Finite Group Databasebeta

← #28335772

Subgroups — #28335772

Order 13696.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293037C212yesmaximal, complemented
2#20293022C211yesmaximal, characteristic
3#20292369C212yesmaximal, complemented
4#20292367C211yesmaximal, characteristic
5#563×C107C211yesmaximal, characteristic
6C2×#13133808—21no—
7C2×#13133806—21no—
8#16812109C2211yescharacteristic
9#16812042—22no—
10#16812040—22no—
11#16812020C2212yes—
12#16812020C2211yescharacteristic
13C4⋊Q8×C107—21no—
14#126×C107—22no—
15#125×C107C2212yes—
16#124×C107C2211yescharacteristic
17#123×C107—21no—
18C22⋊Q8×C107—22no—
19C2×Dic214—21no—
20#13133808—42no—
21#13133806—42no—
22C2×C107⋊C8—22no—
23C2×C107⋊C8—21no—
24C4⋊Dic107—22no—
25C2×Q8×C107—21no—
26C2×Q8×C107D411yescharacteristic
27C2×D4×C107D411yescharacteristic
28C22×C4×C107—21no—
29C4⋊C4×C107—22no—
30C4⋊C4×C107—22no—
31C4⋊C4×C107—22no—
32C4⋊C4×C107D412yes—
33C4⋊C4×C107D412yes—
34C22⋊C4×C107—22no—
35C22⋊C4×C107—21no—
36C22⋊C4×C107—22no—
37C42×C107—21no—
38C42×C107C2311yescharacteristic
39C23×C107—21no—
40Q8×C107—41no—
41Q8×C107—81no—
42D4×C107—41no—
43C2×C4×C107—22no—
44C2×C4×C107—21no—
45C2×C4×C107—22no—
46C2×C4×C107—81no—
47C2×C4×C107C2×D412yes—
48C2×C4×C107C2×D411yescharacteristic
49C2×Dic107—41no—
50Dic214—41no—
51Dic214—41no—
52C107⋊C8—24no—
53C107⋊C8—22no—
54C22×C107—41no—
55C22×C107—42no—
56C22×C107#12011yescharacteristic
57C4×C107—21no—
58C4×C107—22no—
59C4×C107—42no—
60C4×C107—41no—
61C4×C107—42no—
62Dic107—81no—
63C2×C107—81no—
64C2×C107#45311yescharacteristic
65C2×C107#45612yes—
66#1660—1071nomaximal, Sylow
67C107#166011yescharacteristic, complemented, Sylow
68#563D10711yescharacteristic
69#501—1072no—
70#497—1071no—
71#330—1072no—
72#329—1071no—
73C2×Q16—2141no—
74C2×SD16—2141no—
75C4⋊Q8—21no—
76C4⋊Q8—1071no—
77#126—22no—
78#125C2×D10712yes—
79#124C2×D10711yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—2142no—
83#106—2142no—
84#97—1072no—
85#97—1071no—
86C2×Q8—21no—
87C2×Q8—2141no—
88C2×Q8C22⋊D10711yescharacteristic
89C2×D4C22⋊D10711yescharacteristic
90C22×C4—21no—
91Q16—4282no—
92SD16—4282no—
93C2×C8—2142no—
94C2×C8—2141no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—2142no—
99C4⋊C4C22⋊D10712yes—
100C4⋊C4C22⋊D10712yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D10711yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—4281no—
110Q8—4281no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—4281no—
117C2×C4C2×C22⋊D10712yes—
118C2×C4C2×C22⋊D10711yescharacteristic
119C8—2142no—
120C8—2144no—
121C22—41no—
122C22—42no—
123C22#1681207811yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—8561no—
130C2—81no—
131C2#2029279511yescharacteristic
132C2#2029281012yes—
subgroupG/H
centerC22#16812078
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#563×C107C2
socleC22×C107#120
prime radicalC2×D4×C107D4
hypercenter#563D107
hypocenterC107#1660
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20293037C2semidirect
3. #20292369C2semidirect
67. C107#1660semidirect

Maximal subgroups

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

subgroupindex
#202930372
#202930222
#202923692
#202923672
#563×C1072
#1660107