Finite Group Databasebeta

← #14542860

Subgroups — #14542860

Order 2176.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763541C213yesmaximal, complemented
2Q8×Dic34C211yesmaximal, characteristic, complemented
3#11763517C213yesmaximal, complemented
4#11763478C211yesmaximal, characteristic, complemented
5#11763470C211yesmaximal, characteristic, complemented
6#11763463C213yesmaximal, complemented
7#11763459C213yesmaximal, complemented
8#11763432C213yesmaximal, complemented
9#11763428C213yesmaximal, complemented
10#11763354C213yesmaximal, complemented
11#11763339C213yesmaximal, complemented
12#11763292C211yesmaximal, characteristic, complemented
13#11763291C211yesmaximal, characteristic, complemented
14C2×Q8×Dic17C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×#63256C2211yescharacteristic
19C2×#63254C2216yes—
20C17×#124C2213yes—
21C17×C22⋊Q8C2213yes—
22C17×#121C2213yes—
23C4×Q8×C17C2211yescharacteristic
24C4×D4×C17C2213yes—
25#10587633C2213yes—
26#10587632C2213yes—
27Q8×Dic17C2211yescharacteristic
28Q8×Dic17C2211yescharacteristic
29Q8×Dic17C2213yes—
30Q8×Dic17C2211yescharacteristic
31Q8×Dic17C2213yes—
32#10587626C2213yes—
33#10587625C2213yes—
34#10587624C2213yes—
35#10587621C2213yes—
36D4×Dic17C2213yes—
37D4×Dic17C2213yes—
38#10587616C2213yes—
39#10587616C2213yes—
40C4×C22⋊D17C2213yes—
41C4×C22⋊D17C2213yes—
42#10587612C2213yes—
43C2×C4⋊Dic17C2213yes—
44C2×C4⋊Dic17C2213yes—
45C2×C4×Dic17C2213yes—
46#10587584C2213yes—
47#10587583C2216yes—
48#10587579C2213yes—
49C4⋊C4×D17C2213yes—
50#10587575C2213yes—
51#10587574C2213yes—
52#10587574C2213yes—
53#10587572C2213yes—
54#10587572C2216yes—
55#10587571C2213yes—
56#10587571C2216yes—
57#10587569C2216yes—
58#10587567C2216yes—
59#10587565C2213yes—
60#10587564C2216yes—
61#10587564C2211yescharacteristic
62C4×D68C2213yes—
63#10587558C2213yes—
64#10587555C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67C2×Q8×C17—24no—
68C2×Q8×C17C2313yes—
69C2×Q8×C17C2311yescharacteristic
70C2×D4×C17C2313yes—
71C2×D4×C17C2313yes—
72C22×C4×C17C2313yes—
73C22×C4×C17C2313yes—
74C2×C22⋊D17C2313yes—
75C2×C22⋊D17C2313yes—
76C2×C22⋊D17C2316yes—
77C22×Dic17C2313yes—
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2316yes—
80#63256—24no—
81#63254—412no—
82C2×D68C2313yes—
83C2×C4×D17C2313yes—
84C2×C4×D17C2316yes—
85C2×Dic34C2311yescharacteristic
86C2×Dic34C2316yes—
87C4⋊C4×C17C2313yes—
88C4⋊C4×C17C2313yes—
89C22⋊C4×C17C2316yes—
90C22⋊C4×C17C2313yes—
91C22⋊C4×C17C2311yescharacteristic
92C42×C17C2313yes—
93C22⋊Dic17C2313yes—
94C22⋊Dic17C2311yescharacteristic
95C22⋊Dic17C2313yes—
96C22⋊Dic17C2313yes—
97#63227C2313yes—
98#63227C2313yes—
99#63227C2316yes—
100#63227C2313yes—
101#63227C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2316yes—
107C4⋊Dic17C2311yescharacteristic
