Finite Group Databasebeta

← #28334607

Subgroups — #28334607

Order 13696.

122 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 23 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#20292794C211yesmaximal, characteristic
2#20292420C211yesmaximal, characteristic, complemented
3#331×C107C211yesmaximal, characteristic, complemented
4#16812154C411yescharacteristic, complemented
5#16812135—21no—
6#16812037—21no—
7#16812026C411yescharacteristic, complemented
8#16812024—21no—
9#127×C107C2211yescharacteristic
10C4⋊C8×C107—21no—
11#13133818—21no—
12#13133817—41no—
13C2×C22⋊D107—41no—
14D4×D107—41no—
15#13133806—41no—
16#13133803—41no—
17#13133802—21no—
18C22⋊Dic107—41no—
19C2×D4×C107—41no—
20C2×D4×C107C2×C411yescharacteristic
21C2×D4×C107D411yescharacteristic
22C2×C8×C107—41no—
23C42×C107D411yescharacteristic
24C23×C107—21no—
25C23×C107—21no—
26C22×D107—41no—
27D4×C107—41no—
28D4×C107—41no—
29D4×C107—81no—
30D4×C107—81no—
31C2×C4×C107—21no—
32C2×C4×C107C22⋊C411yescharacteristic
33C22⋊D107—42no—
34C22⋊D107—41no—
35C2×Dic107—41no—
36C2×Dic107—81no—
37D428—21no—
38C4×D107—41no—
39Dic214—21no—
40C8×C107—81no—
41C107⋊C8—41no—
42C22×C107—41no—
43C22×C107—41no—
44C22×C107—81no—
45C22×C107—81no—
46C22×C107#9911yescharacteristic
47C2×D107—41no—
48C2×D107—161no—
49C4×C107—21no—
50C4×C107—21no—
51C4×C107—21no—
52Dic107—41no—
53Dic107—161no—
54C2×C107—21no—
55C2×C107—81no—
56C2×C107—81no—
57C2×C107AΣL(1,𝔽16)11yescharacteristic
58D107—81no—
59#1402—1071nomaximal, Sylow
60C107#140211yescharacteristic, complemented, Sylow
61#453—1071no—
62#353—1071no—
63#331D10711yescharacteristic, complemented
64H(5,𝔽2)—1071no—
65AGL(1,ℤ/8)—2141no—
66#127C2×D10711yescharacteristic
67C4⋊C8—21no—
68#104—2141no—
69#100—1071no—
70#99—2141no—
71#41—4281no—
72#41—2141no—
73C2×D4—41no—
74C2×D4—4281no—
75C2×D4—4281no—
76C2×D4C4×D10711yescharacteristic
77C2×D4D42811yescharacteristic
78SD16—4281no—
79D8—4281no—
80M16—2141no—
81C2×C8—41no—
82C22⋊C4—4281no—
83C42C22⋊D10711yescharacteristic
84C23—21no—
85C23—21no—
86C23—4281no—
87Q8—2141no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—4282no—
93D4—4281no—
94D4—2141no—
95C2×C4—21no—
96C2×C4—8561no—
97C2×C4—4281no—
98C2×C4—4281no—
99C2×C4(C2×D107)⋊C411yescharacteristic
100C8—81no—
101C8—4281no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—4281no—
107C22—17121no—
108C22#1681202311yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—4281no—
113C4—17121no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—8561no—
118C2#2029241511yescharacteristic
subgroupG/H
centerC2#20292415
commutatorC2×D4×C107C2×C4
Frattini#127C2×D107
Fitting#331×C107C2
socleC2×C107AΣL(1,𝔽16)
prime radical#20292794C2
cosocle#127×C107C22
hypercenter#331D107
hypocenterC107#1402
LCSG ▷ C2×D4×C107 ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20292420C2semidirect
3. #331×C107C2semidirect
4. #16812154C4semidirect
7. #16812026C4semidirect
60. C107#1402semidirect
63. #331D107semidirect

Maximal subgroups

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

subgroupindex
#202927942
#202924202
#331×C1072
#1402107