Finite Group Databasebeta

← #27195671

Subgroups — #27195671

Order 13184.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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#20268355C211yesmaximal, characteristic
2#20267797C212yesmaximal, complemented
3#20267765C211yesmaximal, characteristic, complemented
4#20267548C212yesmaximal, complemented
5#568×C103C211yesmaximal, characteristic, complemented
6C22×D412C2211yescharacteristic
7#15693017C412yescomplemented
8#15693010C412yescomplemented
9#15692995—22no—
10#15692989—22no—
11#15692980C2212yes—
12#15692980C2211yescharacteristic
13#15692943—24no—
14#15692936—22no—
15C2×#41×C103C2211yescharacteristic
16#131×C103—24no—
17C2×M16×C103—22no—
18C2×M16×C103C2212yes—
19C23×D103—21no—
20C2×D412—22no—
21C2×D412—22no—
22C2×D412—21no—
23C2×D412C2×C412yes—
24C2×C22⋊D103—24no—
25C4⋊Dic103—21no—
26C4⋊Dic103—22no—
27C22⋊Dic103—21no—
28C22⋊Dic103—22no—
29(C2×D103)⋊C4—24no—
30C4×Dic103—22no—
31C4×Dic103C2×C412yes—
32#41×C103—22no—
33#41×C103—22no—
34C2×Q8×C103C2×C411yescharacteristic
35C2×D4×C103C2×C411yescharacteristic
36C2×D4×C103D412yes—
37C22×C4×C103D412yes—
38C22×C4×C103C2311yescharacteristic
39M16×C103—24no—
40M16×C103—22no—
41M16×C103—22no—
42C2×C8×C103—24no—
43C2×C8×C103D414yescomplemented
44C23×C103C22⋊C412yes—
45C23×C103C2×D411yescharacteristic
46C22×D103—22no—
47C22×D103—42no—
48C22×D103—82no—
49Q8×C103—22no—
50D4×C103—24no—
51D4×C103—22no—
52C2×C4×C103—22no—
53C2×C4×C103—22no—
54C2×C4×C103C22⋊C412yes—
55C2×C4×C103C22×C411yescharacteristic
56C2×C4×C103C2×D412yes—
57C2×C4×C103C2×D411yescharacteristic
58C2×C4×C103#4112yes—
59C22⋊D103—84no—
60C2×Dic103—22no—
61C2×Dic103—42no—
62D412—24no—
63D412—41no—
64D412—41no—
65C8×C103—24no—
66C22×C103—22no—
67C22×C103—41no—
68C22×C103—42no—
69C22×C103C2×C22⋊C411yescharacteristic
70C22×C103#12112yes—
71C2×D103—44no—
72C2×D103—42no—
73C2×D103—84no—
74C4×C103—22no—
75C4×C103C4×D412yes—
76C4×C103#12011yescharacteristic
77C4×C103#12311yescharacteristic
78Dic103—42no—
79Dic103—82no—
80C2×C103—21no—
81C2×C103—22no—
82C2×C103—42no—
83C2×C103#38611yescharacteristic
84D103—82no—
85#1910—1031nomaximal, Sylow
86C103#191011yescharacteristic, complemented, Sylow
87#568D10311yescharacteristic, complemented
88#535—1031no—
89#418—1032no—
90#410—1031no—
91H(3,ℤ/4)—1032no—
92C2×#41C2×D10311yescharacteristic
93C22×D4—1031no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D10312yes—
97#127—1032no—
98#124—1032no—
99#121—2062no—
100#120—2062no—
101#117—1031no—
102#117—1032no—
103#102—2064no—
104#99—2062no—
105C24—2061no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D10311yescharacteristic
109C2×D4—1032no—
110C2×D4—2064no—
111C2×D4—2061no—
112C2×D4—2062no—
113C2×D4—2062no—
114C2×D4C4×D10311yescharacteristic
115C2×D4D41212yes—
116C22×C4C22⋊D10312yes—
117C22×C4C22×D10311yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D10314yescomplemented
123C4⋊C4—2062no—
124C4⋊C4—2061no—
125C22⋊C4—2062no—
126C22⋊C4—2061no—
127C22⋊C4—2064no—
128C42—1032no—
129C42—2062no—
130C23—8242no—
131C23—4122no—
132C23—2062no—
133C23(C2×D103)⋊C412yes—
134C23C2×D41211yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—8244no—
139D4—4121no—
140D4—4121no—
141D4—2064no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—4122no—
145C2×C4—2062no—
146C2×C4(C2×D103)⋊C412yes—
147C2×C4C2×C4×D10311yescharacteristic
148C2×C4C2×C22⋊D10312yes—
149C2×C4C2×C22⋊D10311yescharacteristic
150C2×C4#1311010512yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—8244no—
156C22—4124no—
157C22—4122no—
158C22C2×(C2×D103)⋊C411yescharacteristic
159C22#1569299512yes—
160C4—22no—
161C4—8242no—
162C4—4122no—
163C4C4×C22⋊D10312yes—
164C4#1569299011yescharacteristic
165C4#1569300411yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—8242no—
170C2#2026767011yescharacteristic
subgroupG/H
centerC2#20267670
commutatorC2×C4×C103C22×C4
FrattiniC22×C4C22×D103
Fitting#568×C103C2
socleC2×C103#386
prime radical#20268355C2
cosocleC22×C4×C103C23
hypercenter#568D103
hypocenterC103#1910
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20267797C2semidirect
3. #20267765C2semidirect
4. #20267548C2semidirect
5. #568×C103C2semidirect
7. #15693017C4semidirect
8. #15693010C4semidirect
43. C2×C8×C103D4semidirect
86. C103#1910semidirect
87. #568D103semidirect
122. C2×C8C22⋊D103semidirect

Maximal subgroups

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

subgroupindex
#202683552
#202677972
#202677652
#202675482
#568×C1032
#1910103