Finite Group Databasebeta

← #14542896

Subgroups — #14542896

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#11763525C213yesmaximal, complemented
3#11763512C213yesmaximal, complemented
4#11763497C213yesmaximal, complemented
5#11763481C211yesmaximal, characteristic, complemented
6#11763471C211yesmaximal, characteristic, complemented
7#11763471C211yesmaximal, characteristic, complemented
8#11763433C213yesmaximal, complemented
9#11763415C213yesmaximal, complemented
10#11763346C213yesmaximal, complemented
11#11763344C213yesmaximal, complemented
12#11763291C211yesmaximal, characteristic, complemented
13#11763285C211yesmaximal, characteristic, complemented
14C4×#63256C211yesmaximal, characteristic, complemented
15C17×#557C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C2×#63256C2211yescharacteristic
18C2×Q8×D17C2211yescharacteristic
19C17×C4⋊Q8C2213yes—
20C17×#125C2213yes—
21C17×#125C2213yes—
22C4×Q8×C17C2211yescharacteristic
23C4×Q8×C17C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×Q8×C17C2213yes—
26#10587633C2213yes—
27#10587633C2216yes—
28#10587632C2213yes—
29Q8×Dic17C2211yescharacteristic
30#10587584C2216yes—
31#10587584C2213yes—
32#10587584C2211yescharacteristic
33#10587583C2213yes—
34#10587583C2213yes—
35#10587581C2213yes—
36#10587581C2213yes—
37#10587581C2213yes—
38#10587581C2213yes—
39#10587581C2216yes—
40#10587581C2213yes—
41#10587580C2216yes—
42#10587580C2213yes—
43#10587580C2213yes—
44#10587580C2216yes—
45#10587579C2216yes—
46#10587579C2213yes—
47#10587579C2213yes—
48#10587578C2213yes—
49#10587578C2216yes—
50#10587578C2213yes—
51C4⋊C4×D17C2213yes—
52#10587576C2213yes—
53#10587573C2213yes—
54#10587561C2213yes—
55#10587561C2213yes—
56#10587560C2213yes—
57C4×D68C2213yes—
58C4×D68C2213yes—
59C4×D68C2213yes—
60C4×D68C2213yes—
61C4×D68C2213yes—
62#10587558C2213yes—
63#10587558C2213yes—
64C42×D17C2213yes—
65C4×Dic34C2213yes—
66C2×Q8×C17C2313yes—
67C2×Q8×C17C2311yescharacteristic
68#63256—24no—
69#63256—24no—
70Q8×D17—24no—
71C2×D68C2313yes—
72C2×D68C2313yes—
73C2×D68C2313yes—
74C2×D68C2313yes—
75C2×D68C2313yes—
76C2×D68C2316yes—
77C2×C4×D17C2313yes—
78C2×C4×D17C2313yes—
79C2×C4×D17C2313yes—
80C2×C4×D17C2313yes—
81C2×C4×D17C2313yes—
82C2×C4×D17C2316yes—
83C2×Dic34C2313yes—
84C4⋊C4×C17C2316yes—
85C4⋊C4×C17C2313yes—
86C4⋊C4×C17C2313yes—
87C4⋊C4×C17C2313yes—
88C4⋊C4×C17C2313yes—
89C4⋊C4×C17C2313yes—
90C4⋊C4×C17C2311yescharacteristic
91C42×C17C2313yes—
92C42×C17C2313yes—
93C42×C17C2313yes—
94#63227C2311yescharacteristic
95#63227C2313yes—
96#63227C2313yes—
97#63227C2313yes—
98#63227C2311yescharacteristic
99#63227C2311yescharacteristic
100#63227C2313yes—
101#63227C2313yes—
102#63227C2313yes—
103#63227C2316yes—
104#63227C2313yes—
105#63227C2316yes—
106#63227C2316yes—
107C4⋊Dic17C2313yes—
108C4⋊Dic17C2311yescharacteristic
109C4⋊Dic17C2313yes—
110C4⋊Dic17C2313yes—
111#63225C2313yes—
112#63225C2311yescharacteristic
113#63225C2313yes—
114#63225C2313yes—
115C4×Dic17C2316yes—
116C4×Dic17C2311yescharacteristic
117C4×Dic17C2313yes—
118C4×Dic17C2313yes—
119C22×D17C2411yescharacteristic
120C22×D17C2413yes—
121C22×D17C2413yes—
122Q8×C17—43no—
123Q8×C17#4114yes—
124C2×C4×C17C2411yescharacteristic
125C2×C4×C17C2411yescharacteristic
126C2×C4×C17C2411yescharacteristic
127C2×C4×C17C2413yes—
128C2×C4×C17C2413yes—
129C2×C4×C17C2413yes—
130C2×C4×C17C2413yes—
131C2×Dic17C2413yes—
132C2×Dic17C2411yescharacteristic
133C2×Dic17C2413yes—
134C2×Dic17C2411yescharacteristic
135C2×Dic17C2411yescharacteristic
136D68—26no—
137D68—26no—
138D68—43no—
139D68—43no—
140D68—43no—
141D68—46no—
142C4×D17—26no—
143C4×D17—26no—
144C4×D17—26no—
145C4×D17—43no—
146C4×D17—46no—
147C4×D17—43no—
148Dic34—26no—
149#3524—171nomaximal, Sylow
150C22×C17C2511yescharacteristic
151C2×D17—21no—
152C2×D17—23no—
153C2×D17—23no—
154C2×D17—23no—
155C2×D17—23no—
156C2×D17—23no—
157C2×D17—21no—
158C2×D17—23no—
159C2×D17—21no—
160C4×C17—21no—
