Finite Group Databasebeta

← #13057587

Subgroups — #13057587

Order 1344.

180 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 35 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×D7×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#66662C211yesmaximal, characteristic, complemented
3C4×C7×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#90267C311yesmaximal, characteristic, complemented, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6D7×SL(2,𝔽3)—21no—
7#66662—21no—
8C3×(C2×D7)⋊C4—41nomaximal
9C2×#1161C2×C311yescharacteristic, complemented
10C2×Q8×D7C2×C311yescharacteristic
11C4×C7×Q8C2×C311yescharacteristic, complemented
12#6460—31no—
13#6458—31no—
14#6438—31no—
15C4×D28—31no—
16#548⋊C3—71nomaximal, Hall
17C22×C3×D7—41no—
18C2×C3×C4×C7—41no—
19C2×C3×Dic7—41no—
20C7×SL(2,𝔽3)D411yescharacteristic, complemented
21C2×C7×Q8C22×C311yescharacteristic
22#1161—21no—
23#1161—61no—
24Q8×D7—21no—
25Q8×D7—61no—
26C2×D28—31no—
27C2×D28—31no—
28C2×C4×D7—31no—
29C2×C4×D7—31no—
30C2×Dic14—31no—
31C7×C4⋊C4—31no—
32C42×C7—31no—
33(C2×D7)⋊C4—32no—
34(C2×D7)⋊C4—31no—
35(C2×D7)⋊C4A411yescharacteristic
36C4⋊Dic7—31no—
37C2×#41⋊C3—71no—
38C22×SL(2,𝔽3)—71no—
39C4×SL(2,𝔽3)D711yescharacteristic, complemented
40C22×C3×C7—41no—
41C2×C3×D7—42no—
42C2×C3×D7—81no—
43C2×C3×D7—81no—
44C3×C4×C7—81no—
45C3×Dic7—81no—
46#548—71noSylow
47C22×D7—31no—
48C22×D7C2×A411yescharacteristic
49C7×Q8—31no—
50C7×Q8C3×D411yescharacteristic
51C2×C4×C7—31no—
52C2×C4×C7—31no—
53C2×C4×C7C2×A411yescharacteristic
54C2×Dic7—31no—
55C2×Dic7C2×A411yescharacteristic
56D28—61no—
57D28—61no—
58D28—62no—
59C4×D7—61no—
60C4×D7—61no—
61C4×D7—61no—
62C4×D7—61no—
63Dic14—61no—
64Dic14—61no—
65#41⋊C3—141no—
66C2×SL(2,𝔽3)—141no—
67C2×SL(2,𝔽3)C2×D711yescharacteristic
68C3×C22⋊C4—281no—
69C2×C3×C7—41no—
70C2×C3×C7—41no—
71C2×C3×C7—41no—
72C3×D7—82no—
73C2×#41—71no—
74C22×Q8—71no—
75#124—211no—
76C22⋊Q8—211no—
77#121—211no—
78C4×Q8C3×D711yescharacteristic, complemented
79C4×D4—211no—
80C22×C7C22×A411yescharacteristic
81C2×D7—21no—
82C2×D7—21no—
83C2×D7—61no—
84C2×D7—61no—
85C2×D7—61no—
86C2×D7#41⋊C312yescomplemented
87C4×C7—21no—
88C4×C7—31no—
89C4×C7—31no—
90C4×C7—61no—
91Dic7—21no—
92Dic7—61no—
93C23×C3—281no—
94C2×C3×C4—41no—
95C2×C3×C4—281no—
96SL(2,𝔽3)D2811yescharacteristic, complemented
97C3×C7—41noHall
98#41—141no—
99#41—421no—
100C2×Q8—141no—
101C2×Q8—211no—
102C2×Q8—421no—
103C2×Q8C2×C3×D711yescharacteristic
104C2×D4—211no—
105C2×D4—211no—
106C22×C4—211no—
107C22×C4—211no—
108C4⋊C4—31no—
109C4⋊C4—211no—
110C22⋊C4—71no—
111C22⋊C4—212no—
112C22⋊C4—211no—
113C42—31no—
114C2×C7D4×A411yescharacteristic
115C2×C7C2×#41⋊C311yescharacteristic
116C2×C72−1+4⋊C311yescharacteristic
117D7—22no—
118D7—121no—
119C22×C3—41no—
120C22×C3—282no—
121C22×C3—561no—
122C22×C3—561no—
123C3×C4—81no—
124C3×C4—561no—
125C23—71no—
126C23—211no—
127Q8—31no—
128Q8—421no—
129Q8—421no—
130Q8C3×D2811yescharacteristic
131D4—421no—
132D4—421no—
133D4—422no—
134C2×C4—31no—
135C2×C4—31no—
136C2×C4—71no—
137C2×C4—211no—
138C2×C4—421no—
139C2×C4—421no—
140C2×C4—421no—
141C2×C4—421no—
142C2×C4A4×D711yescharacteristic
143C7#548⋊C311yescharacteristic, complemented, Sylow
144C2×C3—41no—
145C2×C3—41no—
146C2×C3—41no—
147C2×C3—562no—
148C22—72no—
149C22—141no—
150C22—141no—
151C22—421no—
152C22—421no—
153C22—421no—
154C22C2×A4×D711yescharacteristic
155C4—21no—
156C4—31no—
157C4—31no—
158C4—61no—
159C4—141no—
160C4—421no—
161C3—41noSylow
162C2—142no—
163C2—841no—
164C2A4×D2811yescharacteristic
165C2#1062152711yescharacteristic
166C2D7×#41⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×D7
commutator, cosocleC2×C7×Q8C22×C3
FittingC4×C7×Q8C2×C3
socleC22×C7C22×A4
hypercenterC2×C4A4×D7
hypocenterC7×Q8C3×D4
LCSG ▷ C2×C7×Q8 ▷ C7×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D7×SL(2,𝔽3)C2semidirect
2. C2×#66662C2semidirect
3. C4×C7×SL(2,𝔽3)C2semidirect
4. #90267C3semidirect
9. C2×#1161C2×C3semidirect
11. C4×C7×Q8C2×C3semidirect
20. C7×SL(2,𝔽3)D4semidirect
39. C4×SL(2,𝔽3)D7semidirect
78. C4×Q8C3×D7semidirect
86. C2×D7#41⋊C3semidirect
96. SL(2,𝔽3)D28semidirect
143. C7#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D7×SL(2,𝔽3)2
C2×#666622
C4×C7×SL(2,𝔽3)2
#902673
C3×(C2×D7)⋊C44
#548⋊C37