Finite Group Databasebeta

← #14534512

Subgroups — #14534512

Order 2176.

816 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 153 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×#63254C211yesmaximal, characteristic
2C2×#10587582C211yesmaximal, characteristic, complemented
3C2×#10587564C212yesmaximal, complemented
4#11762730C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762725C212yesmaximal, complemented
7#11762724C212yesmaximal, complemented
8#11762723C211yesmaximal, characteristic, complemented
9#11762668C211yesmaximal, characteristic, complemented
10#11762667C212yesmaximal, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C23×Dic17C2212yes—
15C2×#63254—28no—
16C2×#63254—24no—
17C22×C4×D17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2212yes—
22C2×C22⋊C4×C17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2212yes—
29C2×#63225C2211yescharacteristic
30C2×#63225C2212yes—
31C2×#63225C2212yes—
32C2×#63225C2211yescharacteristic
33#10587582—24no—
34#10587564—28no—
35C17×#95C2211yescharacteristic
36#10587502C2211yescharacteristic
37#10587502C2212yes—
38#10587502C2212yes—
39C24×C17C2312yes—
40C23×D17C2311yescharacteristic
41C2×D4×C17—24no—
42C2×D4×C17—22no—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C22×Dic17—24no—
51C22×Dic17—24no—
52C22×Dic17D418yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2312yes—
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2312yes—
57C22×Dic17C2311yescharacteristic
58#63254—232no—
59C2×C4×D17—24no—
60C2×C4×D17D414yes—
61C2×Dic34—22no—
62C2×Dic34—24no—
63C2×Dic34—24no—
64C2×Dic34—22no—
65C4⋊C4×C17—22no—
66C22⋊C4×C17—24no—
67C22⋊C4×C17—42no—
68C22⋊Dic17—24no—
69C22⋊Dic17—42no—
70#63227—22no—
71#63227—22no—
72#63227—42no—
73C4⋊Dic17—24no—
74#63225—22no—
75#63225—24no—
76#63225—24no—
77#63225—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17C2×D414yes—
85C23×C17C2411yescharacteristic
86C22×D17—22no—
87C22×D17—21no—
88C22×D17—22no—
89C22×D17—21no—
90C22×D17C2×D412yes—
91D4×C17—28no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17C2×D412yes—
102C22⋊D17—216no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—216no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—21no—
114C2×Dic17C2×D414yes—
115C2×Dic17C2×D414yes—
116C2×Dic17C2×D412yes—
117C4×D17—28no—
118Dic34—28no—
119Dic34—44no—
120#2616—171nomaximal, Sylow
121C22×C17—24no—
122C22×C17—24no—
123C22×C17—24no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17C22×D411yescharacteristic
129C22×C17C22×D412yes—
130C22×C17H(5,𝔽2)11yescharacteristic
131C22×C172−1+412yes—
132C22×C172−1+411yescharacteristic
133C2×D17—22no—
134C2×D17—22no—
135C2×D17—22no—
136C2×D17—42no—
137C2×D17—42no—
138C4×C17—22no—
139C4×C17—41no—
140C4×C17—42no—
141C4×C17—41no—
142Dic17—24no—
143Dic17—41no—
144Dic17—42no—
145Dic17—41no—
146Dic17#12014yes—
147C22×#41—171no—
148C2×C22⋊Q8—171no—
149C2×C22⋊Q8—172no—
150#395—171no—
151#395—171no—
152#394—171no—
153#394—172no—
154#394—172no—
155#394D1711yescharacteristic, complemented
156#380—172no—
157#380—171no—
158C2×C17—44no—
159C2×C17C2×#12011yescharacteristic
160C2×C17#53611yescharacteristic
161C2×C17#53612yes—
162C2×C17#53712yes—
163C2×C17#53711yescharacteristic
164D17—42no—
165C2×#41—348no—
166C2×#41—344no—
167C22×Q8—171no—
168C22×Q8—171no—
169C22×D4—172no—
170C22×D4C2×D1711yescharacteristic
171C23×C4—172no—
172C23×C4—171no—
173C22⋊Q8—348no—
174C22⋊Q8—344no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—172no—
177C2×C4⋊C4—172no—
178C2×C4⋊C4—171no—
179C2×C4⋊C4—172no—
180C2×C4⋊C4C2×D1711yescharacteristic
181C2×C22⋊C4—172no—
182C2×C22⋊C4—171no—
183C2×C22⋊C4—171no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4C2×D1712yes—
187C2×C22⋊C4C2×D1712yes—
188#95—172no—
189#95—172no—
190#95—171no—
191#95C2×D1711yescharacteristic
192C17#261611yescharacteristic, complemented, Sylow
193C24—171no—
194C24C22×D1712yes—
195#41—3432no—
196C2×Q8—344no—
197C2×Q8—344no—
198C2×Q8—342no—
199C2×Q8—342no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—344no—
203C2×D4—344no—
204C2×D4—344no—
205C22×C4—171no—
206C22×C4—178no—
207C22×C4—172no—
208C22×C4—174no—
209C22×C4—172no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4—344no—
213C22×C4—344no—
214C22×C4—344no—
215C22×C4C22×D1711yescharacteristic
216C22×C4C22×D1712yes—
217C22×C4C22×D1711yescharacteristic
218C22×C4C22×D1711yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—344no—
221C4⋊C4—344no—
222C4⋊C4—342no—
223C4⋊C4—344no—
224C4⋊C4—342no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—342no—
228C22⋊C4—342no—
229C22⋊C4—344no—
230C22⋊C4—682no—
231C22⋊C4—682no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—172no—
239C23—342no—
240C23—341no—
241C23—342no—
242C23—341no—
243C23D4×D1714yes—
244C23C23×D1711yescharacteristic
245Q8—348no—
246Q8—684no—
247D4—28no—
248D4—3416no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—172no—
259C2×C4—174no—
260C2×C4—174no—
261C2×C4—348no—
262C2×C4—3416no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—341no—
267C2×C4—341no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—341no—
271C2×C4—342no—
272C2×C4—341no—
273C2×C4D4×D1712yes—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—342no—
282C22—342no—
283C22—342no—
284C22—682no—
285C22—682no—
286C22C2×D4×D1711yescharacteristic
287C22C2×D4×D1712yes—
288C22#1058769111yescharacteristic
289C22#1058769712yes—
290C22#1058769711yescharacteristic
291C4—22no—
292C4—41no—
293C4—42no—
294C4—41no—
295C4—174no—
296C4—344no—
297C4—681no—
298C4—682no—
299C4—681no—
300C2—44no—
301C2—682no—
302C2#120×D1711yescharacteristic
303C2#1176337911yescharacteristic
304C2#1176338112yes—
305C2#1176338612yes—
306C2#1176338811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×#63254C2
hypercenter#394D17
hypocenterC17#2616
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10587582C2semidirect
3. C2×#10587564C2semidirect
4. #11762730C2semidirect
5. #11762729C2semidirect
6. #11762725C2semidirect
7. #11762724C2semidirect
8. #11762723C2semidirect
9. #11762668C2semidirect
10. #11762667C2semidirect
11. C17×#394C2semidirect
155. #394D17semidirect
192. C17#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#632542
C2×#105875822
C2×#105875642
#117627302
#117627292
#117627252
#117627242
#117627232
#117626682
#117626672
C17×#3942
#261617