Finite Group Databasebeta

← #90236

Subgroups — #90236

Order 448.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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×D4×D7C211yesmaximal, characteristic
2C4×C7×D4C211yesmaximal, characteristic, complemented
3D4×Dic7C211yesmaximal, characteristic, complemented
4C4×C22⋊D7C212yesmaximal, complemented
5C2×(C2×D7)⋊C4C212yesmaximal, complemented
6#6456C211yesmaximal, characteristic, complemented
7#6455C211yesmaximal, characteristic, complemented
8(C22⋊D7)⋊C4C212yesmaximal, complemented
9D7×C22⋊C4C212yesmaximal, complemented
10C4×D28C211yesmaximal, characteristic, complemented
11#6435C211yesmaximal, characteristic, complemented
12C23×D7C2212yes—
13C2×C7×D4C2211yescharacteristic
14C22×C4×C7C2212yescomplemented
15C2×C22⋊D7C2212yes—
16C22×Dic7C2212yescomplemented
17D4×D7C418yescomplemented
18C2×D28C2211yescharacteristic
19C2×C4×D7C2211yescharacteristic
20C2×C4×D7C2212yescomplemented
21C2×C4×D7C2212yescomplemented
22C7×C4⋊C4C2211yescharacteristic, complemented
23C7×C22⋊C4C2212yescomplemented
24C42×C7C2211yescharacteristic, complemented
25C22⋊Dic7C2212yescomplemented
26(C2×D7)⋊C4C2211yescharacteristic
27(C2×D7)⋊C4C2211yescharacteristic
28(C2×D7)⋊C4C2212yes—
29(C2×D7)⋊C4C2212yescomplemented
30(C2×D7)⋊C4C2212yescomplemented
31C4⋊Dic7C2211yescharacteristic, complemented
32Dic7⋊C4C2211yescharacteristic, complemented
33Dic7⋊C4C2211yescharacteristic, complemented
34C4×Dic7C2211yescharacteristic, complemented
35C4×Dic7C2212yescomplemented
36#518—71nomaximal, Sylow
37C23×C7C2312yes—
38C22×D7—42no—
39C22×D7—42no—
40C22×D7C2×C418yescomplemented
41C22×D7C2311yescharacteristic
42C22×D7C2312yes—
43C7×D4C2×C414yescomplemented
44C2×C4×C7—42no—
45C2×C4×C7C2312yes—
46C2×C4×C7C2311yescharacteristic
47C2×C4×C7C2311yescharacteristic
48C2×C4×C7C2311yescharacteristic
49C22⋊D7C2×C418yescomplemented
50C2×Dic7—42no—
51C2×Dic7C2311yescharacteristic
52C2×Dic7C2311yescharacteristic
53C2×Dic7C2312yes—
54C2×Dic7C2311yescharacteristic
55D28C2×C414yescomplemented
56C4×D7—42no—
57C4×D7—42no—
58C4×D7C2×C414yescomplemented
59C22×D4—71no—
60C4×D4—71no—
61C4×D4—72no—
62C4×D4—71no—
63C4×D4—72no—
64C4×D4—71no—
65C4×D4D711yescharacteristic, complemented
66#117—71no—
67#117—71no—
68C2×C22⋊C4—72no—
69C2×C22⋊C4—72no—
70C22×C7—22no—
71C22×C7—22no—
72C22×C7C22×C414yes—
73C22×C7C2411yescharacteristic
74C2×D7—22no—
75C2×D7—22no—
76C2×D7—21no—
77C2×D7—21no—
78C2×D7—48no—
79C2×D7C22×C412yes—
80C2×D7C22×C414yes—
81C4×C7—21no—
82C4×C7—22no—
83C4×C7—21no—
84C4×C7C22×C412yes—
85Dic7—21no—
86Dic7—22no—
87Dic7—21no—
88Dic7C22×C412yes—
89C24—72no—
90C2×D4—72no—
91C2×D4—78no—
92C2×D4—71no—
93C2×D4C2×D711yescharacteristic
94C22×C4—71no—
