Finite Group Databasebeta

← #90520

Subgroups — #90520

Order 448.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×C7×#41C211yesmaximal, characteristic, complemented
2C2×D4×D7C211yesmaximal, characteristic, complemented
3#6510C211yesmaximal, characteristic, complemented
4#6509C211yesmaximal, characteristic, complemented
5#6502C212yesmaximal, complemented
6#6500C212yesmaximal, complemented
7D4×Dic7C211yesmaximal, characteristic, complemented
8#6493C211yesmaximal, characteristic, complemented
9#6492C212yesmaximal, complemented
10C4×C22⋊D7C211yesmaximal, characteristic, complemented
11C2×(C2×D7)⋊C4C212yesmaximal, complemented
12C23×D7C2212yes—
13C7×#41—24no—
14C2×C7×Q8C2211yescharacteristic, complemented
15C2×C7×D4C2211yescharacteristic
16C2×C7×D4C2212yescomplemented
17C22×C4×C7C2212yescomplemented
18C22×C4×C7C2211yescharacteristic, complemented
19C2×C22⋊D7C2211yescharacteristic
20C2×C22⋊D7C2212yes—
21C2×C22⋊D7C2212yescomplemented
22C22×Dic7C2212yescomplemented
23D4×D7—24no—
24C2×D28C2211yescharacteristic
25C2×C4×D7C2211yescharacteristic
26C22⋊Dic7C2211yescharacteristic, complemented
27C22⋊Dic7C2212yescomplemented
28(C2×D7)⋊C4C2211yescharacteristic
29(C2×D7)⋊C4C2212yes—
30(C2×D7)⋊C4C2212yescomplemented
31(C2×D7)⋊C4C2212yescomplemented
32(C2×D7)⋊C4C2212yescomplemented
33C4⋊Dic7C2211yescharacteristic, complemented
34Dic7⋊C4C2211yescharacteristic, complemented
35Dic7⋊C4C2212yescomplemented
36C4×Dic7C2211yescharacteristic, complemented
37#546—71nomaximal, Sylow
38C23×C7C2311yescharacteristic
39C23×C7C2312yes—
40C22×D7—24no—
41C22×D7—24no—
42C22×D7—42no—
43C22×D7C2312yes—
44C22×D7C2311yescharacteristic
45C7×Q8—22no—
46C7×D4—24no—
47C7×D4D414yescomplemented
48C2×C4×C7—24no—
49C2×C4×C7—22no—
50C2×C4×C7C2311yescharacteristic
51C2×C4×C7C2312yes—
52C2×C4×C7C2311yescharacteristic
53C22⋊D7—24no—
54C22⋊D7—22no—
55C22⋊D7—42no—
56C2×Dic7—42no—
57C2×Dic7C2311yescharacteristic
58C2×Dic7C2311yescharacteristic
59C2×Dic7C2311yescharacteristic
60C2×Dic7C2312yes—
61D28—22no—
62C4×D7—22no—
63C2×#41D711yescharacteristic, complemented
64C22×D4—71no—
65#124—71no—
66#123—72no—
67C22⋊Q8—71no—
68#121—72no—
69#121—71no—
70#120—72no—
71C4×D4—71no—
72C4×D4—71no—
73C2×C22⋊C4—72no—
74C22×C7—22no—
75C22×C7—22no—
76C22×C7—21no—
77C22×C7—21no—
78C22×C7—21no—
79C22×C7C2×D414yes—
80C22×C7C2411yescharacteristic
81C2×D7—21no—
82C2×D7—21no—
83C2×D7—22no—
84C2×D7—22no—
85C2×D7—22no—
86C2×D7—48no—
87C2×D7#4112yescomplemented
88C4×C7—21no—
89C4×C7—22no—
90C4×C7C2×D412yes—
91Dic7—21no—
92Dic7—21no—
93Dic7—21no—
94Dic7—22no—
95C24—72no—
96#41—24no—
97C2×Q8C2×D711yescharacteristic, complemented
98C2×D4—71no—
