Finite Group Databasebeta

← #28335782

Subgroups — #28335782

Order 13696.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293375C211yesmaximal, characteristic
2#20292989C212yesmaximal, complemented
3#20292369C211yesmaximal, characteristic, complemented
4#20292366C212yesmaximal, complemented
5#491×C107C211yesmaximal, characteristic, complemented
6#16812097C2212yes—
7#16812094—22no—
8#16812087—22no—
9#16812084—22no—
10#16812032—22no—
11#16812029—22no—
12#16812020C2212yes—
13C4⋊Q8×C107C2211yescharacteristic
14#125×C107C2211yescharacteristic
15#103×C107—22no—
16#97×C107C2211yescharacteristic
17C2×Dic214D412yes—
18C2×D428—22no—
19C2×D428D412yes—
20C2×C4×D107—22no—
21C2×C107⋊C8—22no—
22Dic107⋊C4—22no—
23(C2×D107)⋊C4—22no—
24(C2×D107)⋊C4—22no—
25(C2×D107)⋊C4—22no—
26C2×Q8×C107—21no—
27C2×C8×C107—21no—
28C4⋊C4×C107—22no—
29C4⋊C4×C107—22no—
30C4⋊C4×C107D412yes—
31C42×C107C2311yescharacteristic
32C22×D107—22no—
33Q8×C107—42no—
34C2×C4×C107—22no—
35C2×C4×C107—41no—
36C2×C4×C107C2×D412yes—
37C2×C4×C107C2×D411yescharacteristic
38C2×Dic107—22no—
39D428—42no—
40D428—82no—
41C4×D107—82no—
42Dic214—42no—
43C8×C107—41no—
44C107⋊C8—42no—
45C22×C107#12011yescharacteristic
46C2×D107—42no—
47C2×D107—42no—
48C2×D107—42no—
49C4×C107—21no—
50C4×C107—22no—
51C4×C107—42no—
52C4×C107—81no—
53Dic107—42no—
54C2×C107#45411yescharacteristic
55C2×C107#45512yes—
56D107—82no—
57#1662—1071nomaximal, Sylow
58C107#166211yescharacteristic, complemented, Sylow
59#562—1071no—
60#491D10711yescharacteristic, complemented
61#489—1072no—
62#330—1071no—
63#329—1072no—
64C4⋊Q8C2×D10711yescharacteristic
65#125C2×D10711yescharacteristic
66#124—1072no—
67#123—2142no—
68C22⋊Q8—2142no—
69#121—2142no—
70#103—22no—
71#103—2142no—
72#102—2142no—
73#97—1072no—
74#97C2×D10711yescharacteristic
75C2×Q8—21no—
76C2×Q8—1072no—
77C2×D4—1072no—
78C2×D4—2142no—
79C22×C4—2142no—
80C2×C8—21no—
81C2×C8—2142no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—2142no—
85C4⋊C4C22⋊D10712yes—
86C22⋊C4—2142no—
87C22⋊C4—2142no—
88C22⋊C4—2142no—
89C42C22×D10711yescharacteristic
90C23—2142no—
91Q8—42no—
92Q8—4282no—
93D4—8562no—
94D4—4282no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—8562no—
98C2×C4—2142no—
99C2×C4D4×D10712yes—
100C2×C4C2×C22⋊D10711yescharacteristic
101C8—41no—
102C8—4282no—
103C22—4282no—
104C22—4282no—
105C22—4282no—
106C22#1681207711yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—4282no—
112C2—8562no—
113C2#2029279911yescharacteristic
114C2#2029280312yes—
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#491×C107C2
socleC22×C107#120
prime radical#20293375C2
hypercenter#491D107
hypocenterC107#1662
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #20292989C2semidirect
3. #20292369C2semidirect
4. #20292366C2semidirect
5. #491×C107C2semidirect
58. C107#1662semidirect
60. #491D107semidirect

Maximal subgroups

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

subgroupindex
#202933752
#202929892
#202923692
#202923662
#491×C1072
#1662107