Finite Group Databasebeta

← C2×#6471⋊C3 #13057480

Subgroups — C2×#6471⋊C3 #13057480

Order 1344.

596 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 192 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×#1157⋊C3C211yesmaximal, characteristic, complemented
2C22×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C2×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#6471⋊C3C218yesmaximal, direct product factor
5C2×D56⋊C3C212yesmaximal, complemented
6C2×#1132⋊C3C212yesmaximal, complemented
7C2×#6471C311yesmaximal, characteristic, complemented, Hall
8C23×AGL(1,𝔽7)—21no—
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C2×(C22⋊D7)⋊C3—21no—
11#1157⋊C3—22no—
12#1157⋊C3C2214yescomplemented
13C2×C4⋊AGL(1,𝔽7)—22no—
14C2×C4⋊AGL(1,𝔽7)—21no—
15C2×C4⋊AGL(1,𝔽7)C2212yes—
16C2×C4⋊AGL(1,𝔽7)C2214yescomplemented
17C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
18C2×C4×AGL(1,𝔽7)—21no—
19C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
20M16×ΓL(1,𝔽8)C2214yescomplemented
21C2×C8×ΓL(1,𝔽8)C2212yescomplemented
22D56⋊C3C2218yescomplemented
23#1132⋊C3C2218yescomplemented
24C2×#1157C2×C311yescharacteristic, complemented
25C22×D28C2×C311yescharacteristic, complemented
26C2×C7×M16C2×C311yescharacteristic, complemented
27#6471C2×C318yescomplemented
28C2×D56C2×C312yescomplemented
29C2×#1132C2×C312yescomplemented
30C2×C3×AGL(1,ℤ/8)—71nomaximal, Hall
31C23×ΓL(1,𝔽8)D411yescharacteristic
32C22×AGL(1,𝔽7)—21no—
33C22×AGL(1,𝔽7)—22no—
34C22×AGL(1,𝔽7)—24no—
35C22×AGL(1,𝔽7)—44no—
36C2×C4×ΓL(1,𝔽8)D411yescharacteristic
37C2×C4×ΓL(1,𝔽8)D412yes—
38C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
39C2×C4×ΓL(1,𝔽8)C2312yes—
40(C22⋊D7)⋊C3—24no—
41C2×ΓL(1,𝔽8)⋊C4—21no—
42C4⋊AGL(1,𝔽7)—22no—
43C4⋊AGL(1,𝔽7)—24no—
44C4⋊AGL(1,𝔽7)—21no—
45C4⋊AGL(1,𝔽7)C2314yes—
46C4⋊AGL(1,𝔽7)C2312yescomplemented
47C4×AGL(1,𝔽7)—24no—
48ΓL(1,𝔽8)⋊Q8—21no—
49ΓL(1,𝔽8)⋊Q8C2312yescomplemented
50C8×ΓL(1,𝔽8)C2314yescomplemented
51C23×D7—21no—
52C22×C4×C7C22×C311yescharacteristic
53C2×C22⋊D7—21no—
54#1157—22no—
55#1157C22×C314yescomplemented
56C2×D28—22no—
57C2×D28—21no—
58C2×D28C22×C312yes—
59C2×D28C22×C314yescomplemented
60C2×D28C22×C311yescharacteristic, complemented
61C2×C4×D7—21no—
62C2×Dic14C22×C311yescharacteristic, complemented
63C7×M16C22×C314yescomplemented
64C2×C7×C8C22×C312yescomplemented
65D56C22×C318yescomplemented
66#1132C22×C318yescomplemented
67C2×C3×#41—71no—
68C22×C3×D4—71no—
69C3×AGL(1,ℤ/8)—78no—
70C2×C3×SD16—72no—
71C2×C3×D8—72no—
72C2×C3×M16—71no—
73C22×ΓL(1,𝔽8)—22no—
74C22×ΓL(1,𝔽8)C2×D412yes—
75C22×ΓL(1,𝔽8)C2×D411yescharacteristic
