Finite Group Databasebeta

← #89907⋊C3 #13052034

Subgroups — #89907⋊C3 #13052034

Order 1344.

260 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 70 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×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#6493⋊C3C212yesmaximal, complemented
3C2×D56⋊C3C211yesmaximal, characteristic, complemented
4#6395⋊C3C212yesmaximal, complemented
5#6391⋊C3C211yesmaximal, characteristic, complemented
6#89907C311yesmaximal, characteristic, complemented, Hall
7C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
8C2×(C22⋊D7)⋊C3—22no—
9C2×C4⋊AGL(1,𝔽7)C2212yes—
10M16×ΓL(1,𝔽8)—22no—
11C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
12C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
13(C2×AGL(1,𝔽7))⋊C4—22no—
14C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
15D56⋊C3—22no—
16C2×C7×M16C2×C311yescharacteristic, complemented
17#6493C2×C312yescomplemented
18C2×D56C2×C311yescharacteristic, complemented
19#6395C2×C312yescomplemented
20#6391C2×C311yescharacteristic, complemented
21C3×#469—71nomaximal, Hall
22C23×ΓL(1,𝔽8)D411yescharacteristic
23C22×AGL(1,𝔽7)—22no—
24C2×C4×ΓL(1,𝔽8)—21no—
25C2×C4×ΓL(1,𝔽8)—21no—
26C2×C4×ΓL(1,𝔽8)D411yescharacteristic
27C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
28(C22⋊D7)⋊C3—44no—
29C2×ΓL(1,𝔽8)⋊C4—22no—
30C4⋊AGL(1,𝔽7)—22no—
31C4⋊AGL(1,𝔽7)—22no—
32C8×ΓL(1,𝔽8)—21no—
33C8×ΓL(1,𝔽8)D412yescomplemented
34C22×C4×C7C22×C311yescharacteristic
35C2×C22⋊D7—22no—
36C2×D28C22×C312yes—
37C7×M16—22no—
38C2×C7×C8C22×C311yescharacteristic
39C2×C7×C8C22×C311yescharacteristic
40(C2×D7)⋊C4—22no—
41C4⋊Dic7C22×C312yes—
42D56—22no—
43C2×C3×D8—71no—
44C2×C3×M16—71no—
45C3×#121—72no—
46C3×#106—71no—
47C3×#102—72no—
48C22×ΓL(1,𝔽8)—21no—
49C22×ΓL(1,𝔽8)—22no—
50C22×ΓL(1,𝔽8)C2×D411yescharacteristic
51C2×AGL(1,𝔽7)—42no—
52C2×AGL(1,𝔽7)—42no—
53C2×AGL(1,𝔽7)—42no—
54C4×ΓL(1,𝔽8)—21no—
55C4×ΓL(1,𝔽8)C2×D411yescharacteristic
56C4×ΓL(1,𝔽8)#4111yescharacteristic
57ΓL(1,𝔽8)⋊C4—42no—
58#469—71noSylow
59C23×C7C3×D411yescharacteristic
60C22×D7—22no—
61C2×C4×C7—21no—
62C2×C4×C7—21no—
63C2×C4×C7C3×D411yescharacteristic
64C2×C4×C7C23×C311yescharacteristic
65C22⋊D7—44no—
66C2×Dic7—22no—
67D28—22no—
68D28—22no—
69C7×C8—21no—
70C7×C8C3×D412yescomplemented
71C2×C3×D4—72no—
72C2×C3×D4—142no—
73C22×C3×C4—71no—
74C3×D8—142no—
75C3×M16—142no—
76C2×C3×C8—71no—
77C2×C3×C8—71no—
78C3×C4⋊C4—72no—
79C3×C22⋊C4—142no—
80C2×ΓL(1,𝔽8)—41no—
81C2×ΓL(1,𝔽8)#12111yescharacteristic
82C2×ΓL(1,𝔽8)AGL(1,ℤ/8)12yes—
83AGL(1,𝔽7)—82no—
84C2×D8—71no—
