Finite Group Databasebeta

← C2×#6461⋊C3 #13057246

Subgroups — C2×#6461⋊C3 #13057246

Order 1344.

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

normal subgroups only

This group has 492 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×C4⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
4C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
5C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
6C2×(ΓL(1,𝔽8)⋊C4)⋊C4C211yesmaximal, characteristic, complemented
7C2×C4×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
8#6461⋊C3C218yesmaximal, direct product factor
9C2×#6461C311yesmaximal, characteristic, complemented, Hall
10C23×AGL(1,𝔽7)C2211yescharacteristic
11C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
13C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
14C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
15C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
16C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
17C4⋊C4×ΓL(1,𝔽8)C2214yescomplemented
18(C2×AGL(1,𝔽7))⋊C4C2214yes—
19(C2×AGL(1,𝔽7))⋊C4C2214yes—
20(C2×AGL(1,𝔽7))⋊C4C2214yes—
21C4⋊(ΓL(1,𝔽8)⋊C4)C2214yescomplemented
22(ΓL(1,𝔽8)⋊C4)⋊C4C2214yescomplemented
23C4×ΓL(1,𝔽8)⋊C4C2214yescomplemented
24C2×C7×C4⋊C4C2×C311yescharacteristic, complemented
25C2×(C2×D7)⋊C4C2×C311yescharacteristic
26C2×(C2×D7)⋊C4C2×C311yescharacteristic
27C2×(C2×D7)⋊C4C2×C311yescharacteristic
28C2×C4⋊Dic7C2×C311yescharacteristic, complemented
29C2×Dic7⋊C4C2×C311yescharacteristic, complemented
30C2×C4×Dic7C2×C311yescharacteristic, complemented
31#6461C2×C318yescomplemented
32C2×C3×#126—71nomaximal, Hall
33C23×ΓL(1,𝔽8)C2311yescharacteristic
34C22×AGL(1,𝔽7)—22no—
35C22×AGL(1,𝔽7)—22no—
36C22×AGL(1,𝔽7)—22no—
37C22×AGL(1,𝔽7)C2312yes—
38C2×C4×ΓL(1,𝔽8)—22no—
39C2×C4×ΓL(1,𝔽8)—22no—
40C2×C4×ΓL(1,𝔽8)—22no—
41C2×C4×ΓL(1,𝔽8)C2312yes—
42C2×C4×ΓL(1,𝔽8)C2312yes—
43C2×C4×ΓL(1,𝔽8)C2312yes—
44C2×ΓL(1,𝔽8)⋊C4—22no—
45C2×ΓL(1,𝔽8)⋊C4—22no—
46C2×ΓL(1,𝔽8)⋊C4—22no—
47C2×ΓL(1,𝔽8)⋊C4C2312yes—
48C2×ΓL(1,𝔽8)⋊C4C2312yes—
49C2×ΓL(1,𝔽8)⋊C4C2312yes—
50C23×D7C22×C311yescharacteristic
51C22×C4×C7C22×C311yescharacteristic
52C22×C4×C7C22×C311yescharacteristic
53C22×C4×C7C22×C311yescharacteristic
54C22×Dic7C22×C311yescharacteristic
55C22×Dic7C22×C311yescharacteristic
56C22×Dic7C22×C311yescharacteristic
57C7×C4⋊C4C22×C314yescomplemented
58(C2×D7)⋊C4C22×C314yes—
59(C2×D7)⋊C4C22×C314yes—
60(C2×D7)⋊C4C22×C314yes—
61C4⋊Dic7C22×C314yescomplemented
62Dic7⋊C4C22×C314yescomplemented
63C4×Dic7C22×C314yescomplemented
