Finite Group Databasebeta

← #14540775

Subgroups — #14540775

Order 2176.

420 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#11763463C211yesmaximal, characteristic, complemented
2#11763168C211yesmaximal, characteristic, complemented
3#11763142C211yesmaximal, characteristic, complemented
4#11763120C211yesmaximal, characteristic, complemented
5#11763104C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763043C211yesmaximal, characteristic, complemented
8#11763021C211yesmaximal, characteristic, complemented
9#11762954C211yesmaximal, characteristic, complemented
10#11762948C211yesmaximal, characteristic, complemented
11#11762907C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762810C211yesmaximal, characteristic, complemented
14C2×#10587506C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×#63254—21no—
19C17×#124—21no—
20C17×#124C2211yescharacteristic
21C17×C22⋊Q8—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×#121—21no—
24C17×#121C2211yescharacteristic
25C4×Q8×C17C2211yescharacteristic
26C4×D4×C17—21no—
27C4×D4×C17C2211yescharacteristic
28C2×#63230C2211yescharacteristic
29C2×#63230C2211yescharacteristic
30C2×#63228C2211yescharacteristic
31#10587616C2211yescharacteristic
32#10587616C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587583—21no—
38#10587572—21no—
39#10587572—21no—
40C4×D68C2211yescharacteristic
41#10587555C2211yescharacteristic
42#10587506C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587506C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587478C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59#41×C17—22no—
60#41×C17—41no—
61C2×Q8×C17—21no—
62C2×Q8×C17—22no—
63C2×Q8×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×Q8×C17C2311yescharacteristic
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17C2311yescharacteristic
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C2×C22⋊D17—21no—
75C2×C22⋊D17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78#63254—44no—
79C2×D68C2311yescharacteristic
80C2×C4×D17—21no—
81C2×Dic34—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17D411yescharacteristic
85C4⋊C4×C17C2311yescharacteristic
86C22⋊C4×C17—21no—
87C22⋊C4×C17—21no—
88C22⋊C4×C17—21no—
89C22⋊C4×C17—21no—
90C22⋊C4×C17D411yescharacteristic
91C22⋊C4×C17D411yescharacteristic
92C42×C17—21no—
93C42×C17C2311yescharacteristic
94#63230—22no—
95#63230—41no—
96#63228—41no—
97#63227—21no—
98#63227—21no—
99#63227—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17C2311yescharacteristic
104C4⋊Dic17C2311yescharacteristic
105C4⋊Dic17C2311yescharacteristic
106#63223—41no—
107C2×#3622—41no—
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C2×#3622C2311yescharacteristic
112C23×C17—21no—
113C23×C17C2×D411yescharacteristic
114C23×C17C2×D411yescharacteristic
115C22×D17—21no—
116Q8×C17—21no—
117Q8×C17—21no—
118Q8×C17—21no—
119Q8×C17—42no—
120Q8×C17#4112yescomplemented
121D4×C17—22no—
122D4×C17—21no—
123D4×C17—21no—
124D4×C17—42no—
125D4×C17—42no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2×D411yescharacteristic
138C2×C4×C17C2×D411yescharacteristic
139C2×C4×C17C2411yescharacteristic
140C22⋊D17—42no—
141C22⋊D17—42no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—42no—
146C2×Dic17—42no—
147D68—21no—
148D68—21no—
149C4×D17—42no—
150Dic34—42no—
151#3622—21no—
152#3622—21no—
153#3622—21no—
154#3622—21no—
155#3308—171nomaximal, Sylow
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17AGL(1,ℤ/8)12yes—
165C22×C17C22×D411yescharacteristic
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—22no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17C2×#4111yescharacteristic
177C4×C17H(5,𝔽2)11yescharacteristic
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181#548D1711yescharacteristic, complemented
182#547—171no—
183#489—171no—
184#484—171no—
185#480—171no—
186#477—171no—
187#468—171no—
188#466—171no—
189#463—171no—
190#449—171no—
191#448—171no—
192#440—171no—
