Finite Group Databasebeta

← D7×#104 #89592

Subgroups — D7×#104 #89592

Order 448.

170 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 53 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×#41C211yesmaximal, characteristic
2D7×M16C211yesmaximal, characteristic, complemented
3C42×D7C211yesmaximal, characteristic, complemented
4C7×#104C211yesmaximal, characteristic, complemented
5#6411C211yesmaximal, characteristic, complemented
6#6399C211yesmaximal, characteristic, complemented
7#6379C211yesmaximal, characteristic, complemented
8C7×#41C2211yescharacteristic
9#1161C411yescharacteristic, complemented
10Q8×D7C411yescharacteristic, complemented
11#1159—21no—
12#1159C411yescharacteristic, complemented
13D4×D7—21no—
14D4×D7C411yescharacteristic, complemented
15#1157—21no—
16#1157C2211yescharacteristic
17C2×C4×D7—21no—
18C2×C4×D7—21no—
19C2×C4×D7C2211yescharacteristic
20C7×M16C2211yescharacteristic, complemented
21C42×C7C2211yescharacteristic, complemented
22C4×Dic7—21no—
23C4×Dic7C2211yescharacteristic, complemented
24#1136C2211yescharacteristic, complemented
25#1131—21no—
26C8×D7—21no—
27C2×#104—71nomaximal, Sylow
28C22×D7—21no—
29C22×D7D411yescharacteristic
30C7×Q8C2×C411yescharacteristic, complemented
31C7×D4—21no—
32C7×D4C2×C411yescharacteristic, complemented
33C2×C4×C7—21no—
34C2×C4×C7—21no—
35C2×C4×C7C2311yescharacteristic
36C22⋊D7—21no—
37C22⋊D7—21no—
38C22⋊D7—21no—
39C2×Dic7—21no—
40C2×Dic7—21no—
41C2×Dic7D411yescharacteristic
42D28—21no—
43D28C2×C411yescharacteristic, complemented
44C4×D7—24no—
45C4×D7—21no—
46C4×D7—21no—
47C4×D7—21no—
48C4×D7C2×C411yescharacteristic
49C4×D7C2×C411yescharacteristic
50C4×D7D411yescharacteristic
51C4×D7D411yescharacteristic
52Dic14—21no—
53Dic14C2×C411yescharacteristic, complemented
54C7×C8—21no—
55C7⋊C8—21no—
56C2×#41—71no—
57C2×M16—71no—
58C2×C42—71no—
59#104—71no—
60#104—71no—
61#104—71no—
62#104D711yescharacteristic, direct product factor
63C22×C7—21no—
64C22×C7C2×D411yescharacteristic
65C2×D7—22no—
66C2×D7—21no—
67C2×D7—42no—
68C2×D7C22⋊C411yescharacteristic
69C2×D7C22⋊C411yescharacteristic
70C4×C7—22no—
71C4×C7—21no—
72C4×C7C22×C411yescharacteristic
73C4×C7C2×D411yescharacteristic
74Dic7—22no—
75Dic7—21no—
76Dic7C22⋊C411yescharacteristic
77Dic7C22⋊C411yescharacteristic
78#41—71no—
79#41—71no—
80#41—71no—
81#41—141no—
82#41—141no—
83#41C2×D711yescharacteristic
84C2×Q8—71no—
85C2×D4—71no—
86C2×D4—141no—
87C22×C4—71no—
88C22×C4—141no—
89C22×C4—141no—
90M16—71no—
91M16—141no—
92M16C2×D711yescharacteristic, complemented
93C2×C8—141no—
94C42—71no—
95C42—141no—
96C42C2×D711yescharacteristic, complemented
97C2×C7—21no—
98C2×C7—41no—
99C2×C7C2×C22⋊C411yescharacteristic
100D7—21no—
101D7—41no—
102D7#10412yesdirect product factor
103C23—71no—
104C23—141no—
105Q8—71no—
106Q8—141no—
107Q8C4×D711yescharacteristic, complemented
108D4—21no—
109D4—71no—
110D4—141no—
111D4—141no—
112D4—141no—
113D4—141no—
114D4C4×D711yescharacteristic, complemented
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—71no—
118C2×C4—71no—
119C2×C4—71no—
120C2×C4—71no—
121C2×C4—71no—
122C2×C4—141no—
123C2×C4—141no—
124C2×C4—144no—
125C2×C4—141no—
126C2×C4—141no—
127C2×C4—141no—
128C2×C4C22×D711yescharacteristic
129C8—21no—
130C8—141no—
131C7C2×#10411yescharacteristic, complemented, Sylow
132C22—21no—
133C22—71no—
134C22—71no—
135C22—142no—
136C22—141no—
137C22—282no—
138C22D4×D711yescharacteristic
139C4—22no—
140C4—21no—
141C4—71no—
142C4—71no—
143C4—142no—
144C4—141no—
145C4C2×C4×D711yescharacteristic
146C4D4×D711yescharacteristic
147C2—21no—
148C2—41no—
149C2—72no—
150C2—141no—
151C2—281no—
152C2D7×C22⋊C411yescharacteristic
subgroupG/H
centerC4D4×D7
commutatorC4×C7C22×C4
FrattiniC2×C4C22×D7
FittingC7×#104C2
socleC2×C7C2×C22⋊C4
prime radicalD7×#41C2
cosocleC2×C4×C7C23
hypercenter#104D7
hypocenterC7C2×#104
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. D7×M16C2semidirect
3. C42×D7C2semidirect
4. C7×#104C2semidirect
5. #6411C2semidirect
6. #6399C2semidirect
7. #6379C2semidirect
9. #1161C4semidirect
10. Q8×D7C4semidirect
12. #1159C4semidirect
14. D4×D7C4semidirect
20. C7×M16C22semidirect
21. C42×C7C22semidirect
23. C4×Dic7C22semidirect
24. #1136C22semidirect
30. C7×Q8C2×C4semidirect
32. C7×D4C2×C4semidirect
43. D28C2×C4semidirect
53. Dic14C2×C4semidirect
62. #104D7direct (N × H)
92. M16C2×D7semidirect
96. C42C2×D7semidirect
102. D7#104direct (N × H)
107. Q8C4×D7semidirect
114. D4C4×D7semidirect
131. C7C2×#104semidirect

Maximal subgroups

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

subgroupindex
D7×#412
D7×M162
C42×D72
C7×#1042
#64112
#63992
#63792
C2×#1047