Finite Group Databasebeta

← #90241

Subgroups — #90241

Order 448.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#1157C211yesmaximal, characteristic, complemented
2C22×D28C211yesmaximal, characteristic, complemented
3C4×C7×D4C211yesmaximal, characteristic, complemented
4#6500C212yesmaximal, complemented
5#6493C211yesmaximal, characteristic, complemented
6#6487C211yesmaximal, characteristic, complemented
7#6457C212yesmaximal, complemented
8#6447C212yesmaximal, complemented
9D7×C22⋊C4C212yesmaximal, complemented
10#6438C211yesmaximal, characteristic, complemented
11C4×D28C211yesmaximal, characteristic, complemented
12C23×D7C2212yes—
13C2×C7×D4C2211yescharacteristic
14C22×C4×C7C2211yescharacteristic
15C22×C4×C7C2211yescharacteristic, complemented
16C2×C22⋊D7C2212yescomplemented
17C2×C22⋊D7C2212yescomplemented
18#1157—24no—
19C2×D28—24no—
20C2×D28C2211yescharacteristic
21C2×D28C2212yes—
22C2×D28C2211yescharacteristic
23C2×C4×D7C2212yescomplemented
24C2×C4×D7C2212yescomplemented
25C2×Dic14C2211yescharacteristic, complemented
26C7×C4⋊C4C2211yescharacteristic, complemented
27C7×C22⋊C4C2211yescharacteristic
28C7×C22⋊C4C2211yescharacteristic, complemented
29C42×C7C2211yescharacteristic, complemented
30C22⋊Dic7C2212yes—
31(C2×D7)⋊C4C2212yescomplemented
32(C2×D7)⋊C4C2212yescomplemented
33(C2×D7)⋊C4C2212yescomplemented
34(C2×D7)⋊C4C2212yescomplemented
35C4⋊Dic7C2211yescharacteristic, complemented
36Dic7⋊C4C2212yescomplemented
37#546—71nomaximal, Sylow
38C23×C7C2311yescharacteristic
39C23×C7C2311yescharacteristic
40C22×D7—24no—
41C22×D7—24no—
42C22×D7—42no—
43C22×D7C2312yes—
44C22×D7C2312yes—
45C7×D4—22no—
46C2×C4×C7—22no—
47C2×C4×C7—22no—
48C2×C4×C7C2311yescharacteristic
49C2×C4×C7C2311yescharacteristic
50C2×C4×C7C2311yescharacteristic
51C2×C4×C7C2311yescharacteristic
52C2×C4×C7C2311yescharacteristic
53C22⋊D7—24no—
54C22⋊D7—42no—
55C2×Dic7C2312yes—
56C2×Dic7C2312yes—
57D28—24no—
58D28—22no—
59D28D414yescomplemented
60C4×D7—24no—
61C4×D7—42no—
62Dic14—22no—
63C2×#41—71no—
64C22×D4—71no—
65#124—71no—
66#123—72no—
67C22⋊Q8—71no—
68#121—71no—
69#121—72no—
70#120—72no—
71C4×D4—71no—
72C4×D4D711yescharacteristic, complemented
73C2×C22⋊C4—72no—
74C22×C7—21no—
75C22×C7—21no—
76C22×C7—21no—
77C22×C7—21no—
78C22×C7—21no—
79C22×C7#4112yescomplemented
80C22×C7C2411yescharacteristic
81C2×D7—22no—
82C2×D7—22no—
83C2×D7—22no—
84C2×D7—22no—
85C2×D7—22no—
86C2×D7—48no—
87C2×D7C2×D414yes—
88C4×C7—21no—
89C4×C7—21no—
90C4×C7—21no—
91C4×C7—21no—
92C4×C7C2×D412yes—
93Dic7—22no—
94Dic7—22no—
95C24—72no—
96#41—144no—
97C2×Q8—71no—
98C2×D4—72no—
99C2×D4—72no—
100C2×D4—71no—
101C2×D4—72no—
102C2×D4—71no—
103C2×D4—144no—
104C2×D4C2×D711yescharacteristic
105C22×C4—72no—