85C2×M16AGL(1,𝔽7)11yescharacteristic, complemented
86#121—72no—
87#106—71no—
88#102—72no—
89C22×C7—21no—
90C22×C7—22no—
91C22×C7C2×C3×D411yescharacteristic
92C2×D7—42no—
93C2×D7—42no—
94C2×D7—42no—
95C4×C7—21no—
96C4×C7C2×C3×D411yescharacteristic
97C4×C7C3×#4111yescharacteristic
98Dic7—42no—
99C23×C3—71no—
100C23×C3—142no—
101C3×D4—142no—
102C3×D4—142no—
103C3×D4—284no—
104C2×C3×C4—71no—
105C2×C3×C4—71no—
106C2×C3×C4—141no—
107C2×C3×C4—141no—
108C2×C3×C4—142no—
109C3×C8—72no—
110C3×C8—141no—
111ΓL(1,𝔽8)#46911yescharacteristic, complemented, Hall
112C2×D4—72no—
113C2×D4—142no—
114C22×C4C2×AGL(1,𝔽7)11yescharacteristic
115D8—142no—
116M16—22no—
117C2×C8C2×AGL(1,𝔽7)11yescharacteristic
118C2×C8C2×AGL(1,𝔽7)11yescharacteristic
119C4⋊C4—72no—
120C22⋊C4—142no—
121C2×C7—41no—
122C2×C7C3×#12111yescharacteristic
123C2×C7C3×AGL(1,ℤ/8)12yes—
124D7—82no—
125C22×C3—71no—
126C22×C3—141no—
127C22×C3—142no—
128C22×C3—282no—
129C22×C3—282no—
130C22×C3—282no—
131C3×C4—71no—
132C3×C4—71no—
133C3×C4—141no—
134C3×C4—282no—
135C23—142no—
136C23C4⋊AGL(1,𝔽7)11yescharacteristic
137D4—142no—
138D4—142no—
139D4—284no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—142no—
143C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
144C2×C4C22×AGL(1,𝔽7)11yescharacteristic
145C8—21no—
146C8(C22⋊D7)⋊C312yescomplemented
147C7C3×#46911yescharacteristic, complemented, Sylow
148C2×C3—72no—
149C2×C3—71no—
150C2×C3—281no—
151C2×C3—562no—
152C22—22no—
153C22—21no—
154C22—282no—
155C22—282no—
156C22—282no—
157C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
158C4—21no—
159C4—282no—
160C4#1157⋊C311yescharacteristic
161C4C2×(C22⋊D7)⋊C311yescharacteristic
162C3—71noSylow
163C2—41no—
164C2—562no—
165C2#6471⋊C312yes—
166C2#6493⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×M16C2×C3
socleC22×C7C2×C3×D4
hypercenterC2×M16AGL(1,𝔽7)
hypocenterC7C3×#469
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×M16×ΓL(1,𝔽8)C2semidirect
2. #6493⋊C3C2semidirect
3. C2×D56⋊C3C2semidirect
4. #6395⋊C3C2semidirect
5. #6391⋊C3C2semidirect
6. #89907C3semidirect
16. C2×C7×M16C2×C3semidirect
17. #6493C2×C3semidirect
18. C2×D56C2×C3semidirect
19. #6395C2×C3semidirect
20. #6391C2×C3semidirect
33. C8×ΓL(1,𝔽8)D4semidirect
70. C7×C8C3×D4semidirect
85. C2×M16AGL(1,𝔽7)semidirect
111. ΓL(1,𝔽8)#469semidirect
146. C8(C22⋊D7)⋊C3semidirect
147. C7C3×#469semidirect

Maximal subgroups

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

subgroupindex
C2×M16×ΓL(1,𝔽8)2
#6493⋊C32
C2×D56⋊C32
#6395⋊C32
#6391⋊C32
#899073
C3×#4697