108C4⋊Dic17C2313yes—
109#63225C2313yes—
110#63225C2313yes—
111#63225C2311yescharacteristic
112#63225C2313yes—
113#63225C2313yes—
114#63225C2311yescharacteristic
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C4×Dic17C2313yes—
120C23×C17C2413yes—
121C23×C17C2411yescharacteristic
122C22×D17C2413yes—
123Q8×C17—26no—
124Q8×C17#4114yescomplemented
125D4×C17—26no—
126D4×C17—26no—
127C2×C4×C17—26no—
128C2×C4×C17—26no—
129C2×C4×C17C2411yescharacteristic
130C2×C4×C17C2413yes—
131C2×C4×C17C2413yes—
132C2×C4×C17C2411yescharacteristic
133C2×C4×C17C2413yes—
134C22⋊D17—26no—
135C22⋊D17—26no—
136C22⋊D17—46no—
137C2×Dic17—26no—
138C2×Dic17—22no—
139C2×Dic17—46no—
140C2×Dic17C2411yescharacteristic
141C2×Dic17C2411yescharacteristic
142C2×Dic17C2411yescharacteristic
143C2×Dic17C2413yes—
144C2×Dic17C2413yes—
145C2×Dic17C2413yes—
146C2×Dic17C2411yescharacteristic
147D68—26no—
148C4×D17—26no—
149C4×D17—46no—
150Dic34—22no—
151Dic34—46no—
152#3522—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—23no—
156C22×C17—23no—
157C22×C17—23no—
158C22×C17H(5,𝔽2)12yes—
159C22×C17C2511yescharacteristic
160C2×D17—23no—
161C2×D17—23no—
162C2×D17—23no—
163C4×C17—21no—
164C4×C17—23no—
165C4×C17—23no—
166C4×C17—21no—
167C4×C17C2×#4116yes—
168Dic17—21no—
169Dic17—23no—
170Dic17—23no—
171Dic17—23no—
172Dic17—21no—
173Dic17—21no—
174Dic172−1+412yes—
175#559—173no—
176Q82—171no—
177#554—173no—
178#550—171no—
179#548—171no—
180#548D1711yescharacteristic, complemented
181#547—173no—
182#547—173no—
183#542—173no—
184#542—173no—
185#532—173no—
186#529—173no—
187#520—171no—
188#520—171no—
189C2×C4×Q8—171no—
190C2×C17—22no—
191C2×C17—43no—
192C2×C17C22×#4111yescharacteristic
193C2×C17C2×H(5,𝔽2)11yescharacteristic
194C2×C17C2×2−1+411yescharacteristic
195D17—43no—
196C2×#41—171no—
197C2×#41—176no—
198C2×#41C2×D1711yescharacteristic
199C22×Q8C2×D1711yescharacteristic
200C4⋊Q8—173no—
201C4⋊Q8—173no—
202C4⋊Q8—173no—
203#127—173no—
204#126—173no—
205#126—173no—
206#125—173no—
207#124—173no—
208#124—173no—
209#124—176no—
210#124C2×D1713yes—
211#123—173no—
212#123—173no—
213#123—176no—
214#123—176no—
215C22⋊Q8—173no—
216C22⋊Q8—176no—
217C22⋊Q8—171no—
218C22⋊Q8—176no—
219C22⋊Q8C2×D1713yes—
220#121—173no—
221#121—173no—
222#121—173no—
223#121—173no—
224#121C2×D1713yes—
225C4×Q8—171no—
226C4×Q8—173no—
227C4×Q8—171no—
228C4×Q8—173no—
229C4×Q8—171no—
230C4×Q8—173no—
231C4×Q8C2×D1711yescharacteristic
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—173no—
236C4×D4—173no—
237C4×D4—176no—
238C4×D4—173no—
239C4×D4C2×D1713yes—
240#117—173no—
241#117—173no—
242C2×C4⋊C4—173no—
243C2×C4⋊C4—173no—
244C2×C4⋊C4—173no—
245C2×C42—173no—
246C17#352211yescharacteristic, complemented, Sylow
247#41—24no—
248#41—344no—
249#41—6812no—
250C2×Q8—24no—
251C2×Q8—171no—
252C2×Q8—176no—
