Finite Group Databasebeta

← #13051038

Subgroups — #13051038

Order 1344.

140 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C7×#936C211yesmaximal, characteristic
2#10620673C211yesmaximal, characteristic
3C3×#6408C211yesmaximal, characteristic
4#89336—31nomaximal, Hall
5C2×C3×C7×D4C2211yescharacteristic
6C2×C7×Dic6C411yescharacteristic, complemented
7#66636—21no—
8C7×#217—21no—
9C7×#217—21no—
10C4×C7×Dic3C411yescharacteristic, complemented
11C3×C22⋊Dic7—21no—
12C7×#124—31no—
13#6408—31no—
14#6408D311yescharacteristic
15#4473—71nomaximal, Hall
16C23×C3×C7D411yescharacteristic
17C23×C3×C7D411yescharacteristic
18C3×C7×D4—41no—
19C2×C3×C4×C7C2×C411yescharacteristic
20C2×Dic21—41no—
21C2×C7×Dic3—21no—
22C2×C7×Dic3—21no—
23C7×Dic6—41no—
24C2×C3×Dic7—41no—
25C2×C7×Q8—31no—
26C2×C7×D4C2×D311yescharacteristic
27C7×C22⋊C4—61no—
28C7×C22⋊C4—61no—
29C42×C7—31no—
30C22⋊Dic7—21no—
31C22⋊Dic7—61no—
32#936D711yescharacteristic
33C3×#99—71no—
34#834—71no—
35C22×C3×C7—21no—
36C22×C3×C7—21no—
37C22×C3×C7—41no—
38C22×C3×C7—41no—
39C22×C3×C7C22⋊C411yescharacteristic
40C3×C4×C7—21no—
41Dic21—81no—
42C3×Dic7—81no—
43C7×Dic3—41no—
44C7×Dic3—41no—
45#354—211noSylow
46C23×C7D1211yescharacteristic
47C23×C7#6711yescharacteristic
48C7×Q8—121no—
49C7×D4—41no—
50C2×C4×C7—61no—
51C2×C4×C7—61no—
52C2×C4×C7C4×D311yescharacteristic
53C2×Dic7—41no—
54C2×Dic7—121no—
55C2×C3×D4C2×D711yescharacteristic
56C2×Dic6Dic711yescharacteristic, complemented
57C3×C22⋊C4—141no—
58#217—21no—
59#217—21no—
60#217—141no—
61C4×Dic3Dic711yescharacteristic, complemented
62C2×C3×C7—21no—
63C2×C3×C7—41no—
64C2×C3×C7—41no—
65C2×C3×C7#9911yescharacteristic
66#124—31no—
67#99—71no—
68#99—211no—
69C22×C7—21no—
70C22×C7—21no—
71C22×C7—41no—
72C22×C7—41no—
73C22×C7(C2×D3)⋊C411yescharacteristic
74C4×C7—21no—
75C4×C7—121no—
76C4×C7—121no—
77Dic7—81no—
78Dic7—241no—
79C23×C3C22⋊D711yescharacteristic
80C23×C3C22⋊D711yescharacteristic
81C3×D4—41no—
82C2×C3×C4—281no—
83C2×C3×C4C2×Dic711yescharacteristic
84C2×Dic3—21no—
85C2×Dic3—21no—
86C2×Dic3—281no—
87Dic6—41no—
88C3×C7#35411yescharacteristic, complemented, Hall
89C2×Q8—31no—
90C2×D4D3×D711yescharacteristic
91C22⋊C4—61no—
92C22⋊C4—61no—
93C22⋊C4—141no—
94C22⋊C4—421no—
95C42—31no—
96C2×C7—21no—
97C2×C7—41no—
98C2×C7—41no—
99C2×C7#80611yescharacteristic
100C22×C3—21no—
101C22×C3—21no—
102C22×C3—41no—
103C22×C3—41no—
104C22×C3C22⋊Dic711yescharacteristic
105C3×C4—21no—
106C3×C4—561no—
107Dic3—41no—
108Dic3—41no—
109Dic3—561no—
110C23#424511yescharacteristic
111C23#424711yescharacteristic
112Q8—121no—
113D4—41no—
114C2×C4—61no—
115C2×C4—61no—
116C2×C4—281no—
117C2×C4—841no—
118C2×C4D3×Dic711yescharacteristic
119C7#447311yescharacteristic, complemented, Sylow
120C2×C3—21no—
121C2×C3—41no—
122C2×C3—41no—
123C2×C3#640811yescharacteristic
124C22—21no—
125C22—21no—
126C22—41no—
127C22—41no—
128C22(C2×D3)⋊Dic711yescharacteristic
129C4—21no—
130C4—121no—
131C4—121no—
132C4—561no—
133C4—1681no—
134C3#8933611yescharacteristic, complemented, Sylow
135C2—21no—
136C2—41no—
137C2—41no—
138C2#1062053111yescharacteristic
subgroupG/H
centerC2#10620531
commutatorC2×C3×C4×C7C2×C4
Frattini, hypercenterC2×D4D3×D7
Fitting, prime radical, cosocleC2×C3×C7×D4C22
socleC2×C3×C7#99
hypocenterC3×C7#354
LCSG ▷ C2×C3×C4×C7 ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
6. C2×C7×Dic6C4semidirect
10. C4×C7×Dic3C4semidirect
56. C2×Dic6Dic7semidirect
61. C4×Dic3Dic7semidirect
88. C3×C7#354semidirect
119. C7#4473semidirect
134. C3#89336semidirect

Maximal subgroups

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

subgroupindex
C7×#9362
#106206732
C3×#64082
#893363
#44737