64C3×#126—78no—
65C2×C3×C4⋊C4—71no—
66C2×C3×C4⋊C4—71no—
67C2×C3×C4⋊C4—71no—
68C2×C3×C22⋊C4—71no—
69C2×C3×C22⋊C4—71no—
70C2×C3×C22⋊C4—71no—
71C2×C3×C42—71no—
72C22×ΓL(1,𝔽8)#4112yes—
73C22×ΓL(1,𝔽8)#4112yes—
74C22×ΓL(1,𝔽8)#4112yes—
75C22×ΓL(1,𝔽8)C2411yescharacteristic
76C2×AGL(1,𝔽7)—22no—
77C2×AGL(1,𝔽7)—22no—
78C2×AGL(1,𝔽7)—22no—
79C2×AGL(1,𝔽7)—44no—
80C4×ΓL(1,𝔽8)—22no—
81C4×ΓL(1,𝔽8)—22no—
82C4×ΓL(1,𝔽8)—22no—
83ΓL(1,𝔽8)⋊C4—22no—
84ΓL(1,𝔽8)⋊C4—22no—
85ΓL(1,𝔽8)⋊C4—22no—
86C2×#126—71noSylow
87C23×C7C23×C311yescharacteristic
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7C23×C312yes—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7C23×C312yes—
96C2×C4×C7C23×C312yes—
97C2×C4×C7C23×C312yes—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7C23×C312yes—
102C2×Dic7C23×C312yes—
103C2×Dic7C23×C312yes—
104C24×C3—71no—
105C22×C3×C4—71no—
106C22×C3×C4—71no—
107C22×C3×C4—71no—
108C22×C3×C4—71no—
109C22×C3×C4—71no—
110C22×C3×C4—71no—
111C3×C4⋊C4—74no—
112C3×C4⋊C4—74no—
113C3×C4⋊C4—74no—
114C3×C22⋊C4—74no—
115C3×C22⋊C4—74no—
116C3×C22⋊C4—74no—
117C3×C42—74no—
118C2×ΓL(1,𝔽8)#12614yescomplemented
119C2×ΓL(1,𝔽8)C2×#4111yescharacteristic
120C2×ΓL(1,𝔽8)C2×#4111yescharacteristic
121C2×ΓL(1,𝔽8)C2×#4111yescharacteristic
122AGL(1,𝔽7)—42no—
123#126—78no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—71no—
126C2×C4⋊C4AGL(1,𝔽7)11yescharacteristic, complemented
127C2×C22⋊C4—71no—
128C2×C22⋊C4—71no—
129C2×C22⋊C4—71no—
130C2×C42—71no—
131C22×C7C3×#4112yes—
132C22×C7C3×#4112yes—
133C22×C7C3×#4112yes—
134C22×C7C24×C311yescharacteristic
135C2×D7—22no—
136C2×D7—22no—
137C2×D7—22no—
138C2×D7—44no—
139C4×C7—22no—
140C4×C7—22no—
141C4×C7—22no—
142Dic7—22no—
143Dic7—22no—
144Dic7—22no—
145C23×C3—72no—
146C23×C3—71no—
147C23×C3—142no—
148C23×C3—142no—
149C23×C3—142no—
150C2×C3×C4—72no—
151C2×C3×C4—72no—
152C2×C3×C4—72no—
153C2×C3×C4—72no—
154C2×C3×C4—72no—
155C2×C3×C4—72no—
156C2×C3×C4—142no—
157C2×C3×C4—142no—
158C2×C3×C4—142no—
159C2×C3×C4—142no—
160C2×C3×C4—142no—
161C2×C3×C4—142no—
162ΓL(1,𝔽8)C2×#12611yescharacteristic, complemented, Hall
163C24—71no—
164C22×C4—71no—
165C22×C4—71no—
166C22×C4—71no—
167C22×C4C2×AGL(1,𝔽7)11yescharacteristic
168C22×C4C2×AGL(1,𝔽7)11yescharacteristic
169C22×C4C2×AGL(1,𝔽7)11yescharacteristic
170C4⋊C4—74no—
171C4⋊C4—74no—
172C4⋊C4C2×AGL(1,𝔽7)14yescomplemented
173C22⋊C4—74no—
174C22⋊C4—74no—
175C22⋊C4—74no—
176C42—74no—
177C2×C7C3×#12614yescomplemented
178C2×C7C2×C3×#4111yescharacteristic
