Finite Group Databasebeta

← #14540933

Subgroups — #14540933

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#11763523C211yesmaximal, characteristic, complemented
2#11763453C211yesmaximal, characteristic, complemented
3#11763167C211yesmaximal, characteristic, complemented
4#11763151C211yesmaximal, characteristic, complemented
5#11763132C211yesmaximal, characteristic, complemented
6#11763087C211yesmaximal, characteristic, complemented
7#11763053C211yesmaximal, characteristic, complemented
8#11763042C211yesmaximal, characteristic, complemented
9#11763022C211yesmaximal, characteristic, complemented
10#11762951C211yesmaximal, characteristic, complemented
11#11762921C211yesmaximal, characteristic, complemented
12#11762883C211yesmaximal, characteristic, complemented
13#11762814C211yesmaximal, characteristic, complemented
14#11762802C211yesmaximal, characteristic, complemented
15C17×#447C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C2×#63254C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C2×D8×C17C2211yescharacteristic
20C17×#121C2211yescharacteristic
21#10587639—21no—
22C2×C22⋊Dic17—21no—
23#10587625—21no—
24#10587625—21no—
25#10587624C2211yescharacteristic
26#10587624C2211yescharacteristic, complemented
27#10587622—21no—
28#10587622C2211yescharacteristic
29#10587621—21no—
30D4×Dic17C2211yescharacteristic
31D4×Dic17C2211yescharacteristic, complemented
32C2×#63229C2211yescharacteristic, complemented
33C4×C22⋊D17C2211yescharacteristic
34#10587612C2211yescharacteristic
35#10587610C2211yescharacteristic
36C2×#63223C2211yescharacteristic
37C2×C8×D17C2211yescharacteristic, complemented
38#10587578C2211yescharacteristic, complemented
39#10587574C2211yescharacteristic, complemented
40#10587571—21no—
41#10587563—21no—
42C17×#102C2211yescharacteristic
43C17×#102C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic, complemented
50#10587491C2211yescharacteristic, complemented
51#10587489C2211yescharacteristic, complemented
52#10587488C2211yescharacteristic, complemented
53#10587487C2211yescharacteristic, complemented
54#10587487C2211yescharacteristic, complemented
55#10587486C2211yescharacteristic, complemented
56#10587485C2211yescharacteristic, complemented
57#10587481C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C24×C17—21no—
60C2×D4×C17—22no—
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17—41no—
64C2×D4×C17C2311yescharacteristic
65C2×D4×C17C2311yescharacteristic
66C2×D4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17D411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75#63254—22no—
76#63254—41no—
77C2×C4×D17C2311yescharacteristic
78C2×Dic34—21no—
79C2×Dic34C2311yescharacteristic
80D8×C17—22no—
81D8×C17—41no—
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊Dic17—21no—
87C22⋊Dic17—21no—
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17D411yescharacteristic
95#63229—41no—
96#63227—21no—
97#63227D411yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99C4⋊Dic17C2311yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101#63225—21no—
102#63225—21no—
103#63225D411yescharacteristic
104C4×Dic17—21no—
105C4×Dic17—21no—
106C4×Dic17C2311yescharacteristic
107#63223—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C8×D17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—81no—
117C23×C17C2×D411yescharacteristic
118C22×D17C2×D411yescharacteristic
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—42no—
125D4×C17—81no—
126D4×C17#4112yes—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—41no—
133C22⋊D17—42no—
134C22⋊D17—81no—
135C22⋊D17—81no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—81no—
143C2×Dic17C2×D411yescharacteristic
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
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—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—84no—
169C22×C17C22×D411yescharacteristic
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—21no—
173C4×C17—21no—
