Finite Group Databasebeta

← #14530134

Subgroups — #14530134

Order 2176.

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

normal subgroups only

This group has 230 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×#382C211yesmaximal, characteristic
6C2×C4⋊C4×C17C2211yescharacteristic
7C2×C4⋊C4×C17C2211yescharacteristic
8C2×C42×C17C2211yescharacteristic
9#10587612C2212yes—
10#10587610—22no—
11C2×#63223C2212yes—
12#10587576—22no—
13#10587574—22no—
14C17×#95—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—
24C42×C17—21no—
25C42×C17Q812yescomplemented
26C22⋊Dic17—22no—
27C4⋊Dic17—22no—
28C4⋊Dic17D412yes—
29C4⋊Dic17D412yes—
30#63225—24no—
31C4×Dic17—22no—
32#63223—22no—
33#63223Q814yescomplemented
34C2×#3622—22no—
35C23×C17C2×D411yescharacteristic
36C2×C4×C17—22no—
37C2×C4×C17—22no—
38C2×C4×C17—41no—
39C2×C4×C17—41no—
40C2×C4×C17—42no—
41C2×C4×C17—42no—
42C2×C4×C17—41no—
43C2×C4×C17C2×D412yes—
44C2×C4×C17C2×Q811yescharacteristic
45C2×C4×C17C2×Q812yes—
46C2×C4×C17#4111yescharacteristic
47C2×Dic17—22no—
48C2×Dic17—22no—
49Dic34—82no—
50#3622—24no—
51#2061—171nomaximal, Sylow
52C22×C17—21no—
53C22×C17—21no—
54C22×C17#12011yescharacteristic
55C22×C17C22⋊Q811yescharacteristic
56C22×C17C4⋊Q811yescharacteristic
57C4×C17—42no—
58C4×C17—42no—
59C4×C17—81no—
60C4×C17C22⋊Q812yes—
61C4×C17C4⋊Q812yes—
62Dic17—42no—
63Dic17—42no—
64#544—171no—
65#428—172no—
66#382D1711yescharacteristic
67#343—171no—
68#339—172no—
69C2×C17—21no—
70C2×C17—21no—
71C2×C17#39511yescharacteristic
72C2×C17#45411yescharacteristic
73C2×C17#45511yescharacteristic
74C2×M16—172no—
75C4⋊Q8—342no—
76#125—342no—
77C22⋊Q8—342no—
78#117—172no—
79C2×C4⋊C4C2×D1711yescharacteristic
80C2×C4⋊C4C2×D1711yescharacteristic
81C2×C42C2×D1711yescharacteristic
82#107—342no—
83#106—342no—
84#95—22no—
85C17#206111yescharacteristic, complemented, Sylow
86C2×Q8—342no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4C22⋊D1712yes—
90C22×C4C22×D1711yescharacteristic
91M16—174no—
92M16—342no—
93C2×C8—342no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—172no—
97C4⋊C4—172no—
98C4⋊C4—342no—
99C4⋊C4—344no—
100C22⋊C4—342no—
101C42—21no—
102C42—342no—
103C42Dic3412yescomplemented
104C23C2×C22⋊D1711yescharacteristic
105Q8—1362no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—342no—
114C2×C4—342no—
115C2×C4C2×Dic3411yescharacteristic
116C2×C4#6325211yescharacteristic
117C2×C4D4×D1712yes—
118C2×C4Q8×D1712yes—
119C8—344no—
120C22—21no—
121C22—21no—
122C22#1058761011yescharacteristic
123C22#1058762311yescharacteristic
124C22#1058763011yescharacteristic
125C4—42no—
126C4—42no—
127C4—81no—
128C4—682no—
129C4—682no—
130C4#1058757412yes—
131C4#1058758212yes—
132C2—21no—
133C2—21no—
134C2#1176272911yescharacteristic
135C2#1176298111yescharacteristic
136C2#1176298611yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#382C2
socleC22×C17#120
prime radicalC23×C17C2×D4
hypercenter#382D17
hypocenterC17#2061
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
25. C42×C17Q8semidirect
33. #63223Q8semidirect
85. C17#2061semidirect
103. 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×#3822
#206117