Finite Group Databasebeta

← #14539785

Subgroups — #14539785

Order 2176.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763363C211yesmaximal, characteristic, complemented
2#124×D17C211yesmaximal, characteristic
3#11763202C211yesmaximal, characteristic, complemented
4#11763200C211yesmaximal, characteristic, complemented
5#11763170C211yesmaximal, characteristic, complemented
6#11763167C211yesmaximal, characteristic, complemented
7#11763032C211yesmaximal, characteristic, complemented
8#11763031C211yesmaximal, characteristic, complemented
9#11763028C211yesmaximal, characteristic, complemented
10#11762966C211yesmaximal, characteristic, complemented
11#11762963C211yesmaximal, characteristic, complemented
12#11762961C211yesmaximal, characteristic
13#11762958C211yesmaximal, characteristic
14#11762865C211yesmaximal, characteristic, complemented
15C17×#464C211yesmaximal, characteristic, complemented
16C2×Q8×D17C2211yescharacteristic
17C2×D4×D17C2211yescharacteristic
18C2×Q16×C17C2211yescharacteristic, complemented
19C2×SD16×C17C2211yescharacteristic
20C17×#124C2211yescharacteristic
21C4×Q8×C17C2211yescharacteristic, complemented
22#10587633C2211yescharacteristic
23#10587632C2211yescharacteristic
24#10587630C2211yescharacteristic, complemented
25C2×#63231C2211yescharacteristic, complemented
26C2×#63231C2211yescharacteristic, complemented
27C2×#63230C2211yescharacteristic
28#10587622C2211yescharacteristic
29C2×#63229C2211yescharacteristic
30C2×Dic68C2211yescharacteristic, complemented
31C2×#63219C2211yescharacteristic
32#10587583C2211yescharacteristic
33#10587582—21no—
34#10587575—21no—
35#10587574C2211yescharacteristic, complemented
36#10587571—21no—
37#10587571—21no—
38C22⋊C4×D17—21no—
39C22⋊C4×D17—21no—
40#10587561C2211yescharacteristic
41#10587558—21no—
42C42×D17C2211yescharacteristic
43C4×Dic34—21no—
44C4×Dic34C2211yescharacteristic, complemented
45C17×C4⋊C8C2211yescharacteristic, complemented
46C17×#103C2211yescharacteristic, complemented
47C17×#102C2211yescharacteristic
48#10587506C2211yescharacteristic, complemented
49#10587503C2211yescharacteristic
50#10587492C2211yescharacteristic
51#10587491C2211yescharacteristic
52#10587491C2211yescharacteristic
53#10587487C2211yescharacteristic, complemented
54#10587486C2211yescharacteristic, complemented
55#10587486C2211yescharacteristic, complemented
56#10587481C2211yescharacteristic, complemented
57#10587480C2211yescharacteristic
58C4⋊#3622C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17C2311yescharacteristic
63C2×C22⋊D17—21no—
64Q8×D17—41no—
65Q8×D17—41no—
66D4×D17—41no—
67D4×D17—41no—
68C2×D68C2311yescharacteristic
69C2×C4×D17—21no—
70C2×C4×D17D411yescharacteristic
71C2×C4×D17D411yescharacteristic
72C2×C4×D17C2311yescharacteristic
73C2×Dic34—21no—
74C2×Dic34—21no—
75C2×Dic34C2311yescharacteristic
76C2×Dic34C2311yescharacteristic
77Q16×C17—41no—
78SD16×C17—41no—
79C2×C8×C17C2311yescharacteristic
80C2×C8×C17C2311yescharacteristic
81C4⋊C4×C17—21no—
82C4⋊C4×C17C2311yescharacteristic
83C22⋊C4×C17—21no—
84C22⋊C4×C17—21no—
85C42×C17—21no—
86C42×C17C2311yescharacteristic
87C22⋊Dic17—21no—
88C22⋊Dic17—21no—
89#63231—41no—
90#63231—41no—
91#63230—41no—
92#63229—41no—
93#63227—21no—
94#63227—21no—
95#63227—21no—
96#63227—21no—
97#63227—21no—
98#63227—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17C2311yescharacteristic
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225—21no—
105#63225—21no—
106C4×Dic17—21no—
107C4×Dic17D411yescharacteristic
108C4×Dic17D411yescharacteristic
109C4×Dic17C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112Dic68—41no—
113#63219—41no—
114C23×C17—21no—
115C22×D17—21no—
116C22×D17—42no—
117C22×D17—42no—
118C22×D17—42no—
119C22×D17C2×D411yescharacteristic
120Q8×C17—21no—
121Q8×C17—21no—
122Q8×C17—21no—
123Q8×C17—21no—
124D4×C17—21no—
125D4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—42no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17C2×D411yescharacteristic
137C2×Dic17C2×D411yescharacteristic
138C2×Dic17C2×D411yescharacteristic
139D68—21no—
140D68—21no—
141C4×D17—22no—
142C4×D17—21no—
143C4×D17—21no—
144C4×D17—41no—
145C4×D17—42no—
146Dic34—21no—
147Dic34—21no—
148Dic34—21no—
149Dic34—21no—
150Dic34—42no—
151Dic34—42no—
152C8×C17—21no—
153C8×C17—21no—
154#3622—21no—
155#3622—21no—
156#3215—171nomaximal, Sylow
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17C22×D411yescharacteristic
161C2×D17—21no—
162C2×D17—21no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—84no—
167C2×D17#13712yes—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17H(5,𝔽2)11yescharacteristic
174C4×C17H(5,𝔽2)11yescharacteristic
175Dic17—21no—
176Dic17—21no—
177Dic17—21no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181#533—171no—
