Finite Group Databasebeta

← #89503⋊C3 #13051631

Subgroups — #89503⋊C3 #13051631

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
2#6487⋊C3C211yesmaximal, characteristic, complemented
3C2×#1132⋊C3C212yesmaximal, complemented
4ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic, complemented
5#6395⋊C3C212yesmaximal, complemented
6#89503C311yesmaximal, characteristic, complemented, Hall
7C23×AGL(1,𝔽7)—21no—
8C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
9C2×C4⋊AGL(1,𝔽7)—21no—
10C2×C4⋊AGL(1,𝔽7)—24no—
11C2×C4⋊AGL(1,𝔽7)C2212yes—
12C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
13C2×C8×ΓL(1,𝔽8)C2212yescomplemented
14(C22⋊Dic7)⋊C3—21no—
15C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic, complemented
16(ΓL(1,𝔽8)⋊C4)⋊C4—21no—
17#1132⋊C3—24no—
18C22×D28C2×C311yescharacteristic
19#6487C2×C311yescharacteristic, complemented
20C2×#1132C2×C312yescomplemented
21C7×C22⋊C8C2×C311yescharacteristic, complemented
22#6395C2×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—
29C2×C4×ΓL(1,𝔽8)—22no—
30C2×C4×ΓL(1,𝔽8)D411yescharacteristic
31C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
32C2×ΓL(1,𝔽8)⋊C4—21no—
33C2×ΓL(1,𝔽8)⋊C4—21no—
34C4⋊AGL(1,𝔽7)—24no—
35C4⋊AGL(1,𝔽7)—22no—
36C4⋊AGL(1,𝔽7)D414yes—
37ΓL(1,𝔽8)⋊Q8—22no—
38C8×ΓL(1,𝔽8)—22no—
39C23×D7—21no—
40C22×C4×C7C22×C311yescharacteristic
41C2×D28—21no—
42C2×D28—24no—
43C2×D28C22×C312yes—
44C2×Dic14C22×C311yescharacteristic, complemented
45C2×C7×C8C22×C312yescomplemented
46C22⋊Dic7—21no—
47C4⋊Dic7C22×C311yescharacteristic, complemented
48Dic7⋊C4—21no—
49#1132—24no—
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)C2×D412yes—
65ΓL(1,𝔽8)⋊C4—41no—
66ΓL(1,𝔽8)⋊C4—41no—
67#450—71noSylow
68C23×C7C3×D411yescharacteristic
69C22×D7—24no—
70C22×D7—22no—
71C22×D7—42no—
72C22×D7—42no—
73C2×C4×C7—22no—
74C2×C4×C7C3×D411yescharacteristic
75C2×C4×C7C23×C311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—21no—
78D28—24no—
79D28—22no—
80D28C3×D414yes—
81Dic14—22no—
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⋊Q8—71no—
102#102—72no—
103C22⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
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×C7C2×C3×D412yes—
114Dic7—41no—
115Dic7—41no—
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×Q8—71no—
134C2×D4—72no—
135C2×D4—144no—
136C2×D4—141no—
137C22×C4C2×AGL(1,𝔽7)11yescharacteristic
138SD16—144no—
139C2×C8C2×AGL(1,𝔽7)12yescomplemented
140C4⋊C4—71no—
141C4⋊C4—141no—
142C22⋊C4—141no—
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—
164C23C4⋊AGL(1,𝔽7)11yescharacteristic
165Q8—142no—
166D4—74no—
167D4—144no—
168D4—142no—
169C2×C4—22no—
170C2×C4—141no—
171C2×C4—141no—
172C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
173C2×C4C22×AGL(1,𝔽7)11yescharacteristic
174C8—22no—
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#1132⋊C312yescomplemented
188C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
189C4—21no—
190C4—281no—
191C4—281no—
192C4D4×AGL(1,𝔽7)12yes—
193C3—71noSylow
194C2—22no—
195C2—284no—
196C2#6445⋊C311yescharacteristic
197C2C2×#1132⋊C311yescharacteristic
198C2#6471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C22⋊C8C2×C3
socleC22×C7C2×C3×D4
prime radicalC22×C4⋊AGL(1,𝔽7)C2
hypercenterC22⋊C8AGL(1,𝔽7)
hypocenterC7C3×#450
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #6487⋊C3C2semidirect
3. C2×#1132⋊C3C2semidirect
4. ΓL(1,𝔽8)×C22⋊C8C2semidirect
5. #6395⋊C3C2semidirect
6. #89503C3semidirect
12. C2×ΓL(1,𝔽8)⋊Q8C22semidirect
13. C2×C8×ΓL(1,𝔽8)C22semidirect
15. C4⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
19. #6487C2×C3semidirect
20. C2×#1132C2×C3semidirect
21. C7×C22⋊C8C2×C3semidirect
22. #6395C2×C3semidirect
44. C2×Dic14C22×C3semidirect
45. C2×C7×C8C22×C3semidirect
47. C4⋊Dic7C22×C3semidirect
57. C22×ΓL(1,𝔽8)SD16semidirect
103. C22⋊C8AGL(1,𝔽7)semidirect
106. C22×C7C3×SD16semidirect
131. ΓL(1,𝔽8)#450semidirect
139. C2×C8C2×AGL(1,𝔽7)semidirect
175. C7C3×#450semidirect
187. C22#1132⋊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
#6487⋊C32
C2×#1132⋊C32
ΓL(1,𝔽8)×C22⋊C82
#6395⋊C32
#895033
C3×#4507