Finite Group Databasebeta

← #28335542

Subgroups — #28335542

Order 13696.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293343C211yesmaximal, characteristic, complemented
2#20292819C212yesmaximal, complemented
3#20292376C211yesmaximal, characteristic, complemented
4#20292356C212yesmaximal, complemented
5#492×C107C211yesmaximal, characteristic, complemented
6C2×#13133803—22no—
7#16812098—22no—
8#16812084—22no—
9C4×D428C2212yes—
10#16812067—22no—
11C4⋊(C107⋊C8)C2212yes—
12#16812030—22no—
13C2×SD16×C107—22no—
14C4⋊Q8×C107C2211yescharacteristic
15#127×C107C2211yescharacteristic
16C4×C8×C107C2211yescharacteristic
17C2×D428—22no—
18C2×D428D412yes—
19C2×C4×D107—22no—
20#13133803—44no—
21C2×C107⋊C8—22no—
22C4⋊Dic107D412yes—
23(C2×D107)⋊C4—24no—
24(C2×D107)⋊C4—22no—
25C4×Dic107—22no—
26C2×Q8×C107D412yes—
27C2×D4×C107—22no—
28C2×D4×C107—21no—
29SD16×C107—44no—
30C2×C8×C107—21no—
31C4⋊C4×C107—22no—
32C42×C107C2311yescharacteristic
33C23×C107—41no—
34C22×D107—22no—
35Q8×C107—42no—
36D4×C107—42no—
37D4×C107—42no—
38D4×C107—42no—
39C2×C4×C107—22no—
40C2×C4×C107C2×D412yes—
41C2×C4×C107C2×D411yescharacteristic
42C2×Dic107—22no—
43D428—22no—
44D428—22no—
45D428—82no—
46C4×D107—42no—
47C4×D107—42no—
48C8×C107—22no—
49C107⋊C8—42no—
50C22×C107—82no—
51C22×C107—81no—
52C22×C107#12011yescharacteristic
53C2×D107—42no—
54C2×D107—42no—
55C2×D107—42no—
56C4×C107—21no—
57C4×C107—42no—
58C4×C107#13512yes—
59C4×C107AGL(1,ℤ/8)12yes—
60Dic107—42no—
61C2×C107—161no—
62C2×C107#44912yes—
63C2×C107#45311yescharacteristic
64D107—82no—
65#1624—1071nomaximal, Sylow
66C107#162411yescharacteristic, complemented, Sylow
67#555—1071no—
68#492D10711yescharacteristic, complemented
69#459—1072no—
70#333—1071no—
71D4⋊C8—1072no—
72C2×SD16—22no—
73C2×D8—2142no—
74C4⋊Q8C2×D10711yescharacteristic
75#127C2×D10711yescharacteristic
76#124—2142no—
77#121—2142no—
78C4×D4—1072no—
79#117—2142no—
80C4⋊C8—1072no—
81#102—2142no—
82C4×C8C2×D10711yescharacteristic
83C2×Q8C22⋊D10712yes—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—1072no—
87C2×D4—2142no—
88C22×C4—2142no—
89SD16—44no—
90D8—4284no—
91C2×C8—21no—
92C2×C8—2142no—
93C4⋊C4—22no—
94C4⋊C4—1072no—
95C22⋊C4—2144no—
96C22⋊C4—2142no—
97C42—2142no—
98C42C22×D10711yescharacteristic
99C23—41no—
100C23—2142no—
101Q8—42no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—8562no—
106D4—2142no—
107D4—2142no—
108C2×C4—22no—
109C2×C4—4282no—
110C2×C4—4282no—
111C2×C4—2142no—
112C2×C4D4×D10712yes—
113C2×C4C2×C22⋊D10711yescharacteristic
114C8—22no—
115C8—4282no—
116C22—82no—
117C22—81no—
118C22—4282no—
119C22—4282no—
120C22—4282no—
121C22#1681207711yescharacteristic
122C4—21no—
123C4—42no—
124C4—4282no—
125C4#1681213012yes—
126C4#1681213612yes—
127C2—161no—
128C2—8562no—
129C2#2029277112yes—
130C2#2029279411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#492×C107C2
socleC22×C107#120
hypercenter#492D107
hypocenterC107#1624
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20293343C2semidirect
2. #20292819C2semidirect
3. #20292376C2semidirect
4. #20292356C2semidirect
5. #492×C107C2semidirect
66. C107#1624semidirect
68. #492D107semidirect

Maximal subgroups

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

subgroupindex
#202933432
#202928192
#202923762
#202923562
#492×C1072
#1624107