Finite Group Databasebeta

← #14542895

Subgroups — #14542895

Order 2176.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763544C213yesmaximal, complemented
2#11763535C211yesmaximal, characteristic, complemented
3#11763522C213yesmaximal, complemented
4#11763503C213yesmaximal, complemented
5#11763500C213yesmaximal, complemented
6#11763481C211yesmaximal, characteristic, complemented
7#11763470C211yesmaximal, characteristic, complemented
8#11763470C211yesmaximal, characteristic, complemented
9#11763431C213yesmaximal, complemented
10#11763414C213yesmaximal, complemented
11#11763346C213yesmaximal, complemented
12#11763337C213yesmaximal, complemented
13#11763291C211yesmaximal, characteristic, complemented
14C4×#63256C211yesmaximal, characteristic, complemented
15C17×#519C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×#63256C2211yescharacteristic
19C4×Q8×C17C2211yescharacteristic
20C4×Q8×C17C2213yes—
21C4×Q8×C17C2211yescharacteristic
22C4×Q8×C17C2213yes—
23C17×#117C2213yes—
24C17×#117C2213yes—
25#10587633C2213yes—
26#10587633C2213yes—
27#10587632C2213yes—
28#10587632C2213yes—
29Q8×Dic17C2211yescharacteristic
30Q8×Dic17C2211yescharacteristic
31#10587616C2213yes—
32#10587616C2213yes—
33C4×C22⋊D17C2213yes—
34C4×C22⋊D17C2213yes—
35#10587584C2213yes—
36#10587582C2213yes—
37#10587581C2213yes—
38#10587581C2213yes—
39#10587580C2216yes—
40#10587580C2213yes—
41#10587580C2216yes—
42#10587579C2213yes—
43#10587579C2213yes—
44#10587578C2213yes—
45C4⋊C4×D17C2213yes—
46#10587576C2213yes—
47#10587575C2213yes—
48#10587574C2213yes—
49#10587571C2213yes—
50#10587570C2216yes—
51#10587570C2213yes—
52#10587569C2213yes—
53#10587565C2211yescharacteristic
54#10587562C2216yes—
55#10587561C2216yes—
56#10587560C2213yes—
57C4×D68C2213yes—
58C4×D68C2216yes—
59C4×D68C2213yes—
60C4×D68C2213yes—
61#10587558C2216yes—
62#10587558C2213yes—
63C42×D17C2213yes—
64#10587556C2213yes—
65C4×Dic34C2213yes—
66C2×Q8×C17—24no—
67C2×Q8×C17C2313yes—
68C2×Q8×C17C2311yescharacteristic
69C22×C4×C17C2313yes—
70C2×C22⋊D17C2313yes—
71C2×C22⋊D17C2313yes—
72#63256—24no—
73#63256—24no—
74C2×D68C2313yes—
75C2×D68C2313yes—
76C2×D68C2313yes—
77C2×D68C2316yes—
78C2×C4×D17C2313yes—
79C2×C4×D17C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2313yes—
82C2×C4×D17C2316yes—
83C2×Dic34C2313yes—
84C4⋊C4×C17C2313yes—
85C4⋊C4×C17C2313yes—
86C4⋊C4×C17C2313yes—
87C4⋊C4×C17C2313yes—
88C22⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17C2313yes—
90C42×C17C2313yes—
91C42×C17C2313yes—
92C42×C17C2316yes—
93C22⋊Dic17C2311yescharacteristic
94C22⋊Dic17C2311yescharacteristic
95#63227C2313yes—
96#63227C2313yes—
97#63227C2313yes—
98#63227C2316yes—
99#63227C2313yes—
100#63227C2316yes—
101#63227C2313yes—
102#63227C2316yes—
103#63227C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2313yes—
106C4⋊Dic17C2313yes—
107C4⋊Dic17C2311yescharacteristic
108#63225C2311yescharacteristic
109#63225C2313yes—
110#63225C2316yes—
111#63225C2311yescharacteristic
112#63225C2313yes—
113#63225C2313yes—
114#63225C2313yes—
115C4×Dic17C2313yes—
116C4×Dic17C2313yes—
117C4×Dic17C2311yescharacteristic
118C4×Dic17C2313yes—
119C23×C17C2411yescharacteristic
120C22×D17C2413yes—
121C22×D17C2413yes—
122Q8×C17—26no—
123Q8×C17#4114yes—
124C2×C4×C17—26no—
125C2×C4×C17C2413yes—
126C2×C4×C17C2411yescharacteristic
127C2×C4×C17C2413yes—
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C22⋊D17—43no—
132C22⋊D17—43no—
133C2×Dic17C2411yescharacteristic
134C2×Dic17C2411yescharacteristic
135C2×Dic17C2411yescharacteristic
136C2×Dic17C2411yescharacteristic
137C2×Dic17C2413yes—
138C2×Dic17C2413yes—
139D68—26no—
140D68—26no—
141D68—46no—
142D68—43no—
143C4×D17—26no—
144C4×D17—26no—
145C4×D17—43no—
146C4×D17—43no—
147C4×D17—46no—
148Dic34—43no—
149#3524—171nomaximal, Sylow
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17C2511yescharacteristic
154C2×D17—23no—
155C2×D17—23no—
156C2×D17—23no—
157C2×D17—23no—
158C2×D17—23no—
159C2×D17—23no—
160C4×C17—23no—
161C4×C17—23no—
162C4×C17—21no—
163C4×C17—23no—
164C4×C17—21no—
165C4×C17C2×#4116yes—
166Dic17—21no—
167Dic17—21no—
