Finite Group Databasebeta

← #89593

Subgroups — #89593

Order 448.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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
3#6435C211yesmaximal, 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×D7C2211yescharacteristic
19C7×M16C2211yescharacteristic, complemented
20C42×C7C2211yescharacteristic, complemented
21(C2×D7)⋊C4—21no—
22Dic7⋊C4—21no—
23C4×Dic7C2211yescharacteristic, complemented
24#1136C2211yescharacteristic, complemented
25#1131—21no—
26C8×D7—21no—
27#421—71nomaximal, Sylow
28C22×D7—21no—
29C22×D7D411yescharacteristic
30C7×Q8C2×C411yescharacteristic
31C7×D4—21no—
32C7×D4C2×C411yescharacteristic
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
44C4×D7—21no—
45C4×D7—21no—
46C4×D7—21no—
47C4×D7C2×C411yescharacteristic
48C4×D7C2×C411yescharacteristic
49C4×D7D411yescharacteristic
50C4×D7D411yescharacteristic
51Dic14—21no—
52Dic14C2×C411yescharacteristic
53C7×C8—21no—
54C7⋊C8—21no—
55C2×#41—71no—
56C2×M16—71no—
57#117—71no—
58#104—71no—
59#104—71no—
60#104—71no—
61#104D711yescharacteristic, complemented
62C22×C7—21no—
63C22×C7C2×D411yescharacteristic
64C2×D7—21no—
65C2×D7—41no—
66C2×D7—42no—
67C2×D7C22⋊C411yescharacteristic
68C2×D7C22⋊C411yescharacteristic
69C4×C7—21no—
70C4×C7—41no—
71C4×C7C22×C411yescharacteristic
72C4×C7C2×D411yescharacteristic
73Dic7—21no—
74Dic7—41no—
75Dic7C22⋊C411yescharacteristic
76Dic7C22⋊C411yescharacteristic
77#41—71no—
78#41—71no—
79#41—71no—
80#41—141no—
81#41—141no—
82#41C2×D711yescharacteristic
83C2×Q8—71no—
84C2×D4—71no—
85C2×D4—141no—
86C22×C4—71no—
87C22×C4—141no—
88M16—71no—
89M16—141no—
90M16C2×D711yescharacteristic, complemented
91C2×C8—141no—
92C4⋊C4—141no—
93C22⋊C4—141no—
94C42—71no—
95C42C2×D711yescharacteristic, complemented
96C2×C7—21no—
97C2×C7—41no—
98C2×C7C2×C22⋊C411yescharacteristic
99D7—21no—
100D7—21no—
101D7—41no—
102C23—71no—
103C23—141no—
104Q8—71no—
105Q8—141no—
106Q8C4×D711yescharacteristic
107D4—21no—
108D4—71no—
109D4—141no—
110D4—141no—
111D4—141no—
112D4—141no—
113D4C4×D711yescharacteristic
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—71no—
117C2×C4—71no—
118C2×C4—71no—
119C2×C4—71no—
120C2×C4—71no—
121C2×C4—141no—
122C2×C4—141no—
123C2×C4—141no—
124C2×C4—141no—
125C2×C4—141no—
126C2×C4C22×D711yescharacteristic
127C8—21no—
128C8—141no—
129C7#42111yescharacteristic, complemented, Sylow
130C22—21no—
131C22—71no—
132C22—71no—
133C22—141no—
134C22—281no—
135C22—282no—
136C22D4×D711yescharacteristic
137C4—21no—
138C4—41no—
139C4—71no—
140C4—71no—
141C4—141no—
142C4—281no—
143C4C2×C4×D711yescharacteristic
144C4D4×D711yescharacteristic
145C2—21no—
146C2—41no—
147C2—141no—
148C2—141no—
149C2—281no—
150C2D7×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
hypocenterC7#421
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. #6435C2semidirect
4. C7×#104C2semidirect
5. #6411C2semidirect
6. #6399C2semidirect
7. #6379C2semidirect
9. #1161C4semidirect
10. Q8×D7C4semidirect
12. #1159C4semidirect
14. D4×D7C4semidirect
19. C7×M16C22semidirect
20. C42×C7C22semidirect
23. C4×Dic7C22semidirect
24. #1136C22semidirect
61. #104D7semidirect
90. M16C2×D7semidirect
95. C42C2×D7semidirect
129. C7#421semidirect

Maximal subgroups

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

subgroupindex
D7×#412
D7×M162
#64352
C7×#1042
#64112
#63992
#63792
#4217