Finite Group Databasebeta

← #28334969

Subgroups — #28334969

Order 13696.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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#20292684C211yesmaximal, characteristic, complemented
2#20292473C211yesmaximal, characteristic, complemented
3#20292361C214yesmaximal, complemented
4#532×C107C211yesmaximal, characteristic
5C2×C4×Dic107—21no—
6C4⋊(C107⋊C8)—21no—
7C22⋊(C107⋊C8)C2212yes—
8#16812020—21no—
9C107⋊#95—21no—
10C107⋊#95C2212yes—
11C2×#41×C107—21no—
12C4⋊Q8×C107C411yescharacteristic, complemented
13#127×C107C411yescharacteristic, complemented
14#124×C107—21no—
15#121×C107—21no—
16#121×C107C2212yes—
17C4×D4×C107—21no—
18C4×D4×C107C412yescomplemented
19C2×C42×C107C2211yescharacteristic
20C22×Dic107—22no—
21C2×C107⋊C8—22no—
22C4×Dic107—24no—
23#41×C107—44no—
24C2×Q8×C107—21no—
25C2×D4×C107—22no—
26C2×D4×C107—21no—
27C2×D4×C107—21no—
28C2×D4×C107C2×C412yes—
29C22×C4×C107—22no—
30C22×C4×C107D411yescharacteristic
31C22×C4×C107D411yescharacteristic
32C22×C4×C107C2311yescharacteristic
33C4⋊C4×C107—21no—
34C4⋊C4×C107C2×C412yes—
35C22⋊C4×C107—22no—
36C22⋊C4×C107—22no—
37C42×C107C2×C411yescharacteristic
38C42×C107C2×C411yescharacteristic
39C42×C107D411yescharacteristic
40C42×C107D411yescharacteristic
41C23×C107—22no—
42C23×C107C2×D411yescharacteristic
43Q8×C107—42no—
44D4×C107—42no—
45D4×C107—42no—
46D4×C107—44no—
47D4×C107—42no—
48C2×C4×C107—22no—
49C2×C4×C107—22no—
50C2×C4×C107—22no—
51C2×C4×C107—41no—
52C2×C4×C107—44no—
53C2×C4×C107C22⋊C411yescharacteristic
54C2×C4×C107C22⋊C411yescharacteristic
55C2×C4×C107C22⋊C411yescharacteristic
56C2×C4×C107C22⋊C411yescharacteristic
57C2×C4×C107C22×C411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×Dic107—24no—
60C2×Dic107—42no—
61C2×Dic107—42no—
62C107⋊C8—42no—
63C22×C107—21no—
64C22×C107—21no—
65C22×C107—42no—
66C22×C107—42no—
67C22×C107—42no—
68C22×C107C2×C22⋊C411yescharacteristic
69C22×C107AGL(1,ℤ/8)12yes—
70C4×C107—21no—
71C4×C107—21no—
72C4×C107—21no—
73C4×C107—21no—
74C4×C107—21no—
75C4×C107—42no—
76C4×C107#10412yes—
77Dic107—44no—
78C2×C107—22no—
79C2×C107—82no—
80C2×C107#41011yescharacteristic
81C2×C107#41811yescharacteristic
82C2×C107C2×#10411yescharacteristic
83#1505—1071nomaximal, Sylow
84C107#150511yescharacteristic, complemented, Sylow
85#532D10711yescharacteristic
86#432—1071no—
87#376—1071no—
88#327—1074no—
89C2×#41—21no—
90C4⋊Q8Dic10711yescharacteristic, complemented
91#127Dic10711yescharacteristic, complemented
92#124—21no—
93#121—21no—
94#121C2×D10712yes—
95C4×D4—21no—
96C4×D4Dic10712yescomplemented
97C2×C42—2141no—
98C2×C42C2×D10711yescharacteristic
99C4⋊C8—2141no—
100C22⋊C8—1072no—
101#97—2141no—
102#95—1072no—
103#95—2141no—
104#41—44no—
105C2×Q8—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4C2×Dic10712yes—
110C22×C4—22no—
111C22×C4—2142no—
112C22×C4C22⋊D10711yescharacteristic
113C22×C4C22⋊D10711yescharacteristic
114C22×C4C22×D10711yescharacteristic
115C2×C8—2142no—
116C4⋊C4—21no—
117C4⋊C4C2×Dic10712yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—2144no—
121C42C2×Dic10711yescharacteristic
122C42C2×Dic10711yescharacteristic
123C42C22⋊D10711yescharacteristic
124C42C22⋊D10711yescharacteristic
125C23—22no—
126C23C2×C22⋊D10711yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—44no—
131D4—42no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—44no—
137C2×C4—4282no—
138C2×C4—4282no—
139C2×C4—2144no—
140C2×C4C22⋊Dic10711yescharacteristic
141C2×C4C22⋊Dic10711yescharacteristic
142C2×C4C22⋊Dic10711yescharacteristic
143C2×C4C22⋊Dic10711yescharacteristic
144C2×C4C22×Dic10711yescharacteristic
145C2×C4C2×C22⋊D10711yescharacteristic
146C8—4282no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22C2×C22⋊Dic10711yescharacteristic
153C22#1681213512yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—42no—
160C4—4284no—
161C4#1681203612yes—
162C2—22no—
163C2—82no—
164C2#2029260111yescharacteristic
165C2#2029263211yescharacteristic
166C2C2×#1681203611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting, prime radical#532×C107C2
socleC22×C107C2×C22⋊C4
cosocleC22×C4×C107C23
hypercenter#532D107
hypocenterC107#1505
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20292684C2semidirect
2. #20292473C2semidirect
3. #20292361C2semidirect
12. C4⋊Q8×C107C4semidirect
13. #127×C107C4semidirect
18. C4×D4×C107C4semidirect
84. C107#1505semidirect
90. C4⋊Q8Dic107semidirect
91. #127Dic107semidirect
96. C4×D4Dic107semidirect

Maximal subgroups

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

subgroupindex
#202926842
#202924732
#202923612
#532×C1072
#1505107