168Dic17—21no—
169Dic17—23no—
170Dic17—21no—
171Dic17—23no—
172#559—173no—
173#557—171no—
174#555—173no—
175#553—173no—
176#552—173no—
177#550—171no—
178#548—171no—
179#548—171no—
180#542—173no—
181#540—173no—
182#529—173no—
183#529—173no—
184#520—171no—
185#519D1711yescharacteristic, complemented
186C4×#41—171no—
187C2×C17—41no—
188C2×C17C22×#4111yescharacteristic
189C2×C17#58511yescharacteristic
190C2×C17#58511yescharacteristic
191D17—43no—
192D17—43no—
193C2×#41—171no—
194C2×#41—171no—
195C22×Q8C2×D1711yescharacteristic
196C4⋊Q8—173no—
197#127—173no—
198#126—173no—
199#126—171no—
200#126—176no—
201#125—173no—
202#125—173no—
203#125—173no—
204#124—173no—
205#124—173no—
206#124—173no—
207#124—176no—
208#123—176no—
209#123—173no—
210#123—176no—
211#123—173no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—173no—
215#121—173no—
216#121—173no—
217#121—173no—
218#121—173no—
219#121—173no—
220#121—176no—
221C4×Q8—171no—
222C4×Q8—171no—
223C4×Q8—173no—
224C4×Q8C2×D1711yescharacteristic
225C4×Q8C2×D1713yes—
226C4×Q8C2×D1711yescharacteristic
227C4×Q8C2×D1713yes—
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—176no—
236#117—173no—
237#117—173no—
238#117—176no—
239#117C2×D1713yes—
240#117C2×D1713yes—
241C2×C4⋊C4—173no—
242C2×C42—173no—
243C17#352411yescharacteristic, complemented, Sylow
244#41—344no—
245#41—344no—
246C2×Q8—24no—
247C2×Q8—173no—
248C2×Q8C22×D1713yes—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—173no—
251C2×D4—173no—
252C2×D4—173no—
253C2×D4—173no—
254C2×D4—176no—
255C2×D4—173no—
256C22×C4—173no—
257C22×C4—173no—
258C22×C4—173no—
259C22×C4—173no—
260C22×C4—176no—
261C22×C4C22×D1713yes—
262C4⋊C4—173no—
263C4⋊C4—173no—
264C4⋊C4—171no—
265C4⋊C4—173no—
266C4⋊C4—173no—
267C4⋊C4—176no—
268C4⋊C4—171no—
269C4⋊C4—173no—
270C4⋊C4—173no—
271C4⋊C4—171no—
272C4⋊C4—173no—
273C4⋊C4C22×D1713yes—
274C4⋊C4C22×D1713yes—
275C4⋊C4C22×D1713yes—
276C4⋊C4C22×D1713yes—
277C22⋊C4—171no—
278C22⋊C4—171no—
279C22⋊C4—173no—
280C22⋊C4—173no—
281C22⋊C4—176no—
282C22⋊C4—173no—
283C22⋊C4—176no—
284C22⋊C4—173no—
285C22⋊C4—173no—
286C22⋊C4—176no—
287C22⋊C4—173no—
288C22⋊C4C22×D1711yescharacteristic
289C22⋊C4C22×D1713yes—
290C42—171no—
291C42—173no—
292C42—173no—
293C42—173no—
294C42C22×D1713yes—
295C42C22×D1716yes—
296C42C22×D1713yes—
297C23—173no—
298C23—173no—
299C23C23×D1711yescharacteristic
300Q8—26no—
301Q8—683no—
302Q8#6325214yes—
303D4—346no—
304D4—346no—
305D4—683no—
306D4—686no—
307D4—683no—
308D4—683no—
309C2×C4—26no—
310C2×C4—171no—
311C2×C4—171no—
312C2×C4—171no—
313C2×C4—173no—
314C2×C4—173no—
315C2×C4—171no—
316C2×C4—346no—
317C2×C4—346no—
318C2×C4—683no—
319C2×C4—686no—
320C2×C4—683no—
321C2×C4C23×D1711yescharacteristic
322C2×C4C23×D1713yes—
323C2×C4C23×D1713yes—
324C2×C4C23×D1711yescharacteristic
325C2×C4C23×D1713yes—
326C2×C4C23×D1713yes—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—343no—
331C22—343no—
332C22—343no—
333C22—343no—
334C22—343no—
335C22—343no—
336C22C24×D1711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—341no—
343C4—341no—
344C4—341no—
345C4—343no—
346C4—343no—
347C4—341no—
348C4C2×#6325216yes—
349C2—41no—
350C2—683no—
351C2—683no—
352C2C22×#6325211yescharacteristic
353C2#1176364611yescharacteristic
354C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#519C2
hypercenter#519D17
hypocenterC17#3524
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #11763544C2semidirect
2. #11763535C2semidirect
3. #11763522C2semidirect
4. #11763503C2semidirect
5. #11763500C2semidirect
6. #11763481C2semidirect
7. #11763470C2semidirect
8. #11763470C2semidirect
9. #11763431C2semidirect
10. #11763414C2semidirect
11. #11763346C2semidirect
12. #11763337C2semidirect
13. #11763291C2semidirect
14. C4×#63256C2semidirect
15. C17×#519C2semidirect
185. #519D17semidirect
243. C17#3524semidirect

Maximal subgroups

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

subgroupindex
#117635442
#117635352
#117635222
#117635032
#117635002
#117634812
#117634702
#117634312
#117634142
#117633462
#117633372
#117632912
C4×#632562
C17×#5192
#352417