Finite Group Databasebeta

← #16927817

Subgroups — #16927817

Order 4352.

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

normal subgroups only

This group has 384 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#14527118C212yesmaximal
2#14526640C214yesmaximal, complemented
3C17×#2298C211yesmaximal, characteristic
4#11762866C2214yes—
5#11762535—22no—
6#11762533—22no—
7C2×C17×#117C2211yescharacteristic
8C17×#388—22no—
9C17×#387—22no—
10C4×C4⋊C4×C17C2212yes—
11C4×C22⋊C4×C17C412yes—
12C17×#376C411yescharacteristic
13C43×C17C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C17×#117—24no—
16C2×C4⋊C4×C17—22no—
17C2×C4⋊C4×C17D412yes—
18C2×C22⋊C4×C17—22no—
19C2×C22⋊C4×C17D412yes—
20C2×C42×C17—22no—
21C2×C42×C17C2×C411yescharacteristic
22C2×C42×C17C2×C412yes—
23C2×C42×C17C2×C412yes—
24C2×C42×C17C2311yescharacteristic
25C17×#95—22no—
26C17×#95—22no—
27C17×#95—22no—
28C22⋊#3622—24no—
29C4⋊#3622—24no—
30#10587474—24no—
31C24×C17C22⋊C411yescharacteristic
32C22×C4×C17—22no—
33C22×C4×C17—22no—
34C22×C4×C17—22no—
35C22×C4×C17—42no—
36C22×C4×C17C22⋊C411yescharacteristic
37C22×C4×C17C22⋊C412yes—
38C22×C4×C17C22×C411yescharacteristic
39C22×C4×C17C2×D412yes—
40C4⋊C4×C17—24no—
41C4⋊C4×C17—44no—
42C22⋊C4×C17—24no—
43C22⋊C4×C17—44no—
44C42×C17—22no—
45C42×C17—42no—
46C42×C17—42no—
47C42×C17—44no—
48C42×C17#4114yes—
49C2×#3622—44no—
50#8744—171nomaximal, Sylow
51C23×C17—21no—
52C23×C17—22no—
53C23×C17—42no—
54C23×C17C2×C22⋊C411yescharacteristic
55C2×C4×C17—22no—
56C2×C4×C17—24no—
57C2×C4×C17—22no—
58C2×C4×C17—21no—
59C2×C4×C17—22no—
60C2×C4×C17—42no—
61C2×C4×C17—44no—
62C2×C4×C17—42no—
63C2×C4×C17—42no—
64C2×C4×C17—42no—
65C2×C4×C17—42no—
66C2×C4×C17—42no—
67C2×C4×C17—42no—
68C2×C4×C17#11712yes—
69C2×C4×C17#12314yes—
70#3622—84no—
71#2298D1711yescharacteristic
72#1464—172no—
73#1285—174no—
74C22×C17—22no—
75C22×C17—41no—
76C22×C17—42no—
77C22×C17—82no—
78C22×C17#38111yescharacteristic
79C22×C17#41312yes—
80C4×C17—22no—
81C4×C17—24no—
82C4×C17—44no—
83C4×C17—41no—
84C4×C17—42no—
85C4×C17—82no—
86C2×#117C2×D1711yescharacteristic
87#432—174no—
88#388—22no—
89#387—22no—
90C4×C4⋊C4C2×D1712yes—
91C4×C22⋊C4Dic1712yes—
92#376Dic1711yescharacteristic
93C43Dic1711yescharacteristic
94#324—342no—
95#323—342no—
96C2×C17—22no—
97C2×C17—81no—
98C2×C17#182211yescharacteristic
99C2×C17#182712yes—
100C23×C4C2×Dic1711yescharacteristic
101#117—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C22⋊D1712yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4C22⋊D1712yes—
106C2×C42—22no—
107C2×C42C2×Dic1711yescharacteristic
108C2×C42C2×Dic1712yes—
109C2×C42C2×Dic1712yes—
110C2×C42C22×D1711yescharacteristic
111C4⋊C8—344no—
112C22⋊C8—344no—
113#97—344no—
114#95—22no—
115#95—22no—
116#95—22no—
117C17#874411yescharacteristic, complemented, Sylow
118C24C22⋊Dic1711yescharacteristic
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4C22⋊Dic1711yescharacteristic
124C22×C4C22⋊Dic1712yes—
125C22×C4C22×Dic1711yescharacteristic
126C22×C4C2×C22⋊D1712yes—
127C2×C8—684no—
128C4⋊C4—24no—
129C4⋊C4—44no—
130C22⋊C4—24no—
131C22⋊C4—44no—
132C42—22no—
133C42—42no—
134C42—42no—
135C42—44no—
136C42#6325214yes—
137C23—21no—
138C23—22no—
139C23—42no—
140C23C2×C22⋊Dic1711yescharacteristic
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4#1058761212yes—
155C2×C4#1058761514yes—
156C8—1364no—
157C22—22no—
158C22—41no—
159C22—42no—
160C22—82no—
161C22#1176267211yescharacteristic
162C22#1176279312yes—
163C4—22no—
164C4—24no—
165C4—44no—
166C4—41no—
167C4—42no—
168C4—82no—
169C2—22no—
170C2—81no—
171C2#1452886711yescharacteristic
172C2#1452888812yes—
subgroupG/H
centerC22#11762672
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2298C2
socleC22×C17#381
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2298D17
hypocenterC17#8744
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2298

Complements

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

NH ≅ G/Ntype
2. #14526640C2semidirect
117. C17#8744semidirect

Maximal subgroups

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

subgroupindex
#145271182
#145266402
C17×#22982
#874417