Finite Group Databasebeta

← #14540940

Subgroups — #14540940

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#11763168C211yesmaximal, characteristic, complemented
3#11763147C211yesmaximal, characteristic, complemented
4#11763130C211yesmaximal, characteristic, complemented
5#11763089C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic, complemented
8#11763021C211yesmaximal, characteristic, complemented
9#11762954C211yesmaximal, characteristic, complemented
10#11762943C211yesmaximal, characteristic, complemented
11#11762919C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762814C211yesmaximal, characteristic, complemented
14#11762805C211yesmaximal, characteristic, complemented
15C17×#546C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C17×#124C2211yescharacteristic
19C17×#123—21no—
20C17×C22⋊Q8C2211yescharacteristic
21C17×#121—21no—
22C17×#121C2211yescharacteristic
23C17×#120—21no—
24C4×D4×C17C2211yescharacteristic
25C4×D4×C17C2211yescharacteristic
26C2×C22⋊C4×C17—21no—
27C2×#63230C2211yescharacteristic, complemented
28#10587624—21no—
29C2×#63228C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic, complemented
33#10587612C2211yescharacteristic
34#10587612C2211yescharacteristic, complemented
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic, complemented
37#10587578—21no—
38#10587571—21no—
39#10587571—21no—
40C4×D68C2211yescharacteristic, complemented
41#10587555C2211yescharacteristic, complemented
42#10587506C2211yescharacteristic, complemented
43#10587506C2211yescharacteristic, complemented
44#10587506C2211yescharacteristic, complemented
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587479C2211yescharacteristic, complemented
55#10587478C2211yescharacteristic, complemented
56#10587478C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C24×C17—21no—
60#41×C17—22no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—22no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—41no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17C2311yescharacteristic
70C2×D4×C17C2311yescharacteristic
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C2×C22⋊D17—21no—
76C2×C22⋊D17—21no—
77C2×D68C2311yescharacteristic
78C2×C4×D17—21no—
79C2×Dic34—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17D411yescharacteristic
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊C4×C17D411yescharacteristic
89C22⋊C4×C17D411yescharacteristic
90C42×C17C2311yescharacteristic
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93#63230—41no—
94#63228—22no—
95#63228—41no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103C4×Dic17—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—81no—
116C23×C17C2×D411yescharacteristic
117C23×C17C2×D411yescharacteristic
118C22×D17—21no—
119Q8×C17—21no—
120Q8×C17—21no—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—41no—
125D4×C17—42no—
126D4×C17—42no—
127D4×C17—81no—
128D4×C17#4112yes—
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—42no—
136C2×C4×C17—81no—
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2×D411yescharacteristic
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2411yescharacteristic
142C22⋊D17—81no—
143C22⋊D17—81no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147D68—21no—
148D68—21no—
149C4×D17—42no—
150Dic34—81no—
151#3622—21no—
152#3622—21no—
153#3622—21no—
154#3622—21no—
155#3319—171nomaximal, Sylow
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—22no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—84no—
169C22×C17C22×D411yescharacteristic
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17C2×#4111yescharacteristic
180C4×C17H(5,𝔽2)11yescharacteristic
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#555—171no—
185#546D1711yescharacteristic, complemented
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—42no—
202C2×C17—81no—
203C2×C17#54611yescharacteristic
204C2×C17#57511yescharacteristic
205C2×C17#57611yescharacteristic
206D17—81no—
207C2×#41C2×D1711yescharacteristic
208C22×D4C2×D1711yescharacteristic
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8—171no—
212C2×M16—171no—
213C22×C8—171no—
214C4⋊Q8—171no—
215#124—341no—
216#124—341no—
217#124C2×D1711yescharacteristic
218#123—21no—
219C22⋊Q8C2×D1711yescharacteristic
220#121—21no—
221#121—171no—
222#121—171no—
223#121—341no—
224#121C2×D1711yescharacteristic
225#120—21no—
226C4×D4—171no—
227C4×D4C2×D1711yescharacteristic
228C4×D4C2×D1711yescharacteristic
229#117—171no—
230#117—171no—
231#117—341no—
232C2×C22⋊C4—21no—
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—21no—
252#41—22no—
253#41—41no—
254C2×Q8—341no—
255C2×Q8C22×D1711yescharacteristic
256C2×D4—22no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—171no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4C22⋊D1711yescharacteristic
265C2×D4C22×D1711yescharacteristic
266C2×D4C22×D1711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
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—171no—
284C4⋊C4—171no—
285C4⋊C4—341no—
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—341no—
301C42—341no—
302C42C22×D1711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—341no—
309C23C2×C22⋊D1711yescharacteristic
310C23C2×C22⋊D1711yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—1361no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—42no—
319D4—42no—
320D4—81no—
321D4—341no—
322D4—341no—
323D4—1361no—
324D4—1361no—
325D4#6325412yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—682no—
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—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—84no—
360C22—681no—
361C22—681no—
362C22—681no—
363C22C22×C22⋊D1711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4C2×#6325411yescharacteristic
374C4#1058769111yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—1361no—
380C2#1176345311yescharacteristic
381C2#1176360711yescharacteristic
382C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#546C2
socleC22×C17C22×D4
hypercenter#546D17
hypocenterC17#3319
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #11763524C2semidirect
2. #11763168C2semidirect
3. #11763147C2semidirect
4. #11763130C2semidirect
5. #11763089C2semidirect
6. #11763057C2semidirect
7. #11763043C2semidirect
8. #11763021C2semidirect
9. #11762954C2semidirect
10. #11762943C2semidirect
11. #11762919C2semidirect
12. #11762881C2semidirect
13. #11762814C2semidirect
14. #11762805C2semidirect
15. C17×#546C2semidirect
27. C2×#63230C22semidirect
32. #10587616C22semidirect
34. #10587612C22semidirect
36. C22×#3622C22semidirect
40. C4×D68C22semidirect
41. #10587555C22semidirect
42. #10587506C22semidirect
43. #10587506C22semidirect
44. #10587506C22semidirect
52. C22⋊#3622C22semidirect
53. #10587480C22semidirect
54. #10587479C22semidirect
55. #10587478C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
185. #546D17semidirect
250. C17#3319semidirect

Maximal subgroups

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

subgroupindex
#117635242
#117631682
#117631472
#117631302
#117630892
#117630572
#117630432
#117630212
#117629542
#117629432
#117629192
#117628812
#117628142
#117628052
C17×#5462
#331917