Finite Group Databasebeta

← #14539878

Subgroups — #14539878

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763430C211yesmaximal, characteristic, complemented
2#11763196C211yesmaximal, characteristic, complemented
3#11763191C211yesmaximal, characteristic, complemented
4#11763168C211yesmaximal, characteristic, complemented
5#11763155C211yesmaximal, characteristic, complemented
6#11763028C211yesmaximal, characteristic, complemented
7#11763021C211yesmaximal, characteristic, complemented
8#11763006C211yesmaximal, characteristic, complemented
9#11763001C211yesmaximal, characteristic, complemented
10#11762971C211yesmaximal, characteristic, complemented
11#11762960C211yesmaximal, characteristic, complemented
12#11762954C211yesmaximal, characteristic, complemented
13#11762943C211yesmaximal, characteristic, complemented
14#11762870C211yesmaximal, characteristic, complemented
15C17×#535C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C17×#127C2211yescharacteristic
19C17×#127C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C17×#124C2211yescharacteristic
22C17×#121—21no—
23C17×#121—21no—
24C17×#120—21no—
25C17×#120—21no—
26C17×#117C2211yescharacteristic
27C2×#63231C2211yescharacteristic, complemented
28C2×#63230C2211yescharacteristic, complemented
29C2×#63229C2211yescharacteristic
30C2×#63229C2211yescharacteristic
31C2×#63229C2211yescharacteristic, complemented
32C2×#63228C2211yescharacteristic
33C2×#63228C2211yescharacteristic
34C2×#63228C2211yescharacteristic, complemented
35#10587616C2211yescharacteristic
36#10587610C2211yescharacteristic
37#10587584—21no—
38#10587582—21no—
39#10587571—21no—
40#10587569—21no—
41#10587561C2211yescharacteristic, complemented
42C4×D68C2211yescharacteristic, complemented
43#10587555C2211yescharacteristic, complemented
44C4×Dic34C2211yescharacteristic, complemented
45#10587506C2211yescharacteristic, complemented
46#10587506C2211yescharacteristic, complemented
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51#10587503C2211yescharacteristic, complemented
52#10587503C2211yescharacteristic, complemented
53C22⋊#3622C2211yescharacteristic
54C22⋊#3622C2211yescharacteristic
55C4⋊#3622C2211yescharacteristic, complemented
56C4⋊#3622C2211yescharacteristic, complemented
57#10587474C2211yescharacteristic, complemented
58C4×#3622C2211yescharacteristic, complemented
59C24×C17—21no—
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17C2311yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—41no—
70C2×D4×C17—41no—
71C2×D4×C17C2311yescharacteristic
72C2×D4×C17C2311yescharacteristic
73C2×D4×C17C2311yescharacteristic
74C22×C4×C17—21no—
75C22×C4×C17C2311yescharacteristic
76C2×C22⋊D17—21no—
77C2×D68C2311yescharacteristic
78C2×C4×D17—21no—
79C2×Dic34—21no—
80C2×Dic34C2311yescharacteristic
81C4⋊C4×C17D411yescharacteristic
82C4⋊C4×C17D411yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C22⋊C4×C17—21no—
86C22⋊C4×C17—21no—
87C22⋊C4×C17D411yescharacteristic
88C22⋊C4×C17D411yescharacteristic
89C42×C17C2311yescharacteristic
90C42×C17C2311yescharacteristic
91C22⋊Dic17—21no—
92#63231—41no—
93#63230—41no—
94#63229—41no—
95#63229—41no—
96#63229—41no—
97#63228—41no—
98#63228—41no—
99#63228—41no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103#63227—21no—
104C4⋊Dic17—21no—
105C4⋊Dic17C2311yescharacteristic
106C4⋊Dic17C2311yescharacteristic
107#63225—21no—
108#63225—21no—
109C4×Dic17—21no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—42no—
118C23×C17—81no—
119C23×C17—81no—
120C23×C17C2×D411yescharacteristic
121C22×D17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—21no—
129D4×C17—21no—
130D4×C17—42no—
131D4×C17—42no—
132D4×C17—81no—
133D4×C17—81no—
134D4×C17—81no—
135D4×C17—81no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—42no—
140C2×C4×C17C2×D411yescharacteristic
141C2×C4×C17C2×D411yescharacteristic
142C2×C4×C17C2×D411yescharacteristic
143C2×C4×C17C2×D411yescharacteristic
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2411yescharacteristic
146C22⋊D17—81no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150D68—21no—
151D68—21no—
152C4×D17—81no—
153Dic34—21no—
154Dic34—21no—
155Dic34—81no—
156#3622—21no—
157#3622—21no—
158#3622—21no—
159#3622—21no—
160#3224—171nomaximal, Sylow
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—84no—
174C22×C17C22×D411yescharacteristic
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—41no—
184C4×C17H(5,𝔽2)11yescharacteristic
185C4×C17H(5,𝔽2)11yescharacteristic
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189#542—171no—
