Finite Group Databasebeta

← #14542863

Subgroups — #14542863

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#11763542C213yesmaximal, complemented
2Q8×Dic34C211yesmaximal, characteristic, complemented
3#11763514C213yesmaximal, complemented
4#11763478C211yesmaximal, characteristic, complemented
5#11763471C211yesmaximal, characteristic, complemented
6#11763466C211yesmaximal, characteristic, complemented
7#11763464C213yesmaximal, complemented
8#11763460C213yesmaximal, complemented
9#11763430C213yesmaximal, complemented
10#11763424C213yesmaximal, complemented
11#11763356C213yesmaximal, complemented
12#11763337C213yesmaximal, complemented
13#11763291C211yesmaximal, characteristic, complemented
14C4×Q8×D17C211yesmaximal, characteristic, complemented
15C17×#520C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C2×#63252C2216yes—
20C4×Q8×C17C2211yescharacteristic
21C4×Q8×C17C2211yescharacteristic
22C4×D4×C17C2213yes—
23C4×D4×C17C2213yes—
24C17×#117C2213yes—
25C2×C4⋊C4×C17C2213yes—
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2211yescharacteristic
28#10587626C2213yes—
29#10587624C2213yes—
30#10587624C2213yes—
31#10587622C2213yes—
32#10587616C2213yes—
33#10587615C2216yes—
34C4×C22⋊D17C2213yes—
35C4×C22⋊D17C2213yes—
36C4×C22⋊D17C2216yes—
37#10587610C2216yes—
38#10587610C2213yes—
39#10587584C2213yes—
40#10587583C2213yes—
41#10587582C2216yes—
42#10587582C2211yescharacteristic
43#10587581C2213yes—
44#10587580C2213yes—
45#10587579C2213yes—
46C4⋊C4×D17C2213yes—
47C4⋊C4×D17C2213yes—
48#10587575C2213yes—
49#10587574C2213yes—
50#10587574C2213yes—
51#10587573C2213yes—
52#10587571C2213yes—
53#10587571C2216yes—
54#10587570C2213yes—
55#10587569C2213yes—
56#10587569C2216yes—
57#10587565C2213yes—
58#10587561C2213yes—
59C4×D68C2213yes—
60C4×D68C2213yes—
61#10587558C2213yes—
62C42×D17C2213yes—
63#10587555C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66#41×C17—24no—
67C2×Q8×C17C2311yescharacteristic
68C2×D4×C17C2313yes—
69C22×C4×C17C2313yes—
70C22×C4×C17C2316yes—
71C2×C22⋊D17C2313yes—
72C2×C22⋊D17C2313yes—
73C2×C22⋊D17C2316yes—
74#63256—24no—
75Q8×D17—24no—
76#63252—412no—
77C2×D68C2313yes—
78C2×D68C2313yes—
79C2×C4×D17C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2313yes—
82C2×C4×D17C2311yescharacteristic
83C2×C4×D17C2316yes—
84C2×Dic34C2313yes—
85C2×Dic34C2311yescharacteristic
86C2×Dic34C2316yes—
87C4⋊C4×C17C2313yes—
88C4⋊C4×C17C2313yes—
89C4⋊C4×C17C2313yes—
90C4⋊C4×C17C2311yescharacteristic
91C22⋊C4×C17C2313yes—
92C22⋊C4×C17C2313yes—
93C42×C17C2313yes—
94C42×C17C2313yes—
95C22⋊Dic17C2313yes—
96C22⋊Dic17C2313yes—
97C22⋊Dic17C2316yes—
98#63227C2313yes—
99#63227C2313yes—
100#63227C2311yescharacteristic
101#63227C2313yes—
102#63227C2316yes—
103#63227C2311yescharacteristic
104#63227C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2313yes—
107C4⋊Dic17C2313yes—
108C4⋊Dic17C2313yes—
109#63225C2313yes—
110#63225C2316yes—
111#63225C2313yes—
112#63225C2311yescharacteristic
113#63225C2313yes—
114#63225C2311yescharacteristic
115#63225C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C4×Dic17C2313yes—
120C23×C17C2413yes—
121C22×D17C2413yes—
122C22×D17C2411yescharacteristic
123Q8×C17#4114yescomplemented
124D4×C17—26no—
125C2×C4×C17—26no—
126C2×C4×C17—46no—
127C2×C4×C17C2411yescharacteristic
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C2×C4×C17C2411yescharacteristic
132C2×C4×C17C2413yes—
133C22⋊D17—26no—
134C22⋊D17—26no—
135C22⋊D17—46no—
136C2×Dic17C2413yes—
137C2×Dic17C2411yescharacteristic
138C2×Dic17C2411yescharacteristic
139C2×Dic17C2413yes—
140C2×Dic17C2413yes—
141C2×Dic17C2411yescharacteristic
142D68—26no—
143D68—26no—
144C4×D17—26no—
145C4×D17—26no—
146C4×D17—26no—
147C4×D17—22no—
148C4×D17—46no—
149Dic34—26no—
150Dic34—22no—
151Dic34—46no—
152#3522—171nomaximal, Sylow
153C22×C17—23no—
154C22×C17—23no—
155C22×C17—23no—
156C22×C17C2511yescharacteristic
157C2×D17—21no—
158C2×D17—21no—
159C2×D17—23no—
160C2×D17—23no—
161C2×D17—23no—
162C2×D17H(5,𝔽2)12yes—
163C4×C17—21no—
164C4×C17—23no—
165C4×C17—23no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17C2×#4116yes—
169Dic17—23no—
170Dic17—21no—
171Dic17—23no—
172Dic17—23no—
