Finite Group Databasebeta

← #13056334

Subgroups — #13056334

Order 1344.

192 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 67 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#10621358C211yesmaximal, characteristic
2#10621353C211yesmaximal, characteristic, complemented
3C3×C4×C7×C8C211yesmaximal, characteristic, complemented
4#10620660C212yesmaximal
5#10620655C212yesmaximal, complemented
6#89469—31nomaximal, Hall
7C2×D84C2211yescharacteristic
8C2×Dic42C2211yescharacteristic
9C2×C3×C7×C8C2212yes—
10C3×C42×C7C2211yescharacteristic
11#66631—22no—
12C4⋊Dic21C2212yes—
13#66629—22no—
14#6438—31no—
15#6433—31no—
16C4×C7×C8D311yescharacteristic, complemented
17#6395—32no—
18#6390—32no—
19#4697—71nomaximal, Hall
20C22×D21—21no—
21C2×C3×C4×C7D412yes—
22C2×C3×C4×C7C2311yescharacteristic
23C2×Dic21—22no—
24C2×Dic21—21no—
25D84—22no—
26Dic42—22no—
27C3×C7×C8—22no—
28C2×D28—31no—
29C2×Dic14—31no—
30C2×C7×C8C2×D312yes—
31C42×C7C2×D311yescharacteristic
32(C2×D7)⋊C4—62no—
33C4⋊Dic7—32no—
34Dic7⋊C4—62no—
35#875—71no—
36#870—71no—
37C3×C4×C8D711yescharacteristic, complemented
38#821—72no—
39#816—72no—
40C22×C3×C7C2×D411yescharacteristic
41C2×D21—41no—
42C2×D21—42no—
43C3×C4×C7—22no—
44C3×C4×C7#4112yes—
45Dic21—41no—
46Dic21—42no—
47#488—211noSylow
48C22×D7—61no—
49C2×C4×C7D1212yes—
50C2×C4×C7C22×D311yescharacteristic
51C2×Dic7—62no—
52C2×Dic7—61no—
53D28—62no—
54Dic14—62no—
55C7×C8—22no—
56C2×D12—71no—
57C2×Dic6—71no—
58C2×C3×C8C2×D712yes—
59C3×C42C2×D711yescharacteristic
60(C2×D3)⋊C4—142no—
61C4⋊Dic3—72no—
62Dic3⋊C4—142no—
63C2×C3×C7#12411yescharacteristic
64C2×C3×C7#13512yes—
65D21—81no—
66#125—211no—
67#124—211no—
68#103—212no—
69#102—212no—
70C4×C8D2111yescharacteristic, complemented
71C22×C7C2×D1211yescharacteristic
72C2×D7—121no—
73C2×D7—122no—
74C4×C7—22no—
75C4×C7#23512yes—
76Dic7—121no—
77Dic7—122no—
78C22×D3—141no—
79C2×C3×C4D2812yes—
80C2×C3×C4C22×D711yescharacteristic
81C2×Dic3—142no—
82C2×Dic3—141no—
83D12—142no—
84Dic6—142no—
85C3×C8—22no—
86C3×C7#48811yescharacteristic, complemented, Hall
87C2×Q8—211no—
88C2×D4—211no—
89C2×C8C2×D2112yes—
90C4⋊C4—212no—
91C4⋊C4—422no—
92C22⋊C4—422no—
93C42C2×D2111yescharacteristic
94C2×C7#87511yescharacteristic
95C2×C7#90412yes—
96D7—241no—
97C22×C3C2×D2811yescharacteristic
98C2×D3—281no—
99C2×D3—282no—
100C3×C4—22no—
101C3×C4#115712yes—
102Dic3—281no—
103Dic3—282no—
104C23—421no—
105Q8—422no—
106D4—422no—
107C2×C4—421no—
108C2×C4—422no—
109C2×C4D8412yes—
110C2×C4C22×D2111yescharacteristic
111C8—22no—
112C7#469711yescharacteristic, complemented, Sylow
113C2×C3#643811yescharacteristic
114C2×C3#646712yes—
115D3—561no—
116C22—841no—
117C22—842no—
118C22C2×D8411yescharacteristic
119C4—22no—
120C4—841no—
121C4—842no—
122C4#6672812yes—
123C3#8946911yescharacteristic, complemented, Sylow
124C2—1681no—
125C2#1062135811yescharacteristic
126C2#1062138712yes—
subgroupG/H
centerC22C2×D84
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C4×C7×C8C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC4×C8D21
hypocenterC3×C7#488
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. #10621353C2semidirect
3. C3×C4×C7×C8C2semidirect
5. #10620655C2semidirect
16. C4×C7×C8D3semidirect
37. C3×C4×C8D7semidirect
70. C4×C8D21semidirect
86. C3×C7#488semidirect
112. C7#4697semidirect
123. C3#89469semidirect

Maximal subgroups

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

subgroupindex
#106213582
#106213532
C3×C4×C7×C82
#106206602
#106206552
#894693
#46977