Finite Group Databasebeta

← #14540980

Subgroups — #14540980

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#11763511C211yesmaximal, characteristic, complemented
2#11763456C211yesmaximal, characteristic, complemented
3#11763162C211yesmaximal, characteristic, complemented
4#11763153C211yesmaximal, characteristic, complemented
5#11763120C211yesmaximal, characteristic, complemented
6#11763108C211yesmaximal, characteristic, complemented
7#11763054C211yesmaximal, characteristic, complemented
8#11763022C211yesmaximal, characteristic, complemented
9#11762952C211yesmaximal, characteristic, complemented
10#11762941C211yesmaximal, characteristic, complemented
11#11762889C211yesmaximal, characteristic, complemented
12#11762885C211yesmaximal, characteristic, complemented
13#11762812C211yesmaximal, characteristic, complemented
14#11762811C211yesmaximal, characteristic, complemented
15C17×#469C211yesmaximal, characteristic, complemented
16C2×D4×D17C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C2×D8×C17C2211yescharacteristic
19C2×M16×C17C2211yescharacteristic, complemented
20C17×#121C2211yescharacteristic
21C17×#121C2211yescharacteristic, complemented
22#10587625—21no—
23#10587624—21no—
24#10587624C2211yescharacteristic
25#10587623—21no—
26#10587622C2211yescharacteristic
27D4×Dic17—21no—
28D4×Dic17C2211yescharacteristic, complemented
29C2×#63229C2211yescharacteristic, complemented
30C2×#63228C2211yescharacteristic
31C4×C22⋊D17C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic, complemented
33C2×#63218C2211yescharacteristic
34#10587583C2211yescharacteristic
35#10587579C2211yescharacteristic
36#10587578C2211yescharacteristic
37#10587576C2211yescharacteristic, complemented
38#10587572—21no—
39#10587569—21no—
40C22⋊C4×D17—21no—
41#10587565—21no—
42C17×#106C2211yescharacteristic, complemented
43C17×#102C2211yescharacteristic
44C17×#102C2211yescharacteristic, complemented
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47C22⋊#3622C2211yescharacteristic, complemented
48#10587492C2211yescharacteristic
49#10587492C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587487C2211yescharacteristic, complemented
52#10587487C2211yescharacteristic, complemented
53#10587485C2211yescharacteristic, complemented
54C8×Dic17C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic
57#10587478C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×D4×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17C2311yescharacteristic
63C2×D4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68C2×C22⋊D17D411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71D4×D17—22no—
72D4×D17—41no—
73#63252—22no—
74#63252—41no—
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17C2311yescharacteristic
79C2×Dic34C2311yescharacteristic
80D8×C17—41no—
81M16×C17—41no—
82C2×C8×C17C2311yescharacteristic
83C2×C8×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17D411yescharacteristic
93#63229—41no—
94#63228—22no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227D411yescharacteristic
99C4⋊Dic17—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17C2311yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63225D411yescharacteristic
106C4×Dic17—21no—
107C4×Dic17C2311yescharacteristic
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
119D4×C17—21no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—42no—
124D4×C17—81no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2411yescharacteristic
131C22⋊D17—41no—
132C22⋊D17—42no—
133C22⋊D17—42no—
134C22⋊D17—81no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—81no—
139C2×Dic17—81no—
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143D68—21no—
144D68#4112yes—
145C4×D17—21no—
146C4×D17—21no—
147C4×D17—42no—
148C4×D17—81no—
149Dic34—21no—
150Dic34—21no—
151C8×C17—21no—
152C8×C17—21no—
153#3622—21no—
154#3622—21no—
155#3322—171nomaximal, Sylow
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—22no—
168C2×D17—21no—
169C2×D17—21no—
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#553—171no—
185#546—171no—
186#488—171no—
187#485—171no—
188#480—171no—
189#478—171no—
190#469D1711yescharacteristic, complemented
191#468—171no—
192#463—171no—
