Finite Group Databasebeta

← #14534657

Subgroups — #14534657

Order 2176.

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

normal subgroups only

This group has 512 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×#10587576C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic
3#11762746C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic, complemented
5#11762743C212yesmaximal, complemented
6#11762735C211yesmaximal, characteristic, complemented
7#11762724C212yesmaximal
8#11762684C212yesmaximal, complemented
9#11762683C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2211yescharacteristic
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2212yes—
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587576—24no—
30#10587567—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—22no—
44C2×C22⋊D17—22no—
45C2×C22⋊D17—22no—
46C22×Dic17—22no—
47C22×Dic17—42no—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2311yescharacteristic
53C2×C4×D17—22no—
54C2×C4×D17—42no—
55C4⋊C4×C17—22no—
56C22⋊C4×C17—24no—
57C22⋊C4×C17—22no—
58C22⋊Dic17—42no—
59#63227—24no—
60#63227—22no—
61#63227—42no—
62C4⋊Dic17—22no—
63C4⋊Dic17—24no—
64C4⋊Dic17—42no—
65#63225—22no—
66#63225—24no—
67#63225—42no—
68C4×Dic17D414yes—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17#4112yes—
74C23×C17C2411yescharacteristic
75C22×D17—21no—
76C22×D17—21no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17#4112yes—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17C2×D412yes—
90C22⋊D17—28no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—44no—
103C2×Dic17C2×D412yes—
104C2×Dic17C2×D412yes—
105C4×D17—44no—
106#2633—171nomaximal, Sylow
107C22×C17—22no—
108C22×C17—24no—
109C22×C17—41no—
110C22×C17—41no—
111C22×C17—42no—
112C22×C17C22×D411yescharacteristic
113C22×C17C2×#4111yescharacteristic
114C22×C17C2×#4111yescharacteristic
115C22×C17H(5,𝔽2)12yes—
116C22×C172−1+412yes—
117C2×D17—22no—
118C2×D17—24no—
119C2×D17—41no—
120C2×D17—41no—
121C2×D17—42no—
122C4×C17—22no—
123C4×C17—42no—
124C4×C17—41no—
125C4×C17—42no—
126Dic17—22no—
127Dic17—22no—
128Dic17—42no—
129Dic17—41no—
130Dic17—42no—
131C2×#125—171no—
132C2×C4×D4—171no—
133#399—171no—
134#399—171no—
135#398—172no—
136#397—171no—
137#397D1711yescharacteristic, complemented
138#394—172no—
139#385—172no—
140#385—172no—
141#383—171no—
142C2×C17—42no—
143C2×C17#52511yescharacteristic
144C2×C17#53511yescharacteristic
145C2×C17#53611yescharacteristic
146C2×C17#53912yes—
147C2×C17#53912yes—
148D17—42no—
149C22×D4—171no—
150C23×C4—171no—
151C23×C4—171no—
152#125—344no—
153C4×D4—344no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—171no—
158C2×C4⋊C4—172no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4C2×D1711yescharacteristic
161C2×C22⋊C4—171no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4C2×D1711yescharacteristic
166C2×C22⋊C4C2×D1712yes—
167C2×C42—171no—
168#95—172no—
169#95—172no—
170#95—171no—
171#95—172no—
172#95C2×D1711yescharacteristic
173#95C2×D1712yes—
174C17#263311yescharacteristic, complemented, Sylow
175C24—171no—
176C24C22×D1711yescharacteristic
177C2×D4—342no—
178C2×D4—342no—
179C2×D4—342no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—172no—
183C22×C4—171no—
184C22×C4—172no—
185C22×C4—342no—
186C22×C4—342no—
187C22×C4—682no—
188C22×C4—682no—
189C22×C4C22×D1712yes—
190C22×C4C22×D1712yes—
191C22×C4C22×D1711yescharacteristic
192C22×C4C22×D1711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—342no—
195C4⋊C4—344no—
196C4⋊C4—344no—
197C4⋊C4—342no—
198C4⋊C4—682no—
199C4⋊C4—682no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—344no—
203C22⋊C4—342no—
204C22⋊C4—682no—
205C22⋊C4—682no—
206C42—174no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—172no—
212C23—341no—
213C23—341no—
214C23—342no—
215C23—342no—
216C23#6325412yes—
217C23C23×D1711yescharacteristic
218D4—348no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—172no—
229C2×C4—172no—
230C2×C4—341no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—684no—
242C2×C4—684no—
243C2×C4D4×D1712yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—342no—
250C22—344no—
251C22—681no—
252C22—681no—
253C22—682no—
254C22C2×D4×D1711yescharacteristic
255C22C2×#6325411yescharacteristic
256C22#41×D1711yescharacteristic
257C22#1058769612yes—
258C22#1058769712yes—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—42no—
263C4—342no—
264C4—342no—
265C4—681no—
266C4—682no—
267C4—682no—
268C2—42no—
269C2—682no—
270C2#1176331911yescharacteristic
271C2#1176337711yescharacteristic
272C2#1176338111yescharacteristic
273C2#1176340012yes—
274C2#1176340312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC22×C22⋊D17C22
hypercenter#397D17
hypocenterC17#2633
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10587576C2semidirect
3. #11762746C2semidirect
4. #11762745C2semidirect
5. #11762743C2semidirect
6. #11762735C2semidirect
8. #11762684C2semidirect
9. #11762683C2semidirect
10. #11762676C2semidirect
11. C17×#397C2semidirect
137. #397D17semidirect
174. C17#2633semidirect

Maximal subgroups

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

subgroupindex
C2×#105875762
C2×#105875672
#117627462
#117627452
#117627432
#117627352
#117627242
#117626842
#117626832
#117626762
C17×#3972
#263317