Finite Group Databasebeta

← #13057182

Subgroups — #13057182

Order 1344.

160 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 27 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×D4×A4C211yesmaximal, characteristic
2#10621496C211yesmaximal, characteristic, complemented
3C7⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#89990—31nomaximal, Hall
5C22×C7×A4—21no—
6C4×C7×A4C2211yescharacteristic
7C7⋊SL(2,ℤ/4)—21no—
8#66633—41nomaximal
9C22×C7×D4D311yescharacteristic
10C2×#1142—31no—
11C2×#1142—31no—
12#6487—31no—
13#6406—31no—
14#6406—31no—
15C22⋊(C7⋊C8)—31no—
16#5414—71nomaximal, Hall
17C2×C7×A4—41no—
18C2×C7×A4D411yescharacteristic
19C3×C7×D4—41no—
20Dic42—41no—
21C3⋊(C7⋊C8)—41no—
22C24×C7—21no—
23C2×C7×D4—31no—
24C2×C7×D4—31no—
25C2×C7×D4—61no—
26C22×C4×C7C2×D311yescharacteristic
27C2×Dic14—31no—
28C22⋊Dic7—61no—
29#1142—62no—
30#1142—61no—
31#1142—61no—
32C4⋊Dic7—31no—
33Dic7⋊C4—61no—
34C2×C7⋊C8—31no—
35C2×C7⋊C8—31no—
36D4×A4D711yescharacteristic
37A4⋊Q8—71no—
38A4⋊C8—71no—
39C22×C3×C7—81no—
40C7×A4SD1611yescharacteristic, complemented
41C3×C4×C7—41no—
42Dic21—81no—
43#450—211noSylow
44C23×C7—41no—
45C23×C7—61no—
46C23×C7—61no—
47C23×C7—121no—
48C23×C7#6711yescharacteristic
49C7×D4—32no—
50C7×D4—31no—
51C7×D4—61no—
52C7×D4S411yescharacteristic
53C2×C4×C7—31no—
54C2×C4×C7—31no—
55C2×Dic7—61no—
56C2×Dic7—61no—
57Dic14—61no—
58Dic14—61no—
59C7⋊C8—61no—
60C7⋊C8—61no—
61C22×A4—21no—
62C4×A4C2×D711yescharacteristic
63SL(2,ℤ/4)—141no—
64#214—281no—
65C2×C3×C7—41no—
66C2×C3×C7—161no—
67C22×D4D2111yescharacteristic
68C2×SD16—211no—
69C2×SD16—211no—
70C22⋊Q8—211no—
71#102—211no—
72#102—211no—
73C22⋊C8—211no—
74C22×C7—21no—
75C22×C7—31no—
76C22×C7—31no—
77C22×C7—61no—
78C22×C7—121no—
79C22×C7—121no—
80C22×C7—121no—
81C22×C7—121no—
82C22×C7#21411yescharacteristic, complemented
83C4×C7—31no—
84C4×C7C2×S411yescharacteristic
85Dic7—121no—
86Dic7—121no—
87C2×A4—41no—
88C2×A4C22⋊D711yescharacteristic
89C3×D4—41no—
90Dic6—281no—
91C3⋊C8—281no—
92C3×C7—41noHall
93C24—21no—
94C2×Q8—211no—
95C2×D4—31no—
96C2×D4—31no—
97C2×D4—61no—
98C22×C4C2×D2111yescharacteristic
99SD16—422no—
100SD16—421no—
101SD16—421no—
102C2×C8—211no—
103C2×C8—211no—
104C4⋊C4—211no—
105C4⋊C4—421no—
106C22⋊C4—421no—
107C2×C7—31no—
108C2×C7—31no—
109C2×C7—41no—
110C2×C7—121no—
111C2×C7GL(2,ℤ/4)11yescharacteristic
112C22×C3—81no—
113A4#114211yescharacteristic, complemented
114C3×C4—41no—
115Dic3—561no—
116C23—41no—
117C23—61no—
118C23—61no—
119C23—121no—
120C23#426811yescharacteristic
121Q8—421no—
122Q8—421no—
123D4—32no—
124D4—31no—
125D4—61no—
126D4C7⋊S411yescharacteristic
127C2×C4—31no—
128C2×C4—31no—
129C2×C4—421no—
130C2×C4—421no—
131C8—421no—
132C8—421no—
133C7#541411yescharacteristic, complemented, Sylow
134C2×C3—41no—
135C2×C3—161no—
136C22—21no—
137C22—31no—
138C22—31no—
139C22—61no—
140C22—121no—
141C22—121no—
142C22—121no—
143C22—121no—
144C22#6663311yescharacteristic, complemented
145C4—31no—
146C4—841no—
147C4—841no—
148C4C2×C7⋊S411yescharacteristic
149C3—41noSylow
150C2—31no—
151C2—31no—
152C2—41no—
153C2—121no—
154C2#1062150611yescharacteristic
subgroupG/H
centerC2#10621506
commutator, cosocleC4×C7×A4C22
FrattiniC4C2×C7⋊S4
FittingC22×C7×D4D3
socleC23×C7#67
prime radicalC7×D4×A4C2
hypercenterD4C7⋊S4
hypocenterC7×A4SD16
LCSG ▷ C4×C7×A4 ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #10621496C2semidirect
3. C7⋊(A4⋊C8)C2semidirect
40. C7×A4SD16semidirect
82. C22×C7#214semidirect
113. A4#1142semidirect
133. C7#5414semidirect
144. C22#66633semidirect

Maximal subgroups

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

subgroupindex
C7×D4×A42
#106214962
C7⋊(A4⋊C8)2
#899903
#666334
#54147