Finite Group Databasebeta

← #14539623

Subgroups — #14539623

Order 2176.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763319C211yesmaximal, characteristic, complemented
2#11763316C211yesmaximal, characteristic, complemented
3#11763151C211yesmaximal, characteristic, complemented
4#11763148C211yesmaximal, characteristic, complemented
5#11763131C211yesmaximal, characteristic, complemented
6#11763130C211yesmaximal, characteristic, complemented
7#11763066C211yesmaximal, characteristic, complemented
8#11763059C211yesmaximal, characteristic, complemented
9#11763054C211yesmaximal, characteristic, complemented
10#11763052C211yesmaximal, characteristic, complemented
11#11762971C211yesmaximal, characteristic, complemented
12#11762968C211yesmaximal, characteristic, complemented
13#11762951C211yesmaximal, characteristic, complemented
14#11762771C211yesmaximal, characteristic, complemented
15C17×#449C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63254C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×SD16×C17C2211yescharacteristic, complemented
20C2×D8×C17C2211yescharacteristic, complemented
21C17×#121C2211yescharacteristic
22#10587632C2211yescharacteristic
23#10587632C2211yescharacteristic
24C2×#63231C2211yescharacteristic, complemented
25#10587624C2211yescharacteristic
26#10587624C2211yescharacteristic
27#10587624C2211yescharacteristic
28#10587623—21no—
29#10587623—21no—
30#10587621—21no—
31D4×Dic17—21no—
32C2×#63229C2211yescharacteristic, complemented
33C4×C22⋊D17—21no—
34C4×C22⋊D17—21no—
35#10587612C2211yescharacteristic
36#10587610C2211yescharacteristic
37C2×#63218C2211yescharacteristic
38C2×#63218C2211yescharacteristic
39#10587583C2211yescharacteristic
40#10587578C2211yescharacteristic
41#10587569—21no—
42#10587567—21no—
43C17×#103C2211yescharacteristic, complemented
44C17×#102C2211yescharacteristic, complemented
45C17×C22⋊C8C2211yescharacteristic
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587503C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic, complemented
50C22⋊#3622C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587489C2211yescharacteristic, complemented
54#10587488C2211yescharacteristic, complemented
55#10587487C2211yescharacteristic, complemented
56#10587487C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C23×D17D411yescharacteristic
60#41×C17—41no—
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⋊D17—21no—
73C22×Dic17—21no—
74C22×Dic17D411yescharacteristic
75#63254—41no—
76#63254—41no—
77C2×C4×D17—41no—
78C2×C4×D17—41no—
79C2×C4×D17D411yescharacteristic
80C2×C4×D17D411yescharacteristic
81C2×C4×D17C2311yescharacteristic
82C2×C4×D17C2311yescharacteristic
83C2×Dic34C2311yescharacteristic
84SD16×C17—41no—
85D8×C17—41no—
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17—21no—
92C22⋊Dic17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95#63231—41no—
96#63229—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17C2311yescharacteristic
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105#63225—21no—
106#63225—21no—
107#63225—21no—
108C4×Dic17—21no—
109C4×Dic17—21no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112#63218—41no—
113#63218—41no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17C2×D411yescharacteristic
117C22×D17—41no—
118C22×D17—41no—
119C22×D17—41no—
120C22×D17C2×D411yescharacteristic
121C22×D17C2×D411yescharacteristic
122Q8×C17—21no—
123Q8×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—42no—
129D4×C17—42no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—42no—
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2411yescharacteristic
137C22⋊D17—42no—
138C22⋊D17—42no—
139C22⋊D17—42no—
140C22⋊D17—42no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—41no—
146C2×Dic17—42no—
147C2×Dic17C2×D411yescharacteristic
148C2×Dic17C2×D411yescharacteristic
149C4×D17—21no—
150C4×D17—21no—
151C4×D17—21no—
152C4×D17—21no—
153C4×D17—41no—
154C4×D17—41no—
155Dic34—21no—
156Dic34—21no—
157C8×C17—21no—
158C8×C17—21no—
159#3622—21no—
160#3622—21no—
161#3197—171nomaximal, Sylow
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17C22×D411yescharacteristic
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—21no—
175C2×D17—21no—
176C2×D17—21no—
177C2×D17—21no—
178C2×D17—44no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17H(5,𝔽2)11yescharacteristic
183C4×C17H(5,𝔽2)11yescharacteristic
184Dic17—21no—
185Dic17—21no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189Dic17—41no—
190#525—171no—
191#525—171no—
192#485—171no—
193#485—171no—
194#482—171no—
195#482—171no—
