Finite Group Databasebeta

← #13057060

Subgroups — #13057060

Order 1344.

134 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 27 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×C7⋊#226C211yesmaximal, characteristic
2C2×C7×#226C211yesmaximal, characteristic
3SL(2,𝔽3)×Dic7C211yesmaximal, characteristic
4#89954—31nomaximal, Hall
5C2×C7×SL(2,𝔽3)C2211yescharacteristic
6C7⋊#226—21no—
7C7×#226—21no—
8Dic7⋊Dic3—41nomaximal
9C2×C7×Q16—31no—
10Q8×Dic7D311yescharacteristic
11#6507—31no—
12C2×#1144—31no—
13#6409—31no—
14#6390—31no—
15Dic7⋊C8—31no—
16C4⋊#226—71nomaximal, Hall
17C2×Dic21—41no—
18C2×C7×Dic3—41no—
19C2×C3×Dic7—41no—
20C7×SL(2,𝔽3)D411yescharacteristic
21C2×C7×Q8—31no—
22C2×C7×Q8C2×D311yescharacteristic
23C2×Dic14—31no—
24C7×Q16—61no—
25C7×Q16—61no—
26C2×C7×C8—31no—
27#1144—61no—
28#1144—61no—
29C4⋊Dic7—31no—
30Dic7⋊C4—61no—
31Dic7⋊C4—61no—
32C4×Dic7—31no—
33C2×C7⋊C8—31no—
34C2×#226—71no—
35C2×#226D711yescharacteristic
36C4×SL(2,𝔽3)—71no—
37C22×C3×C7—41no—
38Dic21—81no—
39C3×Dic7—42no—
40C7×Dic3—81no—
41#462—211noSylow
42C7×Q8—31no—
43C7×Q8—61no—
44C7×Q8—61no—
45C7×Q8D1211yescharacteristic
46C2×C4×C7—31no—
47C2×C4×C7—61no—
48C2×Dic7—31no—
49C2×Dic7—61no—
50C2×Dic7S411yescharacteristic
51Dic14—61no—
52Dic14—61no—
53C7×C8—61no—
54C7⋊C8—61no—
55C2×SL(2,𝔽3)C2×D711yescharacteristic
56#226—21no—
57#226—141no—
58C4⋊Dic3—281no—
59C2×C3×C7—41no—
60C2×C3×C7—41no—
61C2×C3×C7—41no—
62C2×Q16—31no—
63C2×Q16—211no—
64C4⋊Q8—211no—
65C4×Q8—71no—
66C4⋊C8—211no—
67#103—211no—
68#103—211no—
69C22×C7C2×S411yescharacteristic
70C4×C7—31no—
71C4×C7—31no—
72C4×C7—121no—
73Dic7—61no—
74Dic7—121no—
75Dic7#22612yescomplemented
76C2×C3×C4—281no—
77C2×Dic3—41no—
78C2×Dic3—281no—
79SL(2,𝔽3)C22⋊D711yescharacteristic
80C3×C7—41noHall
81C2×Q8—31no—
82C2×Q8—211no—
83C2×Q8D3×D711yescharacteristic
84Q16—61no—
85Q16—61no—
86Q16—421no—
87Q16—421no—
88C2×C8—31no—
89C2×C8—211no—
90C4⋊C4—211no—
91C4⋊C4—421no—
92C4⋊C4—421no—
93C42—211no—
94C2×C7C4⋊S411yescharacteristic
95C2×C7C2×#22611yescharacteristic
96C2×C7#98411yescharacteristic
97C22×C3—41no—
98C3×C4—282no—
99Dic3—81no—
100Dic3—561no—
101Q8—31no—
102Q8—61no—
103Q8—61no—
104Q8—421no—
105Q8—421no—
106Q8#424711yescharacteristic
107C2×C4—31no—
108C2×C4—61no—
109C2×C4—71no—
110C2×C4—211no—
111C2×C4—421no—
112C8—61no—
113C8—421no—
114C7C4⋊#22611yescharacteristic, complemented, Sylow
115C2×C3—41no—
116C2×C3—41no—
117C2×C3—41no—
118C22D7×S411yescharacteristic
119C4—31no—
120C4—31no—
121C4—72no—
122C4—121no—
123C4—421no—
124C4—841no—
125C3—41noSylow
126C2#1062146911yescharacteristic
127C2D7×#22611yescharacteristic
128C2#1062148011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D7×S4
commutator, prime radical, 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
75. Dic7#226semidirect
114. C7C4⋊#226semidirect

Maximal subgroups

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

subgroupindex
C2×C7⋊#2262
C2×C7×#2262
SL(2,𝔽3)×Dic72
#899543
Dic7⋊Dic34
C4⋊#2267