Finite Group Databasebeta

← #14532434

Subgroups — #14532434

Order 2176.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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×D4×D17C211yesmaximal, characteristic
2C2×#10587561C211yesmaximal, characteristic, complemented
3C2×#10587622C211yesmaximal, characteristic, complemented
4C2×#10587583C211yesmaximal, characteristic, complemented
5#11762725C214yesmaximal, complemented
6#11762710C211yesmaximal, characteristic, complemented
7#11762664C214yesmaximal, complemented
8C4×#63227C211yesmaximal, characteristic, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C24×D17C2212yes—
11C22×D4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C2×D4×D17—24no—
14C2×D4×D17—24no—
15C2×D4×D17—24no—
16C22×D68C2211yescharacteristic
17C22×C4×D17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic, complemented
19C2×C4⋊C4×C17C2211yescharacteristic, complemented
20C2×C22⋊C4×C17C2214yescomplemented
21C2×C42×C17C2211yescharacteristic, complemented
22C2×C22⋊Dic17C2214yescomplemented
23#10587622—24no—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2214yescomplemented
27C2×#63227C2214yescomplemented
28C2×C4⋊Dic17C2212yescomplemented
29C2×C4×Dic17C2211yescharacteristic, complemented
30#10587583—24no—
31#10587561—24no—
32#10587502C2212yescomplemented
33C24×C17C2312yes—
34C23×D17—24no—
35C23×D17—24no—
36C23×D17—24no—
37C23×D17—28no—
38C23×D17—44no—
39C23×D17C2312yes—
40C23×D17C2311yescharacteristic
41C2×D4×C17—22no—
42C2×D4×C17—22no—
43C2×D4×C17D414yes—
44C22×C4×C17C2312yes—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C22×Dic17C2312yes—
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53D4×D17—232no—
54C2×D68—22no—
55C2×D68—22no—
56C2×D68D414yes—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17D414yes—
60C2×Dic34—22no—
61C2×Dic34—22no—
62C2×Dic34—22no—
63C4⋊C4×C17—22no—
64C22⋊C4×C17—44no—
65C42×C17—22no—
66C22⋊Dic17—28no—
67#63227—28no—
68#63227—24no—
69#63227—24no—
70#63227—44no—
71C4⋊Dic17—24no—
72C4×Dic17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—24no—
76C23×C17—24no—
77C23×C17C2×D414yes—
78C23×C17C2411yescharacteristic
79C22×D17—21no—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—216no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—24no—
86C22×D17—24no—
87C22×D17—416no—
88C22×D17—416no—
89C22×D17—48no—
90C22×D17—48no—
91C22×D17C2×D414yes—
92C22×D17C2×D412yes—
93C22×D17#4114yes—
94D4×C17—28no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—24no—
100C2×C4×C17C2×D412yes—
101C2×C4×C17C2×D412yes—
102C2×C4×C17C2×D412yes—
103C22⋊D17—216no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17C2×D412yes—
113D68—28no—
114C4×D17—28no—
115Dic34—44no—
116#2435—171nomaximal, Sylow
117C22×C17—24no—
118C22×C17—28no—
119C22×C17—42no—
120C22×C17—42no—
121C22×C17—44no—
122C22×C17C22×D411yescharacteristic
123C22×C17C22×D411yescharacteristic
124C22×C17C22×D411yescharacteristic
125C22×C17C2×#4112yes—
126C22×C17H(5,𝔽2)12yes—
127C2×D17—22no—
128C2×D17—22no—
129C2×D17—24no—
130C2×D17—24no—
131C2×D17—24no—
132C2×D17—28no—
133C2×D17—432no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—44no—
137C2×D17#12414yescomplemented
138C4×C17—42no—
139C4×C17—42no—
140C4×C17#12014yes—
141Dic17—22no—
142Dic17—42no—
143Dic17—42no—
144C23×D4—171no—
145C2×#124—171no—
146C2×#124—171no—
147C2×C22⋊Q8—171no—
148#394—174no—
149#390—171no—
150#390D1711yescharacteristic, complemented
151#379—174no—
152C4×C22⋊C4—171no—
153C2×C17—44no—
154C2×C17C2×#12011yescharacteristic
155C2×C17C2×#12411yescharacteristic
156C2×C17#53511yescharacteristic
