Finite Group Databasebeta

← #14540936

Subgroups — #14540936

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#11763522C211yesmaximal, characteristic, complemented
2#11763447C211yesmaximal, characteristic, complemented
3#11763170C211yesmaximal, characteristic, complemented
4#11763149C211yesmaximal, characteristic, complemented
5#11763133C211yesmaximal, characteristic, complemented
6#11763087C211yesmaximal, characteristic, complemented
7#11763058C211yesmaximal, characteristic, complemented
8#11763045C211yesmaximal, characteristic, complemented
9#11763027C211yesmaximal, characteristic, complemented
10#11762954C211yesmaximal, characteristic, complemented
11#11762944C211yesmaximal, characteristic, complemented
12#11762918C211yesmaximal, characteristic, complemented
13#11762880C211yesmaximal, characteristic, complemented
14#11762812C211yesmaximal, characteristic, complemented
15C17×#416C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C22×C8×C17C2211yescharacteristic, complemented
19C17×#117C2211yescharacteristic, complemented
20#10587633C2211yescharacteristic
21#10587632C2211yescharacteristic
22C2×#63230C2211yescharacteristic, complemented
23#10587625—21no—
24#10587623—21no—
25D4×Dic17C2211yescharacteristic
26C2×#63228C2211yescharacteristic
27#10587616C2211yescharacteristic
28#10587616C2211yescharacteristic, complemented
29#10587615—21no—
30C4×C22⋊D17C2211yescharacteristic
31C2×#63227—21no—
32C2×D136C2211yescharacteristic
33C2×#63218C2211yescharacteristic
34#10587581C2211yescharacteristic
35#10587579C2211yescharacteristic, complemented
36#10587578C2211yescharacteristic
37#10587574C2211yescharacteristic, complemented
38#10587571—21no—
39#10587570—21no—
40#10587561—21no—
41#10587558—21no—
42C17×#103C2211yescharacteristic, complemented
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic
45C17×#102C2211yescharacteristic
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic, complemented
49#10587492C2211yescharacteristic
50#10587492C2211yescharacteristic, complemented
51#10587491C2211yescharacteristic
52#10587489C2211yescharacteristic, complemented
53#10587486C2211yescharacteristic, complemented
54#10587485C2211yescharacteristic, complemented
55#10587480C2211yescharacteristic
56#10587480C2211yescharacteristic
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17—21no—
60#41×C17—22no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17C2311yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17C2311yescharacteristic
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69C2×C22⋊D17—21no—
70C2×C22⋊D17D411yescharacteristic
71C22×Dic17—21no—
72D4×D17—22no—
73D4×D17—41no—
74C2×D68—21no—
75C2×D68C2311yescharacteristic
76C2×D68C2311yescharacteristic
77C2×C4×D17—21no—
78C2×C4×D17C2311yescharacteristic
79C2×Dic34—21no—
80C2×C8×C17—41no—
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17C2311yescharacteristic
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C42×C17—21no—
87C22⋊Dic17—21no—
88C22⋊Dic17D411yescharacteristic
89#63230—41no—
90#63228—22no—
91#63227—21no—
92#63227—21no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99#63227D411yescharacteristic
100C4⋊Dic17C2311yescharacteristic
101C4⋊Dic17C2311yescharacteristic
102#63225—21no—
103#63225—21no—
104#63225D411yescharacteristic
105C4×Dic17—21no—
106C4×Dic17C2311yescharacteristic
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109D136—41no—
110#63218—41no—
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
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—42no—
123D4×C17#4112yes—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—42no—
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—81no—
139C2×Dic17C2×D411yescharacteristic
140C2×Dic17C2×D411yescharacteristic
141C2×Dic17C2×D411yescharacteristic
142D68—21no—
143D68—21no—
144D68—21no—
145D68—21no—
146D68—81no—
147C4×D17—21no—
148C4×D17—21no—
149C4×D17—42no—
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×C17C22×D411yescharacteristic
163C2×D17—21no—
164C2×D17—21no—
165C2×D17—21no—
