Finite Group Databasebeta

← C3×C4×C22⋊D7 #10621239

Subgroups — C3×C4×C22⋊D7 #10621239

Order 672.

188 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 90 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×C4×C7C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊D7C211yesmaximal, characteristic
3C2×C3×C4×D7C211yesmaximal, characteristic, complemented
4C3×C22⋊Dic7C211yesmaximal, characteristic, complemented
5C3×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
6C3×Dic7⋊C4C211yesmaximal, characteristic, complemented
7C3×C4×Dic7C211yesmaximal, characteristic, complemented
8C4×C22⋊D7C311yesmaximal, characteristic, direct product factor, Hall
9C23×C3×C7C2211yescharacteristic
10C22×C3×D7C2211yescharacteristic
11C2×C3×C4×C7—22no—
12C2×C3×C4×C7C2211yescharacteristic
13C2×C3×C4×C7C2211yescharacteristic
14C3×C22⋊D7C414yesdirect product factor
15C2×C3×Dic7C2211yescharacteristic
16C2×C3×Dic7C2211yescharacteristic
17C2×C3×Dic7C2211yescharacteristic
18C3×C4×D7—22no—
19C22×C4×C7C2×C311yescharacteristic, complemented
20C2×C22⋊D7C2×C311yescharacteristic
21C2×C4×D7C2×C311yescharacteristic, complemented
22C22⋊Dic7C2×C311yescharacteristic, complemented
23(C2×D7)⋊C4C2×C311yescharacteristic, complemented
24Dic7⋊C4C2×C311yescharacteristic, complemented
25C4×Dic7C2×C311yescharacteristic, complemented
26C3×C4×D4—71nomaximal, Hall
27C22×C3×C7—21no—
28C22×C3×C7—21no—
29C22×C3×C7C2×C412yescomplemented
30C22×C3×C7C2311yescharacteristic
31C2×C3×D7—21no—
32C2×C3×D7—21no—
33C2×C3×D7C2×C412yescomplemented
34C3×C4×C7—21no—
35C3×C4×C7D412yescomplemented
36C3×Dic7—21no—
37C3×Dic7—21no—
38C3×Dic7C2×C412yescomplemented
39C23×C7C22×C311yescharacteristic
40C22×D7C22×C311yescharacteristic
41C2×C4×C7—22no—
42C2×C4×C7C22×C311yescharacteristic
43C2×C4×C7C22×C311yescharacteristic
44C22⋊D7C3×C414yesdirect product factor
45C2×Dic7C22×C311yescharacteristic
46C2×Dic7C22×C311yescharacteristic
47C2×Dic7C22×C311yescharacteristic
48C4×D7—22no—
49C2×C3×D4—71no—
50C22×C3×C4—71no—
51C22×C3×C4D711yescharacteristic, complemented
52C3×C4⋊C4—71no—
53C3×C22⋊C4—71no—
54C3×C22⋊C4—71no—
55C3×C42—71no—
56C2×C3×C7—22no—
57C2×C3×C7C22×C411yescharacteristic
58C2×C3×C7C2×D411yescharacteristic
59C2×C3×C7#4111yescharacteristic
60C3×D7—22no—
61C4×D4—71noSylow
62C22×C7—21no—
63C22×C7—21no—
64C22×C7C2×C3×C412yescomplemented
65C22×C7C23×C311yescharacteristic
66C2×D7—21no—
67C2×D7—21no—
68C2×D7C2×C3×C412yescomplemented
69C4×C7—21no—
70C4×C7C3×D412yescomplemented
71Dic7—21no—
72Dic7—21no—
73Dic7C2×C3×C412yescomplemented
74C23×C3—71no—
75C23×C3C2×D711yescharacteristic
76C3×D4—74no—
77C2×C3×C4—22no—
78C2×C3×C4—71no—
79C2×C3×C4—71no—
80C2×C3×C4—71no—
81C2×C3×C4—142no—
82C2×C3×C4C2×D711yescharacteristic
