Finite Group Databasebeta

← #13057197

Subgroups — #13057197

Order 1344.

156 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 23 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×#994C211yesmaximal, characteristic, complemented
2#10621504C211yesmaximal, characteristic, complemented
3C7⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#90013—31nomaximal, Hall
5C7×#41⋊C3—21no—
6C7×#41⋊C3C2211yescharacteristic
7C7⋊GL(2,𝔽3)—21no—
8#66634—41nomaximal
9C7×H(5,𝔽2)D311yescharacteristic, complemented
10#6512—31no—
11#6512—31no—
12#6503—31no—
13#6411—31no—
14#6411—31no—
15#6407—31no—
16#5429—71nomaximal, Hall
17C3×C7×Q8—41no—
18D84—41no—
19C7×SL(2,𝔽3)D411yescharacteristic, complemented
20C3⋊(C7⋊C8)—41no—
21C7×#41—21no—
22C7×#41—31no—
23C7×#41C2×D311yescharacteristic
24C2×C7×D4—31no—
25C2×C7×D4—61no—
26C2×C22⋊D7—62no—
27C2×D28—31no—
28#1143—61no—
29#1143—61no—
30#1141—61no—
31#1141—61no—
32C4×Dic7—31no—
33#1136—31no—
34#1136—31no—
35#994D711yescharacteristic, complemented
36#986—71no—
37GU(2,𝔽3)—71no—
38C2×D21—81no—
39C3×C4×C7—41no—
40C3×C4×C7—81no—
41#453—211noSylow
42C23×C7—32no—
43C22×D7—61no—
44C7×Q8#6711yescharacteristic
45C7×Q8S411yescharacteristic
46C7×D4—31no—
47C7×D4—31no—
48C7×D4—61no—
49C7×D4—61no—
50C2×C4×C7—31no—
51C2×C4×C7—61no—
52C22⋊D7—124no—
53C2×Dic7—61no—
54D28—61no—
55D28—61no—
56C7⋊C8—61no—
57C7⋊C8—61no—
58#41⋊C3—21no—
59#41⋊C3C2×D711yescharacteristic
60GL(2,𝔽3)—141no—
61#215—281no—
62C2×C3×C7—41no—
63D21—161no—
64H(5,𝔽2)D2111yescharacteristic, complemented
65AGL(1,ℤ/8)—211no—
66AGL(1,ℤ/8)—211no—
67#127—211no—
68#104—211no—
69#104—211no—
70#100—211no—
71C22×C7—31no—
72C22×C7—61no—
73C22×C7—122no—
74C2×D7—122no—
75C2×D7—121no—
76C4×C7—21no—
77C4×C7—31no—
78C4×C7C2×S411yescharacteristic
79Dic7—62no—
80C3×Q8—41no—
81D12—281no—
82SL(2,𝔽3)C22⋊D711yescharacteristic, complemented
83C3⋊C8—281no—
84C3×C7—41noHall
85#41—21no—
86#41—31no—
87#41C2×D2111yescharacteristic
88C2×D4—31no—
89C2×D4—61no—
90C2×D4—211no—
91C2×D4—422no—
92SD16—421no—
93SD16—421no—
94D8—421no—
95D8—421no—
96M16—211no—
97M16—211no—
98C42—211no—
99C2×C7—61no—
100C2×C7—121no—
101C2×C7GL(2,ℤ/4)11yescharacteristic
102D7—241no—
103C2×D3—561no—
104C3×C4—41no—
105C3×C4—81no—
106C23—32no—
107C23—421no—
108Q8#426811yescharacteristic
109Q8C7⋊S411yescharacteristic
110D4—31no—
111D4—31no—
112D4—61no—
113D4—61no—
114D4—421no—
115D4—421no—
116D4—844no—
117C2×C4—31no—
118C2×C4—61no—
119C2×C4—421no—
120C8—421no—
121C8—421no—
122C7#542911yescharacteristic, complemented, Sylow
123C2×C3—41no—
124D3—1121no—
125C22—31no—
126C22—61no—
127C22—122no—
128C22—842no—
129C22—841no—
130C4—21no—
131C4—31no—
132C4—422no—
133C4C2×C7⋊S411yescharacteristic
134C3—41noSylow
135C2—61no—
136C2—121no—
137C2—1681no—
138C2#1062150611yescharacteristic
subgroupG/H
centerC2#10621506
commutator, cosocleC7×#41⋊C3C22
FrattiniC4C2×C7⋊S4
FittingC7×H(5,𝔽2)D3
socleC2×C7GL(2,ℤ/4)
hypercenterQ8C7⋊S4
hypocenterC7×SL(2,𝔽3)D4
LCSG ▷ C7×#41⋊C3 ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C7×#994C2semidirect
2. #10621504C2semidirect
3. C7⋊GU(2,𝔽3)C2semidirect
9. C7×H(5,𝔽2)D3semidirect
19. C7×SL(2,𝔽3)D4semidirect
35. #994D7semidirect
64. H(5,𝔽2)D21semidirect
82. SL(2,𝔽3)C22⋊D7semidirect
122. C7#5429semidirect

Maximal subgroups

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

subgroupindex
C7×#9942
#106215042
C7⋊GU(2,𝔽3)2
#900133
#666344
#54297