196#470—171no—
197#469—171no—
198#468—171no—
199#468—171no—
200#452—171no—
201#452—171no—
202#449—171no—
203#449D1711yescharacteristic, complemented
204#407—171no—
205C2×C17—41no—
206C2×C17—81no—
207C2×C17—81no—
208C2×C17#53411yescharacteristic
209C2×C17#57511yescharacteristic
210C2×C17#57511yescharacteristic
211D17—41no—
212D17—41no—
213C2×#41—171no—
214C2×#41C2×D1711yescharacteristic
215C23×C4—171no—
216C2×Q16—171no—
217C2×SD16—171no—
218C2×SD16C2×D1711yescharacteristic, complemented
219C2×D8C2×D1711yescharacteristic, complemented
220C2×M16—171no—
221C2×M16—171no—
222#123—341no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226C22⋊Q8—171no—
227C22⋊Q8—171no—
228#121—171no—
229#121—171no—
230#121—171no—
231#121C2×D1711yescharacteristic
232#120—341no—
233#120—341no—
234C4×D4—341no—
235C4×D4—341no—
236C4×D4—341no—
237C4×D4—341no—
238#117—171no—
239#117—171no—
240#107—171no—
241#107—171no—
242#106—171no—
243#106—171no—
244#103—171no—
245#103—171no—
246#103—171no—
247#103C2×D1711yescharacteristic, complemented
248#102—171no—
249#102—171no—
250#102—171no—
251#102C2×D1711yescharacteristic, complemented
252C22⋊C8—171no—
253C22⋊C8—171no—
254C22⋊C8—171no—
255C22⋊C8C2×D1711yescharacteristic
256C17#319711yescharacteristic, complemented, Sylow
257C24—171no—
258#41—41no—
259#41—41no—
260#41—681no—
261#41—681no—
262C2×Q8—171no—
263C2×Q8C22×D1711yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4C22×D1711yescharacteristic
271C2×D4C22×D1711yescharacteristic
272C22×C4—21no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—171no—
278C22×C4—341no—
279C22×C4—681no—
280C22×C4—681no—
281C22×C4C22×D1711yescharacteristic
282Q16—681no—
283SD16—41no—
284SD16—681no—
285D8—41no—
286M16—681no—
287M16—681no—
288C2×C8—171no—
289C2×C8—171no—
290C2×C8C22×D1711yescharacteristic
291C2×C8C22×D1711yescharacteristic
292C4⋊C4—171no—
293C4⋊C4—171no—
294C4⋊C4—171no—
295C4⋊C4—341no—
296C4⋊C4—341no—
297C4⋊C4—341no—
298C4⋊C4—341no—
299C4⋊C4C22×D1711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4—341no—
307C22⋊C4—341no—
308C22⋊C4—341no—
309C22⋊C4—341no—
310C42—341no—
311C42—341no—
312C23—21no—
313C23—21no—
314C23—171no—
315C23—171no—
316C23—681no—
317C23—681no—
318C23—681no—
319C23D4×D1711yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—341no—
323Q8—341no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—42no—
330D4—682no—
331D4—682no—
332D4—682no—
333D4—682no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—171no—
340C2×C4—171no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4—682no—
353C2×C4D4×D1711yescharacteristic
354C2×C4C23×D1711yescharacteristic
355C8—21no—
356C8—21no—
357C8—341no—
358C8—341no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—341no—
369C22—341no—
370C22—341no—
371C22—341no—
372C22—341no—
373C22—341no—
374C22—684no—
375C22C2×D4×D1711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—341no—
380C4—341no—
381C4—681no—
382C4—681no—
383C4—681no—
384C4—681no—
385C4#1058769111yescharacteristic
386C4#1058769111yescharacteristic
387C2—41no—
388C2—81no—
389C2—81no—
390C2—681no—
391C2—681no—
392C2#1176336911yescharacteristic
393C2#1176360511yescharacteristic
394C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#449C2
socleC22×C17C22×D4
hypercenter#449D17
hypocenterC17#3197
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. #11763319C2semidirect
2. #11763316C2semidirect
3. #11763151C2semidirect
4. #11763148C2semidirect
5. #11763131C2semidirect
6. #11763130C2semidirect
7. #11763066C2semidirect
8. #11763059C2semidirect
9. #11763054C2semidirect
10. #11763052C2semidirect
11. #11762971C2semidirect
12. #11762968C2semidirect
13. #11762951C2semidirect
14. #11762771C2semidirect
15. C17×#449C2semidirect
19. C2×SD16×C17C22semidirect
20. C2×D8×C17C22semidirect
24. C2×#63231C22semidirect
32. C2×#63229C22semidirect
43. C17×#103C22semidirect
44. C17×#102C22semidirect
46. #10587506C22semidirect
47. #10587503C22semidirect
48. #10587503C22semidirect
49. #10587503C22semidirect
53. #10587489C22semidirect
54. #10587488C22semidirect
55. #10587487C22semidirect
56. #10587487C22semidirect
57. #10587479C22semidirect
58. #10587478C22semidirect
203. #449D17semidirect
218. C2×SD16C2×D17semidirect
219. C2×D8C2×D17semidirect
247. #103C2×D17semidirect
251. #102C2×D17semidirect
256. C17#3197semidirect

Maximal subgroups

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

subgroupindex
#117633192
#117633162
#117631512
#117631482
#117631312
#117631302
#117630662
#117630592
#117630542
#117630522
#117629712
#117629682
#117629512
#117627712
C17×#4492
#319717