Finite Group Databasebeta

← #13057081

Subgroups — #13057081

Order 1344.

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

normal subgroups only

This group has 124 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
1#10621503C211yesmaximal, characteristic
2#10620795C211yesmaximal, characteristic, complemented
3C7×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#89595—31nomaximal, Hall
5C7×#41⋊C3C2211yescharacteristic
6C7⋊GL(2,𝔽3)C411yescharacteristic, complemented
7C7⋊#226C411yescharacteristic, complemented
8#66559—41nomaximal
9#6513—31no—
10#6511D311yescharacteristic, complemented
11#6470—31no—
12C7×#104—31no—
13#6396—31no—
14#6379—31no—
15#6377—31no—
16#5405—71nomaximal, Hall
17C4×D21—41no—
18C7×SL(2,𝔽3)C2×C411yescharacteristic
19C3×C7⋊C8—41no—
20C7×C3⋊C8—41no—
21C7×#41C2×D311yescharacteristic
22#1157—31no—
23C7×M16—31no—
24C42×C7—31no—
25#1144—31no—
26#1143—31no—
27#1142—31no—
28#1141—31no—
29#1136—31no—
30C2×C7⋊C8—31no—
31C2×C7⋊C8—31no—
32#1131—61no—
33C8×D7—61no—
34#985—71no—
35#131⋊C3—71no—
36GU(2,𝔽3)D711yescharacteristic, complemented
37C2×D21—41no—
38C3×C4×C7—41no—
39Dic21—41no—
40#444—211noSylow
41C7×Q8C4×D311yescharacteristic
42C7×D4—31no—
43C2×C4×C7—31no—
44C2×C4×C7—61no—
45C22⋊D7—61no—
46D28—31no—
47C4×D7—61no—
48Dic14—31no—
49C7×C8—61no—
50C7⋊C8—31no—
51C7⋊C8—31no—
52C7⋊C8—31no—
53C7⋊C8S411yescharacteristic
54#41⋊C3C2×D711yescharacteristic
55GL(2,𝔽3)—71no—
56#226—71no—
57ΓL(1,𝔽25)—281no—
58C2×C3×C7—41no—
59D21—81no—
60#135—211no—
61#131—71no—
62#131—211no—
63#108—211no—
64#104—31no—
65#104—211no—
66#97—211no—
67C22×C7—31no—
68C2×D7—61no—
69C4×C7—31no—
70C4×C7—121no—
71C4×C7C2×S411yescharacteristic
72Dic7—61no—
73C4×D3—281no—
74SL(2,𝔽3)C4×D711yescharacteristic
75C3×C8—281no—
76C3⋊C8—41no—
77C3×C7—41noHall
78#41—211no—
79#41D3×D711yescharacteristic
80Q16—211no—
81SD16—211no—
82SD16—211no—
83D8—211no—
84M16—31no—
85M16—211no—
86M16—421no—
87C2×C8—211no—
88C2×C8—211no—
89C2×C8—421no—
90C42—31no—
91C2×C7—61no—
92C2×C7C4×S411yescharacteristic
93D7—121no—
94C2×D3—281no—
95C3×C4—41no—
96Dic3—281no—
97Q8—211no—
98Q8D21⋊C411yescharacteristic
99D4—31no—
100D4—211no—
101D4—421no—
102C2×C4—31no—
103C2×C4—61no—
104C2×C4—421no—
105C8—61no—
106C8—71no—
107C8—211no—
108C8—211no—
109C8—211no—
110C7#540511yescharacteristic, complemented, Sylow
111C2×C3—41no—
112D3—561no—
113C22—31no—
114C22—421no—
115C4—31no—
116C4—121no—
117C4—421no—
118C4D7×S411yescharacteristic
119C3—41noSylow
120C2—61no—
121C2—841no—
122C2(C7⋊S4)⋊C411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D7×S4
commutator, hypocenterC7×SL(2,𝔽3)C2×C4
FittingC7×#41C2×D3
socleC2×C7C4×S4
prime radical#10621503C2
cosocleC7×#41⋊C3C22
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #10620795C2semidirect
3. C7×GU(2,𝔽3)C2semidirect
6. C7⋊GL(2,𝔽3)C4semidirect
7. C7⋊#226C4semidirect
10. #6511D3semidirect
36. GU(2,𝔽3)D7semidirect
110. C7#5405semidirect

Maximal subgroups

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

subgroupindex
#106215032
#106207952
C7×GU(2,𝔽3)2
#895953
#665594
#54057