Finite Group Databasebeta

← #14540987

Subgroups — #14540987

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#11763505C211yesmaximal, characteristic, complemented
2#11763455C211yesmaximal, characteristic, complemented
3#11763164C211yesmaximal, characteristic, complemented
4#11763151C211yesmaximal, characteristic, complemented
5#11763123C211yesmaximal, characteristic, complemented
6#11763110C211yesmaximal, characteristic, complemented
7#11763062C211yesmaximal, characteristic, complemented
8#11763055C211yesmaximal, characteristic, complemented
9#11763025C211yesmaximal, characteristic, complemented
10#11762956C211yesmaximal, characteristic, complemented
11#11762945C211yesmaximal, characteristic, complemented
12C4×D136C211yesmaximal, characteristic, complemented
13#11762815C211yesmaximal, characteristic, complemented
14#11762811C211yesmaximal, characteristic, complemented
15C17×#436C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C2×M16×C17C2211yescharacteristic
19C2×M16×C17C2211yescharacteristic, complemented
20C4×D4×C17C2211yescharacteristic
21#10587625—21no—
22#10587623—21no—
23#10587616C2211yescharacteristic
24#10587616C2211yescharacteristic
25#10587616C2211yescharacteristic, complemented
26#10587612C2211yescharacteristic
27C2×C4⋊Dic17C2211yescharacteristic, complemented
28#10587610C2211yescharacteristic
29C2×D136C2211yescharacteristic
30C2×D136C2211yescharacteristic
31C2×#63219C2211yescharacteristic, complemented
32#10587584—21no—
33#10587580—21no—
34#10587570—21no—
35#10587569—21no—
36#10587567—21no—
37C22⋊C4×D17—21no—
38#10587561C2211yescharacteristic
39C4×D68C2211yescharacteristic
40C4×D68C2211yescharacteristic, complemented
41#10587556C2211yescharacteristic, complemented
42C17×C4⋊C8C2211yescharacteristic, complemented
43C17×C22⋊C8C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic, complemented
45C4×C8×C17C2211yescharacteristic, complemented
46#10587492C2211yescharacteristic
47#10587492C2211yescharacteristic
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587492C2211yescharacteristic
52#10587492C2211yescharacteristic, complemented
53#10587489C2211yescharacteristic, complemented
54#10587489C2211yescharacteristic, complemented
55#10587488C2211yescharacteristic, complemented
56#10587487C2211yescharacteristic, complemented
57#10587487C2211yescharacteristic, complemented
58#10587487C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17D411yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C22×Dic17—21no—
68#63252—22no—
69#63252—41no—
70C2×D68—22no—
71C2×D68—21no—
72C2×D68—41no—
73C2×D68C2311yescharacteristic
74C2×D68C2311yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×Dic34C2311yescharacteristic
80M16×C17—41no—
81M16×C17—41no—
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C2×C8×C17C2311yescharacteristic
85C2×C8×C17C2311yescharacteristic
86C4⋊C4×C17D411yescharacteristic
87C22⋊C4×C17D411yescharacteristic
88C22⋊C4×C17D411yescharacteristic
89C42×C17C2311yescharacteristic
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102C4⋊Dic17C2311yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63225—21no—
106#63225—21no—
107C4×Dic17—21no—
108D136—22no—
109D136—41no—
110#63219—41no—
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—81no—
119D4×C17—41no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17C2×D411yescharacteristic
125C2×C4×C17C2×D411yescharacteristic
126C2×C4×C17C2×D411yescharacteristic
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C22⋊D17—42no—
130C22⋊D17—42no—
131C22⋊D17—81no—
132C22⋊D17—81no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—81no—
139D68—21no—
140D68—21no—
141D68—21no—
142D68—21no—
143D68—21no—
144D68—42no—
145D68#4112yes—
146C4×D17—42no—
147C4×D17—81no—
148C4×D17—81no—
149Dic34—21no—
150Dic34—21no—
151C8×C17—21no—
152C8×C17—21no—
153C8×C17—21no—
154C8×C17—21no—
155#3322—171nomaximal, Sylow
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17C22×D411yescharacteristic
163C2×D17—22no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
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×C17C2×#4111yescharacteristic
178C4×C17H(5,𝔽2)11yescharacteristic
179Dic17—41no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184#553—171no—
185#546—171no—
186#488—171no—
187#485—171no—
188#480—171no—
189#478—171no—
190#469—171no—
191#468—171no—
192#463—171no—
193#449—171no—
