Finite Group Databasebeta

← #14533239

Subgroups — #14533239

Order 2176.

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

normal subgroups only

This group has 484 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×#10587578C211yesmaximal, characteristic
3#11762736C214yesmaximal
4#11762716C211yesmaximal, characteristic, complemented
5#11762692C214yesmaximal, complemented
6#11762679C211yesmaximal, characteristic, complemented
7#11762674C211yesmaximal, characteristic, complemented
8C4×#63227C211yesmaximal, characteristic
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×C4×D17C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14#10587632—24no—
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×C4⋊Dic17C2214yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×C4×Dic17C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24#10587578—24no—
25C17×#95C2214yes—
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×D17D414yes—
44C4⋊C4×C17—22no—
45C42×C17—22no—
46#63227—24no—
47#63227—24no—
48#63227—24no—
49C4⋊Dic17—22no—
50C4⋊Dic17—22no—
51C4⋊Dic17—28no—
52#63225—24no—
53C4×Dic17—22no—
54C4×Dic17—24no—
55C23×C17C2411yescharacteristic
56C22×D17—21no—
57C22×D17—21no—
58C22×D17—22no—
59C22×D17—22no—
60C22×D17C2×D412yes—
61Q8×C17—44no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—24no—
69C2×C4×C17C2×D412yes—
70C2×C4×C17#4112yes—
71C2×C4×C17#4112yes—
72C2×C4×C17#4114yes—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—24no—
79C2×Dic17—24no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17C2×D412yes—
83C4×D17—28no—
84#2499—171nomaximal, Sylow
85C22×C17C22×D411yescharacteristic
86C22×C17C2×#4111yescharacteristic
87C22×C17C2×#4111yescharacteristic
88C22×C17C2×#4112yes—
89C22×C172−1+412yes—
90C2×D17—22no—
91C2×D17—22no—
92C2×D17—22no—
93C2×D17—42no—
94C2×D17—42no—
95C4×C17—24no—
96C4×C17—42no—
97C4×C17—42no—
98C4×C17#12314yes—
99Dic17—22no—
100Dic17—42no—
101Dic17—42no—
102Dic17—42no—
103C2×C22⋊Q8—171no—
104C2×#117—171no—
105#397—174no—
106#391—171no—
107#391D1711yescharacteristic, complemented
108#387—174no—
109#384—171no—
110#382—171no—
111C4×C22⋊C4—171no—
112C2×C17C2×#12311yescharacteristic
113C2×C17#53211yescharacteristic
114C2×C17#53611yescharacteristic
115C2×C17#55212yes—
116C2×C17#55612yes—
117D17—42no—
118C22×Q8C2×D1711yescharacteristic
119C23×C4—171no—
120C22⋊Q8—344no—
121#117—344no—
122C2×C4⋊C4—171no—
123C2×C4⋊C4—171no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—174no—
126C2×C4⋊C4C2×D1711yescharacteristic
127C2×C22⋊C4—172no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4—172no—
130C2×C42—172no—
131C2×C42—171no—
132C2×C42C2×D1711yescharacteristic
133#95—172no—
134#95—174no—
135#95—174no—
136#95C2×D1714yes—
137C17#249911yescharacteristic, complemented, Sylow
138C24—171no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4—171no—
143C22×C4—174no—
144C22×C4—172no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—342no—
148C22×C4—342no—
149C22×C4C22×D1712yes—
150C22×C4C22×D1712yes—
151C22×C4C22×D1712yes—
152C22×C4C22×D1711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—344no—
155C4⋊C4—342no—
156C4⋊C4—342no—
157C4⋊C4—348no—
158C22⋊C4—344no—
159C22⋊C4—344no—
160C22⋊C4—344no—
161C42—22no—
162C42—344no—
163C42—342no—
164C23—172no—
165C23—341no—
166C23—341no—
167C23—342no—
168C23—342no—
169C23C23×D1711yescharacteristic
170Q8—44no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—172no—
179C2×C4—341no—
180C2×C4—341no—
181C2×C4—348no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—344no—
187C2×C4—344no—
188C2×C4—342no—
189C2×C4#6325212yes—
190C2×C4D4×D1712yes—
191C2×C4#6325412yes—
192C2×C4#6325614yes—
193C22—342no—
194C22—342no—
195C22—342no—
196C22—682no—
197C22—682no—
198C22C2×#6325211yescharacteristic
199C22C2×D4×D1711yescharacteristic
200C22C2×#6325411yescharacteristic
201C22C2×#6325612yes—
202C22#1058769712yes—
203C4—24no—
204C4—42no—
205C4—42no—
206C4—342no—
207C4—682no—
208C4—682no—
209C4—682no—
210C4#1058756914yes—
211C2—682no—
212C2C2×#1058756911yescharacteristic
213C2#1176336011yescharacteristic
214C2#1176337911yescharacteristic
215C2#1176350112yes—
216C2#1176353212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC23×D17C23
hypercenter#391D17
hypocenterC17#2499
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #11762716C2semidirect
5. #11762692C2semidirect
6. #11762679C2semidirect
7. #11762674C2semidirect
9. C17×#391C2semidirect
107. #391D17semidirect
137. C17#2499semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
C2×#105875782
#117627362
#117627162
#117626922
#117626792
#117626742
C4×#632272
C17×#3912
#249917