106C22×C4—72no—
107C22×C4C2×D711yescharacteristic
108C22×C4C2×D711yescharacteristic, complemented
109C4⋊C4—72no—
110C4⋊C4—71no—
111C4⋊C4C2×D711yescharacteristic, complemented
112C22⋊C4—72no—
113C22⋊C4—72no—
114C22⋊C4—72no—
115C22⋊C4—72no—
116C22⋊C4—72no—
117C22⋊C4C2×D711yescharacteristic
118C22⋊C4C2×D711yescharacteristic, complemented
119C42C2×D711yescharacteristic, complemented
120C2×C7—22no—
121C2×C7—41no—
122C2×C7C22×D411yescharacteristic
123C2×C7C2×#4111yescharacteristic
124C2×C7H(5,𝔽2)11yescharacteristic
125D7—24no—
126D7—42no—
127C23—72no—
128C23—72no—
129C23—144no—
130C23—144no—
131C23—282no—
132C23C22×D711yescharacteristic
133C23C22×D711yescharacteristic
134Q8—142no—
135D4—22no—
136D4—74no—
137D4—144no—
138D4—144no—
139D4—142no—
140D4—282no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—72no—
144C2×C4—72no—
145C2×C4—144no—
146C2×C4—282no—
147C2×C4C22×D711yescharacteristic
148C2×C4C22×D711yescharacteristic
149C2×C4C22×D711yescharacteristic
150C2×C4C22×D711yescharacteristic
151C2×C4C22×D711yescharacteristic
152C7#54611yescharacteristic, complemented, Sylow
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—74no—
159C22—142no—
160C22—142no—
161C22—142no—
162C22—142no—
163C22—142no—
164C22—288no—
165C22#115712yescomplemented
166C22C23×D711yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—142no—
172C4—142no—
173C4D4×D712yes—
174C2—22no—
175C2—41no—
176C2—144no—
177C2—282no—
178C2C2×#115711yescharacteristic
179C2C2×D4×D711yescharacteristic
180C2#655311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC4×C7×D4C2
hypercenterC4×D4D7
hypocenterC7#546
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#1157C2semidirect
2. C22×D28C2semidirect
3. C4×C7×D4C2semidirect
4. #6500C2semidirect
5. #6493C2semidirect
6. #6487C2semidirect
7. #6457C2semidirect
8. #6447C2semidirect
9. D7×C22⋊C4C2semidirect
10. #6438C2semidirect
11. C4×D28C2semidirect
15. C22×C4×C7C22semidirect
16. C2×C22⋊D7C22semidirect
17. C2×C22⋊D7C22semidirect
23. C2×C4×D7C22semidirect
24. C2×C4×D7C22semidirect
25. C2×Dic14C22semidirect
26. C7×C4⋊C4C22semidirect
28. C7×C22⋊C4C22semidirect
29. C42×C7C22semidirect
31. (C2×D7)⋊C4C22semidirect
32. (C2×D7)⋊C4C22semidirect
33. (C2×D7)⋊C4C22semidirect
34. (C2×D7)⋊C4C22semidirect
35. C4⋊Dic7C22semidirect
36. Dic7⋊C4C22semidirect
59. D28D4semidirect
72. C4×D4D7semidirect
79. C22×C7#41semidirect
108. C22×C4C2×D7semidirect
111. C4⋊C4C2×D7semidirect
118. C22⋊C4C2×D7semidirect
119. C42C2×D7semidirect
152. C7#546semidirect
165. C22#1157semidirect

Maximal subgroups

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

subgroupindex
C2×#11572
C22×D282
C4×C7×D42
#65002
#64932
#64872
#64572
#64472
D7×C22⋊C42
#64382
C4×D282
#5467