—341no—
317C23C2×C22⋊D1711yescharacteristic
318C23C2×C22⋊D1711yescharacteristic
319C23C2×C22⋊D1711yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—341no—
323Q8—341no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—42no—
328D4—42no—
329D4—341no—
330D4—341no—
331D4—682no—
332D4—682no—
333D4—682no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—682no—
351C2×C4C2×C22⋊D1711yescharacteristic
352C2×C4C2×C22⋊D1711yescharacteristic
353C2×C4C2×C22⋊D1711yescharacteristic
354C2×C4C23×D1711yescharacteristic
355C8—341no—
356C8—341no—
357C8—341no—
358C8—341no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22C22×C22⋊D1711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—681no—
383C4—681no—
384C4—681no—
385C4#1058769111yescharacteristic
386C4#1058769111yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—81no—
391C2—1361no—
392C2#1176336811yescharacteristic
393C2#1176360711yescharacteristic
394C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#525C2
socleC22×C17C22×D4
hypercenter#525D17
hypocenterC17#3197
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #11763320C2semidirect
2. #11763150C2semidirect
3. #11763147C2semidirect
4. #11763133C2semidirect
5. #11763130C2semidirect
6. #11763065C2semidirect
7. #11763060C2semidirect
8. #11763057C2semidirect
9. #11763053C2semidirect
10. #11762971C2semidirect
11. #11762969C2semidirect
12. #11762954C2semidirect
13. #11762952C2semidirect
14. #11762773C2semidirect
15. C17×#525C2semidirect
29. C2×#63231C22semidirect
30. C2×#63230C22semidirect
31. C2×#63229C22semidirect
32. C2×#63228C22semidirect
43. #10587506C22semidirect
44. #10587506C22semidirect
45. #10587503C22semidirect
46. #10587503C22semidirect
51. #10587481C22semidirect
52. #10587481C22semidirect
53. #10587480C22semidirect
54. #10587480C22semidirect
55. #10587479C22semidirect
56. #10587479C22semidirect
57. #10587478C22semidirect
58. #10587478C22semidirect
191. #525D17semidirect
256. C17#3197semidirect

Maximal subgroups

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

subgroupindex
#117633202
#117631502
#117631472
#117631332
#117631302
#117630652
#117630602
#117630572
#117630532
#117629712
#117629692
#117629542
#117629522
#117627732
C17×#5252
#319717