Finite Group Databasebeta

← #14534709

Subgroups — #14534709

Order 2176.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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×#10587583C211yesmaximal, characteristic, complemented
2C2×#10587579C211yesmaximal, characteristic
3#11762738C212yesmaximal, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762726C212yesmaximal
7#11762722C211yesmaximal, characteristic, complemented
8#11762695C212yesmaximal, complemented
9#11762683C212yesmaximal, complemented
10#11762671C211yesmaximal, characteristic, complemented
11C17×#398C211yesmaximal, characteristic, complemented
12C22×D68C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2212yes—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2211yescharacteristic
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587583—24no—
29#10587579—24no—
30C17×#95C2211yescharacteristic
31C17×#95C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C2×D68—22no—
49C2×D68—22no—
50C2×D68—22no—
51C2×C4×D17—22no—
52C2×C4×D17—42no—
53C2×C4×D17—42no—
54C2×C4×D17D414yes—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—24no—
60#63227—22no—
61#63227—24no—
62#63227—24no—
63#63227—22no—
64#63227—42no—
65#63227—42no—
66C4⋊Dic17—24no—
67#63225—42no—
68C4×Dic17—22no—
69C23×C17C2411yescharacteristic
70C22×D17—21no—
71C22×D17—21no—
72C22×D17—21no—
73C22×D17—21no—
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17C2×D412yes—
79C22×D17#4112yes—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17C2×D412yes—
92C2×C4×C17#4112yes—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—21no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17C2×D412yes—
103D68—28no—
104C4×D17—44no—
105C4×D17—44no—
106Dic34—44no—
107#2638—171nomaximal, Sylow
108C22×C17C22×D411yescharacteristic
109C22×C17C2×#4111yescharacteristic
110C22×C17C2×#4111yescharacteristic
111C22×C17H(5,𝔽2)12yes—
112C22×C172−1+412yes—
113C2×D17—22no—
114C2×D17—22no—
115C2×D17—22no—
116C2×D17—22no—
117C2×D17—24no—
118C2×D17—41no—
119C2×D17—41no—
120C2×D17—42no—
121C2×D17—42no—
122C2×D17—42no—
123C4×C17—22no—
124C4×C17—22no—
125C4×C17—42no—
126C4×C17—41no—
127C4×C17—42no—
128Dic17—22no—
129Dic17—42no—
130Dic17—41no—
131Dic17—42no—
132C2×C22⋊Q8—171no—
133C2×C4×D4—171no—
134#398D1711yescharacteristic, complemented
135#397—171no—
136#397—172no—
137#395—171no—
138#394—172no—
139#393—171no—
140#387—172no—
141#385—172no—
142#381—171no—
143C2×C17#52511yescharacteristic
144C2×C17#53411yescharacteristic
145C2×C17#53611yescharacteristic
146C2×C17#53912yes—
147C2×C17#54212yes—
148D17—42no—
149D17—42no—
150C22×Q8—171no—
151C22×D4—171no—
152C23×C4—171no—
153C23×C4—171no—
154C22⋊Q8—344no—
155C4×D4—344no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—172no—
158C2×C4⋊C4C2×D1711yescharacteristic
159C2×C4⋊C4C2×D1711yescharacteristic
160C2×C4⋊C4C2×D1712yes—
161C2×C22⋊C4—171no—
162C2×C22⋊C4—171no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C42—171no—
168#95—172no—
169#95—171no—
170#95—172no—
171#95—172no—
172#95—172no—
173#95C2×D1711yescharacteristic
174#95C2×D1712yes—
175C17#263811yescharacteristic, complemented, Sylow
176C24—171no—
177C24—171no—
178C2×Q8—342no—
179C2×Q8—682no—
180C2×D4—342no—
181C2×D4—342no—
182C2×D4—342no—
183C22×C4—171no—
184C22×C4—174no—
185C22×C4—171no—
186C22×C4—172no—
187C22×C4—172no—
188C22×C4—342no—
189C22×C4—682no—
190C22×C4—682no—
191C22×C4C22×D1712yes—
192C22×C4C22×D1712yes—
193C22×C4C22×D1711yescharacteristic
194C22×C4C22×D1711yescharacteristic
195C22×C4C22×D1711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—344no—
200C4⋊C4—682no—
201C22⋊C4—342no—
202C22⋊C4—344no—
203C22⋊C4—344no—
204C22⋊C4—342no—
205C22⋊C4—682no—
206C22⋊C4—682no—
207C42—342no—
208C23—172no—
209C23—172no—
210C23—341no—
211C23—341no—
212C23—341no—
213C23—342no—
214C23—342no—
215C23—341no—
216C23—342no—
217C23—342no—
218C23C23×D1711yescharacteristic
219Q8—684no—
220D4—348no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—172no—
233C2×C4—342no—
234C2×C4—341no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—684no—
243C2×C4—684no—
244C2×C4D4×D1712yes—
245C2×C4#6325412yes—
246C22—342no—
247C22—342no—
248C22—342no—
249C22—342no—
250C22—344no—
251C22—681no—
252C22—681no—
253C22—682no—
254C22—682no—
255C22—682no—
256C22C2×D4×D1711yescharacteristic
257C22C2×#6325411yescharacteristic
258C22#1058769412yes—
259C22#41×D1711yescharacteristic
260C22#1058769612yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—41no—
265C4—42no—
266C4—342no—
267C4—682no—
268C4—681no—
269C4—682no—
270C2—682no—
271C2—682no—
272C2#1176331911yescharacteristic
273C2#1176337011yescharacteristic
274C2#1176338411yescharacteristic
275C2#1176340712yes—
276C2#1176343212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC22×D68C22
hypercenter#398D17
hypocenterC17#2638
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#10587583C2semidirect
3. #11762738C2semidirect
4. #11762735C2semidirect
5. #11762729C2semidirect
7. #11762722C2semidirect
8. #11762695C2semidirect
9. #11762683C2semidirect
10. #11762671C2semidirect
11. C17×#398C2semidirect
134. #398D17semidirect
175. C17#2638semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×#105875792
#117627382
#117627352
#117627292
#117627262
#117627222
#117626952
#117626832
#117626712
C17×#3982
#263817