Finite Group Databasebeta

← #14542609

Subgroups — #14542609

Order 2176.

988 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763538C211yesmaximal, characteristic, complemented
2#11763535C211yesmaximal, characteristic, complemented
3#11763535C211yesmaximal, characteristic, complemented
4#11763520C213yesmaximal, complemented
5#11763519C213yesmaximal, complemented
6#11763515C213yesmaximal, complemented
7#11763515C213yesmaximal, complemented
8#11763439C213yesmaximal, complemented
9#11763436C213yesmaximal, complemented
10#11763363C213yesmaximal, complemented
11#125×D17C213yesmaximal, complemented
12#11763291C211yesmaximal, characteristic
13#11763291C211yesmaximal, characteristic
14C4×#63256C211yesmaximal, characteristic
15C17×#557C211yesmaximal, characteristic, complemented
16C2×#63256C2211yescharacteristic
17C17×C4⋊Q8C2213yes—
18C17×#125C2213yes—
19C17×#125C2213yes—
20C4×Q8×C17C2213yes—
21C4×Q8×C17C2211yescharacteristic
22C4×Q8×C17C2211yescharacteristic
23C4×Q8×C17C2211yescharacteristic
24#10587632C2216yes—
25#10587632C2213yes—
26Q8×Dic17C2211yescharacteristic
27Q8×Dic17C2211yescharacteristic
28Q8×Dic17C2211yescharacteristic
29#10587630C2213yes—
30#10587583C2213yes—
31#10587583C2216yes—
32#10587582C2213yes—
33#10587582C2216yes—
34#10587582C2216yes—
35#10587582C2213yes—
36#10587579C2213yes—
37#10587579C2213yes—
38#10587579C2213yes—
39#10587578C2213yes—
40C4⋊C4×D17C2216yes—
41C4⋊C4×D17C2216yes—
42C4⋊C4×D17C2213yes—
43C4⋊C4×D17C2213yes—
44#10587576C2213yes—
45#10587576C2213yes—
46#10587576C2213yes—
47#10587576C2213yes—
48#10587575C2213yes—
49#10587575C2216yes—
50#10587575C2211yescharacteristic
51#10587574C2213yes—
52#10587574C2213yes—
53#10587574C2213yes—
54C4×D68C2213yes—
55C4×D68C2213yes—
56C4×D68C2213yes—
57#10587558C2213yes—
58#10587558C2213yes—
59C42×D17C2213yes—
60#10587556C2213yes—
61#10587556C2213yes—
62#10587555C2213yes—
63C4×Dic34C2213yes—
64C4×Dic34C2213yes—
65C4×Dic34C2213yes—
66C2×Q8×C17C2313yes—
67C2×Q8×C17C2311yescharacteristic
68#63256Q818yescomplemented
69C2×D68C2313yes—
70C2×C4×D17C2313yes—
71C2×C4×D17C2313yes—
72C2×C4×D17C2316yes—
73C2×C4×D17C2316yes—
74C2×C4×D17C2313yes—
75C2×Dic34C2313yes—
76C2×Dic34C2313yes—
77C2×Dic34C2313yes—
78C4⋊C4×C17C2316yes—
79C4⋊C4×C17C2313yes—
80C4⋊C4×C17C2313yes—
81C4⋊C4×C17C2313yes—
82C4⋊C4×C17C2313yes—
83C4⋊C4×C17C2313yes—
84C4⋊C4×C17C2311yescharacteristic
85C42×C17C2313yes—
86C42×C17C2313yes—
87C42×C17C2313yes—
88#63227C2313yes—
89#63227C2313yes—
90#63227C2313yes—
91#63227C2313yes—
92#63227C2316yes—
93C4⋊Dic17C2311yescharacteristic
94C4⋊Dic17C2313yes—
95C4⋊Dic17C2313yes—
96C4⋊Dic17C2313yes—
97C4⋊Dic17C2313yes—
98C4⋊Dic17C2316yes—
