Finite Group Databasebeta

← #16927808

Subgroups — #16927808

Order 4352.

418 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#14531282C211yesmaximal, characteristic
2#14528861C211yesmaximal, characteristic
3#14526638C214yesmaximal
4C17×#1822C211yesmaximal, characteristic
5#11762702—22no—
6#11762695—22no—
7#11762672C2212yes—
8C4×C4⋊Dic17C412yescomplemented
9C4×C22⋊Dic17C2212yes—
10#11762655C411yescharacteristic, complemented
11C42×Dic17C411yescharacteristic, complemented
12#11762576—22no—
13C2×C17×#117C2211yescharacteristic
14C17×#407C2212yes—
15C17×#343—22no—
16C23×C4×C17C2311yescharacteristic
17C2×M16×C17—22no—
18C17×#117—24no—
19C2×C4⋊C4×C17D412yes—
20C2×C22⋊C4×C17D412yes—
21C2×C42×C17C2×C411yescharacteristic
22C2×C42×C17C2×C411yescharacteristic
23C2×C22⋊Dic17—22no—
24C2×C22⋊Dic17—22no—
25C2×C4⋊Dic17—22no—
26C2×C4×Dic17—22no—
27C2×C4×Dic17C2×C412yes—
28C2×C4×Dic17C2×C412yes—
29C17×C22⋊C8—24no—
30#10587502—22no—
31#10587502—22no—
32#10587502—22no—
33#10587502—24no—
34C24×C17C2×D411yescharacteristic
35C22×C4×C17—42no—
36C22×C4×C17C22⋊C412yes—
37C22×C4×C17C22⋊C412yes—
38C22×C4×C17C22×C411yescharacteristic
39C22×C4×C17C2×D411yescharacteristic
40C22×C4×C17#4114yes—
41C22×Dic17—22no—
42C22×Dic17—22no—
43C22×Dic17—42no—
44M16×C17—44no—
45C2×C8×C17—42no—
46C4⋊C4×C17—24no—
47C22⋊C4×C17—24no—
48C42×C17—22no—
49C42×C17—22no—
50C22⋊Dic17—44no—
51C22⋊Dic17—44no—
52C4⋊Dic17—44no—
53C4×Dic17—44no—
54C4×Dic17—42no—
55C4×Dic17—42no—
56#8741—171nomaximal, Sylow
57C23×C17—22no—
58C23×C17—42no—
59C23×C17C2×C22⋊C411yescharacteristic
60C23×C17#12312yes—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—44no—
66C2×C4×C17—42no—
67C2×C4×C17—42no—
68C2×C4×C17#11712yes—
69C2×C4×C17#12312yes—
70C2×Dic17—24no—
71C2×Dic17—42no—
72C2×Dic17—42no—
73C2×Dic17—42no—
74C2×Dic17—42no—
75C2×Dic17—42no—
76C2×Dic17—82no—
77C2×Dic17—84no—
78C8×C17—82no—
79#2298—171no—
80#1822D1711yescharacteristic
81#1820—171no—
82#1284—174no—
83C22×C17—22no—
84C22×C17—22no—
85C22×C17—24no—
86C22×C17—82no—
87C22×C17#38111yescharacteristic
88C22×C17#41012yes—
89C4×C17—22no—
90C4×C17—22no—
91C4×C17—42no—
92C4×C17—42no—
93Dic17—44no—
94Dic17—82no—
95Dic17—162no—
96C2×#117C2×D1711yescharacteristic
97#407C2×D1712yes—
98#388—342no—
99#387—342no—
100#381—172no—
101C4×C4⋊C4—172no—
102C4×C22⋊C4—172no—
103#376—171no—
104C43—171no—
105#343—22no—
106#342—342no—
107C2×C17—22no—
108C2×C17—42no—
109C2×C17#182011yescharacteristic
110C2×C17#182712yes—
111C23×C4C22×D1711yescharacteristic
112C2×M16—22no—
113#117—24no—
114C2×C4⋊C4—342no—
115C2×C4⋊C4C22⋊D1712yes—
116C2×C22⋊C4—342no—
117C2×C22⋊C4—342no—
118C2×C22⋊C4D6812yes—
119C2×C42—172no—
120C2×C42—172no—
121C2×C42—342no—
122C2×C42C4×D1711yescharacteristic
123C2×C42C4×D1711yescharacteristic
124C22⋊C8—24no—
125#95—342no—
126#95—342no—
127#95—342no—
128#95—344no—
129C17#874111yescharacteristic, complemented, Sylow
130C24C2×D6811yescharacteristic
131C22×C4—42no—
132C22×C4—342no—
133C22×C4—342no—
134C22×C4—682no—
135C22×C4#6322712yes—
136C22×C4#6322712yes—
137C22×C4C2×C4×D1711yescharacteristic
138C22×C4#6325414yes—
139C22×C4C2×C22⋊D1711yescharacteristic
140M16—44no—
141C2×C8—42no—
142C4⋊C4—24no—
143C4⋊C4—684no—
144C22⋊C4—24no—
145C22⋊C4—684no—
146C22⋊C4—684no—
147C42—22no—
148C42—22no—
149C42—682no—
150C42—682no—
151C42—684no—
152C23—22no—
153C23—42no—
154C23#1058757212yes—
155C23C2×#6322711yescharacteristic
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—344no—
164C2×C4—682no—
165C2×C4—682no—
166C2×C4—682no—
167C2×C4—682no—
168C2×C4—682no—
169C2×C4—1362no—
170C2×C4—1364no—
171C2×C4#1058756312yes—
172C2×C4#1058762112yes—
173C8—82no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#1176267011yescharacteristic
179C22#1176278212yes—
180C4—22no—
181C4—22no—
182C4—42no—
183C4—42no—
184C4—2722no—
185C4—684no—
186C4—1362no—
187C2—22no—
188C2—42no—
189C2#1452885911yescharacteristic
190C2#1452888912yes—
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1822C2
socleC22×C17#381
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1822D17
hypocenterC17#8741
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
8. C4×C4⋊Dic17C4semidirect
10. #11762655C4semidirect
11. C42×Dic17C4semidirect
129. C17#8741semidirect

Maximal subgroups

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

subgroupindex
#145312822
#145288612
#145266382
C17×#18222
#874117