Finite Group Databasebeta

← #90367

Subgroups — #90367

Order 448.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#1161C211yesmaximal, characteristic, complemented
2C2×D4×D7C211yesmaximal, characteristic, complemented
3C7×#124C211yesmaximal, characteristic, complemented
4#6509C211yesmaximal, characteristic, complemented
5#6501C211yesmaximal, characteristic, complemented
6#6447C212yesmaximal, complemented
7#6446C212yesmaximal, complemented
8#6445C212yesmaximal, complemented
9D7×C22⋊C4C212yesmaximal, complemented
10C4×D28C211yesmaximal, characteristic, complemented
11C4×D28C211yesmaximal, characteristic, complemented
12C23×D7C2212yes—
13C2×C7×Q8C2211yescharacteristic, complemented
14C2×C7×D4C2211yescharacteristic
15C2×C22⋊D7C2212yes—
16C2×C22⋊D7C2212yescomplemented
17#1161—24no—
18D4×D7—24no—
19C2×D28C2211yescharacteristic
20C2×D28C2211yescharacteristic
21C2×D28C2212yescomplemented
22C2×C4×D7C2211yescharacteristic
23C2×C4×D7C2211yescharacteristic, complemented
24C2×C4×D7C2212yescomplemented
25C2×C4×D7C2212yescomplemented
26C7×C22⋊C4C2212yescomplemented
27C7×C22⋊C4C2212yescomplemented
28C42×C7C2211yescharacteristic, complemented
29C22⋊Dic7C2212yescomplemented
30(C2×D7)⋊C4C2211yescharacteristic
31(C2×D7)⋊C4C2212yes—
32(C2×D7)⋊C4C2212yescomplemented
33(C2×D7)⋊C4C2211yescharacteristic, complemented
34C4⋊Dic7C2211yescharacteristic, complemented
35C4⋊Dic7C2211yescharacteristic, complemented
36Dic7⋊C4C2212yescomplemented
37#546—71nomaximal, Sylow
38C23×C7C2312yes—
39C22×D7—24no—
40C22×D7—24no—
41C22×D7—42no—
42C22×D7C2311yescharacteristic
43C22×D7C2312yes—
44C22×D7C2311yescharacteristic
45C7×Q8—22no—
46C7×D4—22no—
47C2×C4×C7C2312yes—
48C2×C4×C7C2311yescharacteristic
49C2×C4×C7C2311yescharacteristic
50C2×C4×C7C2311yescharacteristic
51C22⋊D7—24no—
52C22⋊D7—42no—
53C2×Dic7C2311yescharacteristic
54C2×Dic7C2312yes—
55C2×Dic7C2311yescharacteristic
56D28—24no—
57D28—22no—
58D28D414yescomplemented
59C4×D7—24no—
60C4×D7—22no—
61C4×D7—22no—
62C4×D7—42no—
63C2×#41—71no—
64C22×D4—71no—
65#124D711yescharacteristic, complemented
66#123—72no—
67C22⋊Q8—71no—
68#121—71no—
69#121—72no—
70#120—72no—
71C4×D4—71no—
72C4×D4—71no—
73C2×C22⋊C4—72no—
74C22×C7—22no—
75C22×C7—22no—
76C22×C7—22no—
77C22×C7C2411yescharacteristic
78C2×D7—21no—
79C2×D7—22no—
80C2×D7—22no—
81C2×D7—21no—
82C2×D7—21no—
83C2×D7—21no—
84C2×D7—21no—
85C2×D7—48no—
86C2×D7C2×D414yes—
87C2×D7#4112yescomplemented
88C4×C7—22no—
89C4×C7—21no—
90C4×C7—21no—
91C4×C7C2×D412yes—
92Dic7—21no—
93Dic7—21no—
94Dic7—22no—
95C24—72no—
96#41—144no—
97C2×Q8C2×D711yescharacteristic, complemented
98C2×D4—72no—
99C2×D4—72no—
