Finite Group Databasebeta

← #14533268

Subgroups — #14533268

Order 2176.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587632C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic
3#11762750C214yesmaximal, complemented
4#11762716C211yesmaximal, characteristic, complemented
5#11762675C211yesmaximal, characteristic, complemented
6#11762674C211yesmaximal, characteristic, complemented
7#11762671C214yesmaximal
8C4×#63227C211yesmaximal, characteristic
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×C4×D17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15#10587632—24no—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2211yescharacteristic
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C4⋊C4×D17—24no—
24C17×#95C2214yes—
25#10587502C2214yes—
26#10587502C2212yes—
27#10587502C2214yes—
28#10587502C2214yes—
29C23×D17C2311yescharacteristic
30C2×Q8×C17—22no—
31C2×Q8×C17—22no—
32C2×Q8×C17—22no—
33C22×C4×C17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2311yescharacteristic
41C2×C4×D17—22no—
42C2×C4×D17—22no—
43C2×C4×D17—24no—
44C2×C4×D17—28no—
45C2×C4×D17D414yes—
46C4⋊C4×C17—22no—
47C42×C17—22no—
48#63227—24no—
49#63227—24no—
50C4⋊Dic17—22no—
51C4⋊Dic17—22no—
52#63225—24no—
53#63225—24no—
54C4×Dic17—22no—
55C23×C17C2411yescharacteristic
56C22×D17—21no—
57C22×D17—21no—
58C22×D17—22no—
59C22×D17C2×D412yes—
60C22×D17#4114yes—
61Q8×C17—44no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—24no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17C2×D412yes—
71C2×C4×C17#4112yes—
72C2×C4×C17#4112yes—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—24no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—24no—
81C2×Dic17—22no—
82C2×Dic17C2×D412yes—
83C4×D17—28no—
84C4×D17—48no—
85#2503—171nomaximal, Sylow
86C22×C17C22×D411yescharacteristic
87C22×C17C2×#4111yescharacteristic
88C22×C17C2×#4111yescharacteristic
89C22×C17C2×#4112yes—
90C22×C172−1+412yes—
91C2×D17—22no—
92C2×D17—22no—
93C2×D17—24no—
94C2×D17—24no—
95C2×D17#12414yes—
96C4×C17—42no—
97C4×C17—42no—
98C4×C17—42no—
99C4×C17#12314yes—
100Dic17—22no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104C2×C22⋊Q8—171no—
105C22×C4⋊C4—171no—
106#400—174no—
107#391—171no—
108#391D1711yescharacteristic, complemented
109#382—171no—
110#382—171no—
111#381—174no—
112C4×C22⋊C4—171no—
113C2×C17C2×#12311yescharacteristic
114C2×C17C2×#12411yescharacteristic
115C2×C17#53611yescharacteristic
116C2×C17#55212yes—
117C2×C17#55212yes—
118D17—24no—
119C22×Q8C2×D1711yescharacteristic
120C23×C4—171no—
121C23×C4—172no—
122C22⋊Q8—344no—
123C2×C4⋊C4—171no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—171no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—344no—
128C2×C4⋊C4C2×D1711yescharacteristic
129C2×C22⋊C4—172no—
130C2×C22⋊C4—172no—
131C2×C42—171no—
132C2×C42C2×D1711yescharacteristic
133#95—172no—
134#95—174no—
135#95—174no—
136#95—174no—
137#95C2×D1714yes—
138C17#250311yescharacteristic, complemented, Sylow
139C24—171no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C22×C4—171no—
144C22×C4—174no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4—344no—
149C22×C4—348no—
150C22×C4—342no—
151C22×C4—342no—
152C22×C4C22×D1712yes—
153C22×C4C22×D1712yes—
154C22×C4C22×D1712yes—
155C22×C4C22×D1711yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—344no—
158C4⋊C4—344no—
159C4⋊C4—342no—
160C4⋊C4—342no—
161C22⋊C4—344no—
162C22⋊C4—344no—
163C42—22no—
164C42—342no—
165C23—172no—
166C23—174no—
167C23—341no—
168C23—341no—
169C23—342no—
170C23C23×D1711yescharacteristic
171Q8—44no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—172no—
181C2×C4—341no—
182C2×C4—341no—
183C2×C4—348no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—344no—
189C2×C4—342no—
190C2×C4—344no—
191C2×C4—688no—
192C2×C4#6325212yes—
193C2×C4D4×D1712yes—
194C2×C4#6325412yes—
195C22—174no—
196C22—344no—
197C22—344no—
198C22—342no—
199C22—342no—
200C22C2×#6325211yescharacteristic
201C22C2×D4×D1711yescharacteristic
202C22C2×#6325411yescharacteristic
203C22#1058769412yes—
204C22#41×D1712yes—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—342no—
209C4—682no—
210C4—682no—
211C4—682no—
212C4#1058756914yes—
213C2—344no—
214C2C2×#1058756911yescharacteristic
215C2#124×D1711yescharacteristic
216C2#1176338411yescharacteristic
217C2#1176349312yes—
218C2#1176350012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC23×D17C23
hypercenter#391D17
hypocenterC17#2503
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #11762750C2semidirect
4. #11762716C2semidirect
5. #11762675C2semidirect
6. #11762674C2semidirect
9. C17×#391C2semidirect
108. #391D17semidirect
138. C17#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
C2×C4⋊C4×D172
#117627502
#117627162
#117626752
#117626742
#117626712
C4×#632272
C17×#3912
#250317