Finite Group Databasebeta

← #31885323

Subgroups — #31885323

Order 18176.

150 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 25 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
1#22587926C211yesmaximal, characteristic
2#22586507C211yesmaximal, characteristic
3C71×#1409C211yesmaximal, characteristic
4#17986397C411yescharacteristic, complemented
5#17986395—21no—
6#17986371—21no—
7#17985798—21no—
8C8×C71⋊C8C411yescharacteristic, complemented
9#492×C71C2211yescharacteristic
10#353×C71—21no—
11C2×#11765853—42no—
12#14556128—21no—
13#14556060—41no—
14#14556059—42no—
15#14556047—41no—
16C4×C71⋊C8—21no—
17C2×SD16×C71—41no—
18C4⋊Q8×C71D411yescharacteristic
19#127×C71D411yescharacteristic
20#99×C71—41no—
21C4×C8×C71C2×C411yescharacteristic
22C2×Dic142—41no—
23#11765853—84no—
24#11765849—81no—
25C2×C71⋊C8—22no—
26C4⋊Dic71—41no—
27C4⋊Dic71—41no—
28C2×Q8×C71—21no—
29C2×D4×C71—21no—
30C2×D4×C71—41no—
31SD16×C71—82no—
32C2×C8×C71—21no—
33C4⋊C4×C71—41no—
34C22⋊C4×C71—81no—
35C42×C71C22⋊C411yescharacteristic
36C23×C71—41no—
37Q8×C71—81no—
38D4×C71—81no—
39D4×C71—82no—
40C2×C4×C71—21no—
41C2×C4×C71—41no—
42C2×C4×C71—161no—
43C2×C4×C71#9911yescharacteristic
44C2×Dic71—81no—
45Dic142—82no—
46C8×C71—22no—
47C71⋊C8—24no—
48C71⋊C8—161no—
49C22×C71—81no—
50C22×C71—161no—
51C22×C71#35311yescharacteristic
52C4×C71—22no—
53C4×C71—21no—
54C4×C71—81no—
55C4×C71—321no—
56Dic71—161no—
57#7535—711nomaximal, Sylow
58C2×C71—21no—
59C2×C71—161no—
60C2×C71#140911yescharacteristic
61#1705—711no—
62#1410—711no—
63#1409D7111yescharacteristic
64C71#753511yescharacteristic, complemented, Sylow
65#498—711no—
66#498—1421no—
67#492—1421no—
68#492C2×D7111yescharacteristic
69#356—1421no—
70#353—21no—
71C82—711no—
72C2×SD16—41no—
73C2×SD16—2842no—
74C4⋊Q8—1421no—
75C4⋊Q8D28411yescharacteristic
76#127C22⋊D7111yescharacteristic
77#106—2842no—
78#106—2841no—
79#101—2841no—
80#99—41no—
81C4×C8—1421no—
82C4×C8C4×D7111yescharacteristic
83C2×Q8—21no—
84C2×Q8—2841no—
85C2×D4—21no—
86C2×D4—41no—
87SD16—82no—
88SD16—5684no—
89M16—5681no—
90C2×C8—21no—
91C2×C8—1422no—
92C4⋊C4—41no—
93C4⋊C4—2841no—
94C4⋊C4—2841no—
95C22⋊C4—81no—
96C42(C2×D71)⋊C411yescharacteristic
97C23—41no—
98Q8—81no—
99Q8—5682no—
100D4—81no—
101D4—82no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—161no—
105C2×C4—5681no—
106C2×C4#1455604211yescharacteristic
107C8—22no—
108C8—11361no—
109C8—1424no—
110C22—81no—
111C22—161no—
112C22#1798579111yescharacteristic
113C4—22no—
114C4—21no—
115C4—81no—
116C4—321no—
117C4—11361no—
118C2—21no—
119C2—161no—
120C2#2258650411yescharacteristic
subgroupG/H
centerC2#22586504
commutatorC4×C8×C71C2×C4
Frattini#492C2×D71
FittingC71×#1409C2
socleC2×C71#1409
prime radical#127×C71D4
cosocle#492×C71C22
hypercenter#1409D71
hypocenterC71#7535
LCSG ▷ C4×C8×C71 ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1409

Complements

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

NH ≅ G/Ntype
4. #17986397C4semidirect
8. C8×C71⋊C8C4semidirect
64. C71#7535semidirect

Maximal subgroups

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

subgroupindex
#225879262
#225865072
C71×#14092
#753571