Finite Group Databasebeta

← #14540944

Subgroups — #14540944

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#11763447C211yesmaximal, characteristic, complemented
3#11763170C211yesmaximal, characteristic, complemented
4#11763148C211yesmaximal, characteristic, complemented
5#11763134C211yesmaximal, characteristic, complemented
6#11763087C211yesmaximal, characteristic, complemented
7#11763052C211yesmaximal, characteristic, complemented
8#11763043C211yesmaximal, characteristic, complemented
9#11763026C211yesmaximal, characteristic, complemented
10#11762941C211yesmaximal, characteristic, complemented
11#11762921C211yesmaximal, characteristic, complemented
12#11762877C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#11762805C211yesmaximal, characteristic, complemented
15C17×#449C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×SD16×C17C2211yescharacteristic, complemented
19C2×D8×C17C2211yescharacteristic
20C17×#121C2211yescharacteristic, complemented
21#10587633C2211yescharacteristic
22#10587632C2211yescharacteristic
23#10587625—21no—
24#10587625—21no—
25#10587624C2211yescharacteristic
26#10587623—21no—
27#10587622—21no—
28D4×Dic17C2211yescharacteristic
29D4×Dic17C2211yescharacteristic, complemented
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587615—21no—
33C4×C22⋊D17C2211yescharacteristic
34C2×#63227—21no—
35#10587612C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic, complemented
37C2×#63218C2211yescharacteristic
38#10587578C2211yescharacteristic
39#10587574C2211yescharacteristic, complemented
40#10587571—21no—
41#10587563—21no—
42C17×#103C2211yescharacteristic, complemented
43C17×#102C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic, complemented
45#10587506C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic, complemented
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587489C2211yescharacteristic, complemented
54#10587488C2211yescharacteristic, complemented
55#10587486C2211yescharacteristic, complemented
56#10587485C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—22no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17C2311yescharacteristic
66C2×D4×C17C2311yescharacteristic
67C22×C4×C17—21no—
68C22×C4×C17C2311yescharacteristic
69C2×C22⋊D17—21no—
70C2×C22⋊D17—21no—
71C2×C22⋊D17—21no—
72C2×C22⋊D17D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75D4×D17—22no—
76D4×D17—41no—
77C2×D68C2311yescharacteristic
78C2×C4×D17C2311yescharacteristic
79C2×Dic34—21no—
80SD16×C17—41no—
81D8×C17—22no—
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⋊Dic17D411yescharacteristic
91#63228—41no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
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
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63218—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C22×D17—21no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—81no—
118C22×D17C2×D411yescharacteristic
119Q8×C17—21no—
120Q8×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×C17—21no—
131C2×C4×C17—42no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—41no—
135C22⋊D17—42no—
136C22⋊D17—81no—
137C22⋊D17—81no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—42no—
142C2×Dic17—81no—
143C2×Dic17C2×D411yescharacteristic
144C2×Dic17C2×D411yescharacteristic
145C2×Dic17C2×D411yescharacteristic
146D68—21no—
147D68—21no—
148C4×D17—21no—
149C4×D17—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×C17C22×D411yescharacteristic
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—41no—
175C4×C17—41no—
176C4×C17C2×#4111yescharacteristic
177C4×C17H(5,𝔽2)11yescharacteristic
178Dic17—21no—
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#449D1711yescharacteristic, complemented
194#447—171no—
195#442—171no—
196#435—171no—
197#418—171no—
198#416—171no—
199C2×C17—42no—
200C2×C17—41no—
201C2×C17—81no—
202C2×C17#54611yescharacteristic
203C2×C17#57511yescharacteristic
204C2×C17#57611yescharacteristic
205D17—41no—
206D17—81no—
207C2×#41C2×D1711yescharacteristic
208C22×D4—171no—
209C2×SD16C2×D1711yescharacteristic, complemented
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—341no—
223#121—341no—
224#121C2×D1711yescharacteristic, complemented
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#103C2×D1711yescharacteristic, complemented
240#102—171no—
241#102—171no—
242#102—171no—
243#102—171no—
244#102—171no—
245#102—171no—
246#102C2×D1711yescharacteristic
247C22⋊C8—171no—
248C22⋊C8C2×D1711yescharacteristic, complemented
249#97—171no—
250C17#331911yescharacteristic, complemented, Sylow
251C24—341no—
252#41—22no—
253#41—41no—
254C2×Q8—341no—
255C2×Q8C22×D1711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—171no—
259C2×D4—171no—
260C2×D4—341no—
261C2×D4—342no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—681no—
265C2×D4C22×D1711yescharacteristic
266C2×D4C22×D1711yescharacteristic
267C22×C4—21no—
268C22×C4—171no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272SD16—41no—
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—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—171no—
306C23—341no—
307C23—682no—
308C23—682no—
309C23—1361no—
310C23D4×D1711yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—1361no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—81no—
319D4—341no—
320D4—341no—
321D4—681no—
322D4—682no—
323D4—1361no—
324D4—1361no—
325D4#6325412yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—171no—
332C2×C4—171no—
333C2×C4—171no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
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—341no—
357C22—341no—
358C22—341no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—1364no—
363C22C2×D4×D1711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—341no—
368C4—341no—
369C4—341no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4C2×#6325411yescharacteristic
374C4#1058769111yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—681no—
379C2—1361no—
380C2#1176344611yescharacteristic
381C2#1176360611yescharacteristic
382C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#449C2
socleC22×C17C22×D4
hypercenter#449D17
hypocenterC17#3319
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #11763523C2semidirect
2. #11763447C2semidirect
3. #11763170C2semidirect
4. #11763148C2semidirect
5. #11763134C2semidirect
6. #11763087C2semidirect
7. #11763052C2semidirect
8. #11763043C2semidirect
9. #11763026C2semidirect
10. #11762941C2semidirect
11. #11762921C2semidirect
12. #11762877C2semidirect
13. #11762812C2semidirect
14. #11762805C2semidirect
15. C17×#449C2semidirect
18. C2×SD16×C17C22semidirect
20. C17×#121C22semidirect
29. D4×Dic17C22semidirect
35. #10587612C22semidirect
36. C22×#3622C22semidirect
39. #10587574C22semidirect
42. C17×#103C22semidirect
44. C17×C22⋊C8C22semidirect
45. #10587506C22semidirect
46. #10587506C22semidirect
49. #10587503C22semidirect
53. #10587489C22semidirect
54. #10587488C22semidirect
55. #10587486C22semidirect
56. #10587485C22semidirect
58. #10587478C22semidirect
193. #449D17semidirect
209. C2×SD16C2×D17semidirect
224. #121C2×D17semidirect
239. #103C2×D17semidirect
248. C22⋊C8C2×D17semidirect
250. C17#3319semidirect

Maximal subgroups

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

subgroupindex
#117635232
#117634472
#117631702
#117631482
#117631342
#117630872
#117630522
#117630432
#117630262
#117629412
#117629212
#117628772
#117628122
#117628052
C17×#4492
#331917