99C2×D4—72no—
100C2×D4—72no—
101C2×D4—71no—
102C2×D4—144no—
103C2×D4C2×D711yescharacteristic
104C2×D4C2×D712yescomplemented
105C22×C4—72no—
106C22×C4—71no—
107C22×C4C2×D712yescomplemented
108C22×C4C2×D711yescharacteristic, complemented
109C4⋊C4—71no—
110C4⋊C4—72no—
111C4⋊C4—71no—
112C22⋊C4—72no—
113C22⋊C4—71no—
114C22⋊C4—71no—
115C22⋊C4—72no—
116C22⋊C4—72no—
117C22⋊C4—72no—
118C22⋊C4—72no—
119C42—71no—
120C2×C7—24no—
121C2×C7—41no—
122C2×C7C22×D411yescharacteristic
123C2×C7C2×#4111yescharacteristic
124C2×C7H(5,𝔽2)11yescharacteristic
125D7—22no—
126D7—42no—
127C23—71no—
128C23—72no—
129C23—144no—
130C23—144no—
131C23—282no—
132C23C22×D711yescharacteristic
133C23C22×D712yes—
134Q8—22no—
135D4—24no—
136D4—142no—
137D4—144no—
138D4—142no—
139D4—282no—
140D4C22⋊D714yescomplemented
141C2×C4—24no—
142C2×C4—22no—
143C2×C4—71no—
144C2×C4—71no—
145C2×C4—72no—
146C2×C4—71no—
147C2×C4—142no—
148C2×C4—282no—
149C2×C4C22×D711yescharacteristic
150C2×C4C22×D712yes—
151C2×C4C22×D711yescharacteristic
152C7#54611yescharacteristic, complemented, Sylow
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—72no—
159C22—141no—
160C22—141no—
161C22—142no—
162C22—142no—
163C22—142no—
164C22—288no—
165C22C2×C22⋊D714yes—
166C22C23×D711yescharacteristic
167C4—21no—
168C4—22no—
169C4—141no—
170C4—141no—
171C4—142no—
172C4—141no—
173C4C2×C22⋊D712yes—
174C2—24no—
175C2—41no—
176C2—142no—
177C2—282no—
178C2D7×#4111yescharacteristic
179C2#655311yescharacteristic
180C2C22×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC2×C7×#41C2
hypercenterC2×#41D7
hypocenterC7#546
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C7×#41C2semidirect
2. C2×D4×D7C2semidirect
3. #6510C2semidirect
4. #6509C2semidirect
5. #6502C2semidirect
6. #6500C2semidirect
7. D4×Dic7C2semidirect
8. #6493C2semidirect
9. #6492C2semidirect
10. C4×C22⋊D7C2semidirect
11. C2×(C2×D7)⋊C4C2semidirect
14. C2×C7×Q8C22semidirect
16. C2×C7×D4C22semidirect
17. C22×C4×C7C22semidirect
18. C22×C4×C7C22semidirect
21. C2×C22⋊D7C22semidirect
22. C22×Dic7C22semidirect
26. C22⋊Dic7C22semidirect
27. C22⋊Dic7C22semidirect
30. (C2×D7)⋊C4C22semidirect
31. (C2×D7)⋊C4C22semidirect
32. (C2×D7)⋊C4C22semidirect
33. C4⋊Dic7C22semidirect
34. Dic7⋊C4C22semidirect
35. Dic7⋊C4C22semidirect
36. C4×Dic7C22semidirect
47. C7×D4D4semidirect
63. C2×#41D7semidirect
87. C2×D7#41semidirect
97. C2×Q8C2×D7semidirect
104. C2×D4C2×D7semidirect
107. C22×C4C2×D7semidirect
108. C22×C4C2×D7semidirect
140. D4C22⋊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×C7×#412
C2×D4×D72
#65102
#65092
#65022
#65002
D4×Dic72
#64932
#64922
C4×C22⋊D72
C2×(C2×D7)⋊C42
#5467