Finite Group Databasebeta

← #13056341

Subgroups — #13056341

Order 1344.

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

normal subgroups only

This group has 200 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#10621352C211yesmaximal, characteristic, complemented
3C3×C7×#97C211yesmaximal, characteristic, complemented
4#10620660C212yesmaximal
5#10620655C212yesmaximal, complemented
6#89486—31nomaximal, Hall
7C2×D84C2211yescharacteristic
8C2×Dic42—21no—
9C2×Dic42C2211yescharacteristic
10C2×C3×C7×C8C2212yes—
11C3×C42×C7C2211yescharacteristic
12#66631—21no—
13#66631—21no—
14C4⋊Dic21—21no—
15C4⋊Dic21C2212yes—
16#6438—31no—
17#6432—31no—
18C7×#97D311yescharacteristic, complemented
19#6395—32no—
20#6390—32no—
21#4714—71nomaximal, Hall
22C22×D21—21no—
23C2×C3×C4×C7D411yescharacteristic
24C2×C3×C4×C7D411yescharacteristic
25C2×C3×C4×C7C2311yescharacteristic
26C2×Dic21—22no—
27C2×Dic21—21no—
28D84—22no—
29Dic42—22no—
30Dic42—42no—
31C3×C7×C8—22no—
32C2×D28—31no—
33C2×Dic14—31no—
34C2×Dic14—61no—
35C2×C7×C8C2×D312yes—
36C42×C7C2×D311yescharacteristic
37(C2×D7)⋊C4—61no—
38(C2×D7)⋊C4—61no—
39C4⋊Dic7—32no—
40C4⋊Dic7—61no—
41#875—71no—
42#869—71no—
43C3×#97D711yescharacteristic, complemented
44#821—72no—
45#816—72no—
46C22×C3×C7C2×D411yescharacteristic
47C2×D21—41no—
48C2×D21—41no—
49C2×D21—41no—
50C3×C4×C7—21no—
51C3×C4×C7—21no—
52C3×C4×C7#4112yes—
53Dic21—41no—
54Dic21—42no—
55#489—211noSylow
56C22×D7—61no—
57C2×C4×C7D1211yescharacteristic
58C2×C4×C7D1211yescharacteristic
59C2×C4×C7C22×D311yescharacteristic
60C2×Dic7—62no—
61C2×Dic7—61no—
62D28—62no—
63Dic14—62no—
64Dic14—122no—
65C7×C8—22no—
66C2×D12—71no—
67C2×Dic6—71no—
68C2×Dic6—141no—
69C2×C3×C8C2×D712yes—
70C3×C42C2×D711yescharacteristic
71(C2×D3)⋊C4—141no—
72(C2×D3)⋊C4—141no—
73C4⋊Dic3—72no—
74C4⋊Dic3—141no—
75C2×C3×C7#12411yescharacteristic
76C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
77C2×C3×C7#13711yescharacteristic
78D21—81no—
79C4⋊Q8—211no—
80#124—211no—
81#103—212no—
82#102—212no—
83#97D2111yescharacteristic, complemented
84C22×C7C2×D1211yescharacteristic
85C2×D7—121no—
86C2×D7—121no—
87C2×D7—121no—
88C4×C7—21no—
89C4×C7—21no—
90C4×C7#23512yes—
91Dic7—121no—
92Dic7—122no—
93C22×D3—141no—
94C2×C3×C4D2811yescharacteristic
95C2×C3×C4D2811yescharacteristic
96C2×C3×C4C22×D711yescharacteristic
97C2×Dic3—142no—
98C2×Dic3—141no—
99D12—142no—
100Dic6—142no—
101Dic6—282no—
102C3×C8—22no—
103C3×C7#48911yescharacteristic, complemented, Hall
104C2×Q8—211no—
105C2×Q8—421no—
106C2×D4—211no—
107C2×C8C2×D2112yes—
108C4⋊C4—212no—
109C4⋊C4—421no—
110C22⋊C4—421no—
111C22⋊C4—421no—
112C42C2×D2111yescharacteristic
113C2×C7#87511yescharacteristic
114C2×C7#90811yescharacteristic
115C2×C7#90911yescharacteristic
116D7—241no—
117C22×C3C2×D2811yescharacteristic
118C2×D3—281no—
119C2×D3—281no—
120C2×D3—281no—
121C3×C4—21no—
122C3×C4—21no—
123C3×C4#115712yes—
124Dic3—281no—
125Dic3—282no—
126C23—421no—
127Q8—422no—
128Q8—842no—
129D4—422no—
130C2×C4—421no—
131C2×C4—422no—
132C2×C4D8411yescharacteristic
133C2×C4D8411yescharacteristic
134C2×C4C22×D2111yescharacteristic
135C8—22no—
136C7#471411yescharacteristic, complemented, Sylow
137C2×C3#643811yescharacteristic
138C2×C3#647111yescharacteristic
139C2×C3#647211yescharacteristic
140D3—561no—
141C22—841no—
142C22—841no—
143C22—841no—
144C22C2×D8411yescharacteristic
145C4—21no—
146C4—21no—
147C4—841no—
148C4—842no—
149C4#6672812yes—
150C3#8948611yescharacteristic, complemented, Sylow
151C2—1681no—
152C2#1062135811yescharacteristic
153C2#1062139111yescharacteristic
154C2#1062139211yescharacteristic
subgroupG/H
centerC22C2×D84
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#97C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenter#97D21
hypocenterC3×C7#489
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #10621352C2semidirect
3. C3×C7×#97C2semidirect
5. #10620655C2semidirect
18. C7×#97D3semidirect
43. C3×#97D7semidirect
83. #97D21semidirect
103. C3×C7#489semidirect
136. C7#4714semidirect
150. C3#89486semidirect

Maximal subgroups

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

subgroupindex
#106213582
#106213522
C3×C7×#972
#106206602
#106206552
#894863
#47147