Finite Group Databasebeta

← #89818⋊C3 #13051950

Subgroups — #89818⋊C3 #13051950

Order 1344.

376 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C22×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic
2ΓL(1,𝔽8)×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#1143⋊C3C212yesmaximal, complemented
4(C22⋊(C7⋊C8))⋊C3C211yesmaximal, characteristic, complemented
5#6383⋊C3C212yesmaximal, complemented
6#89818C311yesmaximal, characteristic, complemented, Hall
7C23×AGL(1,𝔽7)—21no—
8C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C2×C4⋊AGL(1,𝔽7)—21no—
11C2×C4⋊AGL(1,𝔽7)—24no—
12C2×C4⋊AGL(1,𝔽7)C2212yes—
13C4⋊C4×ΓL(1,𝔽8)—21no—
14C4⋊C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
15C22⋊C4×ΓL(1,𝔽8)—21no—
16#1143⋊C3—24no—
17C2×ΓL(1,𝔽8)⋊C8C2212yescomplemented
18C22×D28C2×C311yescharacteristic
19C7×C22⋊Q8C2×C311yescharacteristic, complemented
20C2×#1143C2×C312yescomplemented
21C22⋊(C7⋊C8)C2×C311yescharacteristic, complemented
22#6383C2×C312yescomplemented
23C3×#450—71nomaximal, Hall
24C23×ΓL(1,𝔽8)D411yescharacteristic
25C22×AGL(1,𝔽7)—24no—
26C22×AGL(1,𝔽7)—22no—
27C22×AGL(1,𝔽7)—42no—
28C22×AGL(1,𝔽7)—42no—
29Q8×ΓL(1,𝔽8)—22no—
30C2×C4×ΓL(1,𝔽8)—22no—
31C2×C4×ΓL(1,𝔽8)—21no—
32C2×C4×ΓL(1,𝔽8)—21no—
33C2×C4×ΓL(1,𝔽8)D411yescharacteristic
34C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
35C4⋊AGL(1,𝔽7)—24no—
36C4⋊AGL(1,𝔽7)—22no—
37C4⋊AGL(1,𝔽7)D414yes—
38ΓL(1,𝔽8)⋊C8—22no—
39C23×D7—21no—
40C2×C7×Q8C22×C311yescharacteristic, complemented
41C22×C4×C7C22×C311yescharacteristic
42C2×D28—21no—
43C2×D28—24no—
44C2×D28C22×C312yes—
45C7×C4⋊C4—21no—
46C7×C4⋊C4C22×C311yescharacteristic, complemented
47C7×C22⋊C4—21no—
48#1143—24no—
49C2×C7⋊C8C22×C312yescomplemented
50C22×C3×D4—71no—
51C2×C3×SD16—72no—
52C3×C22⋊Q8—71no—
53C3×#102—72no—
54C3×C22⋊C8—71no—
55C22×ΓL(1,𝔽8)—21no—
56C22×ΓL(1,𝔽8)—21no—
57C22×ΓL(1,𝔽8)SD1612yescomplemented
58C22×ΓL(1,𝔽8)C2×D411yescharacteristic
59C2×AGL(1,𝔽7)—24no—
60C2×AGL(1,𝔽7)—48no—
61C2×AGL(1,𝔽7)—42no—
62C2×AGL(1,𝔽7)—42no—
63C4×ΓL(1,𝔽8)—21no—
64C4×ΓL(1,𝔽8)—41no—
65C4×ΓL(1,𝔽8)—41no—
66C4×ΓL(1,𝔽8)C2×D412yes—
67#450—71noSylow
68C23×C7C3×D411yescharacteristic
69C22×D7—24no—
70C22×D7—22no—
71C22×D7—42no—
72C22×D7—42no—
73C7×Q8—22no—
74C2×C4×C7—22no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7C3×D411yescharacteristic
78C2×C4×C7C23×C311yescharacteristic
79D28—24no—
80D28—22no—
81D28C3×D414yes—
82C7⋊C8—22no—
83C24×C3—141no—
84C2×C3×Q8—71no—
85C2×C3×D4—72no—
86C2×C3×D4—141no—
87C2×C3×D4—144no—
88C22×C3×C4—71no—
89C3×SD16—144no—
90C2×C3×C8—72no—
91C3×C4⋊C4—71no—
92C3×C4⋊C4—141no—
93C3×C22⋊C4—141no—
94C2×ΓL(1,𝔽8)—22no—
95C2×ΓL(1,𝔽8)#12011yescharacteristic
96C2×ΓL(1,𝔽8)C2×SD1611yescharacteristic
97C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
98AGL(1,𝔽7)—44no—
99C22×D4—71no—
100C2×SD16—72no—
101C22⋊Q8AGL(1,𝔽7)11yescharacteristic, complemented
102#102—72no—
103C22⋊C8—71no—
104C22×C7—21no—
105C22×C7—21no—
106C22×C7C3×SD1612yescomplemented
