Finite Group Databasebeta

← #28335544

Subgroups — #28335544

Order 13696.

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

normal subgroups only

This group has 240 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#20293001C211yesmaximal, characteristic, complemented
2#20292822C212yesmaximal, complemented
3#20292376C211yesmaximal, characteristic, complemented
4D4⋊(C107⋊C8)C212yesmaximal, complemented
5#555×C107C211yesmaximal, characteristic, complemented
6C2×#13133805—22no—
7C2×#13133803—22no—
8#16812106C2211yescharacteristic
9C4⋊(C107⋊C8)C2212yes—
10#16812029—22no—
11C4×C107⋊C8C2211yescharacteristic
12C4⋊Q8×C107C2211yescharacteristic
13#124×C107—22no—
14#121×C107—22no—
15C4×D4×C107C2212yes—
16#117×C107—22no—
17C2×D428—22no—
18C2×D428—21no—
19#13133805—44no—
20#13133803—44no—
21C2×C107⋊C8—21no—
22C2×C107⋊C8—22no—
23C2×Q8×C107D412yes—
24C2×D4×C107—22no—
25C2×D4×C107D412yes—
26C22×C4×C107—22no—
27C4⋊C4×C107—22no—
28C4⋊C4×C107D412yes—
29C22⋊C4×C107—24no—
30C22⋊C4×C107—22no—
31C42×C107—22no—
32C42×C107C2311yescharacteristic
33C23×C107—22no—
34C22×D107—41no—
35Q8×C107—42no—
36D4×C107—22no—
37D4×C107—22no—
38D4×C107—82no—
39C2×C4×C107—22no—
40C2×C4×C107—22no—
41C2×C4×C107—42no—
42C2×C4×C107—42no—
43C2×C4×C107C2×D411yescharacteristic
44C2×C4×C107C2×D412yes—
45D428—42no—
46D428—42no—
47D428—42no—
48C107⋊C8—22no—
49C107⋊C8—42no—
50C22×C107—42no—
51C22×C107—42no—
52C22×C107—42no—
53C22×C107#12011yescharacteristic
54C2×D107—82no—
55C2×D107—81no—
56C4×C107—21no—
57C4×C107—42no—
58C4×C107—42no—
59C4×C107#13512yes—
60C4×C107AGL(1,ℤ/8)12yes—
61C2×C107—82no—
62C2×C107#44912yes—
63C2×C107#45311yescharacteristic
64D107—161no—
65#1624—1071nomaximal, Sylow
66C107#162411yescharacteristic, complemented, Sylow
67#555D10711yescharacteristic, complemented
68#492—1071no—
69#459—1072no—
70#333—1071no—
71D4⋊C8—1072no—
72C2×SD16—2142no—
73C2×D8—2142no—
74C4⋊Q8C2×D10711yescharacteristic
75#127—1071no—
76#124—22no—
77#121—22no—
78C4×D4C2×D10712yes—
79#117—22no—
80C4⋊C8—1072no—
81#102—2142no—
82C4×C8—1071no—
83C2×Q8C22⋊D10712yes—
84C2×D4—22no—
85C2×D4—2142no—
86C2×D4—2141no—
87C2×D4C22⋊D10712yes—
88C22×C4—22no—
89SD16—4284no—
90D8—4284no—
91C2×C8—2142no—
92C2×C8—2141no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D10712yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D10711yescharacteristic
99C23—22no—
100C23—4281no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—4282no—
106D4—4282no—
107D4—4282no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D10711yescharacteristic
113C2×C4C2×C22⋊D10712yes—
114C8—4282no—
115C8—2142no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—8562no—
120C22—8561no—
121C22#1681207811yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1681212912yes—
126C4#1681213412yes—
127C2—82no—
128C2—17121no—
129C2#2029277212yes—
130C2#2029279511yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#555×C107C2
socleC22×C107#120
hypercenter#555D107
hypocenterC107#1624
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20293001C2semidirect
2. #20292822C2semidirect
3. #20292376C2semidirect
4. D4⋊(C107⋊C8)C2semidirect
5. #555×C107C2semidirect
66. C107#1624semidirect
67. #555D107semidirect

Maximal subgroups

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

subgroupindex
#202930012
#202928222
#202923762
D4⋊(C107⋊C8)2
#555×C1072
#1624107