Finite Group Databasebeta

← #14540945

Subgroups — #14540945

Order 2176.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763524C211yesmaximal, characteristic, complemented
2#11763454C211yesmaximal, characteristic, complemented
3#11763169C211yesmaximal, characteristic, complemented
4#11763153C211yesmaximal, characteristic, complemented
5#11763135C211yesmaximal, characteristic, complemented
6#11763089C211yesmaximal, characteristic, complemented
7#11763054C211yesmaximal, characteristic, complemented
8#11763044C211yesmaximal, characteristic, complemented
9#11763024C211yesmaximal, characteristic, complemented
10#11762951C211yesmaximal, characteristic, complemented
11#11762941C211yesmaximal, characteristic, complemented
12#11762882C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#11762802C211yesmaximal, characteristic, complemented
15C17×#442C211yesmaximal, characteristic, complemented
16C2×#63254C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C4×D4×C17C2211yescharacteristic
20C4×D4×C17C2211yescharacteristic, complemented
21#10587624C2211yescharacteristic
22#10587624C2211yescharacteristic, complemented
23C2×#63229C2211yescharacteristic, complemented
24C2×#63228C2211yescharacteristic
25#10587616—21no—
26#10587616—21no—
27C2×#63227—21no—
28#10587612—21no—
29C2×#63218C2211yescharacteristic
30C2×C8×D17C2211yescharacteristic, complemented
31#10587583C2211yescharacteristic
32#10587581C2211yescharacteristic
33#10587578C2211yescharacteristic
34#10587578C2211yescharacteristic, complemented
35#10587572—21no—
36#10587571—21no—
37#10587571—21no—
38#10587568—21no—
39#10587561C2211yescharacteristic
40C4×D68C2211yescharacteristic
41#10587555C2211yescharacteristic, complemented
42C17×#106C2211yescharacteristic, complemented
43C17×#102C2211yescharacteristic
44C17×#102C2211yescharacteristic, complemented
45C17×#97C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587487C2211yescharacteristic, complemented
54#10587481C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587478C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17C2311yescharacteristic
61C2×D4×C17C2311yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C22×Dic17—21no—
69#63254—22no—
70#63254—41no—
71D4×D17—22no—
72D4×D17—41no—
73C2×D68—21no—
74C2×D68D411yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17C2311yescharacteristic
77C2×C4×D17C2311yescharacteristic
78C2×Dic34—21no—
79C2×Dic34C2311yescharacteristic
80D8×C17—22no—
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90#63229—41no—
91#63228—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227D411yescharacteristic
99#63227D411yescharacteristic
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17D411yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105C4×Dic17—21no—
106C4×Dic17—21no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109#63218—41no—
110C8×D17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C22×D17—21no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—81no—
117C22×D17C2×D411yescharacteristic
118C22×D17C2×D411yescharacteristic
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17#4112yes—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—42no—
126C2×C4×C17—81no—
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2411yescharacteristic
130C22⋊D17—42no—
131C22⋊D17—42no—
132C22⋊D17—81no—
133C22⋊D17—81no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—42no—
138C2×Dic17C2×D411yescharacteristic
139C2×Dic17C2×D411yescharacteristic
140D68—21no—
141D68—21no—
142D68—41no—
143D68—81no—
144C4×D17—21no—
145C4×D17—21no—
146C4×D17—21no—
147C4×D17—21no—
148Dic34—21no—
149Dic34—21no—
150Dic34—81no—
151C8×C17—21no—
152C8×C17—21no—
153#3622—21no—
154#3622—21no—
155#3319—171nomaximal, Sylow
156C22×C17—22no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17C22×D411yescharacteristic
163C2×D17—21no—
164C2×D17—21no—
165C2×D17—21no—
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—84no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17C2×#4111yescharacteristic
178C4×C17H(5,𝔽2)11yescharacteristic
179Dic17—21no—
180Dic17—21no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#555—171no—
185#546—171no—
186#489—171no—
187#485—171no—
188#482—171no—
189#474—171no—
190#468—171no—
191#466—171no—
192#463—171no—
193#449—171no—
194#447—171no—
195#442D1711yescharacteristic, complemented
196#435—171no—
197#418—171no—
198#416—171no—
199C2×C17—42no—
200C2×C17—81no—