193#435—171no—
194#417—171no—
195C2×#103—171no—
196C2×C17—22no—
197C2×C17—41no—
198C2×C17—81no—
199C2×C17#54611yescharacteristic
200C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
201C2×C17#57811yescharacteristic
202D17—81no—
203C2×#41—341no—
204C2×#41C2×D1711yescharacteristic
205C22×Q8C2×D1711yescharacteristic
206C2×SD16—171no—
207C2×SD16—171no—
208C2×D8—171no—
209C2×M16—171no—
210C22×C8—171no—
211C4⋊Q8—171no—
212#124—21no—
213#124C2×D1711yescharacteristic
214#123—341no—
215#123—341no—
216C22⋊Q8—21no—
217C22⋊Q8—341no—
218C22⋊Q8C2×D1711yescharacteristic
219#121—21no—
220#121—171no—
221#121—171no—
222#121C2×D1711yescharacteristic
223C4×Q8C2×D1711yescharacteristic
224C4×D4—21no—
225C4×D4—171no—
226C4×D4C2×D1711yescharacteristic
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
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#103—171no—
238#103—171no—
239#102—171no—
240#102—171no—
241#102—171no—
242#102—171no—
243C22⋊C8—171no—
244C22⋊C8—171no—
245#97—171no—
246C17#330811yescharacteristic, complemented, Sylow
247#41—22no—
248#41—41no—
249#41—684no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—341no—
254C2×Q8C22×D1711yescharacteristic
255C2×Q8C22×D1711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—171no—
259C2×D4—341no—
260C2×D4—341no—
261C2×D4C22⋊D1711yescharacteristic
262C2×D4C22×D1711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4—341no—
268C22×C4C22×D1711yescharacteristic
269C22×C4C22×D1711yescharacteristic
270SD16—342no—
271SD16—681no—
272D8—681no—
273M16—681no—
274C2×C8—171no—
275C2×C8—171no—
276C2×C8—171no—
277C2×C8—171no—
278C2×C8—681no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—171no—
282C4⋊C4—171no—
283C4⋊C4—171no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4C22⋊D1711yescharacteristic
288C4⋊C4C22×D1711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—341no—
294C22⋊C4—341no—
295C22⋊C4—341no—
296C22⋊C4C22⋊D1711yescharacteristic
297C22⋊C4C22⋊D1711yescharacteristic
298C42—21no—
299C42C22×D1711yescharacteristic
300C23—21no—
301C23—341no—
302C23C2×C22⋊D1711yescharacteristic
303C23C2×C22⋊D1711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—682no—
309Q8#6325412yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—341no—
316D4—341no—
317D4—682no—
318D4—682no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—682no—
332C2×C4—682no—
333C2×C4—682no—
334C2×C4C2×C22⋊D1711yescharacteristic
335C2×C4C2×C22⋊D1711yescharacteristic
336C2×C4C2×C22⋊D1711yescharacteristic
337C2×C4C2×C22⋊D1711yescharacteristic
338C2×C4C23×D1711yescharacteristic
339C8—341no—
340C8—341no—
341C8—341no—
342C8—341no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—681no—
352C22—681no—
353C22—681no—
354C22#1058763512yes—
355C22C22×C22⋊D1711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—681no—
364C4—681no—
365C4—681no—
366C4C2×#6325411yescharacteristic
367C4#1058769111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—1361no—
372C2#1176345311yescharacteristic
373C2C2×#1058763511yescharacteristic
374C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#548C2
socleC22×C17C22×D4
hypercenter#548D17
hypocenterC17#3308
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763463C2semidirect
2. #11763168C2semidirect
3. #11763142C2semidirect
4. #11763120C2semidirect
5. #11763104C2semidirect
6. #11763057C2semidirect
7. #11763043C2semidirect
8. #11763021C2semidirect
9. #11762954C2semidirect
10. #11762948C2semidirect
11. #11762907C2semidirect
12. #11762881C2semidirect
13. #11762810C2semidirect
14. C2×#10587506C2semidirect
15. C17×#548C2semidirect
120. Q8×C17#41semidirect
181. #548D17semidirect
246. C17#3308semidirect
309. Q8#63254semidirect

Maximal subgroups

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

subgroupindex
#117634632
#117631682
#117631422
#117631202
#117631042
#117630572
#117630432
#117630212
#117629542
#117629482
#117629072
#117628812
#117628102
C2×#105875062
C17×#5482
#330817