190#535D1711yescharacteristic, complemented
191#496—171no—
192#495—171no—
193#489—171no—
194#486—171no—
195#464—171no—
196#463—171no—
197#460—171no—
198#459—171no—
199#452—171no—
200#450—171no—
201#449—171no—
202#447—171no—
203#433—171no—
204C2×C17—41no—
205C2×C17—81no—
206C2×C17—81no—
207C2×C17—81no—
208C2×C17#53511yescharacteristic
209C2×C17#57611yescharacteristic
210C2×C17#57711yescharacteristic
211D17—81no—
212C2×#41C2×D1711yescharacteristic
213C22×D4C2×D1711yescharacteristic
214C2×Q16—171no—
215C2×SD16—171no—
216C2×SD16—171no—
217C2×SD16—171no—
218C2×SD16—171no—
219C2×D8—171no—
220C2×D8—171no—
221C2×D8—171no—
222C4⋊Q8—171no—
223#127C2×D1711yescharacteristic
224#127C2×D1711yescharacteristic
225#126—341no—
226#124—171no—
227#124—341no—
228#124C2×D1711yescharacteristic
229#124C2×D1711yescharacteristic
230#123—341no—
231C22⋊Q8—171no—
232C22⋊Q8—341no—
233#121—21no—
234#121—21no—
235#121—171no—
236#120—21no—
237#120—21no—
238C4×Q8—171no—
239C4×D4—171no—
240#117C2×D1711yescharacteristic
241C4⋊C8—171no—
242C4⋊C8—171no—
243#103—171no—
244#103—171no—
245#102—171no—
246#102—171no—
247#102—171no—
248#102—171no—
249#102—171no—
250#102—171no—
251C22⋊C8—171no—
252C22⋊C8—171no—
253#97—171no—
254C4×C8—171no—
255C17#322411yescharacteristic, complemented, Sylow
256C24—21no—
257#41—41no—
258#41—41no—
259C2×Q8—171no—
260C2×Q8—341no—
261C2×Q8C22×D1711yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—171no—
271C2×D4—341no—
272C2×D4C22×D1711yescharacteristic
273C2×D4C22×D1711yescharacteristic
274C2×D4C22×D1711yescharacteristic
275C22×C4—21no—
276C22×C4—341no—
277C22×C4C22×D1711yescharacteristic
278Q16—681no—
279SD16—681no—
280SD16—681no—
281SD16—681no—
282SD16—681no—
283D8—681no—
284D8—681no—
285D8—681no—
286C2×C8—171no—
287C2×C8—171no—
288C2×C8—171no—
289C2×C8—171no—
290C4⋊C4—171no—
291C4⋊C4—171no—
292C4⋊C4—341no—
293C4⋊C4—341no—
294C4⋊C4—341no—
295C4⋊C4C22⋊D1711yescharacteristic
296C4⋊C4C22⋊D1711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—341no—
302C22⋊C4—341no—
303C22⋊C4—341no—
304C22⋊C4—341no—
305C22⋊C4—341no—
306C22⋊C4C22⋊D1711yescharacteristic
307C22⋊C4C22⋊D1711yescharacteristic
308C42—341no—
309C42C22×D1711yescharacteristic
310C42C22×D1711yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—341no—
318C23C2×C22⋊D1711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—341no—
322Q8—341no—
323Q8—1361no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—341no—
337D4—341no—
338D4—1361no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—1361no—
347C2×C4C2×C22⋊D1711yescharacteristic
348C2×C4C2×C22⋊D1711yescharacteristic
349C2×C4C2×C22⋊D1711yescharacteristic
350C2×C4C2×C22⋊D1711yescharacteristic
351C2×C4C2×C22⋊D1711yescharacteristic
352C2×C4C23×D1711yescharacteristic
353C8—341no—
354C8—341no—
355C8—341no—
356C8—341no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—84no—
370C22—681no—
371C22—681no—
372C22—681no—
373C22C22×C22⋊D1711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—681no—
381C4—681no—
382C4—681no—
383C4#1058769111yescharacteristic
384C4#1058769111yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—1361no—
390C2#1176337611yescharacteristic
391C2#1176361211yescharacteristic
392C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#535C2
socleC22×C17C22×D4
hypercenter#535D17
hypocenterC17#3224
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #11763430C2semidirect
2. #11763196C2semidirect
3. #11763191C2semidirect
4. #11763168C2semidirect
5. #11763155C2semidirect
6. #11763028C2semidirect
7. #11763021C2semidirect
8. #11763006C2semidirect
9. #11763001C2semidirect
10. #11762971C2semidirect
11. #11762960C2semidirect
12. #11762954C2semidirect
13. #11762943C2semidirect
14. #11762870C2semidirect
15. C17×#535C2semidirect
27. C2×#63231C22semidirect
28. C2×#63230C22semidirect
31. C2×#63229C22semidirect
34. C2×#63228C22semidirect
41. #10587561C22semidirect
42. C4×D68C22semidirect
43. #10587555C22semidirect
44. C4×Dic34C22semidirect
45. #10587506C22semidirect
46. #10587506C22semidirect
51. #10587503C22semidirect
52. #10587503C22semidirect
55. C4⋊#3622C22semidirect
56. C4⋊#3622C22semidirect
57. #10587474C22semidirect
58. C4×#3622C22semidirect
190. #535D17semidirect
255. C17#3224semidirect

Maximal subgroups

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

subgroupindex
#117634302
#117631962
#117631912
#117631682
#117631552
#117630282
#117630212
#117630062
#117630012
#117629712
#117629602
#117629542
#117629432
#117628702
C17×#5352
#322417