Finite Group Databasebeta

← #13057198

Subgroups — #13057198

Order 1344.

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

normal subgroups only

This group has 138 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
1C7×2−1+4⋊C3C211yesmaximal, characteristic
2#10621502C211yesmaximal, characteristic, complemented
3C7⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#90018—31nomaximal, Hall
5C7×#41⋊C3C2211yescharacteristic
6C2×C7×SL(2,𝔽3)—21no—
7C7⋊#226—21no—
8#66633—41nomaximal
9C7×2−1+4D311yescharacteristic
10#6514—31no—
11#6514—31no—
12#6507—31no—
13#6411—31no—
14#6411—31no—
15#6410—31no—
16#5430—71nomaximal, Hall
17C3×C7×D4—41no—
18Dic42—41no—
19C7×SL(2,𝔽3)D411yescharacteristic
20C3⋊(C7⋊C8)—41no—
21C7×#41—31no—
22C7×#41—61no—
23C7×#41C2×D311yescharacteristic
24C2×C7×Q8—21no—
25C2×C7×Q8—31no—
26C2×Dic14—31no—
27#1144—61no—
28#1144—61no—
29#1142—61no—
30#1142—61no—
31Dic7⋊C4—62no—
32C4×Dic7—31no—
33#1136—31no—
34#1136—31no—
352−1+4⋊C3D711yescharacteristic
36#984—71no—
37GU(2,𝔽3)—71no—
38C22×C3×C7—81no—
39C3×C4×C7—41no—
40Dic21—81no—
41#456—211noSylow
42C7×Q8—31no—
43C7×Q8—61no—
44C7×Q8#6711yescharacteristic
45C7×D4—31no—
46C7×D4—61no—
47C7×D4S411yescharacteristic
48C2×C4×C7—31no—
49C2×C4×C7—32no—
50C2×C4×C7—61no—
51C2×Dic7—61no—
52C2×Dic7—61no—
53Dic14—61no—
54Dic14—61no—
55C7⋊C8—61no—
56C7⋊C8—61no—
57#41⋊C3C2×D711yescharacteristic
58C2×SL(2,𝔽3)—21no—
59#226—141no—
60#214—281no—
61C2×C3×C7—41no—
62C2×C3×C7—161no—
632−1+4D2111yescharacteristic
64#137—211no—
65#137—211no—
66C4⋊Q8—211no—
67#104—211no—
68#104—211no—
69#101—211no—
70C22×C7—21no—
71C22×C7—31no—
72C4×C7—31no—
73C4×C7—61no—
74C4×C7C2×S411yescharacteristic
75Dic7—62no—
76Dic7—121no—
77C3×D4—41no—
78Dic6—281no—
79SL(2,𝔽3)C22⋊D711yescharacteristic
80C3⋊C8—281no—
81C3×C7—41noHall
82#41—31no—
83#41—61no—
84#41C2×D2111yescharacteristic
85C2×Q8—21no—
86C2×Q8—31no—
87C2×Q8—211no—
88Q16—421no—
89Q16—421no—
90SD16—421no—
91SD16—421no—
92M16—211no—
93M16—211no—
94C4⋊C4—422no—
95C42—211no—
96C2×C7—41no—
97C2×C7—61no—
98C2×C7GL(2,ℤ/4)11yescharacteristic
99C22×C3—81no—
100C3×C4—41no—
101Dic3—561no—
102Q8—31no—
103Q8—61no—
104Q8—421no—
105Q8—421no—
106Q8#426811yescharacteristic
107D4—31no—
108D4—61no—
109D4C7⋊S411yescharacteristic
110C2×C4—31no—
111C2×C4—32no—
112C2×C4—61no—
113C2×C4—421no—
114C2×C4—421no—
115C8—421no—
116C8—421no—
117C7#543011yescharacteristic, complemented, Sylow
118C2×C3—41no—
119C2×C3—161no—
120C22—21no—
121C22—31no—
122C4—31no—
123C4—61no—
124C4—422no—
125C4—841no—
126C4C2×C7⋊S411yescharacteristic
127C3—41noSylow
128C2—41no—
129C2—61no—
130C2#1062150611yescharacteristic
subgroupG/H
centerC2#10621506
commutator, cosocleC7×#41⋊C3C22
FrattiniC4C2×C7⋊S4
FittingC7×2−1+4D3
socleC2×C7GL(2,ℤ/4)
prime radicalC7×2−1+4⋊C3C2
hypercenterD4C7⋊S4
hypocenterC7×SL(2,𝔽3)D4
LCSG ▷ C7×#41⋊C3 ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #10621502C2semidirect
3. C7⋊GU(2,𝔽3)C2semidirect
117. C7#5430semidirect

Maximal subgroups

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

subgroupindex
C7×2−1+4⋊C32
#106215022
C7⋊GU(2,𝔽3)2
#900183
#666334
#54307