Finite Group Databasebeta

← C3×C7×#451 #13056962

Subgroups — C3×C7×#451 #13056962

Order 1344.

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

normal subgroups only

This group has 296 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×Q8C211yesmaximal, characteristic
2C2×C3×C7×Q16C212yesmaximal, complemented
3C3×C7×C22⋊Q8C211yesmaximal, characteristic
4C3×C7×#103C212yesmaximal, complemented
5C3×C7×C22⋊C8C211yesmaximal, characteristic
6C7×#451C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×Q8—24no—
8C2×C3×C7×Q8—21no—
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C3×C7×Q8C2212yes—
11C22×C3×C4×C7—21no—
12C22×C3×C4×C7C2211yescharacteristic
13C3×C7×Q16—24no—
14C2×C3×C7×C8C2212yes—
15C3×C7×C4⋊C4—21no—
16C3×C7×C4⋊C4C2211yescharacteristic
17C3×C7×C22⋊C4—21no—
18C22×C7×Q8C2×C311yescharacteristic
19C2×C7×Q16C2×C312yescomplemented
20C7×C22⋊Q8C2×C311yescharacteristic
21C7×#103C2×C312yescomplemented
22C7×C22⋊C8C2×C311yescharacteristic
23C3×#451C711yesmaximal, characteristic, direct product factor, Hall
24C23×C3×C7D411yescharacteristic
25C3×C7×Q8—24no—
26C3×C7×Q8—22no—
27C3×C7×Q8—22no—
28C3×C7×Q8D414yes—
29C2×C3×C4×C7—24no—
30C2×C3×C4×C7—22no—
31C2×C3×C4×C7—22no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7D411yescharacteristic
35C2×C3×C4×C7C2311yescharacteristic
36C3×C7×C8—22no—
37C2×C7×Q8—24no—
38C2×C7×Q8—21no—
39C2×C7×Q8C22×C311yescharacteristic
40C2×C7×Q8C22×C312yes—
41C22×C4×C7—21no—
42C22×C4×C7C22×C311yescharacteristic
43C7×Q16—24no—
44C2×C7×C8C22×C312yes—
45C7×C4⋊C4—21no—
46C7×C4⋊C4C22×C311yescharacteristic
47C7×C22⋊C4—21no—
48C22×C3×Q8C2×C711yescharacteristic
49C2×C3×Q16C2×C712yescomplemented
50C3×C22⋊Q8C2×C711yescharacteristic
51C3×#103C2×C712yescomplemented
52C3×C22⋊C8C2×C711yescharacteristic
53C22×C3×C7—21no—
54C22×C3×C7—21no—
55C22×C3×C7Q1612yescomplemented
56C22×C3×C7C2×D411yescharacteristic
57C3×C4×C7—24no—
58C3×C4×C7—21no—
59C3×C4×C7—41no—
60C3×C4×C7—41no—
61C3×C4×C7C2×D412yes—
62#451C3×C711yescharacteristic, direct product factor, Sylow
63C23×C7C3×D411yescharacteristic
64C7×Q8—24no—
65C7×Q8—22no—
66C7×Q8—22no—
67C7×Q8C3×D414yes—
68C2×C4×C7—24no—
69C2×C4×C7—22no—
70C2×C4×C7—22no—
71C2×C4×C7—21no—
72C2×C4×C7—21no—
73C2×C4×C7C3×D411yescharacteristic
74C2×C4×C7C23×C311yescharacteristic
75C7×C8—22no—
76C2×C3×Q8—24no—
77C2×C3×Q8—21no—
78C2×C3×Q8C22×C711yescharacteristic
79C2×C3×Q8C22×C712yes—
80C22×C3×C4—21no—
81C22×C3×C4C22×C711yescharacteristic
82C3×Q16—24no—
83C2×C3×C8C22×C712yes—
84C3×C4⋊C4—21no—
85C3×C4⋊C4C22×C711yescharacteristic
86C3×C22⋊C4—21no—
87C2×C3×C7—22no—
88C2×C3×C7#12011yescharacteristic
89C2×C3×C7C2×Q1611yescharacteristic
90C2×C3×C7#13711yescharacteristic
91C22×Q8C2×C3×C711yescharacteristic
92C2×Q16C2×C3×C712yescomplemented