157C2×C17#54612yes—
158C2×C17#54612yes—
159D17—24no—
160D17—44no—
161C25—172no—
162C22×Q8—171no—
163C22×D4—172no—
164C22×D4—171no—
165C22×D4—344no—
166C22×D4—344no—
167C22×D4—344no—
168C22×D4C2×D1711yescharacteristic
169C23×C4—171no—
170#124—344no—
171#124—344no—
172C22⋊Q8—344no—
173C2×C4⋊C4—172no—
174C2×C4⋊C4C2×D1711yescharacteristic, complemented
175C2×C22⋊C4—174no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—174no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4—174no—
180C2×C22⋊C4C2×D1714yescomplemented
181C2×C42—171no—
182C2×C42C2×D1711yescharacteristic, complemented
183#95—172no—
184C17#243511yescharacteristic, complemented, Sylow
185C24—171no—
186C24—172no—
187C24—344no—
188C24—344no—
189C24—348no—
190C24—344no—
191C24—684no—
192C24C22×D1712yes—
193C2×Q8—342no—
194C2×Q8—342no—
195C2×Q8—342no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—174no—
199C2×D4—344no—
200C2×D4—3432no—
201C2×D4—344no—
202C2×D4—344no—
203C2×D4—342no—
204C2×D4—342no—
205C2×D4D6814yes—
206C22×C4—171no—
207C22×C4—174no—
208C22×C4—172no—
209C22×C4—172no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4C22×D1712yes—
213C22×C4C22×D1712yes—
214C22×C4C22×D1711yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—344no—
217C22⋊C4—44no—
218C22⋊C4—344no—
219C22⋊C4—348no—
220C22⋊C4—344no—
221C22⋊C4—348no—
222C22⋊C4—684no—
223C42—22no—
224C42—342no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—172no—
230C23—174no—
231C23—174no—
232C23—3416no—
233C23—341no—
234C23—341no—
235C23—342no—
236C23—342no—
237C23—342no—
238C23—344no—
239C23—344no—
240C23—6816no—
241C23—688no—
242C23—6816no—
243C23—688no—
244C23C2×D6814yes—
245C23C23×D1711yescharacteristic
246Q8—684no—
247D4—28no—
248D4—3416no—
249D4—348no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—172no—
256C2×C4—341no—
257C2×C4—341no—
258C2×C4—348no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4C2×D6812yes—
266C2×C4D4×D1712yes—
267C2×C4D4×D1712yes—
268C22—24no—
269C22—28no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—174no—
274C22—344no—
275C22—342no—
276C22—342no—
277C22—344no—
278C22—348no—
279C22—344no—
280C22—6832no—
281C22—682no—
282C22—682no—
283C22—684no—
284C22C22×D6811yescharacteristic
285C22C2×D4×D1711yescharacteristic
286C22C2×D4×D1711yescharacteristic
287C22#1058769112yes—
288C22#41×D1712yes—
289C4—42no—
290C4—42no—
291C4—342no—
292C4—682no—
293C4—682no—
294C4#1058756814yes—
295C2—44no—
296C2—344no—
297C2—684no—
298C2C2×#1058756811yescharacteristic
299C2#124×D1711yescharacteristic
300C2#1176337211yescharacteristic
301C2#1176345412yes—
302C2#1176345612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×D17C2
hypercenter#390D17
hypocenterC17#2435
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#10587561C2semidirect
3. C2×#10587622C2semidirect
4. C2×#10587583C2semidirect
5. #11762725C2semidirect
6. #11762710C2semidirect
7. #11762664C2semidirect
8. C4×#63227C2semidirect
9. C17×#390C2semidirect
18. C22×Dic34C22semidirect
19. C2×C4⋊C4×C17C22semidirect
20. C2×C22⋊C4×C17C22semidirect
21. C2×C42×C17C22semidirect
22. C2×C22⋊Dic17C22semidirect
26. C2×#63227C22semidirect
27. C2×#63227C22semidirect
28. C2×C4⋊Dic17C22semidirect
29. C2×C4×Dic17C22semidirect
32. #10587502C22semidirect
137. C2×D17#124semidirect
150. #390D17semidirect
174. C2×C4⋊C4C2×D17semidirect
180. C2×C22⋊C4C2×D17semidirect
182. C2×C42C2×D17semidirect
184. C17#2435semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D172
C2×#105875612
C2×#105876222
C2×#105875832
#117627252
#117627102
#117626642
C4×#632272
C17×#3902
#243517