Finite Group Databasebeta

← #14539636

Subgroups — #14539636

Order 2176.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763387C211yesmaximal, characteristic, complemented
2#11763150C211yesmaximal, characteristic, complemented
3#11763133C211yesmaximal, characteristic, complemented
4#11763125C211yesmaximal, characteristic, complemented
5#11763120C211yesmaximal, characteristic, complemented
6#11763060C211yesmaximal, characteristic, complemented
7#11763053C211yesmaximal, characteristic, complemented
8#11763043C211yesmaximal, characteristic, complemented
9#11763036C211yesmaximal, characteristic, complemented
10#11762960C211yesmaximal, characteristic, complemented
11#11762952C211yesmaximal, characteristic, complemented
12#11762952C211yesmaximal, characteristic, complemented
13#11762943C211yesmaximal, characteristic, complemented
14#11762776C211yesmaximal, characteristic, complemented
15C17×#535C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C2×#63252C2211yescharacteristic, complemented
19C17×#127—21no—
20C17×#124—21no—
21C17×#121C2211yescharacteristic
22C17×#121C2211yescharacteristic
23C17×#121C2211yescharacteristic
24C17×#121C2211yescharacteristic
25C17×#120—21no—
26C17×#120—21no—
27C17×#117C2211yescharacteristic
28#10587632—21no—
29#10587625—21no—
30#10587625—21no—
31#10587624—21no—
32C2×#63229C2211yescharacteristic
33C2×#63229C2211yescharacteristic
34C2×#63228C2211yescharacteristic
35C2×#63228C2211yescharacteristic
36#10587616C2211yescharacteristic
37#10587616C2211yescharacteristic, complemented
38C2×C4⋊Dic17C2211yescharacteristic, complemented
39#10587610C2211yescharacteristic
40#10587610C2211yescharacteristic, complemented
41C2×#63223C2211yescharacteristic, complemented
42C22×#3622C2211yescharacteristic, complemented
43#10587503C2211yescharacteristic
44#10587503C2211yescharacteristic
45#10587503C2211yescharacteristic
46#10587503C2211yescharacteristic
47C22⋊#3622C2211yescharacteristic
48C22⋊#3622C2211yescharacteristic
49C22⋊#3622C2211yescharacteristic, complemented
50C22⋊#3622C2211yescharacteristic, complemented
51#10587481C2211yescharacteristic, complemented
52#10587481C2211yescharacteristic, complemented
53#10587480C2211yescharacteristic, complemented
54#10587480C2211yescharacteristic, complemented
55#10587479C2211yescharacteristic, complemented
56#10587479C2211yescharacteristic, complemented
57#10587478C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C24×C17—21no—
60#41×C17—41no—
61#41×C17—41no—
62C2×Q8×C17D411yescharacteristic
63C2×D4×C17—21no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—41no—
69C2×D4×C17—41no—
70C2×D4×C17D411yescharacteristic
71C2×D4×C17D411yescharacteristic
72C2×D4×C17D411yescharacteristic
73C2×D4×C17C2311yescharacteristic
74C2×D4×C17C2311yescharacteristic
75C22×C4×C17C2311yescharacteristic
76C22×C4×C17C2311yescharacteristic
77C22×C4×C17C2311yescharacteristic
78C2×C22⋊D17—21no—
79C2×C22⋊D17—21no—
80C2×C22⋊D17—21no—
81C22×Dic17—21no—
82#63252—41no—
83#63252—41no—
84C2×D68C2311yescharacteristic
85C2×C4×D17—21no—
86C2×Dic34C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊C4×C17—21no—
93C22⋊C4×C17—21no—
94C42×C17—21no—
95C22⋊Dic17—21no—
96C22⋊Dic17—21no—
97#63229—41no—
98#63229—41no—
99#63228—41no—
100#63228—41no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106#63225—21no—
107#63225—21no—
108#63223—41no—
109C2×#3622—41no—
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C2×#3622C2311yescharacteristic
113C2×#3622C2311yescharacteristic
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—42no—
117C23×C17—81no—
118C23×C17—81no—
119C23×C17C2×D411yescharacteristic
120C23×C17C2×D411yescharacteristic
121C23×C17C2×D411yescharacteristic
122C22×D17—21no—
123Q8×C17—41no—
124D4×C17—21no—
125D4×C17—21no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—41no—
129D4×C17—41no—
130D4×C17—41no—
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—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17C2×D411yescharacteristic
145C2×C4×C17C2×D411yescharacteristic
146C2×C4×C17C2×D411yescharacteristic
147C2×C4×C17C2411yescharacteristic
148C22⋊D17—42no—
149C22⋊D17—81no—
150C22⋊D17—81no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—81no—
155D68—21no—
156D68—21no—
157C4×D17—42no—
158Dic34—21no—
159Dic34—21no—
160#3622—21no—
161#3622—21no—
162#3622—21no—
163#3622—21no—
164#3198—171nomaximal, Sylow
165C22×C17—21no—
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—21no—
173C22×C17—21no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—84no—
181C22×C17C22×D411yescharacteristic
182C2×D17—41no—
183C2×D17—41no—
184C2×D17—41no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17H(5,𝔽2)11yescharacteristic
