Finite Group Databasebeta

← #13057087

Subgroups — #13057087

Order 1344.

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

normal subgroups only

This group has 144 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
1D7×#41⋊C3C211yesmaximal, characteristic
2C7⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
3C7×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#89593—31nomaximal, Hall
5C7×#41⋊C3C2211yescharacteristic
6D7×SL(2,𝔽3)C411yescharacteristic, complemented
7#66662C411yescharacteristic, complemented
8#66557—41nomaximal
9D7×#41D311yescharacteristic
10D7×M16—31no—
11#6435—31no—
12C7×#104—31no—
13#6411—31no—
14#6399—31no—
15#6379—31no—
16#5423—71nomaximal, Hall
17C3×C4×D7—41no—
18C7×SL(2,𝔽3)C2×C411yescharacteristic
19C3⋊(C7⋊C8)—41no—
20C7×C3⋊C8—41no—
21C7×#41C2×D311yescharacteristic
22#1161Dic311yescharacteristic
23Q8×D7Dic311yescharacteristic
24#1159—31no—
25D4×D7—31no—
26#1157—31no—
27C2×C4×D7—31no—
28C7×M16—31no—
29C42×C7—31no—
30(C2×D7)⋊C4—61no—
31Dic7⋊C4—61no—
32C4×Dic7—31no—
33#1136—31no—
34#1131—61no—
35C8×D7—61no—
36C2×#41⋊C3—71no—
37GU(2,𝔽3)—71no—
38GU(2,𝔽3)D711yescharacteristic, complemented
39C2×C3×D7—41no—
40C3×C4×C7—41no—
41C3×Dic7—41no—
42#421—211noSylow
43C22×D7—31no—
44C7×Q8C2×Dic311yescharacteristic
45C7×D4—31no—
46C2×C4×C7—31no—
47C2×C4×C7—61no—
48C22⋊D7—61no—
49C2×Dic7—31no—
50C2×Dic7—61no—
51D28—31no—
52C4×D7—31no—
53C4×D7—31no—
54C4×D7—31no—
55C4×D7S411yescharacteristic
56Dic14—31no—
57C7×C8—61no—
58C7⋊C8—61no—
59#41⋊C3—71no—
60#41⋊C3C2×D711yescharacteristic
61C2×SL(2,𝔽3)—71no—
62#208—281no—
63C2×C3×C7—41no—
64C3×D7—81no—
65C2×#41—71no—
66C2×M16—211no—
67#117—211no—
68#104—31no—
69#104—211no—
70#104—211no—
71#104—211no—
72C22×C7—31no—
73C2×D7—31no—
74C2×D7—121no—
75C2×D7SL(2,ℤ/4)11yescharacteristic
76C4×C7—31no—
77C4×C7—121no—
78C4×C7C2×S411yescharacteristic
79Dic7—31no—
80Dic7—121no—
81Dic7SL(2,ℤ/4)11yescharacteristic
82C2×C3×C4—281no—
83SL(2,𝔽3)C4×D711yescharacteristic
84C3⋊C8—41no—
85C3⋊C8—281no—
86C3×C7—41noHall
87#41—71no—
88#41—211no—
89#41—211no—
90#41D3×D711yescharacteristic
91C2×Q8—71no—
92C2×D4—211no—
93C22×C4—211no—
94M16—31no—
95M16—211no—
96M16—421no—
97C2×C8—421no—
98C4⋊C4—421no—
99C22⋊C4—421no—
100C42—31no—
101C42—211no—
102C2×C7—61no—
103C2×C7C2×SL(2,ℤ/4)11yescharacteristic
104D7—21no—
105D7—61no—
106C22×C3—281no—
107C3×C4—41no—
108C3×C4—281no—
109C23—211no—
110Q8—211no—
111Q8Dic3×D711yescharacteristic
112D4—31no—
113D4—211no—
114D4—421no—
115C2×C4—31no—
116C2×C4—61no—
117C2×C4—71no—
118C2×C4—211no—
119C2×C4—211no—
120C2×C4—211no—
121C2×C4—211no—
122C2×C4—421no—
123C8—61no—
124C8—421no—
125C7#542311yescharacteristic, complemented, Sylow
126C2×C3—41no—
127C2×C3—561no—
128C22—31no—
129C22—71no—
130C22—211no—
131C22—841no—
132C4—31no—
133C4—71no—
134C4—121no—
135C4—211no—
136C4—841no—
137C4D7×S411yescharacteristic
138C3—41noSylow
139C2—61no—
140C2—141no—
141C2—421no—
142C2D7×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D7×S4
commutator, hypocenterC7×SL(2,𝔽3)C2×C4
FittingC7×#41C2×D3
socleC2×C7C2×SL(2,ℤ/4)
prime radicalD7×#41⋊C3C2
cosocleC7×#41⋊C3C22
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C7⋊GU(2,𝔽3)C2semidirect
3. C7×GU(2,𝔽3)C2semidirect
6. D7×SL(2,𝔽3)C4semidirect
7. #66662C4semidirect
38. GU(2,𝔽3)D7semidirect
125. C7#5423semidirect

Maximal subgroups

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

subgroupindex
D7×#41⋊C32
C7⋊GU(2,𝔽3)2
C7×GU(2,𝔽3)2
#895933
#665574
#54237