Finite Group Databasebeta

← #14540859

Subgroups — #14540859

Order 2176.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763455C211yesmaximal, characteristic
2#11763171C211yesmaximal, characteristic, complemented
3#11763144C211yesmaximal, characteristic, complemented
4#11763122C211yesmaximal, characteristic, complemented
5#11763086C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763048C211yesmaximal, characteristic, complemented
8#11763031C211yesmaximal, characteristic, complemented
9#11762969C211yesmaximal, characteristic, complemented
10#11762959C211yesmaximal, characteristic, complemented
11#11762907C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762808C211yesmaximal, characteristic, complemented
14C2×#10587480C211yesmaximal, characteristic, complemented
15C17×#554C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C22×D68C2211yescharacteristic
18C17×C4⋊Q8C2211yescharacteristic, complemented
19C17×#125—21no—
20C17×C22⋊Q8—21no—
21C17×C22⋊Q8—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×C22⋊Q8C2211yescharacteristic, complemented
24C4×Q8×C17C2211yescharacteristic, complemented
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×C4⋊C4×C17C2211yescharacteristic
28C2×C4⋊C4×C17C2211yescharacteristic, complemented
29C2×#63231C2211yescharacteristic, complemented
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32#10587623—21no—
33#10587616C2211yescharacteristic
34#10587610C2211yescharacteristic
35C2×#63223C2211yescharacteristic, complemented
36C22×#3622C2211yescharacteristic
37#10587580—21no—
38#10587570—21no—
39C22⋊C4×D17—21no—
40#10587561C2211yescharacteristic
41C4×D68C2211yescharacteristic
42#10587506C2211yescharacteristic, complemented
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic, complemented
45#10587481C2211yescharacteristic, complemented
46#10587481C2211yescharacteristic, complemented
47#10587481C2211yescharacteristic, complemented
48#10587480C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587478C2211yescharacteristic, complemented
55#10587478C2211yescharacteristic, complemented
56#10587478C2211yescharacteristic, complemented
57C4⋊#3622C2211yescharacteristic, complemented
58#10587474C2211yescharacteristic, complemented
59C23×D17—21no—
60C2×Q8×C17—21no—
61C2×Q8×C17C2311yescharacteristic
62C2×D4×C17D411yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69#63252—22no—
70#63252—41no—
71C2×D68—22no—
72C2×D68—21no—
73C2×D68—41no—
74C2×D68C2311yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×Dic34C2311yescharacteristic
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17D411yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17D411yescharacteristic
92C22⋊C4×C17D411yescharacteristic
93C42×C17—21no—
94C42×C17C2311yescharacteristic
95C22⋊Dic17—21no—
96#63231—41no—
97#63230—22no—
98#63230—41no—
99#63227—21no—
100#63227—21no—
101#63227—21no—
102#63227—21no—
103C4⋊Dic17C2311yescharacteristic
104#63225—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—81no—
118Q8×C17—21no—
119Q8×C17—21no—
120Q8×C17—81no—
121D4×C17—22no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17C2×D411yescharacteristic
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2411yescharacteristic
137C22⋊D17—42no—
138C22⋊D17—81no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141D68—21no—
142D68—21no—
143D68—21no—
144D68—42no—
145D68#4112yes—
146C4×D17—42no—
147C4×D17—81no—
148Dic34—21no—
149Dic34—21no—
150#3622—21no—
151#3622—21no—
152#3622—21no—
153#3622—21no—
154#3314—171nomaximal, Sylow
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17#13712yes—
161C22×C17C22×D411yescharacteristic
162C2×D17—22no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—84no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17C2×#4111yescharacteristic
178C4×C17H(5,𝔽2)11yescharacteristic
179Dic17—41no—
180Dic17—41no—
181#554D1711yescharacteristic, complemented
182#546—171no—
183#489—171no—
184#484—171no—
185#480—171no—
186#474—171no—
187#468—171no—
188#467—171no—
189#464—171no—
190#452—171no—
191#450—171no—
192#440—171no—
193#435—171no—
194#417—171no—
195C2×#102—171no—
196C2×C17—22no—
197C2×C17—41no—
198C2×C17#54611yescharacteristic
199C2×C17C2×#13711yescharacteristic
200C2×C17#57611yescharacteristic
201D17—42no—
202D17—81no—
203C2×#41—171no—
204C22×D4—171no—
205C2×Q16—171no—
