Finite Group Databasebeta

← #14534728

Subgroups — #14534728

Order 2176.

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

normal subgroups only

This group has 584 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×#10587626C211yesmaximal, characteristic
2C2×#10587576C211yesmaximal, characteristic, complemented
3#11762725C214yesmaximal, complemented
4#11762724C214yesmaximal, complemented
5#11762712C211yesmaximal, characteristic, complemented
6#11762711C211yesmaximal, characteristic, complemented
7#11762710C211yesmaximal, characteristic, complemented
8#11762655C211yesmaximal, characteristic, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2214yes—
16C2×C42×C17C2211yescharacteristic
17C2×C22⋊Dic17C2214yes—
18#10587626—24no—
19C2×#63227C2214yes—
20C2×#63227C2214yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2212yes—
23C2×#63225C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25C2×C4×Dic17C2211yescharacteristic
26#10587576—24no—
27#10587502C2212yes—
28#10587502C2212yes—
29C24×C17C2312yes—
30C23×D17C2312yes—
31C2×D4×C17—22no—
32C2×D4×C17—22no—
33C2×D4×C17—22no—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C2×C22⋊D17—24no—
38C2×C22⋊D17—24no—
39C2×C22⋊D17—44no—
40C2×C22⋊D17—44no—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2311yescharacteristic
45C2×D68—22no—
46C2×D68—22no—
47C2×D68—22no—
48C4⋊C4×C17—22no—
49C22⋊C4×C17—44no—
50C42×C17—22no—
51C22⋊Dic17—44no—
52#63227—44no—
53#63227—44no—
54C4⋊Dic17—22no—
55C4⋊Dic17—24no—
56#63225—24no—
57C4×Dic17—22no—
58C4×Dic17D414yes—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—24no—
63C23×C17—24no—
64C23×C17C2411yescharacteristic
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—24no—
69C22×D17—24no—
70D4×C17—44no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17—22no—
76C2×C4×C17C2×D412yes—
77C2×C4×C17#4112yes—
78C2×C4×C17#4112yes—
79C22⋊D17—48no—
80C22⋊D17—48no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17C2×D412yes—
89C2×Dic17C2×D412yes—
90D68—44no—
91#2641—171nomaximal, Sylow
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—44no—
96C22×C17—44no—
97C22×C17C22×D411yescharacteristic
98C22×C17C2×#4111yescharacteristic
99C22×C17C2×#4111yescharacteristic
100C22×C17H(5,𝔽2)12yes—
101C22×C17H(5,𝔽2)12yes—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C2×D17—44no—
106C2×D17—44no—
107C4×C17—42no—
108C4×C17—42no—
109C4×C17#12414yes—
110Dic17—22no—
111Dic17—22no—
112Dic17—42no—
113Dic17—42no—
114C2×#127—171no—
115C2×#125—171no—
116#394—174no—
117#394—174no—
118#390—171no—
119#390—171no—
120#390—171no—
121#390D1711yescharacteristic, complemented
122#376—171no—
123C2×C17—82no—
124C2×C17C2×#12411yescharacteristic
125C2×C17#53511yescharacteristic
126C2×C17#53511yescharacteristic
127C2×C17#54012yes—
128C2×C17#54012yes—
129D17—82no—
130C22×D4—172no—
131C22×D4—172no—
132C22×D4—171no—
133C22×D4C2×D1711yescharacteristic
134#127—344no—
135#125—344no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4C2×D1711yescharacteristic
140C2×C22⋊C4—174no—
141C2×C22⋊C4—174no—
142C2×C22⋊C4—174no—
143C2×C22⋊C4C2×D1714yes—
144C2×C42—171no—
145C2×C42—171no—
146C2×C42C2×D1711yescharacteristic
147#95—172no—
148#95—172no—
149C17#264111yescharacteristic, complemented, Sylow
150C24—172no—
151C24C22×D1712yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—344no—
156C2×D4—344no—
157C2×D4—342no—
158C2×D4—342no—
159C2×D4—342no—
160C2×D4—684no—
161C2×D4—684no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—172no—
165C22×C4—172no—
166C22×C4C22×D1712yes—
167C22×C4C22×D1712yes—
168C22×C4C22×D1711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—344no—
171C4⋊C4—344no—
172C4⋊C4—342no—
173C22⋊C4—44no—
174C22⋊C4—684no—
175C22⋊C4—684no—
176C22⋊C4—684no—
177C42—22no—
178C42—174no—
179C42—342no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—24no—
185C23—342no—
186C23—342no—
187C23—342no—
188C23—344no—
189C23—344no—
190C23C23×D1711yescharacteristic
191D4—44no—
192D4—688no—
193D4—688no—
194D4—684no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—172no—
201C2×C4—172no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—344no—
208C2×C4—342no—
209C2×C4#6325212yes—
210C2×C4D4×D1712yes—
211C2×C4#6325412yes—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22—44no—
217C22—682no—
218C22—682no—
219C22—682no—
220C22—684no—
221C22—684no—
222C22C2×#6325211yescharacteristic
223C22C2×D4×D1711yescharacteristic
224C22C2×#6325411yescharacteristic
225C22#1058769112yes—
226C22#1058769612yes—
227C4—42no—
228C4—42no—
229C4—342no—
230C4—342no—
231C4—682no—
232C4—682no—
233C4#1058757114yes—
234C2—82no—
235C2—1362no—
236C2C2×#1058757111yescharacteristic
237C2#1176337211yescharacteristic
238C2#1176337711yescharacteristic
239C2#1176341012yes—
240C2#1176341212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC2×#10587626C2
hypercenter#390D17
hypocenterC17#2641
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#10587576C2semidirect
3. #11762725C2semidirect
4. #11762724C2semidirect
5. #11762712C2semidirect
6. #11762711C2semidirect
7. #11762710C2semidirect
8. #11762655C2semidirect
9. C17×#390C2semidirect
121. #390D17semidirect
149. C17#2641semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
C2×#105875762
#117627252
#117627242
#117627122
#117627112
#117627102
#117626552
C17×#3902
#264117