83C2×C3×C4C2×D711yescharacteristic
84C3×C7C4×D411yescharacteristic, complemented, Hall
85C2×D4—71no—
86C22×C4—71no—
87C22×C4C3×D711yescharacteristic, complemented
88C4⋊C4—71no—
89C22⋊C4—71no—
90C22⋊C4—71no—
91C42—71no—
92C2×C7—22no—
93C2×C7C22×C3×C411yescharacteristic
94C2×C7C2×C3×D411yescharacteristic
95C2×C7C3×#4111yescharacteristic
96D7—22no—
97C22×C3—21no—
98C22×C3—21no—
99C22×C3—72no—
100C22×C3—141no—
101C22×C3—141no—
102C22×C3C4×D712yescomplemented
103C22×C3C22×D711yescharacteristic
104C3×C4—21no—
105C3×C4—72no—
106C3×C4—141no—
107C3×C4—141no—
108C3×C4C22⋊D712yesdirect product factor
109C23—71no—
110C23C2×C3×D711yescharacteristic
111D4—74no—
112C2×C4—22no—
113C2×C4—71no—
114C2×C4—71no—
115C2×C4—71no—
116C2×C4—142no—
117C2×C4C2×C3×D711yescharacteristic
118C2×C4C2×C3×D711yescharacteristic
119C7C3×C4×D411yescharacteristic, complemented, Sylow
120C2×C3—22no—
121C2×C3—142no—
122C2×C3C2×C4×D711yescharacteristic
123C2×C3#115711yescharacteristic
124C2×C3C2×C22⋊D711yescharacteristic
125C22—21no—
126C22—21no—
127C22—72no—
128C22—141no—
129C22—141no—
130C22C3×C4×D712yescomplemented
131C22C22×C3×D711yescharacteristic
132C4—21no—
133C4—72no—
134C4—141no—
135C4—141no—
136C4C3×C22⋊D712yesdirect product factor
137C3C4×C22⋊D711yescharacteristic, direct product factor, Sylow
138C2—22no—
139C2—142no—
140C2C2×C3×C4×D711yescharacteristic
141C2C3×#115711yescharacteristic
142C2C2×C3×C22⋊D711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D7
commutatorC2×C7C22×C3×C4
FrattiniC22C22×C3×D7
FittingC22×C3×C4×C7C2
socleC22×C3×C7C23
prime radicalC2×C3×C22⋊D7C2
cosocleC22×C7C23×C3
hypercenterC22×C3×C4D7
hypocenterC7C3×C4×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×C7C2semidirect
3. C2×C3×C4×D7C2semidirect
4. C3×C22⋊Dic7C2semidirect
5. C3×(C2×D7)⋊C4C2semidirect
6. C3×Dic7⋊C4C2semidirect
7. C3×C4×Dic7C2semidirect
8. C4×C22⋊D7C3direct (N × H)
14. C3×C22⋊D7C4direct (N × H)
19. C22×C4×C7C2×C3semidirect
21. C2×C4×D7C2×C3semidirect
22. C22⋊Dic7C2×C3semidirect
23. (C2×D7)⋊C4C2×C3semidirect
24. Dic7⋊C4C2×C3semidirect
25. C4×Dic7C2×C3semidirect
29. C22×C3×C7C2×C4semidirect
33. C2×C3×D7C2×C4semidirect
35. C3×C4×C7D4semidirect
38. C3×Dic7C2×C4semidirect
44. C22⋊D7C3×C4direct (N × H)
51. C22×C3×C4D7semidirect
64. C22×C7C2×C3×C4semidirect
68. C2×D7C2×C3×C4semidirect
70. C4×C7C3×D4semidirect
73. Dic7C2×C3×C4semidirect
84. C3×C7C4×D4semidirect
87. C22×C4C3×D7semidirect
102. C22×C3C4×D7semidirect
108. C3×C4C22⋊D7direct (N × H)
119. C7C3×C4×D4semidirect
130. C22C3×C4×D7semidirect
136. C4C3×C22⋊D7direct (N × H)
137. C3C4×C22⋊D7direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C72
C2×C3×C22⋊D72
C2×C3×C4×D72
C3×C22⋊Dic72
C3×(C2×D7)⋊C42
C3×Dic7⋊C42
C3×C4×Dic72
C4×C22⋊D73
C3×C4×D47