182C2×#124—171no—
183#497—171no—
184#497—171no—
185#489—171no—
186#489—171no—
187#464—171no—
188#464—171no—
189#464—171no—
190#464D1711yescharacteristic, complemented
191#451—171no—
192#451—171no—
193#450—171no—
194#450—171no—
195#432—171no—
196C2×C17—81no—
197C2×C17#53511yescharacteristic
198C2×C17C2×#13711yescharacteristic
199C2×C17#57511yescharacteristic
200D17—22no—
201D17—81no—
202C22×Q8—171no—
203C22×D4—171no—
204C2×Q16—171no—
205C2×Q16—171no—
206C2×Q16—171no—
207C2×Q16C2×D1711yescharacteristic, complemented
208C2×SD16—171no—
209C2×SD16—171no—
210C2×SD16—171no—
211C2×SD16C2×D1711yescharacteristic
212C4⋊Q8—171no—
213C4⋊Q8—171no—
214#125—341no—
215#124—171no—
216#124—171no—
217#124—171no—
218#124—341no—
219#124—341no—
220#124C2×D1711yescharacteristic
221C22⋊Q8—171no—
222C22⋊Q8—171no—
223C22⋊Q8—341no—
224C4×Q8—171no—
225C4×Q8—341no—
226C4×Q8C2×D1711yescharacteristic, complemented
227#117—341no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C42—171no—
231C4⋊C8—171no—
232C4⋊C8C2×D1711yescharacteristic, complemented
233#103—171no—
234#103—171no—
235#103—171no—
236#103C2×D1711yescharacteristic, complemented
237#102—171no—
238#102—171no—
239#102—171no—
240#102C2×D1711yescharacteristic
241C22⋊C8—171no—
242C22⋊C8—171no—
243#97—171no—
244#97—171no—
245C17#321511yescharacteristic, complemented, Sylow
246C24—341no—
247C2×Q8—171no—
248C2×Q8—171no—
249C2×Q8—341no—
250C2×Q8—341no—
251C2×Q8—681no—
252C2×Q8—681no—
253C2×Q8C22×D1711yescharacteristic
254C2×Q8C22×D1711yescharacteristic
255C2×D4—171no—
256C2×D4—341no—
257C2×D4—681no—
258C2×D4—681no—
259C2×D4C22×D1711yescharacteristic
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—341no—
264Q16—41no—
265Q16—681no—
266Q16—681no—
267Q16—681no—
268SD16—41no—
269SD16—681no—
270SD16—681no—
271SD16—681no—
272C2×C8—171no—
273C2×C8—171no—
274C2×C8C22×D1711yescharacteristic
275C2×C8C22×D1711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—171no—
278C4⋊C4—341no—
279C4⋊C4—341no—
280C4⋊C4—341no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—341no—
284C4⋊C4C22×D1711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—341no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C42—21no—
296C42—171no—
297C42—171no—
298C42—171no—
299C42—341no—
300C42C22×D1711yescharacteristic
301C23—21no—
302C23—171no—
303C23—341no—
304C23—682no—
305C23—682no—
306C23—682no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—341no—
312Q8—341no—
313Q8—341no—
314Q8—341no—
315Q8—682no—
316Q8—682no—
317D4—21no—
318D4—21no—
319D4—341no—
320D4—341no—
321D4—682no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—171no—
326C2×C4—171no—
327C2×C4—171no—
328C2×C4—342no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—681no—
335C2×C4—682no—
336C2×C4D4×D1711yescharacteristic
337C2×C4D4×D1711yescharacteristic
338C2×C4C23×D1711yescharacteristic
339C8—21no—
340C8—21no—
341C8—341no—
342C8—341no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—172no—
347C22—341no—
348C22—341no—
349C22—681no—
350C22—681no—
351C22—681no—
352C22—1364no—
353C22C2×D4×D1711yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—341no—
360C4—341no—
361C4—341no—
362C4—681no—
363C4—681no—
364C4—681no—
365C4#1058769111yescharacteristic
366C4#1058769611yescharacteristic
367C2—81no—
368C2—342no—
369C2—1361no—
370C2#1176337511yescharacteristic
371C2#137×D1711yescharacteristic
372C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#464C2
socleC22×C17C22×D4
prime radicalC2×D4×D17C22
hypercenter#464D17
hypocenterC17#3215
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #11763363C2semidirect
3. #11763202C2semidirect
4. #11763200C2semidirect
5. #11763170C2semidirect
6. #11763167C2semidirect
7. #11763032C2semidirect
8. #11763031C2semidirect
9. #11763028C2semidirect
10. #11762966C2semidirect
11. #11762963C2semidirect
14. #11762865C2semidirect
15. C17×#464C2semidirect
18. C2×Q16×C17C22semidirect
21. C4×Q8×C17C22semidirect
24. #10587630C22semidirect
25. C2×#63231C22semidirect
26. C2×#63231C22semidirect
30. C2×Dic68C22semidirect
35. #10587574C22semidirect
44. C4×Dic34C22semidirect
45. C17×C4⋊C8C22semidirect
46. C17×#103C22semidirect
48. #10587506C22semidirect
53. #10587487C22semidirect
54. #10587486C22semidirect
55. #10587486C22semidirect
56. #10587481C22semidirect
58. C4⋊#3622C22semidirect
190. #464D17semidirect
207. C2×Q16C2×D17semidirect
226. C4×Q8C2×D17semidirect
232. C4⋊C8C2×D17semidirect
236. #103C2×D17semidirect
245. C17#3215semidirect

Maximal subgroups

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

subgroupindex
#117633632
#124×D172
#117632022
#117632002
#117631702
#117631672
#117630322
#117630312
#117630282
#117629662
#117629632
#117629612
#117629582
#117628652
C17×#4642
#321517