193#449—171no—
194#447—171no—
195C4×D8—171no—
196#436—171no—
197#418—171no—
198#417—171no—
199C2×C17—41no—
200C2×C17—81no—
201C2×C17—81no—
202C2×C17#54611yescharacteristic
203C2×C17#57511yescharacteristic
204C2×C17#57611yescharacteristic
205D17—41no—
206D17—42no—
207C2×#41—171no—
208C22×D4—171no—
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8C2×D1711yescharacteristic
212C2×M16—171no—
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—341no—
222#121—341no—
223#121C2×D1711yescharacteristic
224#121C2×D1711yescharacteristic, complemented
225#120—341no—
226C4×D4—171no—
227C4×D4—171no—
228C4×D4—171no—
229C4×D4—341no—
230#117—171no—
231C2×C4⋊C4—171no—
232C2×C22⋊C4—341no—
233#107—171no—
234#107—171no—
235#106C2×D1711yescharacteristic, complemented
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, complemented
247C22⋊C8—171no—
248C22⋊C8—171no—
249C4×C8—171no—
250C17#332211yescharacteristic, complemented, Sylow
251C24—341no—
252#41—342no—
253#41—681no—
254C2×Q8—171no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—171no—
258C2×D4—171no—
259C2×D4—341no—
260C2×D4—342no—
261C2×D4—341no—
262C2×D4—341no—
263C2×D4—681no—
264C2×D4C22×D1711yescharacteristic
265C2×D4C22×D1711yescharacteristic
266C22×C4—171no—
267C22×C4—341no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272SD16—681no—
273D8—41no—
274D8—342no—
275M16—41no—
276M16—681no—
277C2×C8—171no—
278C2×C8—171no—
279C2×C8C22×D1711yescharacteristic
280C2×C8C22×D1711yescharacteristic
281C4⋊C4—171no—
282C4⋊C4—171no—
283C4⋊C4—171no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4—341no—
288C4⋊C4C22×D1711yescharacteristic
289C4⋊C4C22×D1711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—171no—
293C22⋊C4—171no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C22⋊C4—341no—
298C22⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C42—171no—
302C42—341no—
303C23—21no—
304C23—21no—
305C23—171no—
306C23—341no—
307C23—682no—
308C23—682no—
309C23—1361no—
310C23D4×D1711yescharacteristic
311Q8—341no—
312Q8—341no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—81no—
319D4—172no—
320D4—341no—
321D4—681no—
322D4—682no—
323D4—682no—
324D4—1361no—
325C2×C4—21no—
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—682no—
338C2×C4—1361no—
339C2×C4—1361no—
340C2×C4—1361no—
341C2×C4D4×D1711yescharacteristic
342C2×C4C23×D1711yescharacteristic
343C8—21no—
344C8—21no—
345C8—341no—
346C8—341no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—341no—
357C22—341no—
358C22—341no—
359C22—342no—
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—
373C4#1058769111yescharacteristic
374C4#41×D1711yescharacteristic
375C2—41no—
376C2—81no—
377C2—81no—
378C2—681no—
379C2—682no—
380C2#1176344811yescharacteristic
381C2#1176360511yescharacteristic
382C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#469C2
socleC22×C17C22×D4
hypercenter#469D17
hypocenterC17#3322
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #11763511C2semidirect
2. #11763456C2semidirect
3. #11763162C2semidirect
4. #11763153C2semidirect
5. #11763120C2semidirect
6. #11763108C2semidirect
7. #11763054C2semidirect
8. #11763022C2semidirect
9. #11762952C2semidirect
10. #11762941C2semidirect
11. #11762889C2semidirect
12. #11762885C2semidirect
13. #11762812C2semidirect
14. #11762811C2semidirect
15. C17×#469C2semidirect
19. C2×M16×C17C22semidirect
21. C17×#121C22semidirect
28. D4×Dic17C22semidirect
29. C2×#63229C22semidirect
32. C2×C4⋊Dic17C22semidirect
37. #10587576C22semidirect
42. C17×#106C22semidirect
44. C17×#102C22semidirect
47. C22⋊#3622C22semidirect
51. #10587487C22semidirect
52. #10587487C22semidirect
53. #10587485C22semidirect
54. C8×Dic17C22semidirect
55. #10587481C22semidirect
57. #10587478C22semidirect
58. #10587478C22semidirect
190. #469D17semidirect
213. C2×M16C2×D17semidirect
224. #121C2×D17semidirect
235. #106C2×D17semidirect
246. #102C2×D17semidirect
250. C17#3322semidirect

Maximal subgroups

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

subgroupindex
#117635112
#117634562
#117631622
#117631532
#117631202
#117631082
#117630542
#117630222
#117629522
#117629412
#117628892
#117628852
#117628122
#117628112
C17×#4692
#332217