Finite Group Databasebeta

← #14540934

Subgroups — #14540934

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#11763455C211yesmaximal, characteristic, complemented
2#11763171C211yesmaximal, characteristic, complemented
3#11763150C211yesmaximal, characteristic, complemented
4#11763133C211yesmaximal, characteristic, complemented
5#11763086C211yesmaximal, characteristic, complemented
6#11763053C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic, complemented
8#11763024C211yesmaximal, characteristic, complemented
9#11762952C211yesmaximal, characteristic, complemented
10#11762942C211yesmaximal, characteristic, complemented
11#11762919C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762813C211yesmaximal, characteristic, complemented
14#11762803C211yesmaximal, characteristic, complemented
15C17×#555C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C17×C4⋊Q8C2211yescharacteristic, complemented
19C17×#124—21no—
20C17×#124—21no—
21C17×#121—21no—
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic, complemented
24C4×D4×C17C2211yescharacteristic
25C4×D4×C17C2211yescharacteristic, complemented
26C17×#117—21no—
27C17×#117C2211yescharacteristic
28C17×#117C2211yescharacteristic, complemented
29#10587623—21no—
30C2×#63229C2211yescharacteristic, complemented
31C2×#63228C2211yescharacteristic
32C2×#63228C2211yescharacteristic
33#10587616C2211yescharacteristic
34#10587610C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic, complemented
37#10587580—21no—
38#10587570—21no—
39C22⋊C4×D17—21no—
40#10587561C2211yescharacteristic
41C4×D68C2211yescharacteristic
42#10587503C2211yescharacteristic, complemented
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic, complemented
45#10587481C2211yescharacteristic, complemented
46#10587481C2211yescharacteristic, complemented
47#10587481C2211yescharacteristic, complemented
48#10587480C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic, complemented
55#10587478C2211yescharacteristic, complemented
56#10587478C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17D411yescharacteristic
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70#63252—22no—
71#63252—41no—
72C2×D68—22no—
73C2×D68—21no—
74C2×D68—41no—
75C2×D68C2311yescharacteristic
76C2×D68C2311yescharacteristic
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×Dic34C2311yescharacteristic
80C4⋊C4×C17—21no—
81C4⋊C4×C17D411yescharacteristic
82C4⋊C4×C17C2311yescharacteristic
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17D411yescharacteristic
91C22⋊C4×C17D411yescharacteristic
92C42×C17—21no—
93C42×C17—21no—
94C42×C17C2311yescharacteristic
95C22⋊Dic17—21no—
96#63229—41no—
97#63228—22no—
98#63228—41no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—81no—
119Q8×C17—81no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17—41no—
123D4×C17—81no—
124D4×C17—81no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2411yescharacteristic
138C22⋊D17—42no—
139C22⋊D17—81no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142D68—21no—
143D68—21no—
144D68—21no—
145D68—42no—
146D68#4112yes—
147C4×D17—42no—
148C4×D17—81no—
149Dic34—21no—
150Dic34—21no—
151#3622—21no—
152#3622—21no—
153#3622—21no—
154#3622—21no—
155#3319—171nomaximal, Sylow
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C2×D17—22no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—84no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17C2×#4111yescharacteristic
181C4×C17H(5,𝔽2)11yescharacteristic
182Dic17—41no—
183Dic17—41no—
184#555D1711yescharacteristic, complemented
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#442—171no—
196#435—171no—
197#418—171no—
198#416—171no—
199C2×C17—41no—
200C2×C17—41no—
201C2×C17—81no—
202C2×C17#54611yescharacteristic
203C2×C17#57511yescharacteristic
204C2×C17#57611yescharacteristic
205D17—42no—
206D17—81no—
207C2×#41—171no—
208C22×D4—171no—
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8—171no—
212C2×M16—171no—
