Finite Group Databasebeta

← #14539622

Subgroups — #14539622

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#11763322C211yesmaximal, characteristic, complemented
2#11763319C211yesmaximal, characteristic, complemented
3#11763152C211yesmaximal, characteristic, complemented
4#11763149C211yesmaximal, characteristic, complemented
5#11763136C211yesmaximal, characteristic, complemented
6#11763131C211yesmaximal, characteristic, complemented
7#11763065C211yesmaximal, characteristic, complemented
8#11763060C211yesmaximal, characteristic, complemented
9#11763055C211yesmaximal, characteristic, complemented
10#11762974C211yesmaximal, characteristic, complemented
11#11762968C211yesmaximal, characteristic, complemented
12#11762955C211yesmaximal, characteristic, complemented
13#11762953C211yesmaximal, characteristic, complemented
14#11762772C211yesmaximal, characteristic, complemented
15C17×#468C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×#63254C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×SD16×C17C2211yescharacteristic, complemented
20C2×M16×C17C2211yescharacteristic
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121C2211yescharacteristic
23#10587632C2211yescharacteristic
24C2×#63231C2211yescharacteristic, complemented
25#10587624C2211yescharacteristic
26#10587623—21no—
27D4×Dic17—21no—
28C2×#63228C2211yescharacteristic, complemented
29#10587616C2211yescharacteristic
30#10587610C2211yescharacteristic
31C2×#63223C2211yescharacteristic
32C2×#63219C2211yescharacteristic, complemented
33#10587583C2211yescharacteristic
34#10587581C2211yescharacteristic
35#10587580—21no—
36#10587579—21no—
37#10587578C2211yescharacteristic
38#10587578C2211yescharacteristic
39#10587569—21no—
40#10587568—21no—
41#10587567—21no—
42#10587567—21no—
43C17×#107C2211yescharacteristic, complemented
44C17×#103C2211yescharacteristic, complemented
45C17×#102C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic, complemented
48#10587492C2211yescharacteristic, complemented
49#10587491C2211yescharacteristic
50#10587491C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587489C2211yescharacteristic, complemented
54#10587487C2211yescharacteristic, complemented
55#10587481C2211yescharacteristic, complemented
56#10587480C2211yescharacteristic, complemented
57#10587479C2211yescharacteristic, complemented
58#10587479C2211yescharacteristic, complemented
59C23×D17D411yescharacteristic
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17—21no—
62C2×D4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C22×Dic17—21no—
68C22×Dic17D411yescharacteristic
69#63256—41no—
70#63256—41no—
71#63254—41no—
72#63254—41no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
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—
85M16×C17—41no—
86C2×C8×C17C2311yescharacteristic
87C2×C8×C17C2311yescharacteristic
88C4⋊C4×C17—21no—
89C4⋊C4×C17C2311yescharacteristic
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊Dic17—21no—
94C22⋊Dic17—21no—
95#63231—41no—
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106#63225—21no—
107#63225—21no—
108C4×Dic17—21no—
109C4×Dic17—21no—
110#63223—41no—
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113#63219—41no—
114C23×C17—21no—
115C23×C17C2×D411yescharacteristic
116C22×D17—21no—
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—42no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2411yescharacteristic
134C22⋊D17—42no—
135C22⋊D17—42no—
136C22⋊D17—42no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—41no—
141C2×Dic17—42no—
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144D68—21no—
145D68—21no—
146D68—42no—
147D68—42no—
148C4×D17—21no—
149C4×D17—21no—
150C4×D17—21no—
151C4×D17—21no—
152C4×D17—41no—
153C4×D17—41no—
154C4×D17—42no—
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×C17C22×D411yescharacteristic
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—21no—
172C2×D17—21no—
173C2×D17—21no—
174C2×D17—21no—
175C2×D17—44no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C4×C17—21no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17H(5,𝔽2)11yescharacteristic