95C22×C4—72no—
96C22×C4—72no—
97C22×C4—72no—
98C22×C4C2×D712yescomplemented
99C4⋊C4—71no—
100C4⋊C4—71no—
101C4⋊C4—71no—
102C4⋊C4C2×D711yescharacteristic, complemented
103C22⋊C4—72no—
104C22⋊C4—72no—
105C22⋊C4—71no—
106C22⋊C4—71no—
107C22⋊C4—72no—
108C22⋊C4—72no—
109C22⋊C4C2×D712yescomplemented
110C42—72no—
111C42—71no—
112C42C2×D711yescharacteristic, complemented
113C2×C7—24no—
114C2×C7C23×C411yescharacteristic
115C2×C7H(5,𝔽2)11yescharacteristic
116C2×C7H(5,𝔽2)11yescharacteristic
117D7—22no—
118D7—24no—
119C23—71no—
120C23—72no—
121C23—78no—
122C23—282no—
123C23—282no—
124C23C22×D712yes—
125D4—78no—
126D4—74no—
127D4C4×D714yescomplemented
128C2×C4—42no—
129C2×C4—71no—
130C2×C4—74no—
131C2×C4—71no—
132C2×C4—72no—
133C2×C4—71no—
134C2×C4—282no—
135C2×C4—282no—
136C2×C4—282no—
137C2×C4C22×D712yes—
138C2×C4C22×D711yescharacteristic
139C2×C4C22×D711yescharacteristic
140C2×C4C22×D711yescharacteristic
141C7#51811yescharacteristic, complemented, Sylow
142C22—22no—
143C22—22no—
144C22—72no—
145C22—74no—
146C22—142no—
147C22—141no—
148C22—142no—
149C22—141no—
150C22—288no—
151C22C2×C4×D714yes—
152C22C23×D711yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—72no—
157C4—141no—
158C4—142no—
159C4—141no—
160C4C2×C4×D712yes—
161C2—24no—
162C2—142no—
163C2—144no—
164C2C22×C4×D711yescharacteristic
165C2#654811yescharacteristic
166C2#655311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutatorC2×C7C23×C4
FittingC4×C7×D4C2
socle, cosocleC22×C7C24
prime radicalC2×D4×D7C2
hypercenterC4×D4D7
hypocenterC7#518
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×C7×D4C2semidirect
3. D4×Dic7C2semidirect
4. C4×C22⋊D7C2semidirect
5. C2×(C2×D7)⋊C4C2semidirect
6. #6456C2semidirect
7. #6455C2semidirect
8. (C22⋊D7)⋊C4C2semidirect
9. D7×C22⋊C4C2semidirect
10. C4×D28C2semidirect
11. #6435C2semidirect
14. C22×C4×C7C22semidirect
16. C22×Dic7C22semidirect
17. D4×D7C4semidirect
20. C2×C4×D7C22semidirect
21. C2×C4×D7C22semidirect
22. C7×C4⋊C4C22semidirect
23. C7×C22⋊C4C22semidirect
24. C42×C7C22semidirect
25. C22⋊Dic7C22semidirect
29. (C2×D7)⋊C4C22semidirect
30. (C2×D7)⋊C4C22semidirect
31. C4⋊Dic7C22semidirect
32. Dic7⋊C4C22semidirect
33. Dic7⋊C4C22semidirect
34. C4×Dic7C22semidirect
35. C4×Dic7C22semidirect
40. C22×D7C2×C4semidirect
43. C7×D4C2×C4semidirect
49. C22⋊D7C2×C4semidirect
55. D28C2×C4semidirect
58. C4×D7C2×C4semidirect
65. C4×D4D7semidirect
98. C22×C4C2×D7semidirect
102. C4⋊C4C2×D7semidirect
109. C22⋊C4C2×D7semidirect
112. C42C2×D7semidirect
127. D4C4×D7semidirect
141. C7#518semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D72
C4×C7×D42
D4×Dic72
C4×C22⋊D72
C2×(C2×D7)⋊C42
#64562
#64552
(C22⋊D7)⋊C42
D7×C22⋊C42
C4×D282
#64352
#5187