Finite Group Databasebeta

← #14532397

Subgroups — #14532397

Order 2176.

914 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#63256C211yesmaximal, characteristic, complemented
2C2×#10587569C211yesmaximal, characteristic, complemented
3C2×#10587570C212yesmaximal, complemented
4#11762728C211yesmaximal, characteristic, complemented
5#11762727C211yesmaximal, characteristic, complemented
6#11762718C211yesmaximal, characteristic, complemented
7#11762687C212yesmaximal, complemented
8#11762685C212yesmaximal, complemented
9#11762683C212yesmaximal, complemented
10#95×D17C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×C22⋊D17C2212yes—
15C2×#63256—28no—
16C2×#63256—24no—
17C22×D68C2211yescharacteristic
18C22×D68C2212yes—
19C22×C4×D17C2212yes—
20C22×C4×D17C2212yes—
21C22×C4×D17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2212yes—
24C2×C22⋊Dic17C2211yescharacteristic
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2211yescharacteristic
28C2×#63227C2212yes—
29C2×C4⋊Dic17C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2212yes—
32#10587570—28no—
33#10587569—24no—
34C17×#95C2211yescharacteristic
35C17×#95C2212yes—
36#10587502C2211yescharacteristic
37#10587502C2212yes—
38#10587502C2212yes—
39C24×C17C2311yescharacteristic
40C23×D17C2312yes—
41C23×D17C2311yescharacteristic
42C2×Q8×C17—24no—
43C2×Q8×C17—22no—
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—22no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—24no—
52C2×C22⋊D17—42no—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2312yes—
57#63256—232no—
58C2×D68—24no—
59C2×D68—24no—
60C2×D68—24no—
61C2×D68—22no—
62C2×D68D418yescomplemented
63C2×C4×D17—24no—
64C2×C4×D17—24no—
65C2×C4×D17—24no—
66C2×C4×D17—24no—
67C2×C4×D17—24no—
68C2×C4×D17—24no—
69C2×C4×D17—42no—
70C2×C4×D17D414yescomplemented
71C22⋊C4×C17—22no—
72C22⋊C4×C17—24no—
73C22⋊Dic17—22no—
74#63227—22no—
75#63227—24no—
76#63227—24no—
77#63227—42no—
78C4⋊Dic17—22no—
79#63225—22no—
80#63225—24no—
81C23×C17—21no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17C2411yescharacteristic
87C22×D17—22no—
88C22×D17—21no—
89C22×D17—21no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—22no—
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17C2×D414yes—
97C22×D17C2×D414yes—
98C22×D17C2×D412yes—
99Q8×C17—28no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17C2×D414yes—
109C2×C4×C17C2×D412yes—
110C22⋊D17—48no—
111C22⋊D17—44no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—22no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17C2×D412yes—
121C2×Dic17#4114yescomplemented
122D68—28no—
123D68—216no—
124C4×D17—216no—
125C4×D17—28no—
126C4×D17—48no—
127#2430—171nomaximal, Sylow
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17C22×D411yescharacteristic
134C22×C17C22×D412yes—
135C22×C17C2×#4112yes—
136C22×C17H(5,𝔽2)11yescharacteristic
137C22×C172−1+411yescharacteristic
138C2×D17—22no—
139C2×D17—22no—
140C2×D17—24no—
141C2×D17—24no—
142C2×D17—24no—
143C2×D17—24no—
144C2×D17—24no—
145C2×D17—42no—
146C2×D17—42no—
147C2×D17—42no—
148C2×D17—42no—
149C2×D17#12014yes—
150C4×C17—24no—
151C4×C17—22no—
152C4×C17—41no—
153C4×C17—42no—
154Dic17—22no—
155Dic17—24no—
156Dic17—41no—
157Dic17—41no—
158C22×#41—171no—
159C2×#123—171no—
160C2×#121—172no—
161#395—171no—
162#394—171no—
163#393D1711yescharacteristic, complemented
164#392—171no—
165#386—172no—
166#386—172no—
167#385—172no—
168C2×#95—171no—
169C2×C17—81no—
170C2×C17C2×#12011yescharacteristic
171C2×C17#53211yescharacteristic
