Finite Group Databasebeta

← #89929⋊C3 #13052057

Subgroups — #89929⋊C3 #13052057

Order 1344.

268 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 72 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×D8×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#6501⋊C3C212yesmaximal, complemented
3C2×C8×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4#6406⋊C3C212yesmaximal, complemented
5#6392⋊C3C211yesmaximal, characteristic, complemented
6#89929C311yesmaximal, characteristic, complemented, Hall
7C2×D4×ΓL(1,𝔽8)C2212yes—
8C2×(C22⋊D7)⋊C3—22no—
9C2×C4×AGL(1,𝔽7)C2211yescharacteristic
10D8×ΓL(1,𝔽8)—22no—
11C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
12(C22⋊Dic7)⋊C3—22no—
13C4⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
14C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
15C8×AGL(1,𝔽7)—22no—
16C2×C7×D8C2×C311yescharacteristic, complemented
17#6501C2×C312yescomplemented
18C2×C8×D7C2×C311yescharacteristic, complemented
19#6406C2×C312yescomplemented
20#6392C2×C311yescharacteristic, complemented
21C3×#466—71nomaximal, Hall
22C23×ΓL(1,𝔽8)—22no—
23C22×AGL(1,𝔽7)D411yescharacteristic
24D4×ΓL(1,𝔽8)—22no—
25D4×ΓL(1,𝔽8)—22no—
26C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
27(C22⋊D7)⋊C3—44no—
28C2×ΓL(1,𝔽8)⋊C4—22no—
29C2×ΓL(1,𝔽8)⋊C4D411yescharacteristic
30C4×AGL(1,𝔽7)—21no—
31C4×AGL(1,𝔽7)—21no—
32C8×ΓL(1,𝔽8)D412yescomplemented
33ΓL(1,𝔽8)⋊C8—21no—
34C2×C7×D4C22×C312yes—
35C2×C22⋊D7—22no—
36C2×C4×D7C22×C311yescharacteristic
37C7×D8—22no—
38C2×C7×C8C22×C311yescharacteristic
39C22⋊Dic7—22no—
40C4⋊Dic7C22×C312yes—
41C2×C7⋊C8C22×C311yescharacteristic
42C8×D7—22no—
43C2×C3×D8—71no—
44C22×C3×C8—71no—
45C3×#121—72no—
46C3×#107—71no—
47C3×#102—72no—
48C22×ΓL(1,𝔽8)—42no—
49C22×ΓL(1,𝔽8)—42no—
50C22×ΓL(1,𝔽8)—42no—
51C22×ΓL(1,𝔽8)C2×D411yescharacteristic
52C2×AGL(1,𝔽7)—21no—
53C2×AGL(1,𝔽7)—21no—
54C2×AGL(1,𝔽7)D812yescomplemented
55C4×ΓL(1,𝔽8)C2×D411yescharacteristic
56C4×ΓL(1,𝔽8)#4111yescharacteristic
57ΓL(1,𝔽8)⋊C4—21no—
58ΓL(1,𝔽8)⋊C4—42no—
59#466—71noSylow
60C23×C7—22no—
61C22×D7C3×D411yescharacteristic
62C7×D4—22no—
63C7×D4—22no—
64C2×C4×C7C23×C311yescharacteristic
65C22⋊D7—44no—
66C2×Dic7—22no—
67C2×Dic7C3×D411yescharacteristic
68C4×D7—21no—
69C4×D7—21no—
70C7×C8C3×D412yescomplemented
71C7⋊C8—21no—
72C2×C3×D4—72no—
73C2×C3×D4—142no—
74C22×C3×C4—71no—
75C3×D8—142no—
76C2×C3×C8—71no—
77C2×C3×C8—71no—
78C2×C3×C8—142no—
79C3×C4⋊C4—72no—
80C3×C22⋊C4—142no—
81C2×ΓL(1,𝔽8)—82no—
82C2×ΓL(1,𝔽8)#12111yescharacteristic
83C2×ΓL(1,𝔽8)C2×D811yescharacteristic
84C2×ΓL(1,𝔽8)#13511yescharacteristic
85AGL(1,𝔽7)—22no—
86C2×D8AGL(1,𝔽7)11yescharacteristic, complemented
