Finite Group Databasebeta

← #14542886

Subgroups — #14542886

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#11763541C213yesmaximal, complemented
2#11763535C211yesmaximal, characteristic, complemented
3#11763523C213yesmaximal, complemented
4#11763505C213yesmaximal, complemented
5#11763501C213yesmaximal, complemented
6#11763478C211yesmaximal, characteristic, complemented
7#11763470C211yesmaximal, characteristic, complemented
8#11763432C213yesmaximal, complemented
9#11763412C213yesmaximal, complemented
10#11763341C213yesmaximal, complemented
11#11763338C213yesmaximal, complemented
12#11763292C211yesmaximal, characteristic, complemented
13#11763287C211yesmaximal, characteristic, complemented
14C4×#63256C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C2×#63256C2211yescharacteristic
19C17×#124C2213yes—
20C17×C22⋊Q8C2213yes—
21C17×#121C2213yes—
22C4×Q8×C17C2211yescharacteristic
23C4×D4×C17C2213yes—
24#10587633C2213yes—
25#10587632C2213yes—
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2211yescharacteristic
28Q8×Dic17C2213yes—
29Q8×Dic17C2211yescharacteristic
30Q8×Dic17C2213yes—
31#10587626C2213yes—
32#10587625C2216yes—
33#10587624C2213yes—
34#10587624C2213yes—
35#10587622C2213yes—
36D4×Dic17C2213yes—
37D4×Dic17C2213yes—
38#10587616C2213yes—
39#10587616C2213yes—
40C4×C22⋊D17C2213yes—
41C4×C22⋊D17C2213yes—
42#10587612C2213yes—
43#10587612C2213yes—
44#10587612C2213yes—
45C2×C4⋊Dic17C2213yes—
46#10587584C2216yes—
47#10587583C2213yes—
48#10587579C2213yes—
49#10587578C2213yes—
50#10587576C2213yes—
51#10587576C2213yes—
52#10587574C2213yes—
53#10587572C2216yes—
54#10587571C2216yes—
55#10587569C2213yes—
56#10587569C2213yes—
57#10587569C2216yes—
58#10587567C2216yes—
59#10587565C2213yes—
60#10587565C2211yescharacteristic
61#10587563C2216yes—
62C4×D68C2213yes—
63C42×D17C2213yes—
64#10587556C2213yes—
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⋊D17C2316yes—
75C2×C22⋊D17C2313yes—
76C2×C22⋊D17C2313yes—
77C22×Dic17C2316yes—
78#63256—24no—
79C2×D68C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2313yes—
82C2×C4×D17C2313yes—
83C2×Dic34C2313yes—
84C4⋊C4×C17C2313yes—
85C4⋊C4×C17C2313yes—
86C22⋊C4×C17C2316yes—
87C22⋊C4×C17C2313yes—
88C22⋊C4×C17C2311yescharacteristic
89C42×C17C2313yes—
90C22⋊Dic17C2313yes—
91C22⋊Dic17C2313yes—
92C22⋊Dic17C2311yescharacteristic
93C22⋊Dic17C2311yescharacteristic
94C22⋊Dic17C2313yes—
95C22⋊Dic17C2313yes—
96#63227C2316yes—
97#63227C2316yes—
98#63227C2313yes—
99#63227C2313yes—
100C4⋊Dic17C2313yes—
101C4⋊Dic17C2313yes—
102C4⋊Dic17C2313yes—
103C4⋊Dic17C2313yes—
104C4⋊Dic17C2313yes—
105C4⋊Dic17C2311yescharacteristic
106C4⋊Dic17C2313yes—
107C4⋊Dic17C2313yes—
108#63225C2311yescharacteristic
109#63225C2313yes—
110#63225C2316yes—
111#63225C2313yes—
112#63225C2311yescharacteristic
113C4×Dic17C2313yes—
114C4×Dic17C2313yes—
115C4×Dic17C2311yescharacteristic
116C4×Dic17C2313yes—
117C4×Dic17C2313yes—
118C4×Dic17C2316yes—
119C23×C17C2411yescharacteristic
120C23×C17C2413yes—
121C22×D17C2413yes—
122Q8×C17—26no—
123Q8×C17#4114yes—
124D4×C17—26no—
125D4×C17—43no—
126C2×C4×C17—26no—
127C2×C4×C17—26no—
128C2×C4×C17C2411yescharacteristic
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C2×C4×C17C2411yescharacteristic
132C2×C4×C17C2413yes—
133C22⋊D17—46no—
134C22⋊D17—43no—
135C22⋊D17—43no—
136C2×Dic17—46no—
137C2×Dic17C2411yescharacteristic
138C2×Dic17C2411yescharacteristic
139C2×Dic17C2411yescharacteristic
140C2×Dic17C2413yes—
141C2×Dic17C2413yes—
142C2×Dic17C2413yes—
143C2×Dic17C2411yescharacteristic
144D68—26no—
145C4×D17—26no—
146C4×D17—43no—
147C4×D17—43no—
148Dic34—43no—
149#3524—171nomaximal, Sylow
150C22×C17—21no—
151C22×C17—23no—
152C22×C17—23no—
153C22×C17—21no—
154C22×C17—23no—
155C22×C17—21no—
156C22×C17C2511yescharacteristic
157C2×D17—23no—
158C2×D17—23no—
159C2×D17—23no—
160C4×C17—21no—
161C4×C17—23no—
162C4×C17—23no—
163C4×C17—21no—
164C4×C17C2×#4116yes—
165Dic17—21no—
166Dic17—21no—
167Dic17—23no—
168Dic17—21no—
169Dic17—23no—
170Dic17—23no—
171Dic17—21no—
172#559—173no—
173#557—171no—
174#555—173no—
