Finite Group Databasebeta

← #16927807

Subgroups — #16927807

Order 4352.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#14528867C211yesmaximal, characteristic
2#14528861C211yesmaximal, characteristic
3#14526637C214yesmaximal
4C17×#2298C211yesmaximal, characteristic
5#11762773C2212yes—
6#11762672C2212yes—
7#11762578—22no—
8#11762576—22no—
9C2×C17×#117C2211yescharacteristic
10C17×#388—22no—
11C17×#387—22no—
12C4×C4⋊C4×C17C412yescomplemented
13C4×C22⋊C4×C17C2212yes—
14C17×#376C411yescharacteristic, complemented
15C43×C17C411yescharacteristic, complemented
16C23×C4×C17C2311yescharacteristic
17C17×#117—24no—
18C2×C4⋊C4×C17—22no—
19C2×C4⋊C4×C17D412yes—
20C2×C22⋊C4×C17—22no—
21C2×C22⋊C4×C17D412yes—
22C2×C42×C17—22no—
23C2×C42×C17C2×C411yescharacteristic
24C2×C42×C17C2×C411yescharacteristic
25C2×C42×C17C2×C412yes—
26C2×C42×C17C2×C412yes—
27C2×C22⋊Dic17—22no—
28C2×#63223—22no—
29C17×#95—22no—
30C17×#95—22no—
31C17×#95—22no—
32#10587502—24no—
33C22⋊#3622—24no—
34C24×C17C2×D411yescharacteristic
35C22×C4×C17—22no—
36C22×C4×C17—22no—
37C22×C4×C17—42no—
38C22×C4×C17C22⋊C412yes—
39C22×C4×C17C22⋊C412yes—
40C22×C4×C17C22×C411yescharacteristic
41C22×C4×C17C2×D411yescharacteristic
42C22×C4×C17#4114yes—
43C22×Dic17—42no—
44C4⋊C4×C17—24no—
45C4⋊C4×C17—44no—
46C22⋊C4×C17—24no—
47C22⋊C4×C17—44no—
48C42×C17—22no—
49C42×C17—22no—
50C42×C17—42no—
51C42×C17—42no—
52C42×C17—44no—
53C22⋊Dic17—44no—
54#63223—44no—
55C2×#3622—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—24no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—44no—
67C2×C4×C17—42no—
68C2×C4×C17—42no—
69C2×C4×C17—42no—
70C2×C4×C17—42no—
71C2×C4×C17—42no—
72C2×C4×C17—42no—
73C2×C4×C17—42no—
74C2×C4×C17#11712yes—
75C2×C4×C17#12312yes—
76C2×Dic17—82no—
77C2×Dic17—84no—
78#3622—82no—
79#2298D1711yescharacteristic
80#1822—171no—
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—44no—
92C4×C17—42no—
93C4×C17—42no—
94C4×C17—82no—
95Dic17—162no—
96C2×#117C2×D1711yescharacteristic
97#407—172no—
98#388—22no—
99#387—22no—
100#381—172no—
101C4×C4⋊C4Dic1712yescomplemented
102C4×C22⋊C4C2×D1712yes—
103#376Dic1711yescharacteristic, complemented
104C43Dic1711yescharacteristic, complemented
105#343—342no—
106#342—342no—
107C2×C17—22no—
108C2×C17—42no—
109C2×C17#182011yescharacteristic
110C2×C17#182712yes—
111C23×C4C22×D1711yescharacteristic
112C2×M16—342no—
113#117—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4C22⋊D1712yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—342no—
118C2×C22⋊C4C22⋊D1712yes—
119C2×C42—22no—
120C2×C42C2×Dic1711yescharacteristic
121C2×C42C2×Dic1711yescharacteristic
122C2×C42C2×Dic1712yes—
123C2×C42C2×Dic1712yes—
124C22⋊C8—344no—
125#95—22no—
126#95—22no—
127#95—22no—
128#95—344no—
129C17#874111yescharacteristic, complemented, Sylow
130C24C2×C22⋊D1711yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—682no—
135C22×C4C22⋊Dic1712yes—
136C22×C4C22⋊Dic1712yes—
137C22×C4#6325214yes—
138C22×C4C22×Dic1711yescharacteristic
139C22×C4C2×C22⋊D1711yescharacteristic
140M16—684no—
141C2×C8—682no—
142C4⋊C4—24no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—684no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—42no—
151C42—44no—
152C23—22no—
153C23—42no—
154C23#1058761512yes—
155C23C2×C22⋊Dic1711yescharacteristic
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—1362no—
170C2×C4—1364no—
171C2×C4#1058761212yes—
172C2×C4#1058761512yes—
173C8—1362no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#1176267211yescharacteristic
179C22#1176278312yes—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—2722no—
187C2—22no—
188C2—42no—
189C2#1452886111yescharacteristic
190C2#1452888812yes—
subgroupG/H
centerC22#11762672
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2298C2
socleC22×C17#381
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2298D17
hypocenterC17#8741
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
12. C4×C4⋊C4×C17C4semidirect
14. C17×#376C4semidirect
15. C43×C17C4semidirect
101. C4×C4⋊C4Dic17semidirect
103. #376Dic17semidirect
104. C43Dic17semidirect
129. C17#8741semidirect

Maximal subgroups

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

subgroupindex
#145288672
#145288612
#145266372
C17×#22982
#874117