Finite Group Databasebeta

← #28347010

Subgroups — #28347010

Order 13696.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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#20293184C211yesmaximal, characteristic
2#20293037C214yesmaximal, complemented
3#20293026C214yesmaximal, complemented
4#20292667C214yesmaximal
5C4×#13133802C211yesmaximal, characteristic
6C2×#125×C107C211yesmaximal, characteristic
7C2×#13133802C2212yes—
8#16812109C2212yes—
9#16812101C2212yes—
10#16812088—22no—
11C4×Dic214—22no—
12#16812068C2212yes—
13#16812042C2218yes—
14#16812040C2218yes—
15#16812020C2212yes—
16C4×C107⋊C8C2212yes—
17#125×C107—22no—
18#125×C107C2212yes—
19#125×C107C2212yes—
20C2×C4⋊C4×C107—22no—
21C2×C4⋊C4×C107C2212yes—
22C2×C42×C107C2211yescharacteristic
23C2×Dic214—22no—
24#13133802Q818yescomplemented
25C2×C107⋊C8C2314yes—
26C4⋊Dic107—22no—
27C4⋊Dic107C2314yes—
28Dic107⋊C4—24no—
29C22⋊Dic107—22no—
30C4×Dic107—22no—
31C22×C4×C107—22no—
32C22×C4×C107D412yes—
33C22×C4×C107C2311yescharacteristic
34C4⋊C4×C107—24no—
35C4⋊C4×C107—24no—
36C4⋊C4×C107—22no—
37C4⋊C4×C107C2314yes—
38C42×C107D412yes—
39C42×C107C2311yescharacteristic
40C42×C107C2311yescharacteristic
41C23×C107C2×D411yescharacteristic
42C2×C4×C107—24no—
43C2×C4×C107—42no—
44C2×C4×C107—44no—
45C2×C4×C107C2×D412yes—
46C2×C4×C107C2×D412yes—
47C2×C4×C107C2×D411yescharacteristic
48C2×C4×C107C2×Q814yes—
49C2×C4×C107C2411yescharacteristic
50C2×Dic107—24no—
51Dic214—44no—
52C107⋊C8C2×Q818yes—
53C22×C107—22no—
54C22×C107C4⋊Q812yes—
55C22×C107C22×D411yescharacteristic
56C4×C107—22no—
57C4×C107—22no—
58C4×C107—44no—
59C4×C107C4⋊Q812yes—
60C4×C107C22×Q812yes—
61Dic107—44no—
62C2×C107—22no—
63C2×C107C2×C4⋊Q811yescharacteristic
64C2×C107#57512yes—
65#3166—1071nomaximal, Sylow
66C107#316611yescharacteristic, complemented, Sylow
67#533—1071no—
68C2×#125D10711yescharacteristic
69#501—1074no—
70#499—1074no—
71#428—1074no—
72C4×M16—1071no—
73C2×M16—1072no—
74C4⋊Q8—1072no—
75#125—22no—
76#125—1072no—
77#125C2×D10712yes—
78#125C2×D10712yes—
79C22⋊Q8—2142no—
80C4×Q8—2142no—
81#117—1072no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4C2×D10712yes—
84C2×C42C2×D10711yescharacteristic
85#107—1078no—
86#106—1078no—
87#97—1072no—
88C4×C8—1072no—
89C2×Q8—2142no—
90C22×C4—22no—
91C22×C4C22⋊D10712yes—
92C22×C4C22×D10711yescharacteristic
93M16—1078no—
94C2×C8—1074no—
95C4⋊C4—24no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—1074no—
99C4⋊C4—2144no—
100C4⋊C4—2142no—
101C4⋊C4C22×D10714yes—
102C22⋊C4—2142no—
103C42—2142no—
104C42C22⋊D10712yes—
105C42C22×D10711yescharacteristic
106C42C22×D10711yescharacteristic
107C23C2×C22⋊D10711yescharacteristic
108Q8—4284no—
109C2×C4—24no—
110C2×C4—42no—
111C2×C4—44no—
112C2×C4—2144no—
113C2×C4C2×C22⋊D10712yes—
114C2×C4C2×C22⋊D10712yes—
115C2×C4C2×C22⋊D10711yescharacteristic
116C2×C4Q8×D10714yes—
117C2×C4C23×D10711yescharacteristic
118C8—1078no—
119C22—22no—
120C22#1681210712yes—
121C22C22×C22⋊D10711yescharacteristic
122C4—22no—
123C4—22no—
124C4—44no—
125C4—4284no—
126C4#1681210712yes—
127C4C2×Q8×D10712yes—
128C2—22no—
129C2C2×#1681210711yescharacteristic
130C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#125×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenterC2×#125D107
hypocenterC107#3166
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20293037C2semidirect
3. #20293026C2semidirect
24. #13133802Q8semidirect
66. C107#3166semidirect

Maximal subgroups

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

subgroupindex
#202931842
#202930372
#202930262
#202926672
C4×#131338022
C2×#125×C1072
#3166107