107C22×C7C2×C3×D411yescharacteristic
108C2×D7—24no—
109C2×D7—48no—
110C2×D7—42no—
111C2×D7—42no—
112C4×C7—21no—
113C4×C7—41no—
114C4×C7—41no—
115C4×C7C2×C3×D412yes—
116C23×C3—71no—
117C23×C3—142no—
118C23×C3—144no—
119C23×C3—282no—
120C23×C3—282no—
121C3×Q8—142no—
122C3×D4—74no—
123C3×D4—142no—
124C3×D4—144no—
125C2×C3×C4—71no—
126C2×C3×C4—71no—
127C2×C3×C4—142no—
128C2×C3×C4—141no—
129C2×C3×C4—141no—
130C3×C8—142no—
131ΓL(1,𝔽8)#45011yescharacteristic, complemented, Hall
132C24—141no—
133C2×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
134C2×D4—72no—
135C2×D4—144no—
136C2×D4—141no—
137C22×C4C2×AGL(1,𝔽7)11yescharacteristic
138SD16—144no—
139C2×C8—72no—
140C4⋊C4—21no—
141C4⋊C4C2×AGL(1,𝔽7)11yescharacteristic, complemented
142C22⋊C4—21no—
143C2×C7—22no—
144C2×C7C3×#12011yescharacteristic
145C2×C7C2×C3×SD1611yescharacteristic
146C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
147D7—44no—
148C22×C3—71no—
149C22×C3—72no—
150C22×C3—141no—
151C22×C3—141no—
152C22×C3—144no—
153C22×C3—282no—
154C22×C3—282no—
155C22×C3—288no—
156C3×C4—72no—
157C3×C4—141no—
158C3×C4—281no—
159C3×C4—281no—
160C23—144no—
161C23—142no—
162C23—282no—
163C23—282no—
164C23(C22⋊D7)⋊C311yescharacteristic
165Q8—22no—
166D4—74no—
167D4—144no—
168D4—142no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4(C22⋊D7)⋊C311yescharacteristic
173C2×C4C22×AGL(1,𝔽7)11yescharacteristic
174C8—142no—
175C7C3×#45011yescharacteristic, complemented, Sylow
176C2×C3—71no—
177C2×C3—71no—
178C2×C3—71no—
179C2×C3—142no—
180C2×C3—284no—
181C22—21no—
182C22—21no—
183C22—144no—
184C22—288no—
185C22—282no—
186C22—282no—
187C22#1143⋊C312yescomplemented
188C22C2×(C22⋊D7)⋊C311yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4D4×AGL(1,𝔽7)12yes—
193C3—71noSylow
194C2—22no—
195C2—284no—
196C2#6500⋊C311yescharacteristic
197C2C2×#1143⋊C311yescharacteristic
198C2#6512⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C22⋊D7)⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C22⋊Q8C2×C3
socleC22×C7C2×C3×D4
prime radicalC22×C4⋊AGL(1,𝔽7)C2
hypercenterC22⋊Q8AGL(1,𝔽7)
hypocenterC7C3×#450
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×C22⋊Q8C2semidirect
3. C2×#1143⋊C3C2semidirect
4. (C22⋊(C7⋊C8))⋊C3C2semidirect
5. #6383⋊C3C2semidirect
6. #89818C3semidirect
8. C2×Q8×ΓL(1,𝔽8)C22semidirect
14. C4⋊C4×ΓL(1,𝔽8)C22semidirect
17. C2×ΓL(1,𝔽8)⋊C8C22semidirect
19. C7×C22⋊Q8C2×C3semidirect
20. C2×#1143C2×C3semidirect
21. C22⋊(C7⋊C8)C2×C3semidirect
22. #6383C2×C3semidirect
40. C2×C7×Q8C22×C3semidirect
46. C7×C4⋊C4C22×C3semidirect
49. C2×C7⋊C8C22×C3semidirect
57. C22×ΓL(1,𝔽8)SD16semidirect
101. C22⋊Q8AGL(1,𝔽7)semidirect
106. C22×C7C3×SD16semidirect
131. ΓL(1,𝔽8)#450semidirect
133. C2×Q8C2×AGL(1,𝔽7)semidirect
141. C4⋊C4C2×AGL(1,𝔽7)semidirect
175. C7C3×#450semidirect
187. C22#1143⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,𝔽7)2
ΓL(1,𝔽8)×C22⋊Q82
C2×#1143⋊C32
(C22⋊(C7⋊C8))⋊C32
#6383⋊C32
#898183
C3×#4507