Finite Group Databasebeta

← #14540869

Subgroups — #14540869

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#11763517C211yesmaximal, characteristic, complemented
2#11763453C211yesmaximal, characteristic
3#11763167C211yesmaximal, characteristic, complemented
4#11763145C211yesmaximal, characteristic, complemented
5#11763121C211yesmaximal, characteristic, complemented
6#11763087C211yesmaximal, characteristic, complemented
7#11763052C211yesmaximal, characteristic, complemented
8#11763048C211yesmaximal, characteristic, complemented
9#11763029C211yesmaximal, characteristic, complemented
10#11762968C211yesmaximal, characteristic, complemented
11#11762910C211yesmaximal, characteristic, complemented
12#11762877C211yesmaximal, characteristic, complemented
13#11762807C211yesmaximal, characteristic, complemented
14C2×#10587503C211yesmaximal, characteristic, complemented
15C17×#450C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C2×#63254C2211yescharacteristic
18C2×SD16×C17C2211yescharacteristic
19C2×SD16×C17C2211yescharacteristic
20C17×C22⋊Q8C2211yescharacteristic
21#10587639—21no—
22C2×C22⋊Dic17—21no—
23#10587632C2211yescharacteristic, complemented
24Q8×Dic17C2211yescharacteristic, complemented
25C2×#63231C2211yescharacteristic, complemented
26#10587625—21no—
27#10587624C2211yescharacteristic
28#10587622C2211yescharacteristic
29#10587621—21no—
30D4×Dic17C2211yescharacteristic
31C4×C22⋊D17C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33#10587610C2211yescharacteristic
34C22×#3622C2211yescharacteristic
35C2×#63218C2211yescharacteristic, complemented
36#10587583—21no—
37C4⋊C4×D17C2211yescharacteristic, complemented
38#10587575—21no—
39#10587574C2211yescharacteristic, complemented
40#10587567—21no—
41#10587564—21no—
42C17×#102C2211yescharacteristic
43C17×#102C2211yescharacteristic
44C17×C22⋊C8C2211yescharacteristic
45#10587506C2211yescharacteristic, complemented
46#10587503C2211yescharacteristic
47#10587503C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587491C2211yescharacteristic, complemented
51#10587489C2211yescharacteristic, complemented
52#10587489C2211yescharacteristic, complemented
53#10587487C2211yescharacteristic, complemented
54#10587487C2211yescharacteristic, complemented
55#10587486C2211yescharacteristic, complemented
56#10587485C2211yescharacteristic, complemented
57#10587481C2211yescharacteristic, complemented
58#10587478C2211yescharacteristic, complemented
59C24×C17—21no—
60C2×Q8×C17C2311yescharacteristic
61C2×D4×C17—22no—
62C2×D4×C17—21no—
63C2×D4×C17—41no—
64C2×D4×C17C2311yescharacteristic
65C2×D4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C2×C22⋊D17—21no—
68C2×C22⋊D17—21no—
69C2×C22⋊D17D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73#63254—22no—
74#63254—41no—
75C2×C4×D17—21no—
76C2×C4×D17C2311yescharacteristic
77C2×Dic34—21no—
78C2×Dic34C2311yescharacteristic
79SD16×C17—22no—
80SD16×C17—41no—
81C2×C8×C17C2311yescharacteristic
82C2×C8×C17C2311yescharacteristic
83C4⋊C4×C17—21no—
84C4⋊C4×C17C2311yescharacteristic
85C22⋊C4×C17—21no—
86C22⋊Dic17—21no—
87C22⋊Dic17—21no—
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91C22⋊Dic17D411yescharacteristic
92#63231—41no—
93#63227—21no—
94#63227D411yescharacteristic
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17C2311yescharacteristic
98C4⋊Dic17C2311yescharacteristic
99C4⋊Dic17C2311yescharacteristic
100#63225—21no—
101#63225—21no—
102#63225—21no—
103#63225—21no—
104#63225D411yescharacteristic
105C4×Dic17—21no—
106C4×Dic17C2311yescharacteristic
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110#63218—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—81no—
116C23×C17C2×D411yescharacteristic
117C22×D17C2×D411yescharacteristic
118Q8×C17—21no—
119Q8×C17—21no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17—21no—
123D4×C17—42no—
124D4×C17#4112yes—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2411yescharacteristic
131C22⋊D17—22no—
132C22⋊D17—42no—
133C22⋊D17—81no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—42no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17C2×D411yescharacteristic
142C2×Dic17C2×D411yescharacteristic
143C2×Dic17C2×D411yescharacteristic
144C4×D17—21no—
145C4×D17—21no—
146C4×D17—81no—
147Dic34—21no—
148Dic34—21no—
149Dic34—81no—
150C8×C17—21no—
151C8×C17—21no—
152#3622—21no—
153#3622—21no—
154#3314—171nomaximal, Sylow
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—22no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—84no—
164C22×C17#13712yes—
165C22×C17C22×D411yescharacteristic