161C4×C17—23no—
162C4×C17—23no—
163C4×C17—21no—
164C4×C17—23no—
165C4×C17—21no—
166C4×C17C2×#4116yes—
167Dic17—23no—
168Dic17—21no—
169Dic17—21no—
170Dic17—21no—
171Dic17—23no—
172#559—173no—
173#557D1711yescharacteristic, complemented
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#519—171no—
186C4×#41—171no—
187C2×C17C22×#4111yescharacteristic
188C2×C17#58511yescharacteristic
189C2×C17#58511yescharacteristic
190D17—41no—
191D17—43no—
192D17—43no—
193C2×#41—171no—
194C2×#41—171no—
195C22×Q8—171no—
196C4⋊Q8C2×D1713yes—
197#127—173no—
198#126—171no—
199#126—173no—
200#126—176no—
201#125—173no—
202#125C2×D1713yes—
203#125C2×D1713yes—
204#124—173no—
205#124—176no—
206#124—173no—
207#124—173no—
208#123—176no—
209#123—173no—
210#123—173no—
211#123—176no—
212C22⋊Q8—173no—
213C22⋊Q8—173no—
214C22⋊Q8—173no—
215#121—173no—
216#121—173no—
217#121—173no—
218#121—173no—
219#121—176no—
220#121—173no—
221C4×Q8—171no—
222C4×Q8—173no—
223C4×Q8—173no—
224C4×Q8C2×D1711yescharacteristic
225C4×Q8C2×D1711yescharacteristic
226C4×Q8C2×D1711yescharacteristic
227C4×Q8C2×D1713yes—
228C4×D4—173no—
229C4×D4—173no—
230C4×D4—176no—
231C4×D4—173no—
232C4×D4—173no—
233C4×D4—173no—
234C4×D4—173no—
235C4×D4—173no—
236#117—173no—
237#117—173no—
238#117—176no—
239#117—173no—
240#117—173no—
241C2×C4⋊C4—173no—
242C2×C42—173no—
243C17#352411yescharacteristic, complemented, Sylow
244#41—344no—
245#41—344no—
246C2×Q8—173no—
247C2×Q8—344no—
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—173no—
261C22×C4—176no—
262C4⋊C4—173no—
263C4⋊C4—173no—
264C4⋊C4—171no—
265C4⋊C4—173no—
266C4⋊C4—173no—
267C4⋊C4—173no—
268C4⋊C4—171no—
269C4⋊C4—173no—
270C4⋊C4C22×D1713yes—
271C4⋊C4C22×D1713yes—
272C4⋊C4C22×D1713yes—
273C4⋊C4C22×D1716yes—
274C4⋊C4C22×D1713yes—
275C4⋊C4C22×D1711yescharacteristic
276C4⋊C4C22×D1713yes—
277C22⋊C4—171no—
278C22⋊C4—173no—
279C22⋊C4—173no—
280C22⋊C4—173no—
281C22⋊C4—176no—
282C22⋊C4—173no—
283C22⋊C4—176no—
284C22⋊C4—171no—
285C22⋊C4—173no—
286C22⋊C4—171no—
287C22⋊C4—173no—
288C22⋊C4—176no—
289C22⋊C4—173no—
290C42—171no—
291C42—173no—
292C42—173no—
293C42—176no—
294C42C22×D1713yes—
295C42C22×D1713yes—
296C42C22×D1713yes—
297C23—171no—
298C23—173no—
299C23—173no—
300Q8—43no—
301Q8—346no—
302Q8#6325614yes—
303D4—346no—
304D4—346no—
305D4—683no—
306D4—686no—
307D4—683no—
308D4—683no—
309C2×C4—171no—
310C2×C4—171no—
311C2×C4—173no—
312C2×C4—173no—
313C2×C4—171no—
314C2×C4—346no—
315C2×C4—346no—
316C2×C4—346no—
317C2×C4—683no—
318C2×C4—683no—
319C2×C4—686no—
320C2×C4C23×D1713yes—
321C2×C4C23×D1711yescharacteristic
322C2×C4C23×D1713yes—
323C2×C4C23×D1711yescharacteristic
324C2×C4C23×D1711yescharacteristic
325C2×C4C23×D1713yes—
326C2×C4C23×D1713yes—
327C22—341no—
328C22—343no—
329C22—343no—
330C22—343no—
331C22—341no—
332C22—343no—
333C22—343no—
334C22—343no—
335C22—341no—
336C22C24×D1711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—341no—
344C4—341no—
345C4—343no—
346C4—341no—
347C4—343no—
348C4C2×#6325616yes—
349C2—681no—
350C2—683no—
351C2—683no—
352C2C22×#6325611yescharacteristic
353C2#1176364511yescharacteristic
354C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#557C2
hypercenter#557D17
hypocenterC17#3524
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #11763544C2semidirect
2. #11763525C2semidirect
3. #11763512C2semidirect
4. #11763497C2semidirect
5. #11763481C2semidirect
6. #11763471C2semidirect
7. #11763471C2semidirect
8. #11763433C2semidirect
9. #11763415C2semidirect
10. #11763346C2semidirect
11. #11763344C2semidirect
12. #11763291C2semidirect
13. #11763285C2semidirect
14. C4×#63256C2semidirect
15. C17×#557C2semidirect
173. #557D17semidirect
243. C17#3524semidirect

Maximal subgroups

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

subgroupindex
#117635442
#117635252
#117635122
#117634972
#117634812
#117634712
#117634332
#117634152
#117633462
#117633442
#117632912
#117632852
C4×#632562
C17×#5572
#352417