Finite Group Databasebeta

← #28335575

Subgroups — #28335575

Order 13696.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#20293262C211yesmaximal, characteristic
2#20292914C212yesmaximal, complemented
3#20292371C211yesmaximal, characteristic, complemented
4#20292356C212yesmaximal, complemented
5#492×C107C211yesmaximal, characteristic, complemented
6C2×D4×D107—22no—
7#16812080—22no—
8#16812077—22no—
9C4×D428C2212yes—
10#16812040—22no—
11C4⋊(C107⋊C8)C2212yescomplemented
12#16812029—22no—
13C2×SD16×C107—22no—
14C4⋊Q8×C107C2211yescharacteristic, complemented
15#127×C107C2211yescharacteristic
16C4×C8×C107C2211yescharacteristic, complemented
17C23×D107—22no—
18C2×D428D412yes—
19C2×C22⋊D107—24no—
20D4×D107—44no—
21D4×D107—44no—
22C2×C4×D107—22no—
23C2×C107⋊C8—22no—
24C4⋊Dic107D412yes—
25C22⋊Dic107—22no—
26(C2×D107)⋊C4—22no—
27C2×Q8×C107—21no—
28C2×D4×C107—22no—
29C2×D4×C107D412yes—
30SD16×C107—44no—
31C2×C8×C107—21no—
32C4⋊C4×C107—22no—
33C42×C107C2311yescharacteristic
34C23×C107—22no—
35C22×D107—22no—
36C22×D107—48no—
37C22×D107—82no—
38Q8×C107—42no—
39D4×C107—42no—
40D4×C107—42no—
41D4×C107—42no—
42C2×C4×C107—41no—
43C2×C4×C107C2×D412yes—
44C2×C4×C107C2×D411yescharacteristic
45C22⋊D107—48no—
46C2×Dic107—22no—
47D428—22no—
48D428SD1614yescomplemented
49C4×D107—42no—
50C4×D107—42no—
51C8×C107—22no—
52C107⋊C8—42no—
53C22×C107—44no—
54C22×C107—42no—
55C22×C107#12011yescharacteristic
56C2×D107—24no—
57C2×D107—42no—
58C2×D107—42no—
59C2×D107—88no—
60C4×C107—21no—
61C4×C107—81no—
62C4×C107C2×SD1612yes—
63C4×C107AGL(1,ℤ/8)12yes—
64Dic107—42no—
65C2×C107—82no—
66C2×C107#45012yes—
67C2×C107#45311yescharacteristic
68D107—44no—
69#1629—1071nomaximal, Sylow
70C107#162911yescharacteristic, complemented, Sylow
71D42—1071no—
72#492D10711yescharacteristic, complemented
73#476—1072no—
74#331—1071no—
75D4⋊C8—1072no—
76C22×D4—2142no—
77C2×SD16—22no—
78C4⋊Q8C2×D10711yescharacteristic, complemented
79#127C2×D10711yescharacteristic
80#121—2142no—
81#120—2142no—
82C4×D4—1072no—
83#106—2142no—
84C4⋊C8—1072no—
85#102—2142no—
86C4×C8C2×D10711yescharacteristic, complemented
87C24—2142no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—1072no—
91C2×D4—4284no—
92C2×D4—4284no—
93C2×D4—2144no—
94C2×D4C22⋊D10712yes—
95C22×C4—2142no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—2142no—
99C4⋊C4—22no—
100C4⋊C4—1072no—
101C22⋊C4—2142no—
102C22⋊C4—2142no—
103C42C22×D10711yescharacteristic
104C23—22no—
105C23—8562no—
106C23—4288no—
107C23—2142no—
108Q8—42no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—1074no—
113D4—4288no—
114D4—2142no—
115C2×C4—41no—
116C2×C4—4282no—
117C2×C4—4282no—
118C2×C4—2142no—
119C2×C4D4×D10712yes—
120C2×C4C2×C22⋊D10711yescharacteristic
121C8—22no—
122C8—4282no—
123C22—44no—
124C22—42no—
125C22—8568no—
126C22—4282no—
127C22—4282no—
128C22—2144no—
129C22#1681207711yescharacteristic
130C4—21no—
131C4—81no—
132C4—4282no—
133C4SD16×D10712yes—
134C4#1681213612yes—
135C2—82no—
136C2—4284no—
137C2#2029277612yes—
138C2#2029279411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#492×C107C2
socleC22×C107#120
prime radical#20293262C2
hypercenter#492D107
hypocenterC107#1629
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20292914C2semidirect
3. #20292371C2semidirect
4. #20292356C2semidirect
5. #492×C107C2semidirect
11. C4⋊(C107⋊C8)C22semidirect
14. C4⋊Q8×C107C22semidirect
16. C4×C8×C107C22semidirect
48. D428SD16semidirect
70. C107#1629semidirect
72. #492D107semidirect
78. C4⋊Q8C2×D107semidirect
86. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932622
#202929142
#202923712
#202923562
#492×C1072
#1629107