99C4⋊Dic17C2313yes—
100#63225C2311yescharacteristic
101#63225C2311yescharacteristic
102#63225C2313yes—
103#63225C2313yes—
104#63225C2313yes—
105#63225C2313yes—
106#63225C2316yes—
107#63225C2316yes—
108#63225C2311yescharacteristic
109#63225C2311yescharacteristic
110#63225C2313yes—
111#63225C2313yes—
112C4×Dic17C2313yes—
113C4×Dic17C2313yes—
114C4×Dic17C2313yes—
115C4×Dic17C2311yescharacteristic
116C4×Dic17C2313yes—
117C22×D17C2413yes—
118Q8×C17—43no—
119Q8×C17C2×Q814yes—
120C2×C4×C17C2411yescharacteristic
121C2×C4×C17C2411yescharacteristic
122C2×C4×C17C2411yescharacteristic
123C2×C4×C17C2413yes—
124C2×C4×C17C2413yes—
125C2×C4×C17C2413yes—
126C2×C4×C17C2413yes—
127C2×Dic17C2411yescharacteristic
128C2×Dic17C2411yescharacteristic
129C2×Dic17C2413yes—
130C2×Dic17C2413yes—
131C2×Dic17C2413yes—
132C2×Dic17C2411yescharacteristic
133C2×Dic17C2411yescharacteristic
134D68C2×Q8112yes—
135C4×D17—43no—
136C4×D17—46no—
137C4×D17—46no—
138C4×D17—43no—
139C4×D17C2×Q8112yes—
140Dic34—43no—
141Dic34—43no—
142Dic34—43no—
143#3504—171nomaximal, Sylow
144C22×C17C2511yescharacteristic
145C2×D17—23no—
146C2×D17—23no—
147C2×D17C22×Q816yes—
148C4×C17—21no—
149C4×C17—23no—
150C4×C17—23no—
151C4×C17—21no—
152C4×C17—23no—
153C4×C17—21no—
154C4×C17C22×Q816yes—
155Dic17—21no—
156Dic17—21no—
157Dic17—23no—
158Dic17—23no—
159Dic17—23no—
160Dic17—21no—
161Dic17C22×Q812yes—
162#557—171no—
163#557—171no—
164#557—171no—
165#557D1711yescharacteristic, complemented
166#554—173no—
167#554—173no—
168#554—173no—
169#554—173no—
170#544—173no—
171#544—173no—
172#533—173no—
173C2×#125—173no—
174#520—171no—
175#520—171no—
176C4×#41—171no—
177C2×C17C23×Q811yescharacteristic
178C2×C17#58511yescharacteristic
179C2×C17#58511yescharacteristic
180D17—26no—
181C2×#41—171no—
182C4⋊Q8—173no—
183C4⋊Q8—173no—
184C4⋊Q8—173no—
185C4⋊Q8—173no—
186C4⋊Q8—173no—
187C4⋊Q8C2×D1713yes—
188#125—173no—
189#125—176no—
190#125—173no—
191#125—173no—
192#125—171no—
193#125—173no—
194#125—173no—
195#125—173no—
196#125—173no—
197#125C2×D1713yes—
198#125C2×D1713yes—
199C22⋊Q8—173no—
200C22⋊Q8—176no—
201C22⋊Q8—173no—
202C22⋊Q8—173no—
203C22⋊Q8—173no—
204C22⋊Q8—176no—
205C22⋊Q8—176no—
206C22⋊Q8—176no—
207C4×Q8—171no—
208C4×Q8—171no—
209C4×Q8—171no—
210C4×Q8—173no—
211C4×Q8—173no—
212C4×Q8—173no—
213C4×Q8C2×D1711yescharacteristic
214C4×Q8C2×D1711yescharacteristic
215C4×Q8C2×D1711yescharacteristic
216C4×Q8C2×D1713yes—
217C4×D4—173no—
218C4×D4—173no—
219C4×D4—173no—
220C4×D4—173no—
221C4×D4—173no—
222C4×D4—173no—
223#117—173no—
224#117—173no—
225#117—173no—
226C2×C4⋊C4—173no—
227C2×C4⋊C4—173no—
228C2×C4⋊C4—176no—
229C2×C4⋊C4—176no—
230C2×C42—173no—
