Finite Group Databasebeta

← #13057684

Subgroups — #13057684

Order 1344.

146 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 26 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
12−1+4⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#10621541C211yesmaximal, characteristic, complemented
3#10620803C211yesmaximal, characteristic, complemented
4#90529C311yesmaximal, characteristic, complemented, Hall
5#41⋊ΓL(1,𝔽8)C2211yescharacteristic
6C2×C7⋊SL(2,𝔽3)—21no—
7D7⋊SL(2,𝔽3)—21no—
8#1141⋊C3—41nomaximal
9C7×2−1+4C2×C311yescharacteristic, complemented
10#6554C2×C311yescharacteristic, complemented
11#6514—31no—
12#6513—31no—
13#6511C2×C311yescharacteristic, complemented
14C2×#1144—31no—
15#6505—31no—
16#578⋊C3—71nomaximal, Hall
17C7⋊SL(2,𝔽3)D411yescharacteristic
18D4×ΓL(1,𝔽8)—41no—
19C4⋊AGL(1,𝔽7)—41no—
20ΓL(1,𝔽8)⋊C8—41no—
21C7×#41—31no—
22C7×#41—61no—
23C7×#41C22×C311yescharacteristic
24C2×C7×Q8—21no—
25C2×C7×Q8—31no—
26Q8×D7—21no—
27#1159—61no—
28#1157—31no—
29C2×Dic14—31no—
30#1144—32no—
31#1144—31no—
32#1143—31no—
33#1142—31no—
34#1141A411yescharacteristic
35#1136—31no—
36C2×C7⋊C8—31no—
372−1+4⋊C3—71no—
382−1+4⋊C3—71no—
39#131⋊C3—71no—
40C22×ΓL(1,𝔽8)—81no—
41C2×AGL(1,𝔽7)—81no—
42C4×ΓL(1,𝔽8)—41no—
43#578—71noSylow
44C7×Q8—31no—
45C7×Q8—61no—
46C7×Q8C3×D411yescharacteristic
47C7×D4—31no—
48C7×D4—61no—
49C7×D4C2×A411yescharacteristic
50C2×C4×C7—31no—
51C2×C4×C7—61no—
52C2×C4×C7—61no—
53C22⋊D7—61no—
54C2×Dic7—61no—
55D28C2×A411yescharacteristic
56C4×D7—61no—
57Dic14—31no—
58Dic14—61no—
59C7⋊C8—31no—
60C7⋊C8C2×A411yescharacteristic
61#41⋊C3—71no—
62C2×SL(2,𝔽3)—141no—
63C2×SL(2,𝔽3)—141no—
64C3×D8—281no—
65C2×ΓL(1,𝔽8)—41no—
66C2×ΓL(1,𝔽8)—161no—
67AGL(1,𝔽7)—161no—
682−1+4—71no—
692−1+4AGL(1,𝔽7)11yescharacteristic, complemented
70#137—211no—
71#137—211no—
72#135—211no—
73C2×Q16—211no—
74#131—71no—
75C22×C7—21no—
76C22×C7—31no—
77C2×D7—21no—
78C4×C7—31no—
79C4×C7—61no—
80C4×C7C22×A411yescharacteristic
81Dic7—61no—
82C3×D4—281no—
83C3×D4—281no—
84SL(2,𝔽3)—71no—
85C3×C8—281no—
86ΓL(1,𝔽8)—41noHall
87#41—31no—
88#41—61no—
89#41—211no—
90#41—421no—
91#41C2×AGL(1,𝔽7)11yescharacteristic
92C2×Q8—21no—
93C2×Q8—31no—
94C2×Q8—141no—
95C2×Q8—211no—
96Q16—211no—
97Q16—212no—
98SD16—211no—
99SD16—211no—
100D8—71no—
101M16—211no—
102C2×C8—211no—
103C2×C7—41no—
104C2×C7—61no—
105C2×C7D4×A411yescharacteristic
106D7—41no—
107C22×C3—561no—
108C22×C3—561no—
109C3×C4—281no—
110Q8—31no—
111Q8—61no—
112Q8—211no—
113Q8—421no—
114Q8(C22⋊D7)⋊C311yescharacteristic
115D4—31no—
116D4—61no—
117D4—71no—
118D4—421no—
119D4D7⋊A411yescharacteristic
120C2×C4—31no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—421no—
124C2×C4—421no—
125C8—71no—
126C8—211no—
127C7#578⋊C311yescharacteristic, complemented, Sylow
128C2×C3—281no—
129C2×C3—1121no—
130C2×C3—1121no—
131C22—21no—
132C22—31no—
133C22—141no—
134C4—31no—
135C4—61no—
136C4—421no—
137C4C2×D7⋊A411yescharacteristic
138C3—281noSylow
139C2—41no—
140C2—61no—
141C2—281no—
142C2(C22⋊D7)⋊A411yescharacteristic
subgroupG/H
centerC2(C22⋊D7)⋊A4
commutator, cosocleC7×#41C22×C3
FrattiniC4C2×D7⋊A4
FittingC7×2−1+4C2×C3
socleC2×C7D4×A4
hypercenterD4D7⋊A4
hypocenterC7×Q8C3×D4
LCSG ▷ C7×#41 ▷ C7×Q8
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊ΓL(1,𝔽8)C2semidirect
2. #10621541C2semidirect
3. #10620803C2semidirect
4. #90529C3semidirect
9. C7×2−1+4C2×C3semidirect
10. #6554C2×C3semidirect
13. #6511C2×C3semidirect
69. 2−1+4AGL(1,𝔽7)semidirect
127. C7#578⋊C3semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊ΓL(1,𝔽8)2
#106215412
#106208032
#905293
#1141⋊C34
#578⋊C37