Finite Group Databasebeta

← #14535162

Subgroups — #14535162

Order 2176.

688 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#10587622C211yesmaximal, characteristic, complemented
2#11762737C212yesmaximal, complemented
3#11762723C212yesmaximal, complemented
4#11762721C212yesmaximal, complemented
5#11762697C212yesmaximal, complemented
6#11762689C212yesmaximal, complemented
7#11762689C211yesmaximal, characteristic, complemented
8#11762665C212yesmaximal, complemented
9C2×C17×#120C211yesmaximal, characteristic
10C25×C17C2211yescharacteristic
11C22×D4×C17C2212yes—
12C22×D4×C17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic, complemented
14C22×Dic34C2211yescharacteristic, complemented
15C17×#120—24no—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yescomplemented
25C2×C22⋊Dic17C2211yescharacteristic, complemented
26#10587622—24no—
27C2×#63225C2212yescomplemented
28C2×C4×Dic17C2211yescharacteristic, complemented
29#10587502C2212yescomplemented
30#10587502C2212yescomplemented
31#10587502C2212yescomplemented
32#10587502C2212yescomplemented
33C24×C17—22no—
34C24×C17—24no—
35C24×C17—42no—
36C24×C17—41no—
37C24×C17—41no—
38C24×C17C2311yescharacteristic
39C24×C17C2311yescharacteristic
40C24×C17C2312yes—
41C2×D4×C17—24no—
42C2×D4×C17—24no—
43C2×D4×C17—41no—
44C2×D4×C17—41no—
45C2×D4×C17—42no—
46C2×D4×C17D414yes—
47C22×C4×C17C2312yes—
48C22×C4×C17C2311yescharacteristic
49C22×Dic17—41no—
50C22×Dic17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C2×Dic34—22no—
56C2×Dic34—41no—
57C2×Dic34—41no—
58C22⋊C4×C17—22no—
59C22⋊C4×C17—24no—
60C22⋊Dic17—24no—
61C22⋊Dic17—24no—
62C22⋊Dic17—24no—
63C22⋊Dic17—42no—
64C22⋊Dic17—42no—
65C22⋊Dic17—42no—
66C22⋊Dic17—42no—
67C22⋊Dic17—41no—
68#63225—42no—
69C4×Dic17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—21no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—21no—
80C23×C17—22no—
81C23×C17—21no—
82C23×C17—21no—
83C23×C17—22no—
84C23×C17—48no—
85C23×C17—44no—
86C23×C17—44no—
87C23×C17—48no—
88C23×C17—44no—
89C23×C17C2×D412yes—
90C23×C17C2×D412yes—
91C23×C17#4114yes—
92C23×C17C2411yescharacteristic
93D4×C17—44no—
94D4×C17—48no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17C2×D412yes—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—44no—
114Dic34—44no—
115#2684—171nomaximal, Sylow
116C22×C17—22no—
117C22×C17—22no—
118C22×C17—24no—
119C22×C17—24no—
120C22×C17—24no—
121C22×C17—24no—
122C22×C17—24no—
123C22×C17—416no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—42no—
134C22×C17C22×D411yescharacteristic
135C22×C17C2×#4112yes—
136C22×C17H(5,𝔽2)11yescharacteristic
137C22×C17H(5,𝔽2)11yescharacteristic
138C22×C17H(5,𝔽2)12yes—
139C4×C17—22no—
140C4×C17—42no—
141Dic17—42no—
142Dic17—42no—
143Dic17—42no—
144Dic17—42no—
145C2×#124—171no—
146C2×#120D1711yescharacteristic
147#397—172no—
148#394—172no—
149#393—172no—
150#388—172no—
151#386—171no—
152#386—172no—
153#379—172no—
154C2×C17—42no—
155C2×C17—42no—
156C2×C17—44no—
157C2×C17#53411yescharacteristic
158C2×C17#53812yes—
159C2×C17#54612yes—
160C2×C17#55111yescharacteristic
161C2×C17H(3,𝔽4)11yescharacteristic
162C25C2×D1711yescharacteristic
163C22×Q8—171no—
164C22×D4C2×D1712yes—
165C22×D4C2×D1711yescharacteristic
166C23×C4—171no—
167#124—344no—
168#120—24no—
169C2×C4⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4—171no—
178C2×C22⋊C4C2×D1712yes—
179C2×C22⋊C4C2×D1711yescharacteristic
180C2×C42—171no—
181#95—172no—
182#95—172no—
183#95—172no—
184#95—172no—
185C17#268411yescharacteristic, complemented, Sylow
186C24—24no—
187C24—22no—
188C24—41no—
189C24—42no—
190C24—41no—
191C24C22×D1712yes—
192C24C22×D1711yescharacteristic
193C24C22×D1711yescharacteristic
194C2×Q8—342no—
195C2×Q8—681no—
196C2×Q8—681no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4C22⋊D1714yes—
203C22×C4—172no—
204C22×C4—172no—
205C22×C4—172no—
206C22×C4—172no—
207C22×C4—681no—
208C22×C4—682no—
209C22×C4C22×D1712yes—
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—682no—
212C22⋊C4—22no—
213C22⋊C4—24no—
214C22⋊C4—344no—
215C22⋊C4—344no—
216C22⋊C4—344no—
217C22⋊C4—681no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C42—342no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—44no—
238C23—48no—
239C23—44no—
240C23—48no—
241C23—44no—
242C23#6325414yes—
243C23C2×C22⋊D1712yes—
244C23C2×C22⋊D1712yes—
245C23C23×D1711yescharacteristic
246Q8—684no—
247D4—44no—
248D4—48no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—684no—
267C2×C4C2×C22⋊D1712yes—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—416no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22C2×#6325412yes—
287C22#1058769111yescharacteristic
288C22#1058769112yes—
289C22#1058769111yescharacteristic
290C22C22×C22⋊D1711yescharacteristic
291C4—22no—
292C4—42no—
293C4—682no—
294C4—682no—
295C4—682no—
296C4—682no—
297C2—44no—
298C2—42no—
299C2—42no—
300C2#1176336811yescharacteristic
301C2#1176339412yes—
302C2#1176345312yes—
303C2#1176349011yescharacteristic
304C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC2×C17×#120C2
hypercenterC2×#120D17
hypocenterC17#2684
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#10587622C2semidirect
2. #11762737C2semidirect
3. #11762723C2semidirect
4. #11762721C2semidirect
5. #11762697C2semidirect
6. #11762689C2semidirect
7. #11762689C2semidirect
8. #11762665C2semidirect
13. C23×Dic17C22semidirect
14. C22×Dic34C22semidirect
24. C2×C22⋊Dic17C22semidirect
25. C2×C22⋊Dic17C22semidirect
27. C2×#63225C22semidirect
28. C2×C4×Dic17C22semidirect
29. #10587502C22semidirect
30. #10587502C22semidirect
31. #10587502C22semidirect
32. #10587502C22semidirect
185. C17#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
#117627372
#117627232
#117627212
#117626972
#117626892
#117626652
C2×C17×#1202
#268417