Finite Group Databasebeta

← C2×#10621501 #13062116

Subgroups — C2×#10621501 #13062116

Order 1344.

306 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C22×C7×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10621501C214yesmaximal, direct product factor
3C2×C7⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C7⋊#226C211yesmaximal, characteristic, complemented
5C2×#6505—31nomaximal, Hall
6C2×#4268—41nomaximal
7C2×C7×SL(2,𝔽3)C2211yescharacteristic
8C2×C7×SL(2,𝔽3)C2212yescomplemented
9C7⋊GL(2,𝔽3)C2212yescomplemented
10C7⋊#226C2212yescomplemented
11C22×C7×Q8D311yescharacteristic, complemented
12C2×#1157—31no—
13C2×#1144—31no—
14C2×#1144—31no—
15#6505—34no—
16#6505—34no—
17C2×#1143—31no—
18C2×#1143—31no—
19C2×#1136—31no—
20C2×#983—71nomaximal, Hall
21C23×C3×C7—41no—
22C22×D21—41no—
23#4268—44no—
24C2×Dic21—41no—
25C7×SL(2,𝔽3)C2311yescharacteristic
26C2×C7×Q8—31no—
27C2×C7×Q8—32no—
28C2×C7×Q8C2×D311yescharacteristic
29C2×C7×Q8C2×D312yescomplemented
30C22×C4×C7—31no—
31C2×C22⋊D7—61no—
32#1157—34no—
33#1157—62no—
34C2×D28—31no—
35C2×C4×D7—61no—
36C2×Dic14—31no—
37#1144—32no—
38#1144—34no—
39#1144—32no—
40#1143—32no—
41#1143—34no—
42#1143—32no—
43#1136—34no—
44C2×C7⋊C8—31no—
45C2×C7⋊C8—31no—
46C22×SL(2,𝔽3)D711yescharacteristic, complemented
47#983—74no—
48C2×GL(2,𝔽3)—71no—
49C2×#226—71no—
50C22×C3×C7—41no—
51C22×C3×C7—42no—
52C22×C3×C7—82no—
53C2×D21—42no—
54C2×D21—82no—
55Dic21—42no—
56C2×#137—211noSylow
57C23×C7S411yescharacteristic
58C22×D7—61no—
59C7×Q8—31no—
60C7×Q8—32no—
61C7×Q8—61no—
62C7×Q8C22×D311yescharacteristic
63C2×C4×C7—31no—
64C2×C4×C7—32no—
65C2×C4×C7—31no—
66C2×C4×C7—32no—
67C22⋊D7—64no—
68C2×Dic7—61no—
69D28—32no—
70D28—61no—
71C4×D7—64no—
72Dic14—32no—
73Dic14—61no—
74C7⋊C8—32no—
75C7⋊C8—32no—
76C2×#67—281no—
77C2×SL(2,𝔽3)C2×D711yescharacteristic
78C2×SL(2,𝔽3)C2×D712yescomplemented
79GL(2,𝔽3)—72no—
80#226—72no—
81C2×C3×C7—42no—
82C2×C3×C7—41no—
83C2×C3×C7—82no—
84D21—82no—
85C2×#41—211no—
86C22×Q8D2111yescharacteristic, complemented
87#137—214no—
88#137—214no—
89C2×Q16—211no—
90C2×Q16—211no—
91C2×SD16—211no—
92C2×SD16—211no—
93C2×M16—211no—
94C22×C7—22no—
95C22×C7C2×S412yes—
96C22×C7C2×S411yescharacteristic
97C2×D7—62no—
98C2×D7—122no—
99C4×C7—31no—
100C4×C7—31no—
101C4×C7—32no—
102Dic7—62no—
103C23×C3—41no—
104C22×D3—281no—
105#67—284no—
106C2×Dic3—281no—
107SL(2,𝔽3)C22×D711yescharacteristic
108C3×C7—41noHall
109#41—214no—
110#41—422no—
111C2×Q8—31no—
112C2×Q8—32no—
113C2×Q8—211no—
114C2×Q8C2×D2111yescharacteristic
115C2×Q8C2×D2112yescomplemented
116C2×D4—211no—
117C2×D4—421no—
118C22×C4—31no—
119C22×C4—421no—
120Q16—212no—
121Q16—214no—
122Q16—212no—
123SD16—212no—
124SD16—214no—
125SD16—212no—
126M16—214no—
127C2×C8—211no—
128C2×C8—211no—
129C2×C7—22no—
130C2×C7#98312yescomplemented
131C2×C7C22×S411yescharacteristic
132D7—122no—
133C22×C3—42no—
134C22×C3—41no—
135C22×C3—82no—
136C2×D3—282no—
137C2×D3—562no—
138Dic3—282no—
139C23—421no—
140C23C7⋊S411yescharacteristic
141Q8—31no—
142Q8—32no—
143Q8—61no—
144Q8—212no—
145Q8—421no—
146Q8C22×D2111yescharacteristic
147D4—212no—
148D4—421no—
149D4—424no—
150C2×C4—32no—
151C2×C4—32no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—424no—
155C2×C4—421no—
156C8—212no—
157C8—212no—
158C7C2×#98311yescharacteristic, complemented, Sylow
159C2×C3—42no—
160C2×C3—41no—
161C2×C3—82no—
162D3—562no—
163C22—22no—
164C22—422no—
165C22—842no—
166C22C2×C7⋊S412yes—
167C22C2×C7⋊S411yescharacteristic
168C4—31no—
169C4—32no—
170C4—31no—
171C4—422no—
172C3—41noSylow
173C2—22no—
174C2—842no—
175C2#1062150112yesdirect product factor
176C2C22×C7⋊S411yescharacteristic
subgroupG/H
centerC22C2×C7⋊S4
commutator, cosocle, hypocenterC7×SL(2,𝔽3)C23
FrattiniC2C22×C7⋊S4
FittingC22×C7×Q8D3
socleC22×C7C2×S4
hypercenterC23C7⋊S4
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C7×SL(2,𝔽3)C2semidirect
2. #10621501C2direct (N × H)
3. C2×C7⋊GL(2,𝔽3)C2semidirect
4. C2×C7⋊#226C2semidirect
8. C2×C7×SL(2,𝔽3)C22semidirect
9. C7⋊GL(2,𝔽3)C22semidirect
10. C7⋊#226C22semidirect
11. C22×C7×Q8D3semidirect
29. C2×C7×Q8C2×D3semidirect
46. C22×SL(2,𝔽3)D7semidirect
78. C2×SL(2,𝔽3)C2×D7semidirect
86. C22×Q8D21semidirect
115. C2×Q8C2×D21semidirect
130. C2×C7#983semidirect
158. C7C2×#983semidirect
175. C2#10621501direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×SL(2,𝔽3)2
#106215012
C2×C7⋊GL(2,𝔽3)2
C2×C7⋊#2262
C2×#65053
C2×#42684
C2×#9837