Finite Group Databasebeta

← #14533054

Subgroups — #14533054

Order 2176.

672 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#63227C211yesmaximal, characteristic
2#11762744C212yesmaximal, complemented
3#11762735C212yesmaximal, complemented
4#11762700C214yesmaximal, complemented
5#11762681C212yesmaximal, complemented
6#11762672C211yesmaximal, characteristic, complemented
7#11762664C212yesmaximal
8C2×C17×#117C211yesmaximal, characteristic, complemented
9C24×D17C2211yescharacteristic
10C23×C4×C17C2211yescharacteristic
11C23×Dic17C2211yescharacteristic
12C17×#117—24no—
13C2×C4⋊C4×C17C2211yescharacteristic, complemented
14C2×C4⋊C4×C17C2211yescharacteristic, complemented
15C2×C22⋊C4×C17C2212yes—
16C2×C42×C17C2212yescomplemented
17C2×C22⋊Dic17C2212yes—
18C2×#63227—24no—
19C2×#63227C2212yes—
20C2×#63227C2214yes—
21C2×#63227C2212yes—
22C2×#63227C2214yes—
23C2×#63225C2214yescomplemented
24#10587502C2212yescomplemented
25#10587502C2214yescomplemented
26#10587502C2212yescomplemented
27C24×C17C2311yescharacteristic
28C23×D17—22no—
29C23×D17—24no—
30C23×D17—24no—
31C23×D17—42no—
32C23×D17C2312yes—
33C22×C4×C17—24no—
34C22×C4×C17D414yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17—22no—
40C22×Dic17—42no—
41C22×Dic17C2312yes—
42C22×Dic17C2314yes—
43C4⋊C4×C17—22no—
44C4⋊C4×C17—22no—
45C22⋊C4×C17—24no—
46C42×C17—24no—
47C22⋊Dic17—24no—
48#63227—28no—
49#63227—24no—
50#63227—24no—
51#63227—44no—
52#63225—44no—
53C23×C17—22no—
54C23×C17—22no—
55C23×C17C2×D412yes—
56C23×C17#4114yes—
57C23×C17C2411yescharacteristic
58C22×D17—28no—
59C22×D17—22no—
60C22×D17—24no—
61C22×D17—22no—
62C22×D17—24no—
63C22×D17—416no—
64C22×D17—44no—
65C22×D17—44no—
66C22×D17#4114yes—
67C2×C4×C17—28no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—24no—
73C2×C4×C17—24no—
74C2×C4×C17C2×D412yes—
75C2×C4×C17C2×D412yes—
76C2×Dic17—22no—
77C2×Dic17—24no—
78C2×Dic17—24no—
79C2×Dic17—24no—
80C2×Dic17—24no—
81C2×Dic17—44no—
82#2482—171nomaximal, Sylow
83C22×C17—22no—
84C22×C17—22no—
85C22×C17—28no—
86C22×C17#12314yes—
87C22×C17C22×D411yescharacteristic
88C22×C17C2×#4112yes—
89C22×C17C2×#4112yes—
90C22×C17H(5,𝔽2)12yes—
91C2×D17—24no—
92C2×D17—28no—
93C2×D17—416no—
94C2×D17—42no—
95C2×D17—42no—
96C2×D17—44no—
97C4×C17—22no—
98C4×C17—22no—
99C4×C17—42no—
100C4×C17—42no—
101Dic17—42no—
102Dic17—44no—
103C2×#117D1711yescharacteristic, complemented
104C22×C22⋊C4—171no—
105#399—172no—
106#397—172no—
107#388—174no—
108#385—172no—
109#381—171no—
110#379—172no—
111C2×C17—24no—
112C2×C17C2×#12311yescharacteristic
113C2×C17#52511yescharacteristic
114C2×C17#53511yescharacteristic
115C2×C17#55114yes—
116D17—44no—
117C25—171no—
118C23×C4—171no—
119C23×C4C2×D1711yescharacteristic
120#117—24no—
121C2×C4⋊C4—174no—
122C2×C4⋊C4C2×D1711yescharacteristic, complemented
123C2×C4⋊C4C2×D1711yescharacteristic, complemented
124C2×C22⋊C4—172no—
125C2×C22⋊C4—172no—
126C2×C22⋊C4—172no—
127C2×C22⋊C4—174no—
128C2×C22⋊C4—174no—
129C2×C22⋊C4—344no—
130C2×C22⋊C4C2×D1712yes—
131C2×C42C2×D1712yescomplemented
132#95—172no—
133#95—172no—
134#95—174no—
135C17#248211yescharacteristic, complemented, Sylow
136C24—172no—
137C24—344no—
138C24—342no—
139C24—344no—
140C24—682no—
141C24C22×D1711yescharacteristic
142C22×C4—24no—
143C22×C4—174no—
144C22×C4—172no—
145C22×C4—342no—
146C22×C4—682no—
147C22×C4C22⋊D1714yes—
148C22×C4C22×D1712yes—
149C22×C4C22×D1712yes—
150C22×C4C22×D1711yescharacteristic
151C22×C4C22×D1711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—684no—
155C22⋊C4—24no—
156C22⋊C4—348no—
157C22⋊C4—344no—
158C22⋊C4—344no—
159C22⋊C4—344no—
160C22⋊C4—684no—
161C42—24no—
162C23—22no—
163C23—22no—
164C23—174no—
165C23—348no—
166C23—344no—
167C23—342no—
168C23—344no—
169C23—342no—
170C23—6816no—
171C23—684no—
172C23—684no—
173C23#6325214yes—
174C23C2×C22⋊D1712yes—
175C23C23×D1711yescharacteristic
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—28no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—344no—
184C2×C4—344no—
185C2×C4—344no—
186C2×C4—344no—
187C2×C4—342no—
188C2×C4—684no—
189C2×C4C2×C22⋊D1712yes—
190C2×C4C2×C22⋊D1712yes—
191C22—28no—
192C22—22no—
193C22—22no—
194C22—348no—
195C22—344no—
196C22—6816no—
197C22—684no—
198C22—682no—
199C22—682no—
200C22#1058761514yes—
201C22C2×#6325212yes—
202C22#41×D1712yes—
203C22#1058769612yes—
204C22C22×C22⋊D1711yescharacteristic
205C4—22no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—684no—
210C4—682no—
211C2—24no—
212C2—684no—
213C2C2×#1058761511yescharacteristic
214C2#1176331611yescharacteristic
215C2#1176337311yescharacteristic
216C2#1176348314yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#117C2
prime radicalC24×D17C22
hypercenterC2×#117D17
hypocenterC17#2482
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #11762744C2semidirect
3. #11762735C2semidirect
4. #11762700C2semidirect
5. #11762681C2semidirect
6. #11762672C2semidirect
8. C2×C17×#117C2semidirect
13. C2×C4⋊C4×C17C22semidirect
14. C2×C4⋊C4×C17C22semidirect
16. C2×C42×C17C22semidirect
23. C2×#63225C22semidirect
24. #10587502C22semidirect
25. #10587502C22semidirect
26. #10587502C22semidirect
103. C2×#117D17semidirect
122. C2×C4⋊C4C2×D17semidirect
123. C2×C4⋊C4C2×D17semidirect
131. C2×C42C2×D17semidirect
135. C17#2482semidirect

Maximal subgroups

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

subgroupindex
C22×#632272
#117627442
#117627352
#117627002
#117626812
#117626722
#117626642
C2×C17×#1172
#248217