201C2×C17#54611yescharacteristic
202C2×C17#57511yescharacteristic
203C2×C17#57611yescharacteristic
204D17—41no—
205D17—41no—
206D17—81no—
207C2×#41—171no—
208C22×D4—171no—
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8C2×D1711yescharacteristic
212C2×M16—171no—
213C22×C8—171no—
214C4⋊Q8—171no—
215#124—171no—
216#124—341no—
217#124—341no—
218#123—341no—
219C22⋊Q8—171no—
220#121—171no—
221#121—171no—
222#121—171no—
223#121—341no—
224#121—341no—
225#120—341no—
226C4×D4—171no—
227C4×D4C2×D1711yescharacteristic
228C4×D4C2×D1711yescharacteristic, complemented
229#117—171no—
230#117—171no—
231#117—341no—
232C2×C22⋊C4—341no—
233#107—171no—
234#107—171no—
235#106C2×D1711yescharacteristic, complemented
236C4⋊C8—171no—
237#103—171no—
238#103—171no—
239#103—171no—
240#102—171no—
241#102—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245#102C2×D1711yescharacteristic
246#102C2×D1711yescharacteristic, complemented
247C22⋊C8—171no—
248C22⋊C8—171no—
249#97C2×D1711yescharacteristic, complemented
250C17#331911yescharacteristic, complemented, Sylow
251C24—341no—
252#41—342no—
253#41—681no—
254C2×Q8—171no—
255C2×Q8—341no—
256C2×D4—171no—
257C2×D4—171no—
258C2×D4—341no—
259C2×D4—341no—
260C2×D4—342no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—681no—
265C2×D4C22×D1711yescharacteristic
266C2×D4C22×D1711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—341no—
272SD16—681no—
273D8—22no—
274D8—681no—
275M16—681no—
276C2×C8—171no—
277C2×C8—171no—
278C2×C8—681no—
279C2×C8C22×D1711yescharacteristic
280C2×C8C22×D1711yescharacteristic
281C4⋊C4—171no—
282C4⋊C4—171no—
283C4⋊C4—171no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4C22×D1711yescharacteristic
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—171no—
291C22⋊C4—171no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C42—341no—
301C42—341no—
302C42C22×D1711yescharacteristic
303C23—21no—
304C23—21no—
305C23—171no—
306C23—171no—
307C23—341no—
308C23—682no—
309C23—682no—
310C23—1361no—
311Q8—341no—
312Q8—341no—
313Q8—1361no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—341no—
318D4—341no—
319D4—681no—
320D4—682no—
321D4—682no—
322D4—1361no—
323D4—1361no—
324D4—1361no—
325D4#6325212yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—171no—
331C2×C4—171no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—682no—
340C2×C4D4×D1711yescharacteristic
341C2×C4D4×D1711yescharacteristic
342C2×C4C23×D1711yescharacteristic
343C8—21no—
344C8—21no—
345C8—341no—
346C8—341no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—341no—
354C22—341no—
355C22—341no—
356C22—341no—
357C22—341no—
358C22—341no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—1364no—
363C22C2×D4×D1711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—341no—
369C4—341no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4C2×#6325211yescharacteristic
374C4#1058769611yescharacteristic
375C2—42no—
376C2—81no—
377C2—681no—
378C2—681no—
379C2—1361no—
380C2#1176345511yescharacteristic
381C2#1176360511yescharacteristic
382C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#442C2
socleC22×C17C22×D4
hypercenter#442D17
hypocenterC17#3319
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #11763524C2semidirect
2. #11763454C2semidirect
3. #11763169C2semidirect
4. #11763153C2semidirect
5. #11763135C2semidirect
6. #11763089C2semidirect
7. #11763054C2semidirect
8. #11763044C2semidirect
9. #11763024C2semidirect
10. #11762951C2semidirect
11. #11762941C2semidirect
12. #11762882C2semidirect
13. #11762812C2semidirect
14. #11762802C2semidirect
15. C17×#442C2semidirect
20. C4×D4×C17C22semidirect
22. #10587624C22semidirect
23. C2×#63229C22semidirect
30. C2×C8×D17C22semidirect
34. #10587578C22semidirect
41. #10587555C22semidirect
42. C17×#106C22semidirect
44. C17×#102C22semidirect
45. C17×#97C22semidirect
51. #10587491C22semidirect
52. #10587487C22semidirect
53. #10587487C22semidirect
54. #10587481C22semidirect
56. #10587478C22semidirect
57. #10587478C22semidirect
58. C4⋊#3622C22semidirect
195. #442D17semidirect
228. C4×D4C2×D17semidirect
235. #106C2×D17semidirect
246. #102C2×D17semidirect
249. #97C2×D17semidirect
250. C17#3319semidirect

Maximal subgroups

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

subgroupindex
#117635242
#117634542
#117631692
#117631532
#117631352
#117630892
#117630542
#117630442
#117630242
#117629512
#117629412
#117628822
#117628122
#117628022
C17×#4422
#331917