100C2×D4—71no—
101C2×D4—71no—
102C2×D4—72no—
103C2×D4—144no—
104C2×D4C2×D711yescharacteristic
105C22×C4—72no—
106C22×C4—71no—
107C22×C4—71no—
108C22×C4—72no—
109C4⋊C4—72no—
110C4⋊C4—71no—
111C4⋊C4—71no—
112C22⋊C4—72no—
113C22⋊C4—71no—
114C22⋊C4—72no—
115C22⋊C4—72no—
116C22⋊C4—71no—
117C22⋊C4C2×D712yescomplemented
118C22⋊C4C2×D712yescomplemented
119C42C2×D711yescharacteristic, complemented
120C2×C7—42no—
121C2×C7C22×D411yescharacteristic
122C2×C7C2×#4111yescharacteristic
123C2×C7H(5,𝔽2)11yescharacteristic
124D7—22no—
125D7—24no—
126D7—41no—
127C23—71no—
128C23—72no—
129C23—71no—
130C23—144no—
131C23—144no—
132C23—282no—
133C23C22×D712yes—
134Q8—22no—
135D4—22no—
136D4—74no—
137D4—142no—
138D4—144no—
139D4—144no—
140D4—282no—
141C2×C4—72no—
142C2×C4—71no—
143C2×C4—71no—
144C2×C4—144no—
145C2×C4—142no—
146C2×C4—142no—
147C2×C4—282no—
148C2×C4C22×D711yescharacteristic
149C2×C4C22×D711yescharacteristic
150C2×C4C22×D712yes—
151C2×C4C22×D711yescharacteristic
152C7#54611yescharacteristic, complemented, Sylow
153C22—22no—
154C22—22no—
155C22—22no—
156C22—74no—
157C22—72no—
158C22—141no—
159C22—142no—
160C22—142no—
161C22—141no—
162C22—141no—
163C22—141no—
164C22—141no—
165C22—288no—
166C22C23×D711yescharacteristic
167C4—22no—
168C4—21no—
169C4—21no—
170C4—141no—
171C4—141no—
172C4—142no—
173C4D4×D712yes—
174C2—42no—
175C2—142no—
176C2—144no—
177C2—281no—
178C2C2×D4×D711yescharacteristic
179C2D7×#4111yescharacteristic
180C2#655311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#124C2
hypercenter#124D7
hypocenterC7#546
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#1161C2semidirect
2. C2×D4×D7C2semidirect
3. C7×#124C2semidirect
4. #6509C2semidirect
5. #6501C2semidirect
6. #6447C2semidirect
7. #6446C2semidirect
8. #6445C2semidirect
9. D7×C22⋊C4C2semidirect
10. C4×D28C2semidirect
11. C4×D28C2semidirect
13. C2×C7×Q8C22semidirect
16. C2×C22⋊D7C22semidirect
21. C2×D28C22semidirect
23. C2×C4×D7C22semidirect
24. C2×C4×D7C22semidirect
25. C2×C4×D7C22semidirect
26. C7×C22⋊C4C22semidirect
27. C7×C22⋊C4C22semidirect
28. C42×C7C22semidirect
29. C22⋊Dic7C22semidirect
32. (C2×D7)⋊C4C22semidirect
33. (C2×D7)⋊C4C22semidirect
34. C4⋊Dic7C22semidirect
35. C4⋊Dic7C22semidirect
36. Dic7⋊C4C22semidirect
58. D28D4semidirect
65. #124D7semidirect
87. C2×D7#41semidirect
97. C2×Q8C2×D7semidirect
117. C22⋊C4C2×D7semidirect
118. C22⋊C4C2×D7semidirect
119. C42C2×D7semidirect
152. C7#546semidirect

Maximal subgroups

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

subgroupindex
C2×#11612
C2×D4×D72
C7×#1242
#65092
#65012
#64472
#64462
#64452
D7×C22⋊C42
C4×D282
#5467