Finite Group Databasebeta

← C3×C7×#450 #13056961

Subgroups — C3×C7×#450 #13056961

Order 1344.

376 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 124 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×C3×C7×D4C211yesmaximal, characteristic
2C2×C3×C7×SD16C212yesmaximal, complemented
3C3×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
4C3×C7×#102C212yesmaximal, complemented
5C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
6C7×#450C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×C7—21no—
8C2×C3×C7×Q8C2211yescharacteristic, complemented
9C2×C3×C7×D4—24no—
10C2×C3×C7×D4—21no—
11C2×C3×C7×D4C2212yes—
12C22×C3×C4×C7C2211yescharacteristic
13C3×C7×SD16—24no—
14C2×C3×C7×C8C2212yescomplemented
15C3×C7×C4⋊C4—21no—
16C3×C7×C4⋊C4C2211yescharacteristic, complemented
17C3×C7×C22⋊C4—21no—
18C22×C7×D4C2×C311yescharacteristic
19C2×C7×SD16C2×C312yescomplemented
20C7×C22⋊Q8C2×C311yescharacteristic, complemented
21C7×#102C2×C312yescomplemented
22C7×C22⋊C8C2×C311yescharacteristic, complemented
23C3×#450C711yesmaximal, characteristic, direct product factor, Hall
24C23×C3×C7—24no—
25C23×C3×C7—22no—
26C23×C3×C7—42no—
27C23×C3×C7—42no—
28C23×C3×C7D411yescharacteristic
29C3×C7×Q8—22no—
30C3×C7×D4—24no—
31C3×C7×D4—22no—
32C3×C7×D4D414yes—
33C2×C3×C4×C7—22no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7C2311yescharacteristic
38C3×C7×C8—22no—
39C24×C7—21no—
40C2×C7×Q8C22×C311yescharacteristic, complemented
41C2×C7×D4—24no—
42C2×C7×D4—21no—
43C2×C7×D4C22×C312yes—
44C22×C4×C7C22×C311yescharacteristic
45C7×SD16—24no—
46C2×C7×C8C22×C312yescomplemented
47C7×C4⋊C4—21no—
48C7×C4⋊C4C22×C311yescharacteristic, complemented
49C7×C22⋊C4—21no—
50C22×C3×D4C2×C711yescharacteristic
51C2×C3×SD16C2×C712yescomplemented
52C3×C22⋊Q8C2×C711yescharacteristic, complemented
53C3×#102C2×C712yescomplemented
54C3×C22⋊C8C2×C711yescharacteristic, complemented
55C22×C3×C7—21no—
56C22×C3×C7—21no—
57C22×C3×C7—24no—
58C22×C3×C7—48no—
59C22×C3×C7—42no—
60C22×C3×C7—42no—
61C22×C3×C7SD1612yescomplemented
62C22×C3×C7C2×D411yescharacteristic
63C3×C4×C7—21no—
64C3×C4×C7—41no—
65C3×C4×C7—41no—
66C3×C4×C7C2×D412yes—
67#450C3×C711yescharacteristic, direct product factor, Sylow
68C23×C7—24no—
69C23×C7—22no—
70C23×C7—42no—
71C23×C7—42no—
72C23×C7C3×D411yescharacteristic
73C7×Q8—22no—
74C7×D4—24no—
75C7×D4—22no—
76C7×D4C3×D414yes—
77C2×C4×C7—22no—
78C2×C4×C7—21no—
79C2×C4×C7—21no—
80C2×C4×C7C3×D411yescharacteristic
81C2×C4×C7C23×C311yescharacteristic
82C7×C8—22no—
83C24×C3—21no—
84C2×C3×Q8C22×C711yescharacteristic, complemented
85C2×C3×D4—24no—
86C2×C3×D4—21no—
87C2×C3×D4C22×C712yes—
88C22×C3×C4C22×C711yescharacteristic
89C3×SD16—24no—
90C2×C3×C8C22×C712yescomplemented
91C3×C4⋊C4—21no—
92C3×C4⋊C4C22×C711yescharacteristic, complemented
93C3×C22⋊C4—21no—
94C2×C3×C7—22no—
95C2×C3×C7—44no—
96C2×C3×C7#12011yescharacteristic
97C2×C3×C7C2×SD1611yescharacteristic
98C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
99C22×D4C2×C3×C711yescharacteristic
100C2×SD16C2×C3×C712yescomplemented
101C22⋊Q8C2×C3×C711yescharacteristic, complemented
102#102C2×C3×C712yescomplemented
103C22⋊C8C2×C3×C711yescharacteristic, complemented
104C22×C7—21no—
105C22×C7—21no—
106C22×C7—24no—
107C22×C7—48no—
108C22×C7—42no—
109C22×C7—42no—
110C22×C7C3×SD1612yescomplemented
111C22×C7C2×C3×D411yescharacteristic
112C4×C7—21no—
113C4×C7—41no—
114C4×C7—41no—
115C4×C7C2×C3×D412yes—