213C22×C8—171no—
214C4⋊Q8C2×D1711yescharacteristic, complemented
215#124—21no—
216#124—21no—
217#124—171no—
218#123—341no—
219C22⋊Q8—171no—
220#121—21no—
221#121—171no—
222#121—341no—
223#121C2×D1711yescharacteristic
224#121C2×D1711yescharacteristic, complemented
225#120—341no—
226C4×D4—171no—
227C4×D4C2×D1711yescharacteristic
228C4×D4C2×D1711yescharacteristic, complemented
229#117—21no—
230#117C2×D1711yescharacteristic
231#117C2×D1711yescharacteristic, complemented
232C2×C22⋊C4—341no—
233#107—171no—
234#107—171no—
235#106—171no—
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#102—171no—
246#102—171no—
247C22⋊C8—171no—
248C22⋊C8—171no—
249#97—171no—
250C17#331911yescharacteristic, complemented, Sylow
251C24—341no—
252#41—342no—
253#41—681no—
254C2×Q8—21no—
255C2×Q8—171no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—171no—
259C2×D4—171no—
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—341no—
269C22×C4—341no—
270C22×C4C22×D1711yescharacteristic
271C22×C4C22×D1711yescharacteristic
272SD16—681no—
273D8—342no—
274D8—681no—
275M16—681no—
276C2×C8—171no—
277C2×C8—171no—
278C2×C8—171no—
279C2×C8—171no—
280C2×C8—681no—
281C4⋊C4—21no—
282C4⋊C4—171no—
283C4⋊C4—341no—
284C4⋊C4C22⋊D1711yescharacteristic
285C4⋊C4C22×D1711yescharacteristic
286C4⋊C4C22×D1711yescharacteristic
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4C22⋊D1711yescharacteristic
299C22⋊C4C22⋊D1711yescharacteristic
300C42—21no—
301C42—21no—
302C42C22×D1711yescharacteristic
303C23—21no—
304C23—341no—
305C23—341no—
306C23—682no—
307C23—682no—
308C23—1361no—
309C23C2×C22⋊D1711yescharacteristic
310C23C2×C22⋊D1711yescharacteristic
311Q8—81no—
312Q8—341no—
313Q8—341no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—81no—
318D4—81no—
319D4—172no—
320D4—341no—
321D4—341no—
322D4—341no—
323D4—682no—
324D4—682no—
325D4—1361no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—682no—
337C2×C4—1361no—
338C2×C4C2×C22⋊D1711yescharacteristic
339C2×C4C2×C22⋊D1711yescharacteristic
340C2×C4C2×C22⋊D1711yescharacteristic
341C2×C4C2×C22⋊D1711yescharacteristic
342C2×C4C23×D1711yescharacteristic
343C8—341no—
344C8—341no—
345C8—341no—
346C8—341no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—342no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—1364no—
363C22C22×C22⋊D1711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—681no—
372C4—681no—
373C4#41×D1711yescharacteristic
374C4#1058769611yescharacteristic
375C2—41no—
376C2—41no—
377C2—81no—
378C2—682no—
379C2—1361no—
380C2#1176344711yescharacteristic
381C2#1176360711yescharacteristic
382C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#555C2
socleC22×C17C22×D4
hypercenter#555D17
hypocenterC17#3319
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #11763455C2semidirect
2. #11763171C2semidirect
3. #11763150C2semidirect
4. #11763133C2semidirect
5. #11763086C2semidirect
6. #11763053C2semidirect
7. #11763043C2semidirect
8. #11763024C2semidirect
9. #11762952C2semidirect
10. #11762942C2semidirect
11. #11762919C2semidirect
12. #11762881C2semidirect
13. #11762813C2semidirect
14. #11762803C2semidirect
15. C17×#555C2semidirect
18. C17×C4⋊Q8C22semidirect
23. C17×#121C22semidirect
25. C4×D4×C17C22semidirect
28. C17×#117C22semidirect
30. C2×#63229C22semidirect
36. C22×#3622C22semidirect
42. #10587503C22semidirect
44. C22⋊#3622C22semidirect
45. #10587481C22semidirect
46. #10587481C22semidirect
47. #10587481C22semidirect
54. #10587479C22semidirect
55. #10587478C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
184. #555D17semidirect
214. C4⋊Q8C2×D17semidirect
224. #121C2×D17semidirect
228. C4×D4C2×D17semidirect
231. #117C2×D17semidirect
250. C17#3319semidirect

Maximal subgroups

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

subgroupindex
#117634552
#117631712
#117631502
#117631332
#117630862
#117630532
#117630432
#117630242
#117629522
#117629422
#117629192
#117628812
#117628132
#117628032
C17×#5552
#331917