179C2×C7C2×C3×#4111yescharacteristic
180C2×C7C2×C3×#4111yescharacteristic
181D7—42no—
182C22×C3—72no—
183C22×C3—72no—
184C22×C3—72no—
185C22×C3—71no—
186C22×C3—142no—
187C22×C3—142no—
188C22×C3—142no—
189C22×C3—284no—
190C3×C4—142no—
191C3×C4—142no—
192C3×C4—142no—
193C3×C4—142no—
194C3×C4—142no—
195C3×C4—142no—
196C23—72no—
197C23—142no—
198C23—142no—
199C23—142no—
200C23C22×AGL(1,𝔽7)11yescharacteristic
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—72no—
205C2×C4—72no—
206C2×C4—72no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—142no—
210C2×C4C22×AGL(1,𝔽7)12yes—
211C2×C4C22×AGL(1,𝔽7)12yes—
212C2×C4C22×AGL(1,𝔽7)12yes—
213C7C2×C3×#12611yescharacteristic, complemented, Sylow
214C2×C3—74no—
215C2×C3—71no—
216C2×C3—71no—
217C2×C3—71no—
218C2×C3—282no—
219C22—142no—
220C22—142no—
221C22—142no—
222C22—284no—
223C22#1157⋊C312yes—
224C22#1159⋊C312yes—
225C22#6665912yes—
226C22C23×AGL(1,𝔽7)11yescharacteristic
227C4—22no—
228C4—22no—
229C4—22no—
230C4—142no—
231C4—142no—
232C4—142no—
233C3—71noSylow
234C2—282no—
235C2#6461⋊C314yesdirect product factor
236C2C2×#1157⋊C311yescharacteristic
237C2C2×#1159⋊C311yescharacteristic
238C2C2×#6665911yescharacteristic
subgroupG/H
centerC23C22×AGL(1,𝔽7)
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×AGL(1,𝔽7)
FittingC2×C7×C4⋊C4C2×C3
socleC23×C7C23×C3
prime radicalC23×AGL(1,𝔽7)C22
hypercenterC2×C4⋊C4AGL(1,𝔽7)
hypocenterC7C2×C3×#126
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×ΓL(1,𝔽8)C2semidirect
5. C2×C4⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
6. C2×(ΓL(1,𝔽8)⋊C4)⋊C4C2semidirect
7. C2×C4×ΓL(1,𝔽8)⋊C4C2semidirect
8. #6461⋊C3C2direct (N × H)
9. C2×#6461C3semidirect
17. C4⋊C4×ΓL(1,𝔽8)C22semidirect
21. C4⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
22. (ΓL(1,𝔽8)⋊C4)⋊C4C22semidirect
23. C4×ΓL(1,𝔽8)⋊C4C22semidirect
24. C2×C7×C4⋊C4C2×C3semidirect
28. C2×C4⋊Dic7C2×C3semidirect
29. C2×Dic7⋊C4C2×C3semidirect
30. C2×C4×Dic7C2×C3semidirect
31. #6461C2×C3semidirect
57. C7×C4⋊C4C22×C3semidirect
61. C4⋊Dic7C22×C3semidirect
62. Dic7⋊C4C22×C3semidirect
63. C4×Dic7C22×C3semidirect
118. C2×ΓL(1,𝔽8)#126semidirect
126. C2×C4⋊C4AGL(1,𝔽7)semidirect
162. ΓL(1,𝔽8)C2×#126semidirect
172. C4⋊C4C2×AGL(1,𝔽7)semidirect
177. C2×C7C3×#126semidirect
213. C7C2×C3×#126semidirect
235. C2#6461⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×ΓL(1,𝔽8)2
C2×(C2×AGL(1,𝔽7))⋊C42
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
C2×(ΓL(1,𝔽8)⋊C4)⋊C42
C2×C4×ΓL(1,𝔽8)⋊C42
#6461⋊C32
C2×#64613
C2×C3×#1267