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17C2×#4111yescharacteristic
173C4×C17H(5,𝔽2)11yescharacteristic
174Dic17—21no—
175Dic17—21no—
176Dic17—21no—
177Dic17—41no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181#554—171no—
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#450D1711yescharacteristic, complemented
192#440—171no—
193#435—171no—
194#417—171no—
195C2×#102—171no—
196C2×C17—22no—
197C2×C17—42no—
198C2×C17—81no—
199C2×C17#54611yescharacteristic
200C2×C17C2×#13711yescharacteristic
201C2×C17#57611yescharacteristic
202D17—41no—
203C2×#41—171no—
204C22×D4C2×D1711yescharacteristic
205C2×Q16—171no—
206C2×SD16C2×D1711yescharacteristic
207C2×SD16C2×D1711yescharacteristic
208C2×M16—171no—
209C22×C8—171no—
210C4⋊Q8—171no—
211#125—341no—
212#124—171no—
213#123—341no—
214C22⋊Q8—171no—
215C22⋊Q8—171no—
216C22⋊Q8—341no—
217C22⋊Q8—341no—
218C22⋊Q8C2×D1711yescharacteristic
219#121—171no—
220#121—341no—
221#120—341no—
222C4×Q8—171no—
223C4×D4—171no—
224C4×D4—171no—
225C4×D4—341no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—171no—
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#102C2×D1711yescharacteristic
242#102C2×D1711yescharacteristic
243C22⋊C8—171no—
244C22⋊C8C2×D1711yescharacteristic
245#97—171no—
246C17#331411yescharacteristic, complemented, Sylow
247C24—21no—
248#41—342no—
249#41—681no—
250C2×Q8—171no—
251C2×Q8—341no—
252C2×Q8C22×D1711yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—171no—
257C2×D4—341no—
258C2×D4—341no—
259C2×D4C22×D1711yescharacteristic
260C2×D4C22×D1711yescharacteristic
261C22×C4—171no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4C22×D1711yescharacteristic
267Q16—681no—
268SD16—22no—
269SD16—41no—
270M16—681no—
271C2×C8—171no—
272C2×C8—171no—
273C2×C8—681no—
274C2×C8C22×D1711yescharacteristic
275C2×C8C22×D1711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—171no—
278C4⋊C4—171no—
279C4⋊C4—171no—
280C4⋊C4—171no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4C22×D1711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—171no—
290C22⋊C4—171no—
291C22⋊C4—341no—
292C22⋊C4—341no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4—341no—
297C42—171no—
298C42—341no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—171no—
305C23D4×D1711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—341no—
309Q8—341no—
310Q8—1361no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—342no—
316D4—682no—
317D4—1361no—
318D4#6325412yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—171no—
324C2×C4—171no—
325C2×C4—171no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—682no—
333C2×C4—1361no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4D4×D1711yescharacteristic
337C2×C4C23×D1711yescharacteristic
338C8—21no—
339C8—21no—
340C8—341no—
341C8—341no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—341no—
352C22—341no—
353C22—341no—
354C22#1058760112yes—
355C22C2×D4×D1711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—341no—
360C4—341no—
361C4—341no—
362C4—681no—
363C4—681no—
364C4—681no—
365C4—681no—
366C4C2×#6325411yescharacteristic
367C4#1058769111yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—681no—
372C2#1176344611yescharacteristic
373C2C2×#1058760111yescharacteristic
374C2#1176361311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#450C2
socleC22×C17C22×D4
prime radical#11763453C2
hypercenter#450D17
hypocenterC17#3314
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #11763517C2semidirect
3. #11763167C2semidirect
4. #11763145C2semidirect
5. #11763121C2semidirect
6. #11763087C2semidirect
7. #11763052C2semidirect
8. #11763048C2semidirect
9. #11763029C2semidirect
10. #11762968C2semidirect
11. #11762910C2semidirect
12. #11762877C2semidirect
13. #11762807C2semidirect
14. C2×#10587503C2semidirect
15. C17×#450C2semidirect
23. #10587632C22semidirect
24. Q8×Dic17C22semidirect
25. C2×#63231C22semidirect
35. C2×#63218C22semidirect
37. C4⋊C4×D17C22semidirect
39. #10587574C22semidirect
45. #10587506C22semidirect
50. #10587491C22semidirect
51. #10587489C22semidirect
52. #10587489C22semidirect
53. #10587487C22semidirect
54. #10587487C22semidirect
55. #10587486C22semidirect
56. #10587485C22semidirect
57. #10587481C22semidirect
58. #10587478C22semidirect
191. #450D17semidirect
246. C17#3314semidirect

Maximal subgroups

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

subgroupindex
#117635172
#117634532
#117631672
#117631452
#117631212
#117630872
#117630522
#117630482
#117630292
#117629682
#117629102
#117628772
#117628072
C2×#105875032
C17×#4502
#331417