175#553—173no—
176#552—173no—
177#550—171no—
178#548—171no—
179#548D1711yescharacteristic, complemented
180#542—173no—
181#540—173no—
182#529—173no—
183#529—173no—
184#520—171no—
185#519—171no—
186C4×#41—171no—
187C2×C17—41no—
188C2×C17—43no—
189C2×C17C22×#4111yescharacteristic
190C2×C17#58511yescharacteristic
191C2×C17#58511yescharacteristic
192D17—43no—
193C2×#41—171no—
194C2×#41C2×D1711yescharacteristic
195C22×Q8C2×D1711yescharacteristic
196C4⋊Q8—173no—
197#127—173no—
198#126—176no—
199#126—171no—
200#126—173no—
201#125—173no—
202#125—173no—
203#125—173no—
204#124—173no—
205#124—173no—
206#124—176no—
207#124C2×D1713yes—
208#123—176no—
209#123—173no—
210#123—173no—
211#123—176no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8C2×D1713yes—
215#121—173no—
216#121—173no—
217#121—176no—
218#121—173no—
219#121—173no—
220#121C2×D1713yes—
221C4×Q8—171no—
222C4×Q8—173no—
223C4×Q8—171no—
224C4×Q8—173no—
225C4×Q8—171no—
226C4×Q8—173no—
227C4×Q8C2×D1711yescharacteristic
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—173no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—176no—
234C4×D4—173no—
235C4×D4C2×D1713yes—
236#117—173no—
237#117—173no—
238#117—173no—
239#117—173no—
240#117—176no—
241C2×C4⋊C4—173no—
242C2×C42—173no—
243C17#352411yescharacteristic, complemented, Sylow
244#41—24no—
245#41—344no—
246C2×Q8—24no—
247C2×Q8—173no—
248C2×Q8C22×D1713yes—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—173no—
251C2×D4—176no—
252C2×D4—173no—
253C2×D4—173no—
254C2×D4C22×D1713yes—
255C2×D4C22×D1713yes—
256C22×C4—176no—
257C22×C4—173no—
258C22×C4—173no—
259C22×C4—173no—
260C22×C4C22×D1713yes—
261C22×C4C22×D1713yes—
262C4⋊C4—173no—
263C4⋊C4—173no—
264C4⋊C4—173no—
265C4⋊C4—171no—
266C4⋊C4—173no—
267C4⋊C4—173no—
268C4⋊C4—173no—
269C4⋊C4—173no—
270C4⋊C4—176no—
271C4⋊C4—171no—
272C4⋊C4—173no—
273C4⋊C4—171no—
274C4⋊C4—173no—
275C4⋊C4C22×D1713yes—
276C4⋊C4C22×D1713yes—
277C22⋊C4—171no—
278C22⋊C4—173no—
279C22⋊C4—173no—
280C22⋊C4—173no—
281C22⋊C4—173no—
282C22⋊C4—171no—
283C22⋊C4—176no—
284C22⋊C4—173no—
285C22⋊C4—176no—
286C22⋊C4—173no—
287C22⋊C4C22×D1716yes—
288C22⋊C4C22×D1713yes—
289C22⋊C4C22×D1711yescharacteristic
290C42—171no—
291C42—173no—
292C42—173no—
293C42—176no—
294C42—173no—
295C42—173no—
296C42C22×D1713yes—
297C23—173no—
298C23C23×D1711yescharacteristic
299C23C23×D1713yes—
300Q8—26no—
301Q8—683no—
302Q8#6325414yes—
303D4—26no—
304D4—43no—
305D4—346no—
306D4—686no—
307D4—683no—
308D4—683no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—171no—
312C2×C4—171no—
313C2×C4—173no—
314C2×C4—171no—
315C2×C4—173no—
316C2×C4—171no—
317C2×C4—173no—
318C2×C4—346no—
319C2×C4—683no—
320C2×C4—683no—
321C2×C4—686no—
322C2×C4C23×D1713yes—
323C2×C4C23×D1711yescharacteristic
324C2×C4C23×D1713yes—
325C2×C4C23×D1711yescharacteristic
326C2×C4C23×D1713yes—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22—343no—
334C22—343no—
335C22—343no—
336C22C24×D1711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—341no—
342C4—341no—
343C4—343no—
344C4—341no—
345C4—343no—
346C4—343no—
347C4—341no—
348C4C2×#6325416yes—
349C2—41no—
350C2—43no—
351C2—683no—
352C2C22×#6325411yescharacteristic
353C2#1176364511yescharacteristic
354C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#548C2
hypercenter#548D17
hypocenterC17#3524
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763541C2semidirect
2. #11763535C2semidirect
3. #11763523C2semidirect
4. #11763505C2semidirect
5. #11763501C2semidirect
6. #11763478C2semidirect
7. #11763470C2semidirect
8. #11763432C2semidirect
9. #11763412C2semidirect
10. #11763341C2semidirect
11. #11763338C2semidirect
12. #11763292C2semidirect
13. #11763287C2semidirect
14. C4×#63256C2semidirect
15. C17×#548C2semidirect
179. #548D17semidirect
243. C17#3524semidirect

Maximal subgroups

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

subgroupindex
#117635412
#117635352
#117635232
#117635052
#117635012
#117634782
#117634702
#117634322
#117634122
#117633412
#117633382
#117632922
#117632872
C4×#632562
C17×#5482
#352417