87C22×C8—71no—
88#121—72no—
89#107—71no—
90#102—72no—
91C22×C7—42no—
92C22×C7—42no—
93C22×C7—42no—
94C22×C7C2×C3×D411yescharacteristic
95C2×D7—21no—
96C2×D7—21no—
97C2×D7C3×D812yescomplemented
98C4×C7C2×C3×D411yescharacteristic
99C4×C7C3×#4111yescharacteristic
100Dic7—21no—
101Dic7—42no—
102C23×C3—71no—
103C23×C3—142no—
104C3×D4—142no—
105C3×D4—142no—
106C3×D4—284no—
107C2×C3×C4—71no—
108C2×C3×C4—71no—
109C2×C3×C4—141no—
110C2×C3×C4—141no—
111C2×C3×C4—142no—
112C3×C8—72no—
113C3×C8—141no—
114ΓL(1,𝔽8)#46611yescharacteristic, complemented, Hall
115C2×D4—142no—
116C2×D4C2×AGL(1,𝔽7)12yes—
117C22×C4—71no—
118D8—22no—
119C2×C8—71no—
120C2×C8—142no—
121C2×C8C2×AGL(1,𝔽7)11yescharacteristic
122C4⋊C4—72no—
123C22⋊C4—142no—
124C2×C7—82no—
125C2×C7C3×#12111yescharacteristic
126C2×C7C2×C3×D811yescharacteristic
127C2×C7C3×#13511yescharacteristic
128D7—22no—
129C22×C3—72no—
130C22×C3—71no—
131C22×C3—141no—
132C22×C3—141no—
133C22×C3—282no—
134C22×C3—282no—
135C22×C3—282no—
136C3×C4—71no—
137C3×C4—71no—
138C3×C4—141no—
139C3×C4—282no—
140C23—22no—
141C23—71no—
142D4—22no—
143D4—22no—
144D4—284no—
145C2×C4—71no—
146C2×C4—141no—
147C2×C4—141no—
148C2×C4—142no—
149C2×C4C22×AGL(1,𝔽7)11yescharacteristic
150C8—141no—
151C8(C22⋊D7)⋊C312yescomplemented
152C7C3×#46611yescharacteristic, complemented, Sylow
153C2×C3—71no—
154C2×C3—71no—
155C2×C3—71no—
156C2×C3—142no—
157C2×C3—562no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22—72no—
162C22—141no—
163C22—141no—
164C22D4×AGL(1,𝔽7)11yescharacteristic
165C4—141no—
166C4—282no—
167C4#1159⋊C311yescharacteristic
168C4C2×(C22⋊D7)⋊C311yescharacteristic
169C3—71noSylow
170C2—82no—
171C2—142no—
172C2D8×AGL(1,𝔽7)11yescharacteristic
173C2#6475⋊C311yescharacteristic
174C2#6501⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×D8C2×C3
socleC22×C7C2×C3×D4
hypercenterC2×D8AGL(1,𝔽7)
hypocenterC7C3×#466
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D8×ΓL(1,𝔽8)C2semidirect
2. #6501⋊C3C2semidirect
3. C2×C8×AGL(1,𝔽7)C2semidirect
4. #6406⋊C3C2semidirect
5. #6392⋊C3C2semidirect
6. #89929C3semidirect
16. C2×C7×D8C2×C3semidirect
17. #6501C2×C3semidirect
18. C2×C8×D7C2×C3semidirect
19. #6406C2×C3semidirect
20. #6392C2×C3semidirect
32. C8×ΓL(1,𝔽8)D4semidirect
54. C2×AGL(1,𝔽7)D8semidirect
70. C7×C8C3×D4semidirect
86. C2×D8AGL(1,𝔽7)semidirect
97. C2×D7C3×D8semidirect
114. ΓL(1,𝔽8)#466semidirect
151. C8(C22⋊D7)⋊C3semidirect
152. C7C3×#466semidirect

Maximal subgroups

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

subgroupindex
C2×D8×ΓL(1,𝔽8)2
#6501⋊C32
C2×C8×AGL(1,𝔽7)2
#6406⋊C32
#6392⋊C32
#899293
C3×#4667