Finite Group Databasebeta

← #30743705

Subgroups — #30743705

Order 17152.

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

normal subgroups only

This group has 138 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#22559808C211yesmaximal, characteristic
2#22558376C211yesmaximal, characteristic
3C67×#1409C211yesmaximal, characteristic
4#16863794C411yescharacteristic, complemented
5#16863748—21no—
6#16863742—21no—
7#16863195—21no—
8C8×C67⋊C8C411yescharacteristic, complemented
9#492×C67C2211yescharacteristic
10#353×C67—21no—
11#14524043—21no—
12#14523975—41no—
13#14523966—42no—
14#14523964—42no—
15#14523962—41no—
16C4×C67⋊C8—21no—
17C2×SD16×C67—41no—
18C4⋊Q8×C67D411yescharacteristic
19#127×C67D411yescharacteristic
20#99×C67—41no—
21C4×C8×C67C2×C411yescharacteristic
22C2×Dic134—41no—
23#11761612—81no—
24C2×C67⋊C8—22no—
25C4⋊Dic67—41no—
26C4⋊Dic67—41no—
27C2×Q8×C67—21no—
28C2×D4×C67—21no—
29C2×D4×C67—41no—
30SD16×C67—82no—
31C2×C8×C67—21no—
32C4⋊C4×C67—41no—
33C22⋊C4×C67—81no—
34C42×C67C22⋊C411yescharacteristic
35C23×C67—41no—
36Q8×C67—81no—
37D4×C67—81no—
38D4×C67—82no—
39C2×C4×C67—21no—
40C2×C4×C67—41no—
41C2×C4×C67—161no—
42C2×C4×C67#9911yescharacteristic
43C2×Dic67—81no—
44Dic134—82no—
45C8×C67—22no—
46C67⋊C8—42no—
47C67⋊C8—161no—
48C22×C67—81no—
49C22×C67—161no—
50C22×C67#35311yescharacteristic
51C4×C67—22no—
52C4×C67—21no—
53C4×C67—81no—
54C4×C67—321no—
55Dic67—161no—
56#7536—671nomaximal, Sylow
57C2×C67—21no—
58C2×C67—161no—
59C2×C67#140911yescharacteristic
60#1708—671no—
61#1410—671no—
62#1409D6711yescharacteristic
63C67#753611yescharacteristic, complemented, Sylow
64#498—671no—
65#492C2×D6711yescharacteristic
66#487—1341no—
67#486—1341no—
68#356—1341no—
69#353—21no—
70C82—671no—
71C2×SD16—41no—
72C4⋊Q8—1341no—
73C4⋊Q8D26811yescharacteristic
74#127C22⋊D6711yescharacteristic
75#106—2681no—
76#103—2682no—
77#102—2682no—
78#101—2681no—
79#99—41no—
80C4×C8—1341no—
81C4×C8C4×D6711yescharacteristic
82C2×Q8—21no—
83C2×Q8—2681no—
84C2×D4—21no—
85C2×D4—41no—
86SD16—82no—
87M16—5361no—
88C2×C8—21no—
89C2×C8—1342no—
90C4⋊C4—41no—
91C4⋊C4—2681no—
92C4⋊C4—2681no—
93C22⋊C4—81no—
94C42(C2×D67)⋊C411yescharacteristic
95C23—41no—
96Q8—81no—
97Q8—5362no—
98D4—81no—
99D4—82no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—161no—
103C2×C4—5361no—
104C2×C4#1452395711yescharacteristic
105C8—22no—
106C8—2682no—
107C8—10721no—
108C22—81no—
109C22—161no—
110C22#1686318811yescharacteristic
111C4—22no—
112C4—21no—
113C4—81no—
114C4—321no—
115C4—10721no—
116C2—21no—
117C2—161no—
118C2#2255837311yescharacteristic
subgroupG/H
centerC2#22558373
commutatorC4×C8×C67C2×C4
Frattini#492C2×D67
FittingC67×#1409C2
socleC2×C67#1409
prime radical#127×C67D4
cosocle#492×C67C22
hypercenter#1409D67
hypocenterC67#7536
LCSG ▷ C4×C8×C67 ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1409

Complements

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

NH ≅ G/Ntype
4. #16863794C4semidirect
8. C8×C67⋊C8C4semidirect
63. C67#7536semidirect

Maximal subgroups

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

subgroupindex
#225598082
#225583762
C67×#14092
#753667