Finite Group Databasebeta

← #16922158

Subgroups — #16922158

Order 4352.

148 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 27 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#14526953C211yesmaximal, characteristic, complemented
2#14526823C211yesmaximal, characteristic, complemented
3C17×#1634C211yesmaximal, characteristic
4#11762630—21no—
5#11762608—21no—
6C17×#555C411yescharacteristic
7C17×#494C2211yescharacteristic
8C17×#476—21no—
9C17×#333C411yescharacteristic
10C17×D4⋊C8—21no—
11C2×Q16×C17—41no—
12C17×C4⋊Q8C2×C411yescharacteristic
13C17×C4⋊Q8D411yescharacteristic
14C17×#124—41no—
15C17×#121—41no—
16C4×D4×C17—21no—
17C17×#117—41no—
18C17×#106—41no—
19C17×C4⋊C8—21no—
20C17×#102—41no—
21C4×C8×C17D411yescharacteristic
22#10587507—41no—
23#10587483—41no—
24C2×Q8×C17—21no—
25C2×Q8×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17—41no—
28C22×C4×C17—41no—
29Q16×C17—81no—
30Q16×C17—81no—
31C2×C8×C17—21no—
32C2×C8×C17—41no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17—41no—
35C4⋊C4×C17—41no—
36C22⋊C4×C17—42no—
37C22⋊C4×C17—41no—
38C42×C17—41no—
39C42×C17C22⋊C411yescharacteristic
40#63223—81no—
41#63214—81no—
42#7521—171nomaximal, Sylow
43C23×C17—41no—
44Q8×C17—81no—
45Q8×C17—81no—
46D4×C17—41no—
47D4×C17—41no—
48D4×C17—161no—
49C2×C4×C17—21no—
50C2×C4×C17—41no—
51C2×C4×C17—41no—
52C2×C4×C17—41no—
53C2×C4×C17—81no—
54C2×C4×C17—81no—
55C2×C4×C17#9911yescharacteristic
56C8×C17—21no—
57C8×C17—21no—
58C8×C17—81no—
59#3622—161no—
60#1634D1711yescharacteristic
61#1411—171no—
62#1358—171no—
63C22×C17—81no—
64C22×C17—81no—
65C22×C17—81no—
66C22×C17AΣL(1,𝔽16)11yescharacteristic
67C4×C17—21no—
68C4×C17—21no—
69C4×C17—21no—
70C4×C17—81no—
71C4×C17—81no—
72C4×C17—81no—
73#555Dic1711yescharacteristic
74#494C2×D1711yescharacteristic
75#476—21no—
76#364—341no—
77#356—341no—
78#333Dic1711yescharacteristic
79D4⋊C8—21no—
80C2×C17—21no—
81C2×C17—161no—
82C2×C17#140211yescharacteristic
83C2×Q16—41no—
84C4⋊Q8C2×Dic1711yescharacteristic
85C4⋊Q8C22⋊D1711yescharacteristic
86#124—41no—
87#121—41no—
88C4×D4—21no—
89#117—41no—
90M32—681no—
91#106—41no—
92C4⋊C8—21no—
93#102—41no—
94#101—681no—
95C4×C8C22⋊D1711yescharacteristic
96C17#752111yescharacteristic, complemented, Sylow
97C2×Q8—21no—
98C2×Q8—21no—
99C2×D4—21no—
100C2×D4—41no—
101C22×C4—41no—
102Q16—81no—
103Q16—81no—
104M16—1361no—
105C2×C8—21no—
106C2×C8—41no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—42no—
111C22⋊C4—41no—
112C42—41no—
113C42C22⋊Dic1711yescharacteristic
114C16—1361no—
115C23—41no—
116Q8—81no—
117Q8—81no—
118D4—41no—
119D4—41no—
120D4—161no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4#1058750511yescharacteristic
128C8—21no—
129C8—21no—
130C8—81no—
131C8—2721no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22#1176259611yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—81no—
140C4—81no—
141C4—81no—
142C2—21no—
143C2—161no—
144C2#1452692911yescharacteristic
subgroupG/H
centerC2#14526929
commutatorC17×C4⋊Q8C2×C4
Frattini#494C2×D17
FittingC17×#1634C2
socleC2×C17#1402
prime radicalC17×#555C4
cosocleC17×#494C22
hypercenter#1634D17
hypocenterC17#7521
LCSG ▷ C17×C4⋊Q8 ▷ C42×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ #1634

Complements

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

NH ≅ G/Ntype
1. #14526953C2semidirect
2. #14526823C2semidirect
96. C17#7521semidirect

Maximal subgroups

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

subgroupindex
#145269532
#145268232
C17×#16342
#752117