Finite Group Databasebeta

← #14535124

Subgroups — #14535124

Order 2176.

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

normal subgroups only

This group has 568 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×#10587580C212yesmaximal
2C2×#10587582C212yesmaximal, complemented
3#11762740C212yesmaximal, complemented
4#11762735C212yesmaximal, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762726C211yesmaximal, characteristic
7#11762700C212yesmaximal, complemented
8#11762680C212yesmaximal, complemented
9C17×#384C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C42×C17C2211yescharacteristic
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26#10587582—28no—
27#10587580—28no—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2211yescharacteristic
32#10587502C2212yes—
33C23×D17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×D68—24no—
43C2×D68—41no—
44C2×C4×D17—24no—
45C2×C4×D17—24no—
46C2×C4×D17—42no—
47C2×Dic34—24no—
48C2×Dic34—41no—
49C4⋊C4×C17—24no—
50C4⋊C4×C17—24no—
51C4⋊C4×C17—41no—
52C4⋊C4×C17—41no—
53C42×C17—22no—
54#63227—24no—
55#63227—24no—
56#63227—24no—
57#63227—24no—
58#63227D418yes—
59#63225—24no—
60#63225—24no—
61#63225—24no—
62C23×C17C2411yescharacteristic
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17C2×D414yes—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17C2×D414yes—
82C2×C4×C17#4112yes—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17C2×D414yes—
92D68—44no—
93C4×D17—48no—
94Dic34—44no—
95#2680—171nomaximal, Sylow
96C22×C17C22×D412yes—
97C22×C17C2×#4111yescharacteristic
98C22×C17H(5,𝔽2)11yescharacteristic
99C22×C172−1+411yescharacteristic
100C22×C172−1+412yes—
101C2×D17—24no—
102C2×D17—24no—
103C2×D17—24no—
104C2×D17—42no—
105C2×D17—42no—
106C2×D17—42no—
107C2×D17—42no—
108C4×C17—22no—
109C4×C17—24no—
110C4×C17—41no—
111C4×C17—42no—
112C4×C17—41no—
113Dic17—24no—
114Dic17—42no—
115Dic17—42no—
116C2×#123—172no—
117C2×C22⋊Q8—172no—
118#398—172no—
119#397—172no—
120#395—171no—
121#394—171no—
122#388—172no—
123#385—172no—
124#384D1711yescharacteristic, complemented
125C2×C17#53612yes—
126C2×C17#53911yescharacteristic
127C2×C17D4211yescharacteristic
128C2×C17#54712yes—
129C2×C17#56311yescharacteristic
130D17—44no—
131C22×Q8—171no—
132C22×D4—171no—
133C23×C4—172no—
134#123—348no—
135C22⋊Q8—348no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4C2×D1712yes—
140C2×C4⋊C4C2×D1712yes—
141C2×C4⋊C4C2×D1711yescharacteristic
142C2×C4⋊C4C2×D1711yescharacteristic
143C2×C22⋊C4—172no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C42C2×D1711yescharacteristic
149#95—172no—
150#95—172no—
151#95—172no—
152#95—171no—
153#95—171no—
154C17#268011yescharacteristic, complemented, Sylow
155C24—172no—
156C2×Q8—344no—
157C2×Q8—681no—
158C2×D4—344no—
159C2×D4—681no—
160C22×C4—172no—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—344no—
164C22×C4—344no—
165C22×C4—682no—
166C22×C4C22×D1711yescharacteristic
167C22×C4C22×D1712yes—
168C22×C4C22×D1711yescharacteristic
169C22×C4C22×D1711yescharacteristic
170C22×C4C22×D1712yes—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—344no—
176C4⋊C4—344no—
177C4⋊C4—344no—
178C22⋊C4—178no—
179C22⋊C4—344no—
180C22⋊C4—344no—
181C22⋊C4—344no—
182C22⋊C4—344no—
183C42—22no—
184C23—174no—
185C23—342no—
186C23—342no—
187C23—342no—
188C23—342no—
189C23—342no—
190C23—342no—
191C23C23×D1711yescharacteristic
192Q8—684no—
193D4—684no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—174no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—688no—
215C2×C4#6325212yes—
216C2×C4D4×D1714yes—
217C22—344no—
218C22—344no—
219C22—344no—
220C22—682no—
221C22—682no—
222C22—682no—
223C22—682no—
224C22C2×#6325211yescharacteristic
225C22C2×D4×D1712yes—
226C22#1058769111yescharacteristic
227C22#1058769412yes—
228C22#1058769411yescharacteristic
229C4—24no—
230C4—22no—
231C4—42no—
232C4—41no—
233C4—41no—
234C4—344no—
235C4—682no—
236C4—682no—
237C2—684no—
238C2#1176338412yes—
239C2#1176340611yescharacteristic
240C2#1176344411yescharacteristic
241C2#1176346012yes—
242C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#384C2
prime radicalC22×D68C22
hypercenter#384D17
hypocenterC17#2680
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#10587582C2semidirect
3. #11762740C2semidirect
4. #11762735C2semidirect
5. #11762729C2semidirect
7. #11762700C2semidirect
8. #11762680C2semidirect
9. C17×#384C2semidirect
124. #384D17semidirect
154. C17#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#105875802
C2×#105875822
#117627402
#117627352
#117627292
#117627262
#117627002
#117626802
C17×#3842
#268017