76C2×AGL(1,𝔽7)—22no—
77C2×AGL(1,𝔽7)—24no—
78C2×AGL(1,𝔽7)—48no—
79C2×AGL(1,𝔽7)—42no—
80C2×AGL(1,𝔽7)—44no—
81C4×ΓL(1,𝔽8)C2×D411yescharacteristic
82C4×ΓL(1,𝔽8)C2×D412yes—
83C4×ΓL(1,𝔽8)C2411yescharacteristic
84ΓL(1,𝔽8)⋊C4—22no—
85C2×AGL(1,ℤ/8)—71noSylow
86C23×C7C3×D411yescharacteristic
87C22×D7—21no—
88C22×D7—22no—
89C22×D7—24no—
90C22×D7—44no—
91C2×C4×C7C3×D411yescharacteristic
92C2×C4×C7C3×D412yes—
93C2×C4×C7C23×C311yescharacteristic
94C2×C4×C7C23×C312yes—
95C22⋊D7—24no—
96C2×Dic7—21no—
97D28—22no—
98D28—24no—
99D28—21no—
100D28C23×C314yes—
101D28C23×C312yescomplemented
102C4×D7—24no—
103Dic14—21no—
104Dic14C23×C312yescomplemented
105C7×C8C23×C314yescomplemented
106C24×C3—141no—
107C3×#41—74no—
108C3×#41—142no—
109C2×C3×Q8—71no—
110C2×C3×D4—72no—
111C2×C3×D4—71no—
112C2×C3×D4—74no—
113C2×C3×D4—141no—
114C2×C3×D4—142no—
115C2×C3×D4—141no—
116C22×C3×C4—71no—
117C22×C3×C4—141no—
118C3×SD16—78no—
119C3×D8—78no—
120C3×M16—74no—
121C2×C3×C8—72no—
122C2×ΓL(1,𝔽8)—22no—
123C2×ΓL(1,𝔽8)AGL(1,ℤ/8)12yescomplemented
124C2×ΓL(1,𝔽8)C22×D411yescharacteristic
125AGL(1,𝔽7)—42no—
126AGL(1,𝔽7)—44no—
127C2×#41—71no—
128C22×D4—71no—
129AGL(1,ℤ/8)—78no—
130C2×SD16—72no—
131C2×D8—72no—
132C2×M16AGL(1,𝔽7)11yescharacteristic, complemented
133C22×C7—22no—
134C22×C7C2×C3×D412yes—
135C22×C7C2×C3×D411yescharacteristic
136C2×D7—22no—
137C2×D7—24no—
138C2×D7—48no—
139C2×D7—44no—
140C2×D7—42no—
141C4×C7C2×C3×D411yescharacteristic
142C4×C7C2×C3×D412yes—
143C4×C7C24×C311yescharacteristic
144Dic7—22no—
145C23×C3—71no—
146C23×C3—141no—
147C23×C3—144no—
148C23×C3—142no—
149C23×C3—284no—
150C3×Q8—72no—
151C3×Q8—141no—
152C3×D4—74no—
153C3×D4—72no—
154C3×D4—144no—
155C3×D4—142no—
156C3×D4—141no—
157C3×D4—144no—
158C2×C3×C4—71no—
159C2×C3×C4—72no—
160C2×C3×C4—71no—
161C2×C3×C4—72no—
162C2×C3×C4—144no—
163C2×C3×C4—141no—
164C3×C8—74no—
165ΓL(1,𝔽8)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
166C24—141no—
167#41—74no—
168#41—142no—
169C2×Q8—71no—
170C2×D4—71no—
171C2×D4—72no—
172C2×D4—74no—
173C2×D4—141no—
174C2×D4—142no—
175C2×D4—141no—
176C22×C4—141no—
177C22×C4C2×AGL(1,𝔽7)11yescharacteristic
178SD16—78no—
179D8—78no—
180M16C2×AGL(1,𝔽7)14yescomplemented
181C2×C8C2×AGL(1,𝔽7)12yescomplemented
182C2×C7—22no—
183C2×C7C3×AGL(1,ℤ/8)12yescomplemented
184C2×C7C22×C3×D411yescharacteristic
185D7—44no—
186D7—42no—
187C22×C3—72no—
188C22×C3—71no—
189C22×C3—142no—
190C22×C3—142no—
191C22×C3—144no—
192C22×C3—284no—
193C22×C3—282no—
194C22×C3—288no—
195C3×C4—72no—
196C3×C4—71no—
197C3×C4—71no—
198C3×C4—142no—
199C23—144no—
200C23—141no—
201C23—142no—
202C23—284no—