116C23×C3—24no—
117C23×C3—22no—
118C23×C3—42no—
119C23×C3—42no—
120C23×C3C7×D411yescharacteristic
121C3×Q8—22no—
122C3×D4—24no—
123C3×D4—22no—
124C3×D4C7×D414yes—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4C7×D411yescharacteristic
129C2×C3×C4C23×C711yescharacteristic
130C3×C8—22no—
131C3×C7#45011yescharacteristic, direct product factor, Hall
132C24—21no—
133C2×Q8C22×C3×C711yescharacteristic, complemented
134C2×D4—24no—
135C2×D4—21no—
136C2×D4C22×C3×C712yes—
137C22×C4C22×C3×C711yescharacteristic
138SD16—24no—
139C2×C8C22×C3×C712yescomplemented
140C4⋊C4—21no—
141C4⋊C4C22×C3×C711yescharacteristic, complemented
142C22⋊C4—21no—
143C2×C7—22no—
144C2×C7—44no—
145C2×C7C3×#12011yescharacteristic
146C2×C7C2×C3×SD1611yescharacteristic
147C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—24no—
151C22×C3—48no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3C7×SD1612yescomplemented
155C22×C3C2×C7×D411yescharacteristic
156C3×C4—21no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4C2×C7×D412yes—
160C23—24no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23C3×C7×D411yescharacteristic
165Q8—22no—
166D4—24no—
167D4—22no—
168D4C3×C7×D414yes—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4C3×C7×D411yescharacteristic
173C2×C4C23×C3×C711yescharacteristic
174C8—22no—
175C7C3×#45011yescharacteristic, direct product factor, Sylow
176C2×C3—22no—
177C2×C3—44no—
178C2×C3C7×#12011yescharacteristic
179C2×C3C2×C7×SD1611yescharacteristic
180C2×C3C7×AGL(1,ℤ/8)11yescharacteristic
181C22—21no—
182C22—21no—
183C22—24no—
184C22—48no—
185C22—42no—
186C22—42no—
187C22C3×C7×SD1612yescomplemented
188C22C2×C3×C7×D411yescharacteristic
189C4—21no—
190C4—41no—
191C4—41no—
192C4C2×C3×C7×D412yes—
193C3C7×#45011yescharacteristic, direct product factor, Sylow
194C2—22no—
195C2—44no—
196C2C3×C7×#12011yescharacteristic
197C2C2×C3×C7×SD1611yescharacteristic
198C2C3×C7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
center, socleC22×C3×C7C2×D4
commutator, Frattini, cosocleC2×C4C23×C3×C7
prime radicalC22×C3×C7×D4C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C7 ◁ C22×C3×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C7×SD16C2semidirect
3. C3×C7×C22⋊Q8C2semidirect
4. C3×C7×#102C2semidirect
5. C3×C7×C22⋊C8C2semidirect
6. C7×#450C3direct (N × H)
8. C2×C3×C7×Q8C22semidirect
14. C2×C3×C7×C8C22semidirect
16. C3×C7×C4⋊C4C22semidirect
19. C2×C7×SD16C2×C3semidirect
20. C7×C22⋊Q8C2×C3semidirect
21. C7×#102C2×C3semidirect
22. C7×C22⋊C8C2×C3semidirect
23. C3×#450C7direct (N × H)
40. C2×C7×Q8C22×C3semidirect
46. C2×C7×C8C22×C3semidirect
48. C7×C4⋊C4C22×C3semidirect
51. C2×C3×SD16C2×C7semidirect
52. C3×C22⋊Q8C2×C7semidirect
53. C3×#102C2×C7semidirect
54. C3×C22⋊C8C2×C7semidirect
61. C22×C3×C7SD16semidirect
67. #450C3×C7direct (N × H)
84. C2×C3×Q8C22×C7semidirect
90. C2×C3×C8C22×C7semidirect
92. C3×C4⋊C4C22×C7semidirect
100. C2×SD16C2×C3×C7semidirect
101. C22⋊Q8C2×C3×C7semidirect
102. #102C2×C3×C7semidirect
103. C22⋊C8C2×C3×C7semidirect
110. C22×C7C3×SD16semidirect
131. C3×C7#450direct (N × H)
133. C2×Q8C22×C3×C7semidirect
139. C2×C8C22×C3×C7semidirect
141. C4⋊C4C22×C3×C7semidirect
154. C22×C3C7×SD16semidirect
175. C7C3×#450direct (N × H)
187. C22C3×C7×SD16semidirect
193. C3C7×#450direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D42
C2×C3×C7×SD162
C3×C7×C22⋊Q82
C3×C7×#1022
C3×C7×C22⋊C82
C7×#4503
C3×#4507