Finite Group Databasebeta

← #13057681

Subgroups — #13057681

Order 1344.

180 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 32 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×#41⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×D7⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3Dic7⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#90522C311yesmaximal, characteristic, complemented, Hall
5#41⋊ΓL(1,𝔽8)—21no—
6C2×C7⋊SL(2,𝔽3)C2211yescharacteristic
7D7⋊SL(2,𝔽3)—21no—
8(C2×AGL(1,𝔽7))⋊C4—41nomaximal
9C2×C7×#41C2×C311yescharacteristic, complemented
10C2×Q8×D7C2×C311yescharacteristic
11Q8×Dic7C2×C311yescharacteristic, complemented
12#6501—31no—
13#6499—31no—
14C4×C22⋊D7—31no—
15#6487—31no—
16#548⋊C3—71nomaximal, Hall
17C22×AGL(1,𝔽7)—41no—
18C7⋊SL(2,𝔽3)D411yescharacteristic, complemented
19C2×C4×ΓL(1,𝔽8)—41no—
20C2×ΓL(1,𝔽8)⋊C4—41no—
21C7×#41—21no—
22C7×#41—61no—
23C2×C7×Q8C22×C311yescharacteristic
24C2×C7×D4—31no—
25C22×C4×C7—31no—
26C2×C22⋊D7—31no—
27Q8×D7—21no—
28Q8×D7—61no—
29C2×C4×D7—31no—
30C2×Dic14—31no—
31C22⋊Dic7—32no—
32C22⋊Dic7—31no—
33(C2×D7)⋊C4A411yescharacteristic
34C4⋊Dic7—31no—
35Dic7⋊C4—31no—
36C4×Dic7—31no—
37C2×#41⋊C3—71no—
38C22×SL(2,𝔽3)—71no—
39C4×SL(2,𝔽3)—71no—
40C22×ΓL(1,𝔽8)—41no—
41C2×AGL(1,𝔽7)—42no—
42C2×AGL(1,𝔽7)—81no—
43C2×AGL(1,𝔽7)—81no—
44C4×ΓL(1,𝔽8)—81no—
45ΓL(1,𝔽8)⋊C4—81no—
46#548—71noSylow
47C23×C7—31no—
48C22×D7C2×A411yescharacteristic
49C7×Q8—31no—
50C7×Q8C3×D411yescharacteristic
51C7×D4—61no—
52C7×D4—61no—
53C2×C4×C7—31no—
54C2×C4×C7—61no—
55C2×C4×C7—61no—
56C2×C4×C7C2×A411yescharacteristic
57C22⋊D7—62no—
58C2×Dic7—31no—
59C2×Dic7—31no—
60C2×Dic7C2×A411yescharacteristic
61C4×D7—61no—
62C4×D7—61no—
63Dic14—61no—
64Dic14—61no—
65#41⋊C3—141no—
66C2×SL(2,𝔽3)—71no—
67C2×SL(2,𝔽3)—141no—
68C3×C22⋊C4—281no—
69C2×ΓL(1,𝔽8)—41no—
70C2×ΓL(1,𝔽8)—41no—
71C2×ΓL(1,𝔽8)—41no—
72AGL(1,𝔽7)—82no—
73C2×#41AGL(1,𝔽7)11yescharacteristic, complemented
74C22×Q8—71no—
75#124—211no—
76C22⋊Q8—211no—
77#121—211no—
78C4×Q8—71no—
79C4×D4—211no—
80C22×C7—61no—
81C22×C7—61no—
82C22×C7—61no—
83C22×C7C22×A411yescharacteristic
84C2×D7—21no—
85C2×D7—21no—
86C2×D7#41⋊C312yescomplemented
87C4×C7—21no—
88C4×C7—31no—
89C4×C7—31no—
90Dic7—21no—
91Dic7—61no—
92Dic7—61no—
93C23×C3—281no—
94C2×C3×C4—281no—
95C2×C3×C4—281no—
96SL(2,𝔽3)—71no—
97ΓL(1,𝔽8)—41noHall
98#41—21no—
99#41—61no—
100C2×Q8—141no—
101C2×Q8—211no—
102C2×Q8—421no—
103C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
104C2×D4—31no—
105C2×D4—211no—
106C22×C4—31no—
107C22×C4—211no—
108C4⋊C4—211no—
109C4⋊C4—211no—
110C22⋊C4—71no—
111C22⋊C4—211no—
112C22⋊C4—212no—
113C42—211no—
114C2×C7—121no—
115C2×C7D4×A411yescharacteristic
116C2×C7C2×#41⋊C311yescharacteristic
117C2×C72−1+4⋊C311yescharacteristic
118D7—22no—
119C22×C3—282no—
120C22×C3—281no—
121C22×C3—561no—
122C22×C3—561no—
123C3×C4—561no—
124C3×C4—561no—
125C23—31no—
126C23—71no—
127Q8—31no—
128Q8—421no—
129Q8—421no—
130Q8(C22⋊D7)⋊C311yescharacteristic
131D4—61no—
132D4—61no—
133D4—422no—
134C2×C4—31no—
135C2×C4—61no—
136C2×C4—61no—
137C2×C4—71no—
138C2×C4—211no—
139C2×C4—211no—
140C2×C4—421no—
141C2×C4—421no—
142C2×C4D7⋊A411yescharacteristic
143C7#548⋊C311yescharacteristic, complemented, Sylow
144C2×C3—281no—
145C2×C3—281no—
146C2×C3—281no—
147C2×C3—562no—
148C22—61no—
149C22—61no—
150C22—61no—
151C22—72no—
152C22—141no—
153C22—141no—
154C22C2×D7⋊A411yescharacteristic
155C4—21no—
156C4—31no—
157C4—31no—
158C4—141no—
159C4—421no—
160C4—421no—
161C3—281noSylow
162C2—121no—
163C2—142no—
164C2D7⋊(#41⋊C3)11yescharacteristic
165C2#1062154111yescharacteristic
166C2(C22⋊D7)⋊A411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D7⋊A4
commutator, cosocleC2×C7×Q8C22×C3
FittingC2×C7×#41C2×C3
socleC22×C7C22×A4
hypercenterC2×C4D7⋊A4
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×#41⋊ΓL(1,𝔽8)C2semidirect
2. C2×D7⋊SL(2,𝔽3)C2semidirect
3. Dic7⋊SL(2,𝔽3)C2semidirect
4. #90522C3semidirect
9. C2×C7×#41C2×C3semidirect
11. Q8×Dic7C2×C3semidirect
18. C7⋊SL(2,𝔽3)D4semidirect
73. C2×#41AGL(1,𝔽7)semidirect
86. C2×D7#41⋊C3semidirect
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×#41⋊ΓL(1,𝔽8)2
C2×D7⋊SL(2,𝔽3)2
Dic7⋊SL(2,𝔽3)2
#905223
(C2×AGL(1,𝔽7))⋊C44
#548⋊C37