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—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17C2×#4111yescharacteristic
178C4×C17H(5,𝔽2)11yescharacteristic
179Dic17—21no—
180Dic17—21no—
181Dic17—21no—
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#447—171no—
195#442—171no—
196#435—171no—
197#418—171no—
198#416D1711yescharacteristic, complemented
199C2×C17—42no—
200C2×C17—41no—
201C2×C17#54611yescharacteristic
202C2×C17#57511yescharacteristic
203C2×C17#57611yescharacteristic
204D17—41no—
205D17—81no—
206D17—81no—
207C2×#41C2×D1711yescharacteristic
208C22×D4—171no—
209C2×SD16—171no—
210C2×D8—171no—
211C2×D8—171no—
212C2×M16—171no—
213C22×C8C2×D1711yescharacteristic, complemented
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—171no—
223#121—341no—
224#121—341no—
225#120—341no—
226C4×D4—171no—
227C4×D4—171no—
228C4×D4—171no—
229#117—171no—
230#117—341no—
231#117C2×D1711yescharacteristic, complemented
232C2×C22⋊C4—341no—
233#107—171no—
234#107—171no—
235#106—171no—
236C4⋊C8—171no—
237#103—171no—
238#103C2×D1711yescharacteristic, complemented
239#103C2×D1711yescharacteristic, complemented
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⋊C8—171no—
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—171no—
258C2×D4—171no—
259C2×D4—171no—
260C2×D4—341no—
261C2×D4—341no—
262C2×D4—342no—
263C2×D4—341no—
264C2×D4—341no—
265C2×D4—681no—
266C2×D4C22×D1711yescharacteristic
267C22×C4—21no—
268C22×C4—171no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4C22×D1711yescharacteristic
272SD16—681no—
273D8—342no—
274D8—681no—
275M16—681no—
276C2×C8—41no—
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⋊C4C22×D1711yescharacteristic
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—21no—
301C42—171no—
302C42—341no—
303C23—21no—
304C23—171no—
305C23—341no—
306C23—341no—
307C23—682no—
308C23—682no—
309C23—1361no—
310C23D4×D1711yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—1361no—
314D4—21no—
315D4—42no—
316D4—341no—
317D4—341no—
318D4—341no—
319D4—341no—
320D4—681no—
321D4—682no—
322D4—1361no—
323D4—1361no—
324D4—1361no—
325D4#6325212yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—171no—
333C2×C4—171no—
334C2×C4—171no—
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—341no—
354C22—341no—
355C22—341no—
356C22—681no—
357C22—681no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—1364no—
363C22C2×D4×D1711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—341no—
369C4—341no—
370C4—341no—
371C4—681no—
372C4—681no—
373C4C2×#6325211yescharacteristic
374C4#1058769111yescharacteristic
375C2—41no—
376C2—42no—
377C2—681no—
378C2—1361no—
379C2—1361no—
380C2#1176344911yescharacteristic
381C2#1176360611yescharacteristic
382C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#416C2
socleC22×C17C22×D4
hypercenter#416D17
hypocenterC17#3319
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #11763522C2semidirect
2. #11763447C2semidirect
3. #11763170C2semidirect
4. #11763149C2semidirect
5. #11763133C2semidirect
6. #11763087C2semidirect
7. #11763058C2semidirect
8. #11763045C2semidirect
9. #11763027C2semidirect
10. #11762954C2semidirect
11. #11762944C2semidirect
12. #11762918C2semidirect
13. #11762880C2semidirect
14. #11762812C2semidirect
15. C17×#416C2semidirect
18. C22×C8×C17C22semidirect
19. C17×#117C22semidirect
22. C2×#63230C22semidirect
28. #10587616C22semidirect
35. #10587579C22semidirect
37. #10587574C22semidirect
42. C17×#103C22semidirect
43. C17×#103C22semidirect
46. #10587506C22semidirect
48. C22⋊#3622C22semidirect
50. #10587492C22semidirect
52. #10587489C22semidirect
53. #10587486C22semidirect
54. #10587485C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
198. #416D17semidirect
213. C22×C8C2×D17semidirect
231. #117C2×D17semidirect
238. #103C2×D17semidirect
239. #103C2×D17semidirect
250. C17#3319semidirect

Maximal subgroups

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

subgroupindex
#117635222
#117634472
#117631702
#117631492
#117631332
#117630872
#117630582
#117630452
#117630272
#117629542
#117629442
#117629182
#117628802
#117628122
C17×#4162
#331917