231C17#350411yescharacteristic, complemented, Sylow
232#41—178no—
233C2×Q8—173no—
234C2×Q8—173no—
235C2×Q8—173no—
236C2×Q8C22×D1713yes—
237C2×Q8C22×D1711yescharacteristic
238C2×D4—173no—
239C22×C4—173no—
240C22×C4—173no—
241C22×C4—173no—
242C22×C4—176no—
243C22×C4—176no—
244C4⋊C4—173no—
245C4⋊C4—173no—
246C4⋊C4—173no—
247C4⋊C4—173no—
248C4⋊C4—173no—
249C4⋊C4—173no—
250C4⋊C4—173no—
251C4⋊C4—176no—
252C4⋊C4—176no—
253C4⋊C4—176no—
254C4⋊C4—171no—
255C4⋊C4—171no—
256C4⋊C4—173no—
257C4⋊C4—171no—
258C4⋊C4—171no—
259C4⋊C4—171no—
260C4⋊C4—173no—
261C4⋊C4—173no—
262C4⋊C4—173no—
263C4⋊C4C22×D1713yes—
264C4⋊C4C22×D1713yes—
265C4⋊C4C22×D1713yes—
266C4⋊C4C22×D1716yes—
267C4⋊C4C22×D1713yes—
268C4⋊C4C22×D1711yescharacteristic
269C4⋊C4C22×D1713yes—
270C22⋊C4—173no—
271C22⋊C4—173no—
272C22⋊C4—176no—
273C22⋊C4—173no—
274C22⋊C4—173no—
275C42—171no—
276C42—173no—
277C42—173no—
278C42—173no—
279C42—173no—
280C42C22×D1713yes—
281C42C22×D1713yes—
282C42C22×D1713yes—
283C23—173no—
284Q8—43no—
285Q8—683no—
286Q8—683no—
287Q8—683no—
288Q8Q8×D1714yes—
289D4—1712no—
290C2×C4—171no—
291C2×C4—1712no—
292C2×C4—171no—
293C2×C4—171no—
294C2×C4—171no—
295C2×C4—173no—
296C2×C4—173no—
297C2×C4—173no—
298C2×C4—686no—
299C2×C4—683no—
300C2×C4—683no—
301C2×C4—686no—
302C2×C4C23×D1711yescharacteristic
303C2×C4C23×D1713yes—
304C2×C4C23×D1711yescharacteristic
305C2×C4C23×D1711yescharacteristic
306C2×C4C23×D1713yes—
307C2×C4C23×D1713yes—
308C2×C4C23×D1713yes—
309C22—176no—
310C22—343no—
311C22—343no—
312C22C24×D1711yescharacteristic
313C4—23no—
314C4—23no—
315C4—21no—
316C4—21no—
317C4—23no—
318C4—21no—
319C4—172no—
320C4—341no—
321C4—343no—
322C4—341no—
323C4—343no—
324C4—343no—
325C4—341no—
326C4C2×Q8×D1716yes—
327C2—346no—
328C2C22×Q8×D1711yescharacteristic
329C2#1176364511yescharacteristic
330C2#1176364611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D17
commutator, socle, cosocleC22×C17C25
FittingC17×#557C2
prime radicalC2×#63256C22
hypercenter#557D17
hypocenterC17#3504
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #11763538C2semidirect
2. #11763535C2semidirect
3. #11763535C2semidirect
4. #11763520C2semidirect
5. #11763519C2semidirect
6. #11763515C2semidirect
7. #11763515C2semidirect
8. #11763439C2semidirect
9. #11763436C2semidirect
10. #11763363C2semidirect
11. #125×D17C2semidirect
15. C17×#557C2semidirect
68. #63256Q8semidirect
165. #557D17semidirect
231. C17#3504semidirect

Maximal subgroups

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

subgroupindex
#117635382
#117635352
#117635202
#117635192
#117635152
#117634392
#117634362
#117633632
#125×D172
#117632912
C4×#632562
C17×#5572
#350417