253C2×Q8C22×D1713yes—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—173no—
256C2×D4—173no—
257C2×D4—176no—
258C2×D4—173no—
259C2×D4C22×D1713yes—
260C2×D4C22×D1713yes—
261C22×C4—171no—
262C22×C4—173no—
263C22×C4—176no—
264C22×C4—173no—
265C22×C4—176no—
266C22×C4C22×D1713yes—
267C22×C4C22×D1713yes—
268C4⋊C4—173no—
269C4⋊C4—173no—
270C4⋊C4—173no—
271C4⋊C4—173no—
272C4⋊C4—173no—
273C4⋊C4—176no—
274C4⋊C4—171no—
275C4⋊C4—171no—
276C4⋊C4—171no—
277C4⋊C4—173no—
278C4⋊C4—173no—
279C4⋊C4—173no—
280C4⋊C4—173no—
281C4⋊C4C22×D1713yes—
282C4⋊C4C22×D1713yes—
283C22⋊C4—173no—
284C22⋊C4—173no—
285C22⋊C4—171no—
286C22⋊C4—173no—
287C22⋊C4—173no—
288C22⋊C4—173no—
289C22⋊C4—173no—
290C22⋊C4—176no—
291C22⋊C4—173no—
292C22⋊C4C22×D1716yes—
293C22⋊C4C22×D1711yescharacteristic
294C22⋊C4C22×D1713yes—
295C42—173no—
296C42—173no—
297C42—173no—
298C42—173no—
299C42—173no—
300C42C22×D1713yes—
301C23—173no—
302C23C23×D1713yes—
303C23C23×D1711yescharacteristic
304Q8—26no—
305Q8—342no—
306Q8—686no—
307Q8#6325414yescomplemented
308D4—26no—
309D4—26no—
310D4—346no—
311D4—346no—
312D4—346no—
313D4—686no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—171no—
317C2×C4—171no—
318C2×C4—171no—
319C2×C4—173no—
320C2×C4—173no—
321C2×C4—173no—
322C2×C4—171no—
323C2×C4—342no—
324C2×C4—346no—
325C2×C4—346no—
326C2×C4—686no—
327C2×C4—686no—
328C2×C4C23×D1713yes—
329C2×C4C23×D1711yescharacteristic
330C2×C4C23×D1713yes—
331C2×C4C23×D1711yescharacteristic
332C2×C4C23×D1713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—343no—
339C22—343no—
340C22—343no—
341C22#1058769612yes—
342C22C24×D1711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—172no—
348C4—341no—
349C4—343no—
350C4—343no—
351C4—343no—
352C4—341no—
353C4—341no—
354C4C2×#6325416yes—
355C2—22no—
356C2—43no—
357C2—683no—
358C2C22×#6325411yescharacteristic
359C2C2×#1058769611yescharacteristic
360C22−1+4×D1711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#548C2
hypercenter#548D17
hypocenterC17#3522
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763541C2semidirect
2. Q8×Dic34C2semidirect
3. #11763517C2semidirect
4. #11763478C2semidirect
5. #11763470C2semidirect
6. #11763463C2semidirect
7. #11763459C2semidirect
8. #11763432C2semidirect
9. #11763428C2semidirect
10. #11763354C2semidirect
11. #11763339C2semidirect
12. #11763292C2semidirect
13. #11763291C2semidirect
14. C2×Q8×Dic17C2semidirect
15. C17×#548C2semidirect
124. Q8×C17#41semidirect
180. #548D17semidirect
246. C17#3522semidirect
307. Q8#63254semidirect

Maximal subgroups

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

subgroupindex
#117635412
Q8×Dic342
#117635172
#117634782
#117634702
#117634632
#117634592
#117634322
#117634282
#117633542
#117633392
#117632922
#117632912
C2×Q8×Dic172
C17×#5482
#352217