206C2×SD16—171no—
207C2×SD16—171no—
208C2×M16—171no—
209C22×C8—171no—
210C4⋊Q8C2×D1711yescharacteristic, complemented
211#125—21no—
212#124—171no—
213#123—341no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—171no—
217C22⋊Q8C2×D1711yescharacteristic
218C22⋊Q8C2×D1711yescharacteristic, complemented
219#121—171no—
220#121—341no—
221#120—341no—
222C4×Q8C2×D1711yescharacteristic, complemented
223C4×D4—21no—
224C4×D4—171no—
225C4×D4C2×D1711yescharacteristic
226C2×C4⋊C4C2×D1711yescharacteristic
227C2×C4⋊C4C2×D1711yescharacteristic, complemented
228C2×C22⋊C4—341no—
229#107—171no—
230#107—171no—
231#107—171no—
232C4⋊C8—171no—
233#103—171no—
234#103—171no—
235#103—171no—
236#103—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240#102—171no—
241#102—171no—
242#102—171no—
243C22⋊C8—171no—
244C22⋊C8—171no—
245#97—171no—
246C17#331411yescharacteristic, complemented, Sylow
247C24—341no—
248#41—342no—
249#41—681no—
250C2×Q8—21no—
251C2×Q8—171no—
252C2×Q8C22×D1711yescharacteristic
253C2×D4—171no—
254C2×D4—171no—
255C2×D4—342no—
256C2×D4—341no—
257C2×D4—341no—
258C2×D4—341no—
259C2×D4—681no—
260C2×D4C22⋊D1711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4C22×D1711yescharacteristic
266C22×C4C22×D1711yescharacteristic
267Q16—681no—
268SD16—342no—
269SD16—681no—
270M16—681no—
271C2×C8—171no—
272C2×C8—171no—
273C2×C8—171no—
274C2×C8—171no—
275C2×C8—681no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—171no—
283C4⋊C4—341no—
284C4⋊C4C22⋊D1711yescharacteristic
285C4⋊C4C22×D1711yescharacteristic
286C4⋊C4C22×D1711yescharacteristic
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—341no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4C22⋊D1711yescharacteristic
296C22⋊C4C22⋊D1711yescharacteristic
297C42—21no—
298C42C22×D1711yescharacteristic
299C23—341no—
300C23—341no—
301C23—682no—
302C23—682no—
303C23—1361no—
304C23C2×C22⋊D1711yescharacteristic
305C23C2×C22⋊D1711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—341no—
310Q8—341no—
311D4—22no—
312D4—172no—
313D4—341no—
314D4—341no—
315D4—341no—
316D4—682no—
317D4—682no—
318D4—1361no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—682no—
332C2×C4—1361no—
333C2×C4C2×C22⋊D1711yescharacteristic
334C2×C4C2×C22⋊D1711yescharacteristic
335C2×C4C2×C22⋊D1711yescharacteristic
336C2×C4C2×C22⋊D1711yescharacteristic
337C2×C4C23×D1711yescharacteristic
338C8—341no—
339C8—341no—
340C8—341no—
341C8—341no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—342no—
348C22—681no—
349C22—681no—
350C22—681no—
351C22—681no—
352C22—681no—
353C22—1364no—
354C22#1058762812yes—
355C22C22×C22⋊D1711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—681no—
365C4—681no—
366C4#41×D1711yescharacteristic
367C4#1058769611yescharacteristic
368C2—22no—
369C2—41no—
370C2—682no—
371C2—1361no—
372C2#1176344711yescharacteristic
373C2C2×#1058762811yescharacteristic
374C2#1176361211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#554C2
socleC22×C17C22×D4
prime radical#11763455C2
hypercenter#554D17
hypocenterC17#3314
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #11763171C2semidirect
3. #11763144C2semidirect
4. #11763122C2semidirect
5. #11763086C2semidirect
6. #11763057C2semidirect
7. #11763048C2semidirect
8. #11763031C2semidirect
9. #11762969C2semidirect
10. #11762959C2semidirect
11. #11762907C2semidirect
12. #11762881C2semidirect
13. #11762808C2semidirect
14. C2×#10587480C2semidirect
15. C17×#554C2semidirect
18. C17×C4⋊Q8C22semidirect
23. C17×C22⋊Q8C22semidirect
24. C4×Q8×C17C22semidirect
28. C2×C4⋊C4×C17C22semidirect
29. C2×#63231C22semidirect
35. C2×#63223C22semidirect
42. #10587506C22semidirect
44. C22⋊#3622C22semidirect
45. #10587481C22semidirect
46. #10587481C22semidirect
47. #10587481C22semidirect
54. #10587478C22semidirect
55. #10587478C22semidirect
56. #10587478C22semidirect
57. C4⋊#3622C22semidirect
58. #10587474C22semidirect
181. #554D17semidirect
210. C4⋊Q8C2×D17semidirect
218. C22⋊Q8C2×D17semidirect
222. C4×Q8C2×D17semidirect
227. C2×C4⋊C4C2×D17semidirect
246. C17#3314semidirect

Maximal subgroups

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

subgroupindex
#117634552
#117631712
#117631442
#117631222
#117630862
#117630572
#117630482
#117630312
#117629692
#117629592
#117629072
#117628812
#117628082
C2×#105874802
C17×#5542
#331417