Finite Group Databasebeta

← #13056599

Subgroups — #13056599

Order 1344.

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

normal subgroups only

This group has 208 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
1C2×C3×C7×C4⋊C4C211yesmaximal, characteristic
2#10621407C211yesmaximal, characteristic
3#10620669C211yesmaximal, characteristic
4#10620649C212yesmaximal, complemented
5#10620646C212yesmaximal, complemented
6#89742—31nomaximal, Hall
7C22×C3×C4×C7—21no—
8C22×C3×C4×C7C2211yescharacteristic
9C2×Dic42C2211yescharacteristic
10C3×C7×C4⋊C4—21no—
11C3×C7×C4⋊C4C2212yes—
12#66636—21no—
13C4⋊Dic21C2211yescharacteristic
14#66629—21no—
15C2×C3⋊(C7⋊C8)C2212yes—
16C2×C7×C4⋊C4D311yescharacteristic
17#6487—31no—
18C22⋊(C7⋊C8)—31no—
19#6384—32no—
20#6381—32no—
21#4970—71nomaximal, Hall
22C23×C3×C7D411yescharacteristic
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—42no—
26C2×C3×C4×C7D411yescharacteristic
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—21no—
29C2×Dic21—21no—
30Dic42—22no—
31C3⋊(C7⋊C8)—22no—
32C22×C4×C7—21no—
33C22×C4×C7C2×D311yescharacteristic
34C2×Dic14—31no—
35C7×C4⋊C4—21no—
36C7×C4⋊C4C2×D312yes—
37C22⋊Dic7—61no—
38C4⋊Dic7—31no—
39Dic7⋊C4—61no—
40C2×C7⋊C8—32no—
41C2×C3×C4⋊C4D711yescharacteristic
42#924—71no—
43#830—71no—
44#810—72no—
45#807—72no—
46C22×C3×C7—21no—
47C22×C3×C7—21no—
48C22×C3×C7Q1612yes—
49C22×C3×C7C2×D411yescharacteristic
50C3×C4×C7—21no—
51C3×C4×C7—42no—
52C3×C4×C7#4112yes—
53Dic21—41no—
54Dic21—41no—
55#484—211noSylow
56C23×C7#6711yescharacteristic
57C2×C4×C7—22no—
58C2×C4×C7—22no—
59C2×C4×C7—42no—
60C2×C4×C7#6711yescharacteristic
61C2×C4×C7C22×D311yescharacteristic
62C2×Dic7—61no—
63C2×Dic7—61no—
64Dic14—62no—
65C7⋊C8—62no—
66C22×C3×C4—21no—
67C22×C3×C4C2×D711yescharacteristic
68C2×Dic6—71no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2×D712yes—
71#217—141no—
72C4⋊Dic3—71no—
73Dic3⋊C4—141no—
74C2×C3⋊C8—72no—
75C2×C3×C7—22no—
76C2×C3×C7#12311yescharacteristic
77C2×C3×C7C2×Q1611yescharacteristic
78C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
79C22⋊Q8—211no—
80C2×C4⋊C4D2111yescharacteristic
81#107—212no—
82#103—212no—
83C22⋊C8—211no—
84C22×C7—21no—
85C22×C7—21no—
86C22×C7#21612yes—
87C22×C7C2×#6711yescharacteristic
88C4×C7—21no—
89C4×C7—42no—
90C4×C7#23512yes—
91Dic7—121no—
92Dic7—121no—
93C23×C3C22⋊D711yescharacteristic
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—42no—
97C2×C3×C4C22⋊D711yescharacteristic
98C2×C3×C4C22×D711yescharacteristic
99C2×Dic3—141no—
100C2×Dic3—141no—
101Dic6—142no—
102C3⋊C8—142no—
103C3×C7#48411yescharacteristic, complemented, Hall
104C2×Q8—211no—
105C22×C4—21no—
106C22×C4C2×D2111yescharacteristic
107C2×C8—212no—
108C4⋊C4—21no—
109C4⋊C4—211no—
110C4⋊C4—421no—
111C4⋊C4C2×D2112yes—
112C22⋊C4—421no—
113C2×C7—22no—
114C2×C7#92911yescharacteristic
115C2×C7#93211yescharacteristic
116C2×C7C2×#21611yescharacteristic
117C22×C3—21no—
118C22×C3—21no—
119C22×C3#114412yes—
120C22×C3C2×C22⋊D711yescharacteristic
121C3×C4—21no—
122C3×C4—42no—
123C3×C4#115712yes—
124Dic3—281no—
125Dic3—281no—
126C23#426811yescharacteristic
127Q8—422no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—421no—
132C2×C4—421no—
133C2×C4#426811yescharacteristic
134C2×C4C22×D2111yescharacteristic
135C8—422no—
136C7#497011yescharacteristic, complemented, Sylow
137C2×C3—22no—
138C2×C3#649211yescharacteristic
139C2×C3#649511yescharacteristic
140C2×C3C2×#114411yescharacteristic
141C22—21no—
142C22—21no—
143C22#6663512yes—
144C22C2×#426811yescharacteristic
145C4—21no—
146C4—42no—
147C4—841no—
148C4—841no—
149C4#6672812yes—
150C3#8974211yescharacteristic, complemented, Sylow
151C2—22no—
152C2#1062141211yescharacteristic
153C2#1062141511yescharacteristic
154C2C2×#6663511yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC2×C3×C7×C4⋊C4C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#484
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #10620649C2semidirect
5. #10620646C2semidirect
103. C3×C7#484semidirect
136. C7#4970semidirect
150. C3#89742semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
#106214072
#106206692
#106206492
#106206462
#897423
#49707