172C2×C17#53711yescharacteristic
173C2×C17#54612yes—
174C2×C17#54712yes—
175D17—24no—
176D17—44no—
177C2×#41—344no—
178C2×#41—348no—
179C22×Q8C2×D1711yescharacteristic
180C22×D4—171no—
181C22×D4—172no—
182C22×D4—172no—
183C22×D4—171no—
184C23×C4—171no—
185C23×C4—172no—
186C23×C4—172no—
187#123—344no—
188#121—348no—
189C2×C4⋊C4—172no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C22⋊C4—171no—
193C2×C22⋊C4—172no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—172no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4C2×D1712yes—
198C2×C22⋊C4C2×D1711yescharacteristic
199#95—172no—
200#95—171no—
201#95—172no—
202#95C2×D1711yescharacteristic
203#95C2×D1712yes—
204C17#243011yescharacteristic, complemented, Sylow
205C24—171no—
206C24—172no—
207C24C22×D1711yescharacteristic
208#41—3432no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×D4—178no—
212C2×D4—344no—
213C2×D4—344no—
214C2×D4—344no—
215C2×D4—344no—
216C2×D4—344no—
217C2×D4—344no—
218C2×D4—342no—
219C2×D4—342no—
220C2×D4—682no—
221C22×C4—172no—
222C22×C4—171no—
223C22×C4—174no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—344no—
227C22×C4—344no—
228C22×C4—344no—
229C22×C4—344no—
230C22×C4—344no—
231C22×C4—344no—
232C22×C4—682no—
233C22×C4C22×D1712yes—
234C22×C4C22×D1711yescharacteristic
235C22×C4C22×D1711yescharacteristic
236C22×C4C22×D1712yes—
237C4⋊C4—342no—
238C4⋊C4—344no—
239C4⋊C4—342no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—344no—
243C22⋊C4—344no—
244C22⋊C4—342no—
245C22⋊C4—342no—
246C22⋊C4—682no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—22no—
251C23—21no—
252C23—172no—
253C23—174no—
254C23—174no—
255C23—341no—
256C23—342no—
257C23—341no—
258C23—342no—
259C23—342no—
260C23—342no—
261C23—342no—
262C23—342no—
263C23—342no—
264C23C23×D1711yescharacteristic
265Q8—28no—
266D4—3416no—
267D4—348no—
268D4—688no—
269D4—684no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—172no—
279C2×C4—174no—
280C2×C4—348no—
281C2×C4—3416no—
282C2×C4—342no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—341no—
286C2×C4—342no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—688no—
291C2×C4D4×D1712yes—
292C2×C4D4×D1714yes—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—174no—
299C22—344no—
300C22—342no—
301C22—342no—
302C22—344no—
303C22—344no—
304C22—344no—
305C22—344no—
306C22—682no—
307C22—682no—
308C22—682no—
309C22—682no—
310C22C2×D4×D1712yes—
311C22C2×D4×D1711yescharacteristic
312C22#1058769411yescharacteristic
313C22#41×D1712yes—
314C22#1058769611yescharacteristic
315C4—22no—
316C4—24no—
317C4—41no—
318C4—42no—
319C4—342no—
320C4—344no—
321C4—681no—
322C4—681no—
323C2—81no—
324C2—344no—
325C2—684no—
326C2#120×D1711yescharacteristic
327C2#1176335811yescharacteristic
328C2#1176338911yescharacteristic
329C2#1176345212yes—
330C2#1176346112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
hypercenter#393D17
hypocenterC17#2430
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#63256C2semidirect
2. C2×#10587569C2semidirect
3. C2×#10587570C2semidirect
4. #11762728C2semidirect
5. #11762727C2semidirect
6. #11762718C2semidirect
7. #11762687C2semidirect
8. #11762685C2semidirect
9. #11762683C2semidirect
10. #95×D17C2semidirect
11. C17×#393C2semidirect
62. C2×D68D4semidirect
70. C2×C4×D17D4semidirect
121. C2×Dic17#41semidirect
163. #393D17semidirect
204. C17#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#632562
C2×#105875692
C2×#105875702
#117627282
#117627272
#117627182
#117626872
#117626852
#117626832
#95×D172
C17×#3932
#243017