Finite Group Databasebeta

← #90497

Subgroups — #90497

Order 448.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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×C7×D4C211yesmaximal, characteristic, complemented
2C2×#1159C211yesmaximal, characteristic, complemented
3#6516C212yesmaximal, complemented
4C2×C22⋊Dic7C212yesmaximal, complemented
5#6502C212yesmaximal, complemented
6#6501C211yesmaximal, characteristic, complemented
7#6499C211yesmaximal, characteristic, complemented
8#6498C212yesmaximal, complemented
9D4×Dic7C211yesmaximal, characteristic, complemented
10C4×C22⋊D7C211yesmaximal, characteristic, complemented
11#6487C211yesmaximal, characteristic, complemented
12C24×C7C2212yes—
13C2×C7×D4—24no—
14C2×C7×D4C2212yes—
15C2×C7×D4C2211yescharacteristic
16C2×C7×D4C2211yescharacteristic
17C22×C4×C7C2211yescharacteristic
18C2×C22⋊D7C2211yescharacteristic
19C2×C22⋊D7C2212yescomplemented
20C2×C22⋊D7C2212yescomplemented
21C22×Dic7C2212yescomplemented
22C22×Dic7C2212yescomplemented
23#1159—24no—
24C2×C4×D7C2211yescharacteristic, complemented
25C2×Dic14C2211yescharacteristic, complemented
26C22⋊Dic7C2212yes—
27C22⋊Dic7C2211yescharacteristic
28C22⋊Dic7C2212yescomplemented
29C22⋊Dic7C2212yescomplemented
30C22⋊Dic7C2212yescomplemented
31C22⋊Dic7C2212yescomplemented
32(C2×D7)⋊C4C2211yescharacteristic, complemented
33C4⋊Dic7C2211yescharacteristic, complemented
34Dic7⋊C4C2212yescomplemented
35Dic7⋊C4C2211yescharacteristic, complemented
36C4×Dic7C2211yescharacteristic, complemented
37#546—71nomaximal, Sylow
38C23×C7—24no—
39C23×C7—24no—
40C23×C7—42no—
41C23×C7C2312yes—
42C23×C7C2312yes—
43C23×C7C2311yescharacteristic
44C22×D7C2311yescharacteristic
45C7×D4—24no—
46C7×D4—22no—
47C7×D4D414yescomplemented
48C2×C4×C7—22no—
49C2×C4×C7C2311yescharacteristic
50C2×C4×C7C2311yescharacteristic
51C22⋊D7—24no—
52C22⋊D7—22no—
53C22⋊D7—42no—
54C2×Dic7—24no—
55C2×Dic7—42no—
56C2×Dic7C2311yescharacteristic
57C2×Dic7C2311yescharacteristic
58C2×Dic7C2312yes—
59C2×Dic7C2312yes—
60C2×Dic7C2311yescharacteristic
61C4×D7—22no—
62Dic14—22no—
63C2×#41—71no—
64C22×D4D711yescharacteristic, complemented
65#124—71no—
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—21no—
75C22×C7—22no—
76C22×C7—22no—
77C22×C7—21no—
78C22×C7—22no—
79C22×C7—22no—
80C22×C7—22no—
81C22×C7—48no—
82C22×C7C2×D414yes—
83C22×C7#4112yescomplemented
84C22×C7C2411yescharacteristic
85C2×D7—21no—
86C2×D7—21no—
87C2×D7—21no—
88C4×C7—21no—
89C4×C7C2×D412yes—
90Dic7—22no—
91Dic7—21no—
92Dic7—21no—
93Dic7—21no—
94Dic7—22no—
95C24C2×D712yes—
96#41—144no—
97C2×Q8—71no—
98C2×D4—24no—
99C2×D4—71no—
100C2×D4—72no—
101C2×D4—72no—
102C2×D4C2×D712yes—
103C2×D4C2×D711yescharacteristic
104C2×D4C2×D711yescharacteristic
105C22×C4—72no—
106C22×C4—72no—
107C22×C4—71no—
108C22×C4C2×D711yescharacteristic
109C4⋊C4—71no—
110C4⋊C4—72no—
111C4⋊C4—71no—
112C22⋊C4—71no—
113C22⋊C4—72no—
114C22⋊C4—72no—
115C22⋊C4—72no—
116C22⋊C4—72no—
117C22⋊C4—72no—
118C22⋊C4—71no—
119C42—71no—
120C2×C7—22no—
121C2×C7—24no—
122C2×C7—42no—
123C2×C7C22×D411yescharacteristic
124C2×C7C2×#4111yescharacteristic
125C2×C7H(5,𝔽2)11yescharacteristic
126D7—41no—
127C23—24no—
128C23—24no—
129C23—42no—
130C23—71no—
131C23C22×D712yes—
132C23C22×D711yescharacteristic
133C23C22×D712yes—
134Q8—142no—
135D4—24no—
136D4—22no—
137D4—142no—
138D4—144no—
139D4—282no—
140D4C22⋊D714yescomplemented
141C2×C4—22no—
142C2×C4—71no—
143C2×C4—71no—
144C2×C4—72no—
145C2×C4—71no—
146C2×C4—72no—
147C2×C4—144no—
148C2×C4—142no—
149C2×C4—282no—
150C2×C4C22×D711yescharacteristic
151C2×C4C22×D711yescharacteristic
152C7#54611yescharacteristic, complemented, Sylow
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—141no—
162C22—141no—
163C22—141no—
164C22#115912yescomplemented
165C22C2×C22⋊D714yes—
166C22C23×D711yescharacteristic
167C4—21no—
168C4—141no—
169C4—141no—
170C4—142no—
171C4—141no—
172C4—142no—
173C4C2×C22⋊D712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—281no—
178C2C2×#115911yescharacteristic
179C2#654811yescharacteristic
180C2C22×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC22×C7×D4C2
hypercenterC22×D4D7
hypocenterC7#546
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C7×D4C2semidirect
2. C2×#1159C2semidirect
3. #6516C2semidirect
4. C2×C22⋊Dic7C2semidirect
5. #6502C2semidirect
6. #6501C2semidirect
7. #6499C2semidirect
8. #6498C2semidirect
9. D4×Dic7C2semidirect
10. C4×C22⋊D7C2semidirect
11. #6487C2semidirect
19. C2×C22⋊D7C22semidirect
20. C2×C22⋊D7C22semidirect
21. C22×Dic7C22semidirect
22. C22×Dic7C22semidirect
24. C2×C4×D7C22semidirect
25. C2×Dic14C22semidirect
28. C22⋊Dic7C22semidirect
29. C22⋊Dic7C22semidirect
30. C22⋊Dic7C22semidirect
31. C22⋊Dic7C22semidirect
32. (C2×D7)⋊C4C22semidirect
33. C4⋊Dic7C22semidirect
34. Dic7⋊C4C22semidirect
35. Dic7⋊C4C22semidirect
36. C4×Dic7C22semidirect
47. C7×D4D4semidirect
64. C22×D4D7semidirect
83. C22×C7#41semidirect
140. D4C22⋊D7semidirect
152. C7#546semidirect
164. C22#1159semidirect

Maximal subgroups

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

subgroupindex
C22×C7×D42
C2×#11592
#65162
C2×C22⋊Dic72
#65022
#65012
#64992
#64982
D4×Dic72
C4×C22⋊D72
#64872
#5467