Finite Group Databasebeta

← #28337006

Subgroups — #28337006

Order 13696.

340 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 85 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#20293394C211yesmaximal, characteristic, complemented
2#20293191C211yesmaximal, characteristic
3C2×#16812025C212yesmaximal, complemented
4#20292398C212yesmaximal, complemented
5#413×C107C211yesmaximal, characteristic, complemented
6C22×D428C2211yescharacteristic
7#16812116—24no—
8C2×#13133802—22no—
9C2×#13133802C2212yes—
10#16812105—21no—
11#16812098—21no—
12#16812083—22no—
13#16812077C414yes—
14#16812068C2211yescharacteristic
15#16812025—24no—
16C2×#41×C107C2211yescharacteristic
17C2×M16×C107C2212yes—
18#101×C107—21no—
19#100×C107—21no—
20C23×D107C2×C412yes—
21C2×D428—21no—
22C2×D428—21no—
23C2×D428—22no—
24C2×D428—22no—
25C2×C22⋊D107—24no—
26#13133802—24no—
27#13133802—22no—
28#13133802D414yescomplemented
29C2×C107⋊C8—24no—
30C2×C107⋊C8—22no—
31C4⋊Dic107—21no—
32C22⋊Dic107C2×C412yes—
33(C2×D107)⋊C4—24no—
34C4×Dic107—21no—
35#41×C107—22no—
36#41×C107—22no—
37C2×Q8×C107D411yescharacteristic
38C2×D4×C107C2×C412yes—
39C2×D4×C107D411yescharacteristic
40C22×C4×C107D412yes—
41C22×C4×C107C2311yescharacteristic
42M16×C107—22no—
43M16×C107—22no—
44C2×C8×C107—22no—
45C23×C107C22⋊C412yes—
46C23×C107C22×C411yescharacteristic
47C22×D107—22no—
48C22×D107—24no—
49C22×D107—82no—
50Q8×C107—22no—
51D4×C107—24no—
52D4×C107—22no—
53C2×C4×C107—22no—
54C2×C4×C107—22no—
55C2×C4×C107C22⋊C412yes—
56C2×C4×C107C2×D412yes—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107#4112yes—
60C22⋊D107—84no—
61C2×Dic107—22no—
62D428—41no—
63D428—42no—
64D428—41no—
65C8×C107—42no—
66C107⋊C8—24no—
67C22×C107—22no—
68C22×C107—41no—
69C22×C107—42no—
70C22×C107C2×C22⋊C411yescharacteristic
71C22×C107C4×D412yes—
72C2×D107—44no—
73C2×D107—42no—
74C2×D107—84no—
75C4×C107—22no—
76C4×C107#12011yescharacteristic
77C4×C107#12112yes—
78C4×C107#12311yescharacteristic
79Dic107—42no—
80C2×C107—21no—
81C2×C107—22no—
82C2×C107—42no—
83C2×C107#38611yescharacteristic
84D107—82no—
85#1914—1071nomaximal, Sylow
86C107#191411yescharacteristic, complemented, Sylow
87#568—1071no—
88#535—1071no—
89#413D10711yescharacteristic, complemented
90C2×#100—1072no—
91#344—1072no—
92C2×#41C2×D10711yescharacteristic
93C22×D4—1071no—
94#131—2144no—
95C2×M16—1072no—
96C2×M16—2142no—
97C2×M16C2×D10712yes—
98#127—2141no—
99#124—2141no—
100#121—2142no—
101#120—1074no—
102#117—1071no—
103#101—21no—
104#100—21no—
105#100—2144no—
106C24—1072no—
107#41—22no—
108#41—22no—
109C2×Q8C22⋊D10711yescharacteristic
110C2×D4—2144no—
111C2×D4—2142no—
112C2×D4—2141no—
113C2×D4—2141no—
114C2×D4—2142no—
115C2×D4C4×D10712yes—
116C2×D4C22⋊D10711yescharacteristic
117C22×C4D42812yes—
118C22×C4C22×D10711yescharacteristic
119M16—22no—
120M16—22no—
121M16—1074no—
122M16—2144no—
123M16—2142no—
124C2×C8—22no—
125C2×C8—2144no—
126C2×C8—2142no—
127C4⋊C4—2141no—
128C22⋊C4—1072no—
129C22⋊C4—2144no—
130C42—2141no—
131C23—8562no—
132C23—2144no—
133C23—2142no—
134C23(C2×D107)⋊C412yes—
135C23C2×C4×D10711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—8564no—
140D4—4281no—
141D4—4282no—
142D4—4281no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—2142no—
146C2×C4(C2×D107)⋊C412yes—
147C2×C4D4×D10712yes—
148C2×C4C2×C22⋊D10711yescharacteristic
149C2×C4C2×D42811yescharacteristic
150C2×C4#1313381712yes—
151C8—42no—
152C8—2144no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—8564no—
157C22—4284no—
158C22—4282no—
159C22C2×(C2×D107)⋊C411yescharacteristic
160C22(C22⋊D107)⋊C412yes—
161C4—22no—
162C4—4282no—
163C4#1681207911yescharacteristic
164C4#1681208112yes—
165C4#1681209311yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—8562no—
170C2#2029250411yescharacteristic
subgroupG/H
centerC2#20292504
commutatorC23×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#413×C107C2
socleC2×C107#386
prime radical#20293191C2
cosocleC22×C4×C107C23
hypercenter#413D107
hypocenterC107#1914
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #20293394C2semidirect
3. C2×#16812025C2semidirect
4. #20292398C2semidirect
5. #413×C107C2semidirect
28. #13133802D4semidirect
86. C107#1914semidirect
89. #413D107semidirect

Maximal subgroups

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

subgroupindex
#202933942
#202931912
C2×#168120252
#202923982
#413×C1072
#1914107