Finite Group Databasebeta

← #14530129

Subgroups — #14530129

Order 2176.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#11763440C211yesmaximal, characteristic
2#11762849C212yesmaximal
3#11762578C211yesmaximal, characteristic
4#11762569C212yesmaximal
5C17×#384C211yesmaximal, characteristic
6C2×C4⋊C4×C17—22no—
7C2×C4⋊C4×C17C2211yescharacteristic
8C2×C4⋊C4×C17C2211yescharacteristic
9C2×C42×C17C2211yescharacteristic
10#10587612C2212yes—
11#10587610—22no—
12C2×#63223C2212yes—
13#10587576—22no—
14#10587574—22no—
15#10587479—22no—
16#10587478—22no—
17C22×C4×C17—21no—
18C22×C4×C17—21no—
19C22×C4×C17D412yes—
20C22×C4×C17C2311yescharacteristic
21C2×Dic34—22no—
22C4⋊C4×C17—22no—
23C4⋊C4×C17—22no—
24C4⋊C4×C17—44no—
25C42×C17—21no—
26C42×C17Q812yescomplemented
27C22⋊Dic17—22no—
28C4⋊Dic17—22no—
29C4⋊Dic17D412yes—
30C4⋊Dic17D412yes—
31#63225—24no—
32C4×Dic17—22no—
33#63223—22no—
34#63223Q814yescomplemented
35C2×#3622—22no—
36C23×C17C2×D411yescharacteristic
37C2×C4×C17—22no—
38C2×C4×C17—22no—
39C2×C4×C17—22no—
40C2×C4×C17—22no—
41C2×C4×C17—42no—
42C2×C4×C17—42no—
43C2×C4×C17—41no—
44C2×C4×C17C2×D412yes—
45C2×C4×C17C2×Q811yescharacteristic
46C2×C4×C17C2×Q812yes—
47C2×C4×C17#4111yescharacteristic
48C2×Dic17—22no—
49C2×Dic17—22no—
50Dic34—82no—
51#3622—24no—
52#2060—171nomaximal, Sylow
53C22×C17—21no—
54C22×C17—21no—
55C22×C17#12011yescharacteristic
56C22×C17C22⋊Q811yescharacteristic
57C22×C17C4⋊Q811yescharacteristic
58C4×C17—24no—
59C4×C17—42no—
60C4×C17—81no—
61C4×C17C22⋊Q812yes—
62C4×C17C4⋊Q812yes—
63Dic17—42no—
64Dic17—42no—
65#544—171no—
66#428—172no—
67#384D1711yescharacteristic
68#343—171no—
69#339—172no—
70C2×C17—21no—
71C2×C17—21no—
72C2×C17#39511yescharacteristic
73C2×C17#45311yescharacteristic
74C2×C17#45611yescharacteristic
75C2×M16—172no—
76C4⋊Q8—342no—
77#125—342no—
78C22⋊Q8—342no—
79#117—172no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4C2×D1711yescharacteristic
82C2×C4⋊C4C2×D1711yescharacteristic
83C2×C42C2×D1711yescharacteristic
84#107—342no—
85#106—342no—
86C17#206011yescharacteristic, complemented, Sylow
87C2×Q8—342no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C22⋊D1712yes—
91C22×C4C22×D1711yescharacteristic
92M16—174no—
93M16—342no—
94C2×C8—342no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—44no—
98C4⋊C4—172no—
99C4⋊C4—172no—
100C4⋊C4—342no—
101C4⋊C4—344no—
102C22⋊C4—342no—
103C42—21no—
104C42—342no—
105C42Dic3412yescomplemented
106C23C2×C22⋊D1711yescharacteristic
107Q8—1362no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—342no—
116C2×C4—342no—
117C2×C4C2×Dic3411yescharacteristic
118C2×C4#6325211yescharacteristic
119C2×C4D4×D1712yes—
120C2×C4Q8×D1712yes—
121C8—344no—
122C22—21no—
123C22—21no—
124C22#1058761011yescharacteristic
125C22#1058762311yescharacteristic
126C22#1058763011yescharacteristic
127C4—24no—
128C4—42no—
129C4—81no—
130C4—682no—
131C4—682no—
132C4#1058757412yes—
133C4#1058758212yes—
134C2—21no—
135C2—21no—
136C2#1176272911yescharacteristic
137C2#1176297611yescharacteristic
138C2#1176299111yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#384C2
socleC22×C17#120
prime radicalC23×C17C2×D4
hypercenter#384D17
hypocenterC17#2060
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
26. C42×C17Q8semidirect
34. #63223Q8semidirect
86. C17#2060semidirect
105. C42Dic34semidirect

Maximal subgroups

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

subgroupindex
#117634402
#117628492
#117625782
#117625692
C17×#3842
#206017