Finite Group Databasebeta

← #30743706

Subgroups — #30743706

Order 17152.

138 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 27 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#22558376C211yesmaximal, characteristic
2#22558374C211yesmaximal, characteristic
3C67×#1708C211yesmaximal, characteristic
4#16863195—21no—
5#16863189—21no—
6#498×C67C411yescharacteristic, complemented
7#492×C67C2211yescharacteristic
8#487×C67—21no—
9#486×C67—21no—
10C82×C67C411yescharacteristic, complemented
11#14523962—41no—
12#14523958—41no—
13C2×SD16×C67—41no—
14C4⋊Q8×C67—21no—
15C4⋊Q8×C67D411yescharacteristic
16#127×C67D411yescharacteristic
17#106×C67—41no—
18#103×C67—42no—
19#102×C67—42no—
20C4×C8×C67—21no—
21C4×C8×C67C2×C411yescharacteristic
22#11761612—81no—
23C22⋊Dic67—81no—
24C2×Q8×C67—21no—
25C2×Q8×C67—41no—
26C2×D4×C67—21no—
27C2×D4×C67—41no—
28SD16×C67—82no—
29C2×C8×C67—21no—
30C2×C8×C67—22no—
31C4⋊C4×C67—41no—
32C4⋊C4×C67—41no—
33C4⋊C4×C67—41no—
34C42×C67C22⋊C411yescharacteristic
35C23×C67—41no—
36Q8×C67—81no—
37Q8×C67—82no—
38D4×C67—81no—
39D4×C67—82no—
40C2×C4×C67—21no—
41C2×C4×C67—41no—
42C2×C4×C67—81no—
43C2×C4×C67#9911yescharacteristic
44C2×Dic67—161no—
45C8×C67—22no—
46C8×C67—42no—
47C67⋊C8—161no—
48C22×C67—81no—
49C22×C67—161no—
50C22×C67#35311yescharacteristic
51C4×C67—22no—
52C4×C67—21no—
53C4×C67—81no—
54C4×C67—161no—
55Dic67—321no—
56#7536—671nomaximal, Sylow
57C2×C67—21no—
58C2×C67—161no—
59C2×C67#140911yescharacteristic
60#1708D6711yescharacteristic
61#1410—671no—
62#1409—671no—
63C67#753611yescharacteristic, complemented, Sylow
64#498Dic6711yescharacteristic, complemented
65#492C2×D6711yescharacteristic
66#487—21no—
67#486—21no—
68#356—1341no—
69#353—1341no—
70C82Dic6711yescharacteristic, complemented
71C2×SD16—41no—
72C4⋊Q8—21no—
73C4⋊Q8C22⋊D6711yescharacteristic
74#127C22⋊D6711yescharacteristic
75#106—41no—
76#103—42no—
77#102—42no—
78#101—2681no—
79#99—2681no—
80C4×C8—21no—
81C4×C8C2×Dic6711yescharacteristic
82C2×Q8—21no—
83C2×Q8—41no—
84C2×D4—21no—
85C2×D4—41no—
86SD16—82no—
87M16—5361no—
88C2×C8—21no—
89C2×C8—22no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C22⋊C4—5361no—
94C42C22⋊Dic6711yescharacteristic
95C23—41no—
96Q8—81no—
97Q8—82no—
98D4—81no—
99D4—82no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—81no—
103C2×C4—10721no—
104C2×C4#1452395811yescharacteristic
105C8—22no—
106C8—42no—
107C8—10721no—
108C22—81no—
109C22—161no—
110C22#1686318911yescharacteristic
111C4—22no—
112C4—21no—
113C4—81no—
114C4—161no—
115C4—21441no—
116C2—21no—
117C2—161no—
118C2#2255837411yescharacteristic
subgroupG/H
centerC2#22558374
commutatorC4×C8×C67C2×C4
Frattini#492C2×D67
FittingC67×#1708C2
socleC2×C67#1409
prime radical#127×C67D4
cosocle#492×C67C22
hypercenter#1708D67
hypocenterC67#7536
LCSG ▷ C4×C8×C67 ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1708

Complements

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

NH ≅ G/Ntype
6. #498×C67C4semidirect
10. C82×C67C4semidirect
63. C67#7536semidirect
64. #498Dic67semidirect
70. C82Dic67semidirect

Maximal subgroups

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

subgroupindex
#225583762
#225583742
C67×#17082
#753667