Finite Group Databasebeta

← #13056372

Subgroups — #13056372

Order 1344.

228 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 75 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#10621413C211yesmaximal, characteristic
2C2×C4⋊Dic21C211yesmaximal, characteristic, complemented
3C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
4#10620660C212yesmaximal, complemented
5#10620656C212yesmaximal, complemented
6#89507—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C2×#4268—21no—
9C22×Dic21—21no—
10C2×D84C2211yescharacteristic
11C2×C3×C7×C8C2212yes—
12#66631—21no—
13C4⋊Dic21—21no—
14C4⋊Dic21C2212yes—
15C4⋊Dic21C2211yescharacteristic
16#6493—31no—
17C2×C4⋊Dic7—31no—
18C7×C22⋊C8D311yescharacteristic, complemented
19#6395—32no—
20#6391—32no—
21#4735—71nomaximal, Hall
22C23×C3×C7D411yescharacteristic
23C22×D21—21no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7D411yescharacteristic
26C2×C3×C4×C7C2311yescharacteristic
27#4268—42no—
28C2×Dic21—22no—
29C2×Dic21—21no—
30C2×Dic21—42no—
31D84—22no—
32C3×C7×C8—22no—
33C22×C4×C7C2×D311yescharacteristic
34C2×C22⋊D7—61no—
35C22×Dic7—61no—
36C2×D28—31no—
37C2×C7×C8C2×D312yes—
38(C2×D7)⋊C4—61no—
39C4⋊Dic7—32no—
40C4⋊Dic7—31no—
41C4⋊Dic7—61no—
42#930—71no—
43C2×C4⋊Dic3—71no—
44C3×C22⋊C8D711yescharacteristic, complemented
45#821—72no—
46#817—72no—
47C22×C3×C7—21no—
48C22×C3×C7—21no—
49C22×C3×C7SD1612yes—
50C22×C3×C7C2×D411yescharacteristic
51C2×D21—41no—
52C2×D21—41no—
53C2×D21—41no—
54C3×C4×C7—21no—
55C3×C4×C7#4112yes—
56Dic21—41no—
57Dic21—42no—
58#481—211noSylow
59C23×C7D1211yescharacteristic
60C22×D7—61no—
61C2×C4×C7—22no—
62C2×C4×C7D1211yescharacteristic
63C2×C4×C7C22×D311yescharacteristic
64C22⋊D7—122no—
65C2×Dic7—62no—
66C2×Dic7—61no—
67C2×Dic7—122no—
68D28—62no—
69C7×C8—22no—
70C22×C3×C4C2×D711yescharacteristic
71C2×#67—141no—
72C22×Dic3—141no—
73C2×D12—71no—
74C2×C3×C8C2×D712yes—
75(C2×D3)⋊C4—141no—
76C4⋊Dic3—71no—
77C4⋊Dic3—72no—
78C4⋊Dic3—141no—
79C2×C3×C7—22no—
80C2×C3×C7#12311yescharacteristic
81C2×C3×C7C2×SD1611yescharacteristic
82C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
83D21—81no—
84#121—211no—
85C2×C4⋊C4—211no—
86#106—212no—
87#102—212no—
88C22⋊C8D2111yescharacteristic, complemented
89C22×C7—21no—
90C22×C7—21no—
91C22×C7#20412yes—
92C22×C7C2×D1211yescharacteristic
93C2×D7—121no—
94C2×D7—121no—
95C2×D7—121no—
96C4×C7—21no—
97C4×C7#23712yes—
98Dic7—121no—
99Dic7—122no—
100C23×C3D2811yescharacteristic
101C22×D3—141no—
102C2×C3×C4—22no—
103C2×C3×C4D2811yescharacteristic
104C2×C3×C4C22×D711yescharacteristic
105#67—282no—
106C2×Dic3—142no—
107C2×Dic3—141no—
108C2×Dic3—282no—
109D12—142no—
110C3×C8—22no—
111C3×C7#48111yescharacteristic, complemented, Hall
112C2×D4—211no—
113C2×D4—421no—
114C22×C4—421no—
115C22×C4C2×D2111yescharacteristic
116C2×C8C2×D2112yes—
117C4⋊C4—212no—
118C4⋊C4—211no—
119C4⋊C4—421no—
120C22⋊C4—421no—
121C2×C7—22no—
122C2×C7#88611yescharacteristic
123C2×C7C2×#20411yescharacteristic
124C2×C7#90811yescharacteristic
125D7—241no—
126C22×C3—21no—
127C22×C3—21no—
128C22×C3#113212yes—
129C22×C3C2×D2811yescharacteristic
130C2×D3—281no—
131C2×D3—281no—
132C2×D3—281no—
133C3×C4—21no—
134C3×C4#115912yes—
135Dic3—281no—
136Dic3—282no—
137C23—421no—
138C23D8411yescharacteristic
139D4—422no—
140D4—842no—
141C2×C4—22no—
142C2×C4—421no—
143C2×C4—422no—
144C2×C4—842no—
145C2×C4D8411yescharacteristic
146C2×C4C22×D2111yescharacteristic
147C8—22no—
148C7#473511yescharacteristic, complemented, Sylow
149C2×C3—22no—
150C2×C3#644911yescharacteristic
151C2×C3C2×#113211yescharacteristic
152C2×C3#647111yescharacteristic
153D3—561no—
154C22—21no—
155C22—21no—
156C22—841no—
157C22—841no—
158C22—841no—
159C22#6662312yes—
160C22C2×D8411yescharacteristic
161C4—21no—
162C4—841no—
163C4—842no—
164C4#6673012yes—
165C3#8950711yescharacteristic, complemented, Sylow
166C2—22no—
167C2—1681no—
168C2#1062136911yescharacteristic
169C2C2×#6662311yescharacteristic
170C2#1062139111yescharacteristic
subgroupG/H
centerC22C2×D84
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C2×D4
prime radical#10621413C2
hypercenterC22⋊C8D21
hypocenterC3×C7#481
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊Dic21C2semidirect
3. C3×C7×C22⋊C8C2semidirect
4. #10620660C2semidirect
5. #10620656C2semidirect
18. C7×C22⋊C8D3semidirect
44. C3×C22⋊C8D7semidirect
88. C22⋊C8D21semidirect
111. C3×C7#481semidirect
148. C7#4735semidirect
165. C3#89507semidirect

Maximal subgroups

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

subgroupindex
#106214132
C2×C4⋊Dic212
C3×C7×C22⋊C82
#106206602
#106206562
#895073
#47357