Finite Group Databasebeta

← #13057065

Subgroups — #13057065

Order 1344.

166 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 25 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
1C2×#66662C211yesmaximal, characteristic, complemented
2C2×C7⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C7×#859C211yesmaximal, characteristic, complemented
4#89583—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6#66662—21no—
7C7⋊GL(2,𝔽3)—21no—
8#66575—41nomaximal
9C2×#1161D311yescharacteristic, complemented
10C2×#1143—31no—
11C2×D56—31no—
12#6458—31no—
13C7×#103—31no—
14(C2×D7)⋊C8—31no—
15#6383—31no—
16#5427—71nomaximal, Hall
17C22×D21—41no—
18C2×C7×Dic3—41no—
19C2×C3×Dic7—41no—
20C7×SL(2,𝔽3)D411yescharacteristic, complemented
21C2×C7×Q8C2×D311yescharacteristic
22#1161—21no—
23#1161—61no—
24C2×D28—31no—
25C2×D28—31no—
26C2×D28—61no—
27C2×C4×D7—31no—
28C2×C7×C8—31no—
29C7×C4⋊C4—31no—
30#1143—61no—
31#1143—61no—
32(C2×D7)⋊C4—61no—
33C2×C7⋊C8—31no—
34D56—62no—
35C2×#41⋊C3—71no—
36C2×GL(2,𝔽3)—71no—
37#859D711yescharacteristic, complemented
38C22×C3×C7—41no—
39C2×D21—42no—
40C2×D21—81no—
41C2×D21—81no—
42C3×Dic7—81no—
43C7×Dic3—81no—
44#449—211noSylow
45C22×D7—31no—
46C22×D7—61no—
47C7×Q8—31no—
48C7×Q8#6711yescharacteristic
49C2×C4×C7—31no—
50C2×C4×C7—61no—
51C2×Dic7S411yescharacteristic
52D28—32no—
53D28—61no—
54D28—61no—
55D28—61no—
56D28—122no—
57C4×D7—61no—
58C4×D7—61no—
59C7×C8—61no—
60C7⋊C8—61no—
61#41⋊C3—141no—
62C2×SL(2,𝔽3)C2×D711yescharacteristic
63GL(2,𝔽3)—141no—
64(C2×D3)⋊C4—281no—
65C2×C3×C7—41no—
66C2×C3×C7—41no—
67C2×C3×C7—41no—
68D21—82no—
69C2×#41—71no—
70C2×SD16—211no—
71C2×D8—211no—
72#121—211no—
73#103—31no—
74#102—211no—
75C22⋊C8—211no—
76C22×C7C2×S411yescharacteristic
77C2×D7—61no—
78C2×D7—61no—
79C2×D7—61no—
80C2×D7—121no—
81C2×D7—121no—
82C2×D7—121no—
83C4×C7—31no—
84C4×C7—31no—
85C4×C7—121no—
86Dic7—21no—
87C22×D3—281no—
88C2×C3×C4—281no—
89C2×Dic3—41no—
90SL(2,𝔽3)D2811yescharacteristic, complemented
91C3×C7—41noHall
92#41—141no—
93#41—421no—
94C2×Q8D3×D711yescharacteristic
95C2×D4—211no—
96C2×D4—211no—
97C2×D4—421no—
98C22×C4—211no—
99SD16—421no—
100SD16—421no—
101D8—422no—
102C2×C8—31no—
103C2×C8—211no—
104C4⋊C4—31no—
105C22⋊C4—421no—
106C2×C7#98511yescharacteristic
107C2×C7#98611yescharacteristic
108C2×C7GL(2,ℤ/4)11yescharacteristic
109D7—121no—
110D7—241no—
111C22×C3—41no—
112C2×D3—282no—
113C2×D3—561no—
114C2×D3—561no—
115C3×C4—561no—
116Dic3—81no—
117C23—211no—
118C23—421no—
119Q8—31no—
120Q8#424611yescharacteristic
121D4—212no—
122D4—421no—
123D4—421no—
124D4—421no—
125D4—842no—
126C2×C4—31no—
127C2×C4—61no—
128C2×C4—71no—
129C2×C4—421no—
130C2×C4—421no—
131C8—61no—
132C8—421no—
133C7#542711yescharacteristic, complemented, Sylow
134C2×C3—41no—
135C2×C3—41no—
136C2×C3—41no—
137D3—562no—
138C22—421no—
139C22—421no—
140C22—421no—
141C22—841no—
142C22—841no—
143C22—841no—
144C22D7×S411yescharacteristic
145C4—31no—
146C4—31no—
147C4—121no—
148C4—141no—
149C3—41noSylow
150C2—841no—
151C2—1681no—
152C2#1062147111yescharacteristic
153C2#1062147211yescharacteristic
154C2#1062148311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, cosocleC2×C7×SL(2,𝔽3)C22
FittingC2×C7×Q8C2×D3
socleC22×C7C2×S4
hypocenterC7×SL(2,𝔽3)D4
LCSG ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#66662C2semidirect
2. C2×C7⋊GL(2,𝔽3)C2semidirect
3. C7×#859C2semidirect
9. C2×#1161D3semidirect
20. C7×SL(2,𝔽3)D4semidirect
37. #859D7semidirect
90. SL(2,𝔽3)D28semidirect
133. C7#5427semidirect

Maximal subgroups

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

subgroupindex
C2×#666622
C2×C7⋊GL(2,𝔽3)2
C7×#8592
#895833
#665754
#54277