Finite Group Databasebeta

← #13051045

Subgroups — #13051045

Order 1344.

120 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 36 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#10621427C211yesmaximal, characteristic
2C7×#836C211yesmaximal, characteristic
3C3×#6410C211yesmaximal, characteristic
4#89273—31nomaximal, Hall
5C2×C3×C7×Q8C2211yescharacteristic
6C2×Dic42C411yescharacteristic
7#66629—21no—
8#66629—21no—
9C4×Dic21C411yescharacteristic
10C7×#208—21no—
11C3×#1136—21no—
12#6507—31no—
13C7×#101—31no—
14#6410D311yescharacteristic
15#4478—71nomaximal, Hall
16C3×C7×Q8—41no—
17C2×C3×C4×C7C2×C411yescharacteristic
18C2×C3×C4×C7D411yescharacteristic
19C2×C3×C4×C7D411yescharacteristic
20C2×Dic21—21no—
21C2×Dic21—21no—
22Dic42—41no—
23C3×C7⋊C8—41no—
24C7×C3⋊C8—41no—
25C2×C7×Q8C2×D311yescharacteristic
26C2×Dic14—31no—
27C7×M16—61no—
28Dic7⋊C4—61no—
29Dic7⋊C4—61no—
30C4×Dic7—31no—
31#1136—21no—
32#944—71no—
33C3×#101—71no—
34#836D711yescharacteristic
35C22×C3×C7C22⋊C411yescharacteristic
36C3×C4×C7—21no—
37C3×C4×C7—21no—
38C3×C4×C7—21no—
39Dic21—22no—
40Dic21—41no—
41#356—211noSylow
42C7×Q8—41no—
43C2×C4×C7C4×D311yescharacteristic
44C2×C4×C7D1211yescharacteristic
45C2×C4×C7#6711yescharacteristic
46C2×Dic7—61no—
47C2×Dic7—61no—
48Dic14—121no—
49C7×C8—121no—
50C7⋊C8—41no—
51C2×C3×Q8C2×D711yescharacteristic
52C2×Dic6—71no—
53C3×M16—141no—
54Dic3⋊C4—141no—
55Dic3⋊C4—141no—
56C4×Dic3—71no—
57#208—21no—
58C2×C3×C7—21no—
59C2×C3×C7#9911yescharacteristic
60C4⋊Q8—211no—
61#101—31no—
62#101—71no—
63C22×C7(C2×D3)⋊C411yescharacteristic
64C4×C7—21no—
65C4×C7—21no—
66C4×C7—21no—
67Dic7—62no—
68Dic7—121no—
69C3×Q8—41no—
70C2×C3×C4C4×D711yescharacteristic
71C2×C3×C4D2811yescharacteristic
72C2×C3×C4C22⋊D711yescharacteristic
73C2×Dic3—141no—
74C2×Dic3—141no—
75Dic6—281no—
76C3×C8—281no—
77C3⋊C8—41no—
78C3×C7#35611yescharacteristic, complemented, Hall
79C2×Q8—211no—
80C2×Q8D3×D711yescharacteristic
81M16—61no—
82M16—141no—
83C4⋊C4—421no—
84C4⋊C4—421no—
85C42—211no—
86C2×C7—21no—
87C2×C7#80611yescharacteristic
88C22×C3(C2×D7)⋊C411yescharacteristic
89C3×C4—21no—
90C3×C4—21no—
91C3×C4—21no—
92Dic3—142no—
93Dic3—281no—
94Q8—41no—
95Q8—841no—
96C2×C4—421no—
97C2×C4—421no—
98C2×C4D21⋊C411yescharacteristic
99C2×C4#424611yescharacteristic
100C2×C4#424711yescharacteristic
101C8—121no—
102C8—281no—
103C7#447811yescharacteristic, complemented, Sylow
104C2×C3—21no—
105C2×C3#638011yescharacteristic
106C22#6657511yescharacteristic
107C4—21no—
108C4—21no—
109C4—21no—
110C4—422no—
111C4—841no—
112C3#8927311yescharacteristic, complemented, Sylow
113C2—21no—
114C2#1062053211yescharacteristic
subgroupG/H
centerC2#10620532
commutatorC2×C3×C4×C7C2×C4
Frattini, hypercenterC2×Q8D3×D7
Fitting, cosocleC2×C3×C7×Q8C22
socleC2×C3×C7#99
prime radicalC22×C3×C7C22⋊C4
hypocenterC3×C7#356
LCSG ▷ C2×C3×C4×C7 ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
78. C3×C7#356semidirect
103. C7#4478semidirect
112. C3#89273semidirect

Maximal subgroups

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

subgroupindex
#106214272
C7×#8362
C3×#64102
#892733
#44787