Finite Group Databasebeta

← #30743704

Subgroups — #30743704

Order 17152.

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#22559795C211yesmaximal, characteristic
2#22558374C211yesmaximal, characteristic
3C67×#1410C211yesmaximal, characteristic
4#16863794C411yescharacteristic, complemented
5#16863792—21no—
6#16863768—21no—
7#16863189—21no—
8C8×C67⋊C8C411yescharacteristic, complemented
9#492×C67C2211yescharacteristic
10#356×C67—21no—
11C2×#11761616—42no—
12#14524043—21no—
13#14523975—41no—
14#14523974—42no—
15#14523958—41no—
16C4×C67⋊C8—21no—
17C2×SD16×C67—41no—
18C4⋊Q8×C67D411yescharacteristic
19#127×C67D411yescharacteristic
20#101×C67—41no—
21C4×C8×C67C2×C411yescharacteristic
22C2×Dic134—41no—
23#11761616—84no—
24C2×C67⋊C8—22no—
25C4⋊Dic67—41no—
26C4⋊Dic67—41no—
27C22⋊Dic67—81no—
28C2×Q8×C67—21no—
29C2×D4×C67—21no—
30C2×D4×C67—41no—
31SD16×C67—82no—
32M16×C67—81no—
33C2×C8×C67—21no—
34C4⋊C4×C67—41no—
35C42×C67C22⋊C411yescharacteristic
36C23×C67—41no—
37Q8×C67—81no—
38D4×C67—81no—
39D4×C67—82no—
40C2×C4×C67—21no—
41C2×C4×C67—41no—
42C2×C4×C67#9911yescharacteristic
43C2×Dic67—81no—
44C2×Dic67—161no—
45Dic134—82no—
46C8×C67—22no—
47C8×C67—161no—
48C67⋊C8—24no—
49C22×C67—81no—
50C22×C67—161no—
51C22×C67#35311yescharacteristic
52C4×C67—22no—
53C4×C67—21no—
54C4×C67—81no—
55Dic67—161no—
56Dic67—321no—
57#7535—671nomaximal, Sylow
58C2×C67—21no—
59C2×C67—161no—
60C2×C67#140911yescharacteristic
61#1705—671no—
62#1410D6711yescharacteristic
63#1409—671no—
64C67#753511yescharacteristic, complemented, Sylow
65#498—671no—
66#498—1341no—
67#492—1341no—
68#492C2×D6711yescharacteristic
69#356—21no—
70#353—1341no—
71C82—671no—
72C2×SD16—41no—
73C2×SD16—2682no—
74C4⋊Q8—1341no—
75C4⋊Q8C22⋊D6711yescharacteristic
76#127D26811yescharacteristic
77#106—2682no—
78#106—2681no—
79#101—41no—
80#99—2681no—
81C4×C8—1341no—
82C4×C8C4×D6711yescharacteristic
83C2×Q8—21no—
84C2×Q8—2681no—
85C2×D4—21no—
86C2×D4—41no—
87SD16—82no—
88SD16—5364no—
89M16—81no—
90C2×C8—21no—
91C2×C8—1342no—
92C4⋊C4—41no—
93C4⋊C4—2681no—
94C4⋊C4—2681no—
95C22⋊C4—5361no—
96C42(C2×D67)⋊C411yescharacteristic
97C23—41no—
98Q8—81no—
99Q8—5362no—
100D4—81no—
101D4—82no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—5361no—
105C2×C4—10721no—
106C2×C4#1452395711yescharacteristic
107C8—22no—
108C8—161no—
109C8—1344no—
110C22—81no—
111C22—161no—
112C22#1686318811yescharacteristic
113C4—22no—
114C4—21no—
115C4—81no—
116C4—10721no—
117C4—21441no—
118C2—21no—
119C2—161no—
120C2#2255837311yescharacteristic
subgroupG/H
centerC2#22558373
commutatorC4×C8×C67C2×C4
Frattini#492C2×D67
FittingC67×#1410C2
socleC2×C67#1409
prime radical#127×C67D4
cosocle#492×C67C22
hypercenter#1410D67
hypocenterC67#7535
LCSG ▷ C4×C8×C67 ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1410

Complements

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

NH ≅ G/Ntype
4. #16863794C4semidirect
8. C8×C67⋊C8C4semidirect
64. C67#7535semidirect

Maximal subgroups

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

subgroupindex
#225597952
#225583742
C67×#14102
#753567