203C23C4⋊AGL(1,𝔽7)11yescharacteristic
204Q8—72no—
205Q8—141no—
206D4—74no—
207D4—72no—
208D4—144no—
209D4—144no—
210D4—142no—
211D4—141no—
212C2×C4—141no—
213C2×C4—144no—
214C2×C4C4⋊AGL(1,𝔽7)12yes—
215C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
216C2×C4C22×AGL(1,𝔽7)12yes—
217C2×C4C22×AGL(1,𝔽7)11yescharacteristic
218C8C22×AGL(1,𝔽7)14yescomplemented
219C7C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
220C2×C3—71no—
221C2×C3—72no—
222C2×C3—142no—
223C2×C3—284no—
224C2×C3—282no—
225C22—22no—
226C22—142no—
227C22—144no—
228C22—288no—
229C22—284no—
230C22—282no—
231C22C2×C4⋊AGL(1,𝔽7)12yes—
232C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
233C4—142no—
234C4C2×C4⋊AGL(1,𝔽7)12yes—
235C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
236C4C23×AGL(1,𝔽7)11yescharacteristic
237C3—71noSylow
238C2—22no—
239C2—284no—
240C2—282no—
241C2#6471⋊C312yesdirect product factor
242C2C22×C4⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×AGL(1,𝔽7)
FittingC2×C7×M16C2×C3
socleC22×C7C2×C3×D4
hypercenterC2×M16AGL(1,𝔽7)
hypocenterC7C2×C3×AGL(1,ℤ/8)
LCSG ▷ 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×#1157⋊C3C2semidirect
2. C22×C4⋊AGL(1,𝔽7)C2semidirect
3. C2×M16×ΓL(1,𝔽8)C2semidirect
4. #6471⋊C3C2direct (N × H)
5. C2×D56⋊C3C2semidirect
6. C2×#1132⋊C3C2semidirect
7. C2×#6471C3semidirect
12. #1157⋊C3C22semidirect
16. C2×C4⋊AGL(1,𝔽7)C22semidirect
17. C2×C4⋊AGL(1,𝔽7)C22semidirect
19. C2×ΓL(1,𝔽8)⋊Q8C22semidirect
20. M16×ΓL(1,𝔽8)C22semidirect
21. C2×C8×ΓL(1,𝔽8)C22semidirect
22. D56⋊C3C22semidirect
23. #1132⋊C3C22semidirect
24. C2×#1157C2×C3semidirect
25. C22×D28C2×C3semidirect
26. C2×C7×M16C2×C3semidirect
27. #6471C2×C3semidirect
28. C2×D56C2×C3semidirect
29. C2×#1132C2×C3semidirect
46. C4⋊AGL(1,𝔽7)C23semidirect
49. ΓL(1,𝔽8)⋊Q8C23semidirect
50. C8×ΓL(1,𝔽8)C23semidirect
55. #1157C22×C3semidirect
59. C2×D28C22×C3semidirect
60. C2×D28C22×C3semidirect
62. C2×Dic14C22×C3semidirect
63. C7×M16C22×C3semidirect
64. C2×C7×C8C22×C3semidirect
65. D56C22×C3semidirect
66. #1132C22×C3semidirect
101. D28C23×C3semidirect
104. Dic14C23×C3semidirect
105. C7×C8C23×C3semidirect
123. C2×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
132. C2×M16AGL(1,𝔽7)semidirect
165. ΓL(1,𝔽8)C2×AGL(1,ℤ/8)semidirect
180. M16C2×AGL(1,𝔽7)semidirect
181. C2×C8C2×AGL(1,𝔽7)semidirect
183. C2×C7C3×AGL(1,ℤ/8)semidirect
218. C8C22×AGL(1,𝔽7)semidirect
219. C7C2×C3×AGL(1,ℤ/8)semidirect
241. C2#6471⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#1157⋊C32
C22×C4⋊AGL(1,𝔽7)2
C2×M16×ΓL(1,𝔽8)2
#6471⋊C32
C2×D56⋊C32
C2×#1132⋊C32
C2×#64713
C2×C3×AGL(1,ℤ/8)7