Finite Group Databasebeta

← #28346864

Subgroups — #28346864

Order 13696.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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#20293177C211yesmaximal, characteristic
2#20292997C212yesmaximal, complemented
3#20292993C212yesmaximal, complemented
4#20292984C214yesmaximal, complemented
5#20292626C214yesmaximal, complemented
6C4×#13133802C211yesmaximal, characteristic, complemented
7C2×#125×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2212yes—
9C2×#13133802C2212yes—
10#16812109C2211yescharacteristic
11#16812106C2211yescharacteristic
12#16812097C2212yes—
13#16812083—22no—
14C4×D428—22no—
15#16812032C2218yes—
16#16812029C2218yes—
17#16812020C2212yes—
18C4×C107⋊C8C2212yes—
19#125×C107—22no—
20#125×C107C2212yes—
21#125×C107C2212yes—
22C2×C4⋊C4×C107—22no—
23C2×C4⋊C4×C107C2212yes—
24C2×C42×C107C2211yescharacteristic
25#13133818—28no—
26C2×Dic214C2312yes—
27C2×D428—22no—
28C2×D428C2312yes—
29C2×C22⋊D107—22no—
30C2×C4×D107—22no—
31#13133802—24no—
32C2×C107⋊C8C2314yes—
33C4⋊Dic107—22no—
34(C2×D107)⋊C4—24no—
35C22×C4×C107—22no—
36C22×C4×C107D412yes—
37C22×C4×C107C2311yescharacteristic
38C4⋊C4×C107—24no—
39C4⋊C4×C107—24no—
40C4⋊C4×C107—22no—
41C4⋊C4×C107C2314yes—
42C42×C107D412yes—
43C42×C107C2311yescharacteristic
44C42×C107C2311yescharacteristic
45C23×C107C2×D411yescharacteristic
46C22×D107—22no—
47C2×C4×C107—24no—
48C2×C4×C107—42no—
49C2×C4×C107—44no—
50C2×C4×C107C2×D412yes—
51C2×C4×C107C2×D412yes—
52C2×C4×C107C2×D411yescharacteristic
53C2×C4×C107#4114yes—
54C2×C4×C107C2411yescharacteristic
55C22⋊D107—44no—
56C2×Dic107—22no—
57D428—24no—
58D428—44no—
59C4×D107—44no—
60Dic214—24no—
61C107⋊C8—24no—
62C22×C107—22no—
63C22×C107#12412yes—
64C22×C107C22×D411yescharacteristic
65C2×D107—42no—
66C2×D107—44no—
67C4×C107—22no—
68C4×C107—22no—
69C4×C107—44no—
70C4×C107#12412yes—
71C4×C107C2×#4112yes—
72Dic107—42no—
73C2×C107—22no—
74C2×C107C2×#12411yescharacteristic
75C2×C107#57512yes—
76D107—82no—
77#3142—1071nomaximal, Sylow
78C107#314211yescharacteristic, complemented, Sylow
79#532—1071no—
80C2×#125D10711yescharacteristic, complemented
81#491—1072no—
82#490—1072no—
83#488—1074no—
84#417—1074no—
85C4×M16—1071no—
86C2×#41—1072no—
87C2×M16—1072no—
88C4⋊Q8—1071no—
89#127—1071no—
90#125—22no—
91#125C2×D10712yes—
92#125C2×D10712yes—
93#124—1072no—
94#121—2142no—
95C4×D4—2142no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4C2×D10712yes—
98C2×C42C2×D10711yescharacteristic
99#103—1078no—
100#102—1078no—
101#97—1072no—
102C4×C8—1072no—
103#41—2148no—
104C2×Q8—1072no—
105C2×D4—1072no—
106C2×D4—2142no—
107C2×D4—2142no—
108C22×C4—22no—
109C22×C4—2142no—
110C22×C4C22⋊D10712yes—
111C22×C4C22×D10711yescharacteristic
112M16—2144no—
113C2×C8—1074no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—2142no—
118C4⋊C4C22×D10714yes—
119C22⋊C4—2144no—
120C42C22⋊D10712yes—
121C42C22×D10711yescharacteristic
122C42C22×D10711yescharacteristic
123C23—2142no—
124C23C2×C22⋊D10711yescharacteristic
125Q8—2144no—
126D4—4284no—
127D4—4284no—
128D4—2144no—
129C2×C4—24no—
130C2×C4—42no—
131C2×C4—44no—
132C2×C4—4284no—
133C2×C4—2142no—
134C2×C4C2×C22⋊D10712yes—
135C2×C4C2×C22⋊D10712yes—
136C2×C4C2×C22⋊D10711yescharacteristic
137C2×C4#1313381914yes—
138C2×C4C23×D10711yescharacteristic
139C8—2144no—
140C22—22no—
141C22—4282no—
142C22—4284no—
143C22#1681209812yes—
144C22C22×C22⋊D10711yescharacteristic
145C4—22no—
146C4—22no—
147C4—44no—
148C4—4282no—
149C4#1681209812yes—
150C4C2×#1313381912yes—
151C2—22no—
152C2—8562no—
153C2C2×#1681209811yescharacteristic
154C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#125×C107C2
socleC22×C107C22×D4
prime radical#20293177C2
hypercenterC2×#125D107
hypocenterC107#3142
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20292997C2semidirect
3. #20292993C2semidirect
4. #20292984C2semidirect
5. #20292626C2semidirect
6. C4×#13133802C2semidirect
7. C2×#125×C107C2semidirect
78. C107#3142semidirect
80. C2×#125D107semidirect

Maximal subgroups

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

subgroupindex
#202931772
#202929972
#202929932
#202929842
#202926262
C4×#131338022
C2×#125×C1072
#3142107