Finite Group Databasebeta

← #28335173

Subgroups — #28335173

Order 13696.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#127×D107C211yesmaximal, characteristic
2#20292683C212yesmaximal, complemented
3#20292371C211yesmaximal, characteristic, complemented
4#20292370C212yesmaximal, complemented
5#331×C107C211yesmaximal, characteristic, complemented
6C2×D4×D107—22no—
7C2×D4×D107C412yes—
8#16812106C2211yescharacteristic
9#16812105—22no—
10#16812105C412yes—
11C42×D107C2211yescharacteristic
12C4⋊(C107⋊C8)C2212yescomplemented
13(C2×D107)⋊C8—22no—
14#16812019—22no—
15#127×C107C2211yescharacteristic
16C4⋊C8×C107C2212yescomplemented
17C23×D107—22no—
18C2×D428—22no—
19C2×D428C2×C412yes—
20C2×C22⋊D107—22no—
21C2×C22⋊D107—24no—
22D4×D107—44no—
23D4×D107—44no—
24D4×D107—44no—
25C2×C4×D107C2×C411yescharacteristic
26C2×C4×D107D412yes—
27C2×C107⋊C8—22no—
28C4×Dic107C2×C411yescharacteristic
29C4×Dic107D412yes—
30C2×D4×C107—22no—
31C2×D4×C107C2×C412yes—
32C2×C8×C107—22no—
33C42×C107C2311yescharacteristic
34C23×C107—22no—
35C22×D107—22no—
36C22×D107—48no—
37C22×D107—44no—
38C22×D107C22⋊C411yescharacteristic
39D4×C107—42no—
40D4×C107—42no—
41D4×C107—42no—
42C2×C4×C107C22×C411yescharacteristic
43C2×C4×C107C2×D412yes—
44C22⋊D107—44no—
45C22⋊D107—48no—
46C2×Dic107C22⋊C412yes—
47C2×Dic107C22⋊C411yescharacteristic
48D428—42no—
49D428—42no—
50D428—42no—
51C4×D107—22no—
52C4×D107—22no—
53C4×D107—22no—
54C8×C107—42no—
55C107⋊C8—42no—
56C22×C107—42no—
57C22×C107—44no—
58C22×C107C2×C22⋊C411yescharacteristic
59C2×D107—22no—
60C2×D107—42no—
61C2×D107—44no—
62C2×D107—88no—
63C2×D107#10012yes—
64C4×C107—21no—
65C4×C107AGL(1,ℤ/8)12yes—
66C4×C107AGL(1,ℤ/8)12yes—
67Dic107—21no—
68Dic107—22no—
69C2×C107—82no—
70C2×C107C2×#10011yescharacteristic
71C2×C107#41812yes—
72D107—22no—
73D107—82no—
74#1555—1071nomaximal, Sylow
75C107#155511yescharacteristic, complemented, Sylow
76C2×#127—1071no—
77#432—1072no—
78#331—1071no—
79#331—1072no—
80#331D10711yescharacteristic, complemented
81C22×D4—1072no—
82C22×D4—2142no—
83#127—1072no—
84#127—1071no—
85#127—2142no—
86#127C2×D10711yescharacteristic
87C2×C42—1071no—
88C4⋊C8—1072no—
89C4⋊C8C2×D10712yescomplemented
90C22⋊C8—2142no—
91#97—2142no—
92C24—2142no—
93C2×D4—22no—
94C2×D4—1072no—
95C2×D4—4284no—
96C2×D4—4284no—
97C2×D4—4284no—
98C2×D4—2144no—
99C2×D4—2142no—
100C2×D4—2142no—
101C2×D4C4×D10712yes—
102C22×C4—1071no—
103C22×C4—1072no—
104C2×C8—22no—
105C2×C8—2142no—
106C42—1071no—
107C42—1072no—
108C42C22×D10711yescharacteristic
109C23—22no—
110C23—1071no—
111C23—4284no—
112C23—4288no—
113C23—2142no—
114D4—42no—
115D4—42no—
116D4—42no—
117D4—4282no—
118D4—4288no—
119D4—4284no—
120D4—4282no—
121D4—4282no—
122C2×C4—1072no—
123C2×C4—1071no—
124C2×C4—2142no—
125C2×C4—2142no—
126C2×C4—2142no—
127C2×C4C2×C4×D10711yescharacteristic
128C2×C4D4×D10712yes—
129C8—42no—
130C8—4282no—
131C22—44no—
132C22—42no—
133C22—8568no—
134C22—1072no—
135C22—4284no—
136C22—4282no—
137C22—2142no—
138C22C22⋊C4×D10711yescharacteristic
139C4—21no—
140C4—2142no—
141C4—2141no—
142C4#1681213312yes—
143C4#1681213612yes—
144C2—82no—
145C2—8562no—
146C2—2142no—
147C2#100×D10711yescharacteristic
148C2#2029263012yes—
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
Fitting#331×C107C2
socleC22×C107C2×C22⋊C4
prime radical#127×D107C2
cosocleC42×C107C23
hypercenter#331D107
hypocenterC107#1555
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20292683C2semidirect
3. #20292371C2semidirect
4. #20292370C2semidirect
5. #331×C107C2semidirect
12. C4⋊(C107⋊C8)C22semidirect
16. C4⋊C8×C107C22semidirect
75. C107#1555semidirect
80. #331D107semidirect
89. C4⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#127×D1072
#202926832
#202923712
#202923702
#331×C1072
#1555107