Finite Group Databasebeta

← #13053845

Subgroups — #13053845

Order 1344.

208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 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#10621407C211yesmaximal, characteristic
2C2×C3×C4⋊Dic7C211yesmaximal, characteristic
3C7×#830C211yesmaximal, characteristic
4#10620522C212yesmaximal, complemented
5#10620509C212yesmaximal, complemented
6#89493—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C2×Dic42C2211yescharacteristic
9C22×C3×Dic7—21no—
10#66636—21no—
11C4⋊Dic21C2211yescharacteristic
12#66629—21no—
13C2×C7×C3⋊C8C2212yes—
14C3×C4⋊Dic7—21no—
15C3×C4⋊Dic7C2212yes—
16C2×C4⋊Dic7D311yescharacteristic
17#6487—31no—
18C7×C22⋊C8—31no—
19#6391—32no—
20#6390—32no—
21#4971—71nomaximal, Hall
22C23×C3×C7D411yescharacteristic
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7D411yescharacteristic
25C2×C3×C4×C7C2311yescharacteristic
26C2×Dic21—21no—
27C2×Dic21—21no—
28Dic42—22no—
29C2×C3×Dic7—22no—
30C2×C3×Dic7—42no—
31C7×C3⋊C8—22no—
32C22×C4×C7C2×D311yescharacteristic
33C22×Dic7—21no—
34C2×Dic14—31no—
35C2×C7×C8—32no—
36C22⋊Dic7—61no—
37C4⋊Dic7—21no—
38C4⋊Dic7—31no—
39C4⋊Dic7C2×D312yes—
40Dic7⋊C4—61no—
41C2×C3×C4⋊C4—71no—
42#924—71no—
43#830D711yescharacteristic
44#810—72no—
45#808—72no—
46C22×C3×C7—21no—
47C22×C3×C7—21no—
48C22×C3×C7SD1612yes—
49C22×C3×C7C2×D411yescharacteristic
50C3×C4×C7—21no—
51C3×C4×C7#4112yes—
52Dic21—41no—
53Dic21—41no—
54C3×Dic7—42no—
55#483—211noSylow
56C23×C7#6711yescharacteristic
57C2×C4×C7—22no—
58C2×C4×C7#6711yescharacteristic
59C2×C4×C7C22×D311yescharacteristic
60C2×Dic7—22no—
61C2×Dic7—42no—
62C2×Dic7—61no—
63C2×Dic7—61no—
64Dic14—62no—
65C7×C8—62no—
66C22×C3×C4—141no—
67C22×C3×C4C2×D711yescharacteristic
68C2×Dic6—71no—
69C3×C4⋊C4—72no—
70C3×C4⋊C4—141no—
71#217—141no—
72C4⋊Dic3—71no—
73Dic3⋊C4—141no—
74C2×C3⋊C8C2×D712yes—
75C2×C3×C7—22no—
76C2×C3×C7#12311yescharacteristic
77C2×C3×C7C2×SD1611yescharacteristic
78C2×C3×C7#13711yescharacteristic
79C22⋊Q8—211no—
80C2×C4⋊C4—71no—
81#106—212no—
82#103—212no—
83C22⋊C8—31no—
84C22×C7—21no—
85C22×C7—21no—
86C22×C7#21412yes—
87C22×C7C2×#6711yescharacteristic
88C4×C7—21no—
89C4×C7#23512yes—
90Dic7—42no—
91Dic7—121no—
92Dic7—121no—
93C23×C3D2811yescharacteristic
94C2×C3×C4—22no—
95C2×C3×C4—142no—
96C2×C3×C4—282no—
97C2×C3×C4D2811yescharacteristic
98C2×C3×C4C22×D711yescharacteristic
99C2×Dic3—141no—
100C2×Dic3—141no—
101Dic6—142no—
102C3⋊C8—22no—
103C3×C7#48311yescharacteristic, complemented, Hall
104C2×Q8—211no—
105C22×C4—141no—
106C22×C4D3×D711yescharacteristic
107C2×C8—32no—
108C4⋊C4—72no—
109C4⋊C4—141no—
110C4⋊C4—211no—
111C4⋊C4—421no—
112C22⋊C4—421no—
113C2×C7—22no—
114C2×C7#92911yescharacteristic
115C2×C7C2×#21411yescharacteristic
116C2×C7#94211yescharacteristic
117C22×C3—21no—
118C22×C3—21no—
119C22×C3#113212yes—
120C22×C3C2×D2811yescharacteristic
121C3×C4—21no—
122C3×C4—282no—
123C3×C4#115912yes—
124Dic3—281no—
125Dic3—281no—
126C23#424611yescharacteristic
127Q8—422no—
128C2×C4—22no—
129C2×C4—142no—
130C2×C4—282no—
131C2×C4—421no—
132C2×C4—421no—
133C2×C4#424611yescharacteristic
134C2×C4C2×D3×D711yescharacteristic
135C8—62no—
136C7#497111yescharacteristic, complemented, Sylow
137C2×C3—22no—
138C2×C3#644911yescharacteristic
139C2×C3C2×#113211yescharacteristic
140C2×C3#647211yescharacteristic
141C22—21no—
142C22—21no—
143C22#6656512yes—
144C22C2×#424611yescharacteristic
145C4—21no—
146C4—282no—
147C4—841no—
148C4—841no—
149C4#6668612yes—
150C3#8949311yescharacteristic, complemented, Sylow
151C2—22no—
152C2C2×#6656511yescharacteristic
153C2#1062087711yescharacteristic
154C2#1062110811yescharacteristic
subgroupG/H
centerC22C2×#4246
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#483
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #10620522C2semidirect
5. #10620509C2semidirect
103. C3×C7#483semidirect
136. C7#4971semidirect
150. C3#89493semidirect

Maximal subgroups

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

subgroupindex
#106214072
C2×C3×C4⋊Dic72
C7×#8302
#106205222
#106205092
#894933
#49717