194#447—171no—
195C4×D8—171no—
196#436D1711yescharacteristic, complemented
197#418—171no—
198#417—171no—
199C2×C17—41no—
200C2×C17—41no—
201C2×C17#54611yescharacteristic
202C2×C17#57511yescharacteristic
203C2×C17#57611yescharacteristic
204D17—42no—
205D17—81no—
206D17—81no—
207C2×#41—171no—
208C22×D4—171no—
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8—171no—
212C2×M16C2×D1711yescharacteristic
213C2×M16C2×D1711yescharacteristic, complemented
214#126—341no—
215#125—171no—
216#124—171no—
217#123—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×D4—171no—
228C4×D4—341no—
229C4×D4C2×D1711yescharacteristic
230#117—171no—
231C2×C4⋊C4—171no—
232C2×C22⋊C4—341no—
233#107—171no—
234#107—171no—
235#106—171no—
236C4⋊C8C2×D1711yescharacteristic, complemented
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⋊C8C2×D1711yescharacteristic
248C22⋊C8C2×D1711yescharacteristic, complemented
249C4×C8C2×D1711yescharacteristic, complemented
250C17#332211yescharacteristic, complemented, Sylow
251C24—341no—
252#41—342no—
253#41—681no—
254C2×Q8—171no—
255C2×D4—171no—
256C2×D4—171no—
257C2×D4—171no—
258C2×D4—342no—
259C2×D4—341no—
260C2×D4—341no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—341no—
264C2×D4—681no—
265C2×D4D6811yescharacteristic
266C22×C4—341no—
267C22×C4—341no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4C22×D1711yescharacteristic
271C22×C4C22×D1711yescharacteristic
272SD16—681no—
273D8—342no—
274D8—681no—
275M16—41no—
276M16—41no—
277C2×C8C22×D1711yescharacteristic
278C2×C8C22×D1711yescharacteristic
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⋊C4—341no—
288C4⋊C4—341no—
289C4⋊C4D6811yescharacteristic
290C22⋊C4—341no—
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⋊C4D6811yescharacteristic
300C22⋊C4D6811yescharacteristic
301C42—341no—
302C42C22×D1711yescharacteristic
303C23—341no—
304C23—341no—
305C23—341no—
306C23—682no—
307C23—682no—
308C23—1361no—
309C23C2×D6811yescharacteristic
310C23C2×D6811yescharacteristic
311Q8—341no—
312Q8—341no—
313D4—41no—
314D4—172no—
315D4—341no—
316D4—341no—
317D4—341no—
318D4—341no—
319D4—341no—
320D4—682no—
321D4—682no—
322D4—682no—
323D4—1361no—
324D4—1361no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—682no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4—1361no—
338C2×C4C2×D6811yescharacteristic
339C2×C4C2×D6811yescharacteristic
340C2×C4C2×D6811yescharacteristic
341C2×C4C2×D6811yescharacteristic
342C2×C4C23×D1711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—342no—
354C22—681no—
355C22—681no—
356C22—681no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—1364no—
363C22C22×D6811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—681no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4#1058769111yescharacteristic
374C4#41×D1711yescharacteristic
375C2—41no—
376C2—41no—
377C2—682no—
378C2—1361no—
379C2—1361no—
380C2#1176345411yescharacteristic
381C2#1176360911yescharacteristic
382C2#1176361411yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#436C2
socleC22×C17C22×D4
hypercenter#436D17
hypocenterC17#3322
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #11763505C2semidirect
2. #11763455C2semidirect
3. #11763164C2semidirect
4. #11763151C2semidirect
5. #11763123C2semidirect
6. #11763110C2semidirect
7. #11763062C2semidirect
8. #11763055C2semidirect
9. #11763025C2semidirect
10. #11762956C2semidirect
11. #11762945C2semidirect
12. C4×D136C2semidirect
13. #11762815C2semidirect
14. #11762811C2semidirect
15. C17×#436C2semidirect
19. C2×M16×C17C22semidirect
25. #10587616C22semidirect
27. C2×C4⋊Dic17C22semidirect
31. C2×#63219C22semidirect
40. C4×D68C22semidirect
41. #10587556C22semidirect
42. C17×C4⋊C8C22semidirect
44. C17×C22⋊C8C22semidirect
45. C4×C8×C17C22semidirect
52. #10587492C22semidirect
53. #10587489C22semidirect
54. #10587489C22semidirect
55. #10587488C22semidirect
56. #10587487C22semidirect
57. #10587487C22semidirect
58. #10587487C22semidirect
196. #436D17semidirect
213. C2×M16C2×D17semidirect
236. C4⋊C8C2×D17semidirect
248. C22⋊C8C2×D17semidirect
249. C4×C8C2×D17semidirect
250. C17#3322semidirect

Maximal subgroups

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

subgroupindex
#117635052
#117634552
#117631642
#117631512
#117631232
#117631102
#117630622
#117630552
#117630252
#117629562
#117629452
C4×D1362
#117628152
#117628112
C17×#4362
#332217