174C4×C17—41no—
175C4×C17C2×#4111yescharacteristic
176C4×C17H(5,𝔽2)11yescharacteristic
177Dic17—21no—
178Dic17—21no—
179Dic17—21no—
180Dic17—41no—
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#447D1711yescharacteristic, complemented
195#442—171no—
196#435—171no—
197#418—171no—
198#416—171no—
199C2×C17—41no—
200C2×C17—42no—
201C2×C17—81no—
202C2×C17—81no—
203C2×C17#54611yescharacteristic
204C2×C17#57511yescharacteristic
205C2×C17#57611yescharacteristic
206D17—41no—
207C2×#41—171no—
208C22×D4C2×D1711yescharacteristic
209C2×SD16—171no—
210C2×D8C2×D1711yescharacteristic
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—341no—
223#121—341no—
224#121C2×D1711yescharacteristic
225#120—341no—
226C4×D4—171no—
227C4×D4—171no—
228C4×D4—171no—
229#117—171no—
230#117—171no—
231#117—341no—
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#102C2×D1711yescharacteristic
246#102C2×D1711yescharacteristic
247C22⋊C8—171no—
248C22⋊C8C2×D1711yescharacteristic
249#97—171no—
250C17#331911yescharacteristic, complemented, Sylow
251C24—21no—
252#41—342no—
253#41—681no—
254C2×Q8—171no—
255C2×Q8—341no—
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—171no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4C22×D1711yescharacteristic
265C2×D4C22×D1711yescharacteristic
266C2×D4C22×D1711yescharacteristic
267C22×C4—171no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272SD16—681no—
273D8—22no—
274D8—41no—
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—171no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—171no—
290C22⋊C4—171no—
291C22⋊C4—341no—
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—171no—
301C42—341no—
302C42—341no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23—171no—
310C23D4×D1711yescharacteristic
311Q8—341no—
312Q8—341no—
313Q8—1361no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—81no—
321D4—681no—
322D4—682no—
323D4—1361no—
324D4—1361no—
325D4#6325412yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—171no—
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—682no—
339C2×C4—682no—
340C2×C4—1361no—
341C2×C4D4×D1711yescharacteristic
342C2×C4C23×D1711yescharacteristic
343C8—21no—
344C8—21no—
345C8—341no—
346C8—341no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—84no—
360C22—341no—
361C22—341no—
362C22—341no—
363C22C2×D4×D1711yescharacteristic
364C4—21no—
365C4—41no—
366C4—341no—
367C4—341no—
368C4—341no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4C2×#6325411yescharacteristic
374C4#1058769111yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—81no—
379C2—681no—
380C2#1176344611yescharacteristic
381C2#1176360511yescharacteristic
382C2#1176361011yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#447C2
socleC22×C17C22×D4
hypercenter#447D17
hypocenterC17#3319
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #11763523C2semidirect
2. #11763453C2semidirect
3. #11763167C2semidirect
4. #11763151C2semidirect
5. #11763132C2semidirect
6. #11763087C2semidirect
7. #11763053C2semidirect
8. #11763042C2semidirect
9. #11763022C2semidirect
10. #11762951C2semidirect
11. #11762921C2semidirect
12. #11762883C2semidirect
13. #11762814C2semidirect
14. #11762802C2semidirect
15. C17×#447C2semidirect
26. #10587624C22semidirect
31. D4×Dic17C22semidirect
32. C2×#63229C22semidirect
37. C2×C8×D17C22semidirect
38. #10587578C22semidirect
39. #10587574C22semidirect
49. #10587503C22semidirect
50. #10587491C22semidirect
51. #10587489C22semidirect
52. #10587488C22semidirect
53. #10587487C22semidirect
54. #10587487C22semidirect
55. #10587486C22semidirect
56. #10587485C22semidirect
57. #10587481C22semidirect
58. #10587478C22semidirect
194. #447D17semidirect
250. C17#3319semidirect

Maximal subgroups

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

subgroupindex
#117635232
#117634532
#117631672
#117631512
#117631322
#117630872
#117630532
#117630422
#117630222
#117629512
#117629212
#117628832
#117628142
#117628022
C17×#4472
#331917