Finite Group Databasebeta

← C22×#6474 #11744239

Subgroups — C22×#6474 #11744239

Order 896.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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×#1159C211yesmaximal, characteristic, complemented
2C22×D4×D7C211yesmaximal, characteristic, complemented
3C22×C7×D8C211yesmaximal, characteristic, complemented
4C22×#1142C211yesmaximal, characteristic, complemented
5C22×#1141C211yesmaximal, characteristic, complemented
6C2×#6474C2124yesmaximal, direct product factor
7C22×#1132C211yesmaximal, characteristic, complemented
8C22×#1131C211yesmaximal, characteristic, complemented
9C24×D7—21no—
10C22×C7×D4C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic, complemented
12C22×C22⋊D7—21no—
13C22×C22⋊D7—21no—
14C23×Dic7—21no—
15C2×#1159—26no—
16C2×#1159C22112yescomplemented
17C2×D4×D7—26no—
18C2×D4×D7C22112yescomplemented
19C22×D28C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic, complemented
22C2×C7×D8C22112yescomplemented
23C22×C7×C8C2211yescharacteristic, complemented
24C2×#1142C22112yescomplemented
25C2×#1141C22112yescomplemented
26C22×C7⋊C8C2211yescharacteristic, complemented
27#6474C22164yesdirect product factor
28C2×#1132C22112yescomplemented
29C2×#1131C22112yescomplemented
30C22×AGL(1,ℤ/8)—71nomaximal, Sylow
31C24×C7—21no—
32C24×C7—21no—
33C23×D7—21no—
34C23×D7—212no—
35C23×D7—48no—
36C23×D7D411yescharacteristic
37C2×C7×D4—23no—
38C2×C7×D4—23no—
39C2×C7×D4C2316yes—
40C2×C7×D4C2316yescomplemented
41C22×C4×C7C2311yescharacteristic
42C2×C22⋊D7—212no—
43C2×C22⋊D7—212no—
44C22×Dic7—21no—
45C22×Dic7—212no—
46C22×Dic7D411yescharacteristic
47#1159—224no—
48#1159C23116yescomplemented
49D4×D7—224no—
50D4×D7C23116yescomplemented
51C2×D28—23no—
52C2×D28C2316yes—
53C2×C4×D7D416yes—
54C2×C4×D7C2316yes—
55C2×Dic14—23no—
56C2×Dic14C2316yescomplemented
57C7×D8C23116yescomplemented
58C2×C7×C8C2316yescomplemented
59#1142C23116yescomplemented
60#1141C23116yescomplemented
61C2×C7⋊C8C2316yescomplemented
62#1132C23116yescomplemented
63#1131C23116yescomplemented
64C22×#41—71no—
65C23×D4—71no—
66C2×AGL(1,ℤ/8)—724no—
67C22×SD16—71no—
68C22×SD16—71no—
69C22×D8—71no—
70C22×D8D711yescharacteristic, complemented
71C22×M16—71no—
72C23×C7—26no—
73C23×C7—26no—
74C23×C7—44no—
75C23×C7—44no—
76C23×C7C2×D411yescharacteristic
77C22×D7—24no—
78C22×D7—26no—
79C22×D7—216no—
80C22×D7—448no—
81C22×D7—44no—
82C22×D7C2×D416yes—
83C7×D4—26no—
84C7×D4—26no—
85C7×D4C2414yes—
86C7×D4C2414yescomplemented
87C2×C4×C7C2×D413yes—
88C2×C4×C7C2413yes—
89C22⋊D7—216no—
90C22⋊D7—216no—
91C2×Dic7—26no—
92C2×Dic7—216no—
93C2×Dic7C2×D416yes—
94D28—26no—
95D28C2414yes—
96C4×D7C2×D4112yes—
97C4×D7C2414yes—
98Dic14—26no—
99Dic14C2414yescomplemented
100C7×C8C2414yescomplemented
101C7⋊C8C2414yescomplemented
102C25—141no—
103C2×#41—712no—
104C2×#41—146no—
105C22×Q8—71no—
106C22×D4—712no—
107C22×D4—71no—
108C22×D4—146no—
109C22×D4—141no—
110C22×D4—141no—
111C22×D4C2×D711yescharacteristic
112C22×D4C2×D711yescharacteristic, complemented
113C23×C4—71no—
114C23×C4—141no—
115AGL(1,ℤ/8)—764no—
116C2×SD16—712no—
117C2×SD16—712no—
118C2×D8—712no—
119C2×D8C2×D7112yescomplemented
120C2×M16—712no—
121C22×C8—71no—
122C22×C8C2×D711yescharacteristic, complemented
123C22×C7—24no—
124C22×C7—24no—
125C22×C7—412no—
126C22×C7—412no—
127C22×C7AGL(1,ℤ/8)14yescomplemented
128C22×C7C22×D413yes—
129C2×D7—212no—
130C2×D7—24no—