173Dic17—21no—
174Dic172−1+412yes—
175#559—173no—
176Q82—171no—
177#554—173no—
178#550—171no—
179#548—171no—
180#548—171no—
181#547—173no—
182#547—173no—
183#542—173no—
184#542—173no—
185#532—173no—
186#529—173no—
187#520—171no—
188#520D1711yescharacteristic, complemented
189C2×C4×Q8—171no—
190C2×C17—43no—
191C2×C17C22×#4111yescharacteristic
192C2×C17C2×H(5,𝔽2)11yescharacteristic
193C2×C17C2×2−1+411yescharacteristic
194D17—22no—
195D17—43no—
196C2×#41—171no—
197C2×#41—176no—
198C2×#41C2×D1711yescharacteristic
199C22×Q8—171no—
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#124—173no—
211#123—176no—
212#123—173no—
213#123—173no—
214#123—176no—
215C22⋊Q8—176no—
216C22⋊Q8—173no—
217C22⋊Q8—171no—
218C22⋊Q8—176no—
219C22⋊Q8—173no—
220#121—173no—
221#121—173no—
222#121—173no—
223#121—173no—
224#121—173no—
225C4×Q8—171no—
226C4×Q8—171no—
227C4×Q8—173no—
228C4×Q8—173no—
229C4×Q8—173no—
230C4×Q8C2×D1711yescharacteristic
231C4×Q8C2×D1711yescharacteristic
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—176no—
235C4×D4—173no—
236C4×D4—173no—
237C4×D4—173no—
238C4×D4C2×D1713yes—
239C4×D4C2×D1713yes—
240#117—173no—
241#117C2×D1713yes—
242C2×C4⋊C4—173no—
243C2×C4⋊C4—173no—
244C2×C4⋊C4C2×D1713yes—
245C2×C42—173no—
246C17#352211yescharacteristic, complemented, Sylow
247#41—24no—
248#41—344no—
249#41—6812no—
250C2×Q8—171no—
251C2×Q8—173no—
252C2×Q8—176no—
253C2×Q8—344no—
254C2×Q8C22×D1711yescharacteristic
255C2×D4—173no—
256C2×D4—173no—
257C2×D4—176no—
258C2×D4—173no—
259C2×D4—173no—
260C2×D4C22×D1713yes—
261C22×C4—171no—
262C22×C4—173no—
263C22×C4—173no—
264C22×C4—173no—
265C22×C4—176no—
266C22×C4C22×D1713yes—
267C22×C4C22×D1716yes—
268C4⋊C4—173no—
269C4⋊C4—173no—
270C4⋊C4—173no—
271C4⋊C4—176no—
272C4⋊C4—171no—
273C4⋊C4—171no—
274C4⋊C4—173no—
275C4⋊C4—173no—
276C4⋊C4—173no—
277C4⋊C4—173no—
278C4⋊C4—173no—
279C4⋊C4C22×D1713yes—
280C4⋊C4C22×D1713yes—
281C4⋊C4C22×D1713yes—
282C4⋊C4C22×D1711yescharacteristic
283C22⋊C4—173no—
284C22⋊C4—176no—
285C22⋊C4—173no—
286C22⋊C4—173no—
287C22⋊C4—173no—
288C22⋊C4—171no—
289C22⋊C4—171no—
290C22⋊C4—173no—
291C22⋊C4—176no—
292C22⋊C4—173no—
293C22⋊C4C22×D1713yes—
294C22⋊C4C22×D1713yes—
295C42—173no—
296C42—173no—
297C42—173no—
298C42—173no—
299C42C22×D1713yes—
300C42C22×D1713yes—
301C23—171no—
302C23—173no—
303C23C23×D1713yes—
304Q8—342no—
305Q8—346no—
306Q8—686no—
307Q8#6325214yescomplemented
308D4—26no—
309D4—346no—
310D4—346no—
311D4—346no—
312D4—346no—
313D4—686no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—171no—
317C2×C4—171no—
318C2×C4—173no—
319C2×C4—173no—
320C2×C4—173no—
321C2×C4—171no—
322C2×C4—342no—
323C2×C4—346no—
324C2×C4—346no—
325C2×C4—346no—
326C2×C4—686no—
327C2×C4C23×D1711yescharacteristic
328C2×C4C23×D1713yes—
329C2×C4C23×D1711yescharacteristic
330C2×C4C23×D1713yes—
331C2×C4C23×D1711yescharacteristic
332C2×C4C23×D1713yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—172no—
337C22—341no—
338C22—341no—
339C22—343no—
340C22—343no—
341C22—343no—
342C22C24×D1711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—172no—
349C4—341no—
350C4—343no—
351C4—343no—
352C4—343no—
353C4—341no—
354C4C2×#6325216yes—
355C2—43no—
356C2—342no—
357C2—683no—
358C2C22×#6325211yescharacteristic
359C2H(5,𝔽2)×D1711yescharacteristic
360C22−1+4×D1711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#520C2
hypercenter#520D17
hypocenterC17#3522
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #11763542C2semidirect
2. Q8×Dic34C2semidirect
3. #11763514C2semidirect
4. #11763478C2semidirect
5. #11763471C2semidirect
6. #11763466C2semidirect
7. #11763464C2semidirect
8. #11763460C2semidirect
9. #11763430C2semidirect
10. #11763424C2semidirect
11. #11763356C2semidirect
12. #11763337C2semidirect
13. #11763291C2semidirect
14. C4×Q8×D17C2semidirect
15. C17×#520C2semidirect
123. Q8×C17#41semidirect
188. #520D17semidirect
246. C17#3522semidirect
307. Q8#63252semidirect

Maximal subgroups

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

subgroupindex
#117635422
Q8×Dic342
#117635142
#117634782
#117634712
#117634662
#117634642
#117634602
#117634302
#117634242
#117633562
#117633372
#117632912
C4×Q8×D172
C17×#5202
#352217