191C4×C17H(5,𝔽2)11yescharacteristic
192Dic17—41no—
193Dic17—41no—
194Dic17—41no—
195#537—171no—
196#535D1711yescharacteristic, complemented
197#485—171no—
198#482—171no—
199#481—171no—
200#480—171no—
201#469—171no—
202#468—171no—
203#466—171no—
204#465—171no—
205#450—171no—
206#449—171no—
207#449—171no—
208#447—171no—
209#408—171no—
210C2×C17—41no—
211C2×C17—41no—
212C2×C17—41no—
213C2×C17—81no—
214C2×C17—81no—
215C2×C17#53411yescharacteristic
216C2×C17#57611yescharacteristic
217C2×C17#57711yescharacteristic
218D17—81no—
219C2×#41—171no—
220C2×#41C2×D1711yescharacteristic
221C22×D4C2×D1711yescharacteristic
222C2×SD16—171no—
223C2×SD16—171no—
224C2×D8—171no—
225C2×D8—171no—
226C2×M16—171no—
227C22×C8—171no—
228#127—21no—
229#124—21no—
230C22⋊Q8—171no—
231C22⋊Q8—171no—
232C22⋊Q8—341no—
233#121—171no—
234#121—171no—
235#121—341no—
236#121—341no—
237#121—341no—
238#121C2×D1711yescharacteristic
239#121C2×D1711yescharacteristic
240#121C2×D1711yescharacteristic
241#121C2×D1711yescharacteristic
242#120—21no—
243#120—21no—
244#117C2×D1711yescharacteristic
245C2×C4⋊C4—171no—
246#107—171no—
247#107—171no—
248#106—171no—
249#106—171no—
250#103—171no—
251#103—171no—
252#102—171no—
253#102—171no—
254#102—171no—
255#102—171no—
256#102—171no—
257#102—171no—
258C22⋊C8—171no—
259C22⋊C8—171no—
260C22⋊C8—171no—
261C22⋊C8—171no—
262C17#319811yescharacteristic, complemented, Sylow
263C24—21no—
264#41—41no—
265#41—41no—
266#41—681no—
267#41—681no—
268C2×Q8—171no—
269C2×Q8C22⋊D1711yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—171no—
278C2×D4—341no—
279C2×D4—341no—
280C2×D4—341no—
281C2×D4C22⋊D1711yescharacteristic
282C2×D4C22⋊D1711yescharacteristic
283C2×D4C22⋊D1711yescharacteristic
284C2×D4C22×D1711yescharacteristic
285C2×D4C22×D1711yescharacteristic
286C22×C4—341no—
287C22×C4—341no—
288C22×C4C22×D1711yescharacteristic
289C22×C4C22×D1711yescharacteristic
290C22×C4C22×D1711yescharacteristic
291SD16—681no—
292SD16—681no—
293D8—681no—
294D8—681no—
295M16—681no—
296C2×C8—171no—
297C2×C8—171no—
298C2×C8—171no—
299C2×C8—171no—
300C2×C8—681no—
301C4⋊C4—171no—
302C4⋊C4—171no—
303C4⋊C4—341no—
304C4⋊C4—341no—
305C4⋊C4—341no—
306C4⋊C4C22×D1711yescharacteristic
307C4⋊C4C22×D1711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—341no—
314C22⋊C4—341no—
315C22⋊C4—341no—
316C22⋊C4—341no—
317C42—21no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—341no—
324C23C2×C22⋊D1711yescharacteristic
325C23C2×C22⋊D1711yescharacteristic
326C23C2×C22⋊D1711yescharacteristic
327Q8—41no—
328Q8—341no—
329Q8—341no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—42no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—341no—
343D4—341no—
344D4—682no—
345D4—1361no—
346D4—1361no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—682no—
359C2×C4—1361no—
360C2×C4C2×C22⋊D1711yescharacteristic
361C2×C4C2×C22⋊D1711yescharacteristic
362C2×C4C2×C22⋊D1711yescharacteristic
363C2×C4C23×D1711yescharacteristic
364C8—341no—
365C8—341no—
366C8—341no—
367C8—341no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—84no—
384C22—681no—
385C22—681no—
386C22—681no—
387C22C22×C22⋊D1711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—681no—
394C4—681no—
395C4—681no—
396C4#1058769111yescharacteristic
397C4#1058769111yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—81no—
403C2—1361no—
404C2#1176336811yescharacteristic
405C2#1176361211yescharacteristic
406C2#1176362011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#535C2
socleC22×C17C22×D4
hypercenter#535D17
hypocenterC17#3198
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #11763387C2semidirect
2. #11763150C2semidirect
3. #11763133C2semidirect
4. #11763125C2semidirect
5. #11763120C2semidirect
6. #11763060C2semidirect
7. #11763053C2semidirect
8. #11763043C2semidirect
9. #11763036C2semidirect
10. #11762960C2semidirect
11. #11762952C2semidirect
12. #11762952C2semidirect
13. #11762943C2semidirect
14. #11762776C2semidirect
15. C17×#535C2semidirect
18. C2×#63252C22semidirect
37. #10587616C22semidirect
38. C2×C4⋊Dic17C22semidirect
40. #10587610C22semidirect
41. C2×#63223C22semidirect
42. C22×#3622C22semidirect
49. C22⋊#3622C22semidirect
50. C22⋊#3622C22semidirect
51. #10587481C22semidirect
52. #10587481C22semidirect
53. #10587480C22semidirect
54. #10587480C22semidirect
55. #10587479C22semidirect
56. #10587479C22semidirect
57. #10587478C22semidirect
58. #10587478C22semidirect
196. #535D17semidirect
262. C17#3198semidirect

Maximal subgroups

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

subgroupindex
#117633872
#117631502
#117631332
#117631252
#117631202
#117630602
#117630532
#117630432
#117630362
#117629602
#117629522
#117629432
#117627762
C17×#5352
#319817