131C2×D7—432no—
132C2×D7—412no—
133C2×D7C22×D414yes—
134C4×C7C22×D413yes—
135C4×C7C2511yescharacteristic
136Dic7—24no—
137Dic7C22×D414yes—
138C24—21no—
139C24—21no—
140C24—71no—
141C24—1412no—
142C24—141no—
143C24—288no—
144#41—716no—
145#41—1424no—
146C2×Q8—76no—
147C2×Q8—143no—
148C2×D4—23no—
149C2×D4—23no—
150C2×D4—716no—
151C2×D4—76no—
152C2×D4—1424no—
153C2×D4—1412no—
154C2×D4—1412no—
155C2×D4—143no—
156C2×D4C22×D716yes—
157C2×D4C22×D716yescomplemented
158C22×C4—71no—
159C22×C4—76no—
160C22×C4—76no—
161C22×C4—1412no—
162C22×C4—141no—
163C22×C4C22×D711yescharacteristic
164SD16—716no—
165SD16—716no—
166D8—716no—
167D8C22×D7116yescomplemented
168M16—716no—
169C2×C8—76no—
170C2×C8C22×D716yescomplemented
171C2×C7—44no—
172C2×C7—44no—
173C2×C7C2×AGL(1,ℤ/8)16yescomplemented
174C2×C7C23×D411yescharacteristic
175D7—24no—
176D7—44no—
177C23—26no—
178C23—26no—
179C23—44no—
180C23—44no—
181C23—76no—
182C23—1416no—
183C23—144no—
184C23—146no—
185C23—2848no—
186C23—284no—
187C23D4×D711yescharacteristic
188Q8—74no—
189Q8—146no—
190D4—26no—
191D4—26no—
192D4—74no—
193D4—1416no—
194D4—1416no—
195D4—146no—
196D4C23×D714yes—
197D4C23×D714yescomplemented
198C2×C4—76no—
199C2×C4—712no—
200C2×C4—74no—
201C2×C4—1416no—
202C2×C4—146no—
203C2×C4D4×D713yes—
204C2×C4C23×D713yes—
205C8—74no—
206C8C23×D714yescomplemented
207C7C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
208C22—24no—
209C22—24no—
210C22—412no—
211C22—412no—
212C22—74no—
213C22—1412no—
214C22—144no—
215C22—2832no—
216C22—2812no—
217C22#647414yesdirect product factor
218C22C2×D4×D713yes—
219C4—74no—
220C4—144no—
221C4C2×D4×D713yes—
222C4C24×D711yescharacteristic
223C2—44no—
224C2—44no—
225C2—144no—
226C2—284no—
227C2C2×#647416yesdirect product factor
228C2C22×D4×D711yescharacteristic
subgroupG/H
centerC23D4×D7
commutator, cosocleC4×C7C25
FrattiniC4C24×D7
FittingC22×C7×D8C2
socleC23×C7C2×D4
hypercenterC22×D8D7
hypocenterC7C22×AGL(1,ℤ/8)
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#1159C2semidirect
2. C22×D4×D7C2semidirect
3. C22×C7×D8C2semidirect
4. C22×#1142C2semidirect
5. C22×#1141C2semidirect
6. C2×#6474C2direct (N × H)
7. C22×#1132C2semidirect
8. C22×#1131C2semidirect
11. C22×C7×D4C22semidirect
16. C2×#1159C22semidirect
18. C2×D4×D7C22semidirect
21. C22×Dic14C22semidirect
22. C2×C7×D8C22semidirect
23. C22×C7×C8C22semidirect
24. C2×#1142C22semidirect
25. C2×#1141C22semidirect
26. C22×C7⋊C8C22semidirect
27. #6474C22direct (N × H)
28. C2×#1132C22semidirect
29. C2×#1131C22semidirect
40. C2×C7×D4C23semidirect
48. #1159C23semidirect
50. D4×D7C23semidirect
56. C2×Dic14C23semidirect
57. C7×D8C23semidirect
58. C2×C7×C8C23semidirect
59. #1142C23semidirect
60. #1141C23semidirect
61. C2×C7⋊C8C23semidirect
62. #1132C23semidirect
63. #1131C23semidirect
70. C22×D8D7semidirect
86. C7×D4C24semidirect
99. Dic14C24semidirect
100. C7×C8C24semidirect
101. C7⋊C8C24semidirect
112. C22×D4C2×D7semidirect
119. C2×D8C2×D7semidirect
122. C22×C8C2×D7semidirect
127. C22×C7AGL(1,ℤ/8)semidirect
157. C2×D4C22×D7semidirect
167. D8C22×D7semidirect
170. C2×C8C22×D7semidirect
173. C2×C7C2×AGL(1,ℤ/8)semidirect
197. D4C23×D7semidirect
206. C8C23×D7semidirect
207. C7C22×AGL(1,ℤ/8)semidirect
217. C22#6474direct (N × H)
227. C2C2×#6474direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#11592
C22×D4×D72
C22×C7×D82
C22×#11422
C22×#11412
C2×#64742
C22×#11322
C22×#11312
C22×AGL(1,ℤ/8)7