Finite Group Databasebeta

← #90318

Subgroups — #90318

Order 448.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C22×C4×D7C211yesmaximal, characteristic, complemented
3C7×C22⋊Q8C211yesmaximal, characteristic, complemented
4#6509C211yesmaximal, characteristic, complemented
5#6493C211yesmaximal, characteristic, complemented
6#6458C211yesmaximal, characteristic, complemented
7#6457C212yesmaximal, complemented
8#6456C212yesmaximal, complemented
9#6455C211yesmaximal, characteristic, complemented
10#6445C212yesmaximal, complemented
11(C22⋊D7)⋊C4C212yesmaximal, complemented
12C23×D7C2211yescharacteristic
13C2×C7×Q8C2211yescharacteristic, complemented
14C22×C4×C7C2211yescharacteristic
15C2×C22⋊D7C2212yes—
16C22×Dic7C2211yescharacteristic
17#1161—24no—
18C2×D28C2212yes—
19C2×D28C2212yes—
20C2×D28C2211yescharacteristic, complemented
21C2×C4×D7—24no—
22C2×C4×D7C2211yescharacteristic
23C2×C4×D7C2211yescharacteristic
24C2×C4×D7C2211yescharacteristic
25C2×C4×D7C2211yescharacteristic
26C2×C4×D7C2212yescomplemented
27C7×C4⋊C4C2211yescharacteristic, complemented
28C7×C4⋊C4C2212yescomplemented
29C7×C22⋊C4C2212yes—
30(C2×D7)⋊C4C2212yes—
31(C2×D7)⋊C4C2212yes—
32(C2×D7)⋊C4C2212yescomplemented
33(C2×D7)⋊C4C2212yescomplemented
34C4⋊Dic7C2211yescharacteristic, complemented
35Dic7⋊C4C2212yescomplemented
36C4×Dic7C2212yescomplemented
37#525—71nomaximal, Sylow
38C23×C7C2311yescharacteristic
39C22×D7—22no—
40C22×D7—22no—
41C22×D7—22no—
42C22×D7C2312yes—
43C22×D7C2311yescharacteristic
44C22×D7C2311yescharacteristic
45C7×Q8—22no—
46C2×C4×C7—22no—
47C2×C4×C7C2312yes—
48C2×C4×C7C2312yes—
49C2×C4×C7C2311yescharacteristic
50C2×C4×C7C2311yescharacteristic
51C22⋊D7—24no—
52C2×Dic7—22no—
53C2×Dic7C2312yes—
54C2×Dic7C2311yescharacteristic
55C2×Dic7C2311yescharacteristic
56D28—24no—
57D28—24no—
58D28—22no—
59C4×D7—22no—
60C4×D7—22no—
61C4×D7—22no—
62C4×D7—24no—
63C4×D7D414yescomplemented
64C2×#41—71no—
65C23×C4—71no—
66#123—72no—
67C22⋊Q8—71no—
68C22⋊Q8D711yescharacteristic, complemented
69#121—71no—
70#121—71no—
71#120—72no—
72C4×D4—72no—
73C4×D4—72no—
74#117—71no—
75C22×C7—21no—
76C22×C7—21no—
77C22×C7#4112yescomplemented
78C22×C7C2411yescharacteristic
79C2×D7—28no—
80C2×D7—21no—
81C2×D7—21no—
82C2×D7—21no—
83C2×D7—21no—
84C2×D7—22no—
85C2×D7—22no—
86C2×D7—22no—
87C2×D7C2×D412yes—
88C2×D7#4112yescomplemented
89C4×C7—21no—
90C4×C7—22no—
91C4×C7—22no—
92C4×C7C2×D412yes—
93Dic7—21no—
94Dic7—22no—
95Dic7C2×D412yes—
96C24—71no—
97#41—144no—
98C2×Q8C2×D711yescharacteristic, complemented
99C2×D4—72no—
100C2×D4—72no—
101C2×D4—72no—
102C2×D4—71no—
103C22×C4—71no—
104C22×C4—71no—
105C22×C4—71no—
106C22×C4—72no—
107C22×C4—71no—
108C22×C4—71no—
109C22×C4—144no—
110C22×C4C2×D711yescharacteristic
111C4⋊C4—72no—
112C4⋊C4—71no—
113C4⋊C4C2×D711yescharacteristic, complemented
114C4⋊C4C2×D712yescomplemented
115C22⋊C4—72no—
116C22⋊C4—72no—
117C22⋊C4—72no—
118C22⋊C4—72no—
119C22⋊C4C2×D712yes—
120C42—72no—
121C2×C7—22no—
122C2×C7C22×D411yescharacteristic
123C2×C7C2×#4111yescharacteristic
124C2×C7C2×#4111yescharacteristic
125D7—22no—
126D7—22no—
127D7—42no—
128C23—71no—
129C23—71no—
130C23—72no—
131C23—142no—
132C23—142no—
133C23—142no—
134C23C22×D711yescharacteristic
135Q8—22no—
136D4—144no—
137D4—144no—
138D4—144no—
139D4—142no—
140C2×C4—22no—
141C2×C4—71no—
142C2×C4—74no—
143C2×C4—72no—
144C2×C4—71no—
145C2×C4—142no—
146C2×C4—144no—
147C2×C4—142no—
148C2×C4—142no—
149C2×C4—142no—
150C2×C4C22×D712yes—
151C2×C4C22×D711yescharacteristic
152C2×C4C22×D712yes—
153C2×C4C22×D711yescharacteristic
154C7#52511yescharacteristic, complemented, Sylow
155C22—21no—
156C22—21no—
157C22—72no—
158C22—72no—
159C22—148no—
160C22—141no—
161C22—141no—
162C22—141no—
163C22—141no—
164C22—142no—
165C22—142no—
166C22—142no—
167C22#116112yescomplemented
168C22C23×D711yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—72no—
173C4—141no—
174C4—142no—
175C4D4×D712yes—
176C2—22no—
177C2—142no—
178C2—142no—
179C2—282no—
180C2C2×D4×D711yescharacteristic
181C2C2×#116111yescharacteristic
182C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×C22⋊Q8C2
hypercenterC22⋊Q8D7
hypocenterC7#525
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#1161C2semidirect
2. C22×C4×D7C2semidirect
3. C7×C22⋊Q8C2semidirect
4. #6509C2semidirect
5. #6493C2semidirect
6. #6458C2semidirect
7. #6457C2semidirect
8. #6456C2semidirect
9. #6455C2semidirect
10. #6445C2semidirect
11. (C22⋊D7)⋊C4C2semidirect
13. C2×C7×Q8C22semidirect
20. C2×D28C22semidirect
26. C2×C4×D7C22semidirect
27. C7×C4⋊C4C22semidirect
28. C7×C4⋊C4C22semidirect
32. (C2×D7)⋊C4C22semidirect
33. (C2×D7)⋊C4C22semidirect
34. C4⋊Dic7C22semidirect
35. Dic7⋊C4C22semidirect
36. C4×Dic7C22semidirect
63. C4×D7D4semidirect
68. C22⋊Q8D7semidirect
77. C22×C7#41semidirect
88. C2×D7#41semidirect
98. C2×Q8C2×D7semidirect
113. C4⋊C4C2×D7semidirect
114. C4⋊C4C2×D7semidirect
154. C7#525semidirect
167. C22#1161semidirect

Maximal subgroups

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

subgroupindex
C2×#11612
C22×C4×D72
C7×C22⋊Q82
#65092
#64932
#64582
#64572
#64562
#64552
#64452
(C22⋊D7)⋊C42
#5257