184C4×C17H(5,𝔽2)11yescharacteristic
185Dic17—21no—
186Dic17—21no—
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#468D1711yescharacteristic, complemented
200#452—171no—
201#452—171no—
202#449—171no—
203#449—171no—
204#407—171no—
205C2×C17—41no—
206C2×C17—81no—
207C2×C17#53411yescharacteristic
208C2×C17#57511yescharacteristic
209C2×C17#57511yescharacteristic
210D17—41no—
211D17—41no—
212D17—81no—
213C2×#41—171no—
214C2×#41—171no—
215C23×C4—171no—
216C2×Q16—171no—
217C2×SD16—171no—
218C2×SD16C2×D1711yescharacteristic, complemented
219C2×D8—171no—
220C2×M16—171no—
221C2×M16C2×D1711yescharacteristic
222#123—341no—
223#123—341no—
224C22⋊Q8—171no—
225C22⋊Q8—171no—
226C22⋊Q8—171no—
227C22⋊Q8C2×D1711yescharacteristic
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#107C2×D1711yescharacteristic, complemented
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⋊C8—171no—
256C17#319711yescharacteristic, complemented, Sylow
257C24—171no—
258#41—681no—
259#41—681no—
260#41—681no—
261#41—681no—
262C2×Q8—171no—
263C2×Q8C22×D1711yescharacteristic
264C2×D4—21no—
265C2×D4—171no—
266C2×D4—341no—
267C2×D4—341no—
268C2×D4—341no—
269C2×D4—341no—
270C2×D4—341no—
271C2×D4C22×D1711yescharacteristic
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—341no—
278C22×C4—341no—
279C22×C4—681no—
280C22×C4—681no—
281C22×C4C22×D1711yescharacteristic
282Q16—681no—
283SD16—41no—
284SD16—681no—
285D8—681no—
286M16—41no—
287M16—681no—
288C2×C8—171no—
289C2×C8—171no—
290C2×C8C22×D1711yescharacteristic
291C2×C8C22×D1711yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—171no—
294C4⋊C4—171no—
295C4⋊C4—341no—
296C4⋊C4—341no—
297C4⋊C4—341no—
298C4⋊C4C22×D1711yescharacteristic
299C4⋊C4C22×D1711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
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—171no—
314C23—171no—
315C23—341no—
316C23—681no—
317C23—681no—
318C23—681no—
319C23D4×D1711yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—341no—
323Q8—341no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—341no—
328D4—341no—
329D4—682no—
330D4—682no—
331D4—682no—
332D4—682no—
333D4—682no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
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—681no—
349C2×C4—681no—
350C2×C4—682no—
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—341no—
366C22—341no—
367C22—341no—
368C22—341no—
369C22—341no—
370C22—341no—
371C22—684no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22C2×D4×D1711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—341no—
381C4—341no—
382C4—681no—
383C4—681no—
384C4—681no—
385C4#1058769111yescharacteristic
386C4#1058769611yescharacteristic
387C2—41no—
388C2—81no—
389C2—681no—
390C2—681no—
391C2—1361no—
392C2#1176336711yescharacteristic
393C2#1176360511yescharacteristic
394C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#468C2
socleC22×C17C22×D4
hypercenter#468D17
hypocenterC17#3197
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #11763322C2semidirect
2. #11763319C2semidirect
3. #11763152C2semidirect
4. #11763149C2semidirect
5. #11763136C2semidirect
6. #11763131C2semidirect
7. #11763065C2semidirect
8. #11763060C2semidirect
9. #11763055C2semidirect
10. #11762974C2semidirect
11. #11762968C2semidirect
12. #11762955C2semidirect
13. #11762953C2semidirect
14. #11762772C2semidirect
15. C17×#468C2semidirect
19. C2×SD16×C17C22semidirect
24. C2×#63231C22semidirect
28. C2×#63228C22semidirect
32. C2×#63219C22semidirect
43. C17×#107C22semidirect
44. C17×#103C22semidirect
45. C17×#102C22semidirect
46. #10587506C22semidirect
47. #10587503C22semidirect
48. #10587492C22semidirect
53. #10587489C22semidirect
54. #10587487C22semidirect
55. #10587481C22semidirect
56. #10587480C22semidirect
57. #10587479C22semidirect
58. #10587479C22semidirect
199. #468D17semidirect
218. C2×SD16C2×D17semidirect
241. #107C2×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
#117633222
#117633192
#117631522
#117631492
#117631362
#117631312
#117630652
#117630602
#117630552
#117629742
#117629682
#117629552
#117629532
#117627722
C17×#4682
#319717