Finite Group Databasebeta

← D7×#490 #11739157

Subgroups — D7×#490 #11739157

Order 896.

488 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 125 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
1D7×#127C211yesmaximal, characteristic
2D7×#125C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic, complemented
4#89843C211yesmaximal, characteristic, complemented
5#89839C211yesmaximal, characteristic, complemented
6D7×#102C214yesmaximal, complemented
7#89536C214yesmaximal, complemented
8#89485C211yesmaximal, characteristic, complemented
9D7×#97C211yesmaximal, characteristic, complemented
10C2×D4×D7—22no—
11C2×D4×D7C2212yes—
12C7×#127C2211yescharacteristic
13C7×#125C2211yescharacteristic, complemented
14#6503—22no—
15#6503C2212yes—
16C2×C8×D7C2212yes—
17D7×C4⋊C4—22no—
18D7×C4⋊C4C2212yes—
19#6453C2212yescomplemented
20#6452—22no—
21#6437C2211yescharacteristic
22C42×D7C2211yescharacteristic
23#6433C2211yescharacteristic, complemented
24C7×#102C2214yescomplemented
25C7×#97C2211yescharacteristic, complemented
26#6406C2214yescomplemented
27#6395C2214yescomplemented
28#6389C2212yescomplemented
29#6383C2214yescomplemented
30#6377C2211yescharacteristic, complemented
31C2×#490—71nomaximal, Sylow
32C23×D7—22no—
33C2×C7×D4—22no—
34C2×C7×D4C2312yes—
35C2×C22⋊D7—22no—
36C2×C22⋊D7—24no—
37D4×D7—28no—
38D4×D7—48no—
39C2×D28—22no—
40C2×D28C2312yes—
41C2×C4×D7—22no—
42C2×C4×D7D412yes—
43C2×C4×D7C2311yescharacteristic
44C2×C7×C8C2312yes—
45C7×C4⋊C4—22no—
46C7×C4⋊C4C2312yes—
47C42×C7C2311yescharacteristic
48C4⋊Dic7—22no—
49C4⋊Dic7C2312yes—
50Dic7⋊C4—22no—
51Dic7⋊C4—22no—
52Dic7⋊C4—22no—
53C4×Dic7D412yes—
54C4×Dic7C2311yescharacteristic
55C2×C7⋊C8C2312yes—
56C8×D7—24no—
57C2×#127—71no—
58C2×#125—71no—
59#490—71no—
60#490—71no—
61#490—74no—
62#490—71no—
63#490D711yescharacteristic, direct product factor
64C2×#102—74no—
65C2×#97—71no—
66C23×C7—22no—
67C22×D7—22no—
68C22×D7—44no—
69C22×D7—48no—
70C22×D7C2×D411yescharacteristic
71C7×D4—24no—
72C7×D4—44no—
73C2×C4×C7—22no—
74C2×C4×C7C2×D412yes—
75C2×C4×C7C2411yescharacteristic
76C22⋊D7—48no—
77C22⋊D7—44no—
78C2×Dic7—22no—
79C2×Dic7C2×D412yes—
80C2×Dic7C2×D411yescharacteristic
81D28—24no—
82D28—44no—
83C4×D7—24no—
84C4×D7—44no—
85C4×D7#4114yes—
86C7×C8—22no—
87C7⋊C8—22no—
88C22×D4—72no—
89C22×D4—142no—
90C22×C8—72no—
91#127—72no—
92#127—71no—
93#127—142no—
94#127C2×D711yescharacteristic
95#125—72no—
96#125—71no—
97#125—142no—
98#125C2×D711yescharacteristic, complemented
99C2×C4⋊C4—72no—
100C2×C4⋊C4—142no—
101C2×C42—71no—
102#102—74no—
103#102—74no—
104#102—74no—
105#102C2×D714yescomplemented
106#97—72no—
107#97—71no—
108#97C2×D711yescharacteristic, complemented
109C22×C7—42no—
110C22×C7—44no—
111C22×C7C22×D411yescharacteristic
112C2×D7—42no—
113C2×D7—44no—
114C2×D7—88no—
115C2×D7#12412yes—
116C2×D7AGL(1,ℤ/8)14yes—
117C4×C7—22no—
118C4×C7—42no—
119C4×C7C2×#4112yes—
120Dic7—22no—
121Dic7—42no—
122Dic7#12412yes—
123C24—142no—
124C2×D4—22no—
125C2×D4—72no—
126C2×D4—148no—
127C2×D4—144no—
128C2×D4—142no—
129C2×D4—142no—
130C2×D4—288no—
131C2×D4C22×D712yes—
132C22×C4—71no—
133C22×C4—72no—
134C22×C4—142no—
135C2×C8—72no—
136C2×C8—144no—
137C2×C8C22×D712yes—
138C4⋊C4—22no—
139C4⋊C4—72no—
140C4⋊C4—142no—
141C4⋊C4—142no—
142C4⋊C4—142no—
143C4⋊C4—142no—
144C4⋊C4C22×D712yes—
145C42—72no—
146C42—71no—
147C42C22×D711yescharacteristic
148C2×C7—82no—
149C2×C7C2×#12411yescharacteristic
150C2×C7C2×AGL(1,ℤ/8)12yes—
151D7—82no—
152D7#49014yesdirect product factor
153C23—22no—
154C23—71no—
155C23—142no—
156C23—284no—
157C23—288no—
158D4—24no—
159D4—44no—
160D4—144no—
161D4—284no—
162D4—284no—
163D4—288no—
164C2×C4—22no—
165C2×C4—71no—
166C2×C4—74no—
167C2×C4—72no—
168C2×C4—144no—
169C2×C4—142no—
170C2×C4—284no—
171C2×C4D4×D712yes—
172C2×C4C23×D711yescharacteristic
173C8—22no—
174C8—142no—
175C7C2×#49011yescharacteristic, complemented, Sylow
176C22—44no—
177C22—42no—
178C22—74no—
179C22—72no—
180C22—284no—
181C22—282no—
182C22—568no—
183C22C2×D4×D711yescharacteristic
184C4—22no—
185C4—42no—
186C4—72no—
187C4—142no—
188C4—282no—
189C4D7×#4112yes—
190C2—74no—
191C2—82no—
192C2—562no—
193C2D7×#12411yescharacteristic
194C2D7×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#490C2
socleC22×C7C22×D4
prime radicalD7×#127C2
hypercenter#490D7
hypocenterC7C2×#490
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. D7×#125C2semidirect
3. C7×#490C2semidirect
4. #89843C2semidirect
5. #89839C2semidirect
6. D7×#102C2semidirect
7. #89536C2semidirect
8. #89485C2semidirect
9. D7×#97C2semidirect
13. C7×#125C22semidirect
19. #6453C22semidirect
23. #6433C22semidirect
24. C7×#102C22semidirect
25. C7×#97C22semidirect
26. #6406C22semidirect
27. #6395C22semidirect
28. #6389C22semidirect
29. #6383C22semidirect
30. #6377C22semidirect
63. #490D7direct (N × H)
98. #125C2×D7semidirect
105. #102C2×D7semidirect
108. #97C2×D7semidirect
152. D7#490direct (N × H)
175. C7C2×#490semidirect

Maximal subgroups

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

subgroupindex
D7×#1272
D7×#1252
C7×#4902
#898432
#898392
D7×#1022
#895362
#894852
D7×#972
C2×#4907