93C22⋊Q8C2×C3×C711yescharacteristic
94#103C2×C3×C712yescomplemented
95C22⋊C8C2×C3×C711yescharacteristic
96C22×C7—21no—
97C22×C7—21no—
98C22×C7C3×Q1612yescomplemented
99C22×C7C2×C3×D411yescharacteristic
100C4×C7—24no—
101C4×C7—21no—
102C4×C7—41no—
103C4×C7—41no—
104C4×C7C2×C3×D412yes—
105C23×C3C7×D411yescharacteristic
106C3×Q8—24no—
107C3×Q8—22no—
108C3×Q8—22no—
109C3×Q8C7×D414yes—
110C2×C3×C4—24no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4C7×D411yescharacteristic
116C2×C3×C4C23×C711yescharacteristic
117C3×C8—22no—
118C3×C7#45111yescharacteristic, direct product factor, Hall
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8C22×C3×C711yescharacteristic
122C2×Q8C22×C3×C712yes—
123C22×C4—21no—
124C22×C4C22×C3×C711yescharacteristic
125Q16—24no—
126C2×C8C22×C3×C712yes—
127C4⋊C4—21no—
128C4⋊C4C22×C3×C711yescharacteristic
129C22⋊C4—21no—
130C2×C7—22no—
131C2×C7C3×#12011yescharacteristic
132C2×C7C2×C3×Q1611yescharacteristic
133C2×C7C3×#13711yescharacteristic
134C22×C3—21no—
135C22×C3—21no—
136C22×C3C7×Q1612yescomplemented
137C22×C3C2×C7×D411yescharacteristic
138C3×C4—24no—
139C3×C4—21no—
140C3×C4—41no—
141C3×C4—41no—
142C3×C4C2×C7×D412yes—
143C23C3×C7×D411yescharacteristic
144Q8—24no—
145Q8—22no—
146Q8—22no—
147Q8C3×C7×D414yes—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4C3×C7×D411yescharacteristic
154C2×C4C23×C3×C711yescharacteristic
155C8—22no—
156C7C3×#45111yescharacteristic, direct product factor, Sylow
157C2×C3—22no—
158C2×C3C7×#12011yescharacteristic
159C2×C3C2×C7×Q1611yescharacteristic
160C2×C3C7×#13711yescharacteristic
161C22—21no—
162C22—21no—
163C22C3×C7×Q1612yescomplemented
164C22C2×C3×C7×D411yescharacteristic
165C4—24no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4C2×C3×C7×D412yes—
170C3C7×#45111yescharacteristic, direct product factor, Sylow
171C2—22no—
172C2C3×C7×#12011yescharacteristic
173C2C2×C3×C7×Q1611yescharacteristic
174C2C3×C7×#13711yescharacteristic
subgroupG/H
center, socleC22×C3×C7C2×D4
commutator, Frattini, cosocleC2×C4C23×C3×C7
prime radicalC23×C3×C7D4
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×Q16C2semidirect
4. C3×C7×#103C2semidirect
6. C7×#451C3direct (N × H)
19. C2×C7×Q16C2×C3semidirect
21. C7×#103C2×C3semidirect
23. C3×#451C7direct (N × H)
49. C2×C3×Q16C2×C7semidirect
51. C3×#103C2×C7semidirect
55. C22×C3×C7Q16semidirect
62. #451C3×C7direct (N × H)
92. C2×Q16C2×C3×C7semidirect
94. #103C2×C3×C7semidirect
98. C22×C7C3×Q16semidirect
118. C3×C7#451direct (N × H)
136. C22×C3C7×Q16semidirect
156. C7C3×#451direct (N × H)
163. C22C3×C7×Q16semidirect
170. C3C7×#451direct (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×Q82
C2×C3×C7×Q162
C3×C7×C22⋊Q82
C3×C7×#1032
C3×C7×C22⋊C82
C7×#4513
C3×#4517