Finite Group Databasebeta

← C137×#7371 #46856757

Subgroups — C137×#7371 #46856757

Order 35072.

190 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 50 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#1844×C137C211yesmaximal, characteristic
2#1325×C137C212yesmaximal
3C2×#106×C137—22no—
4C2×C4×C8×C137C411yescharacteristic
5#384×C137C411yescharacteristic
6#384×C137C2211yescharacteristic
7#324×C137—22no—
8C22×C8×C137—21no—
9C2×C4⋊C4×C137—21no—
10C2×C4⋊C4×C137—41no—
11C2×C4⋊C4×C137—41no—
12C2×C4⋊C4×C137D412yes—
13C2×C42×C137C2×C411yescharacteristic
14#106×C137—44no—
15C22⋊C8×C137—42no—
16C4×C8×C137—21no—
17C4×C8×C137C812yescomplemented
18C22×C4×C137—21no—
19C22×C4×C137—22no—
20C22×C4×C137—41no—
21C22×C4×C137C22⋊C411yescharacteristic
22C2×C8×C137—22no—
23C2×C8×C137—22no—
24C2×C8×C137—22no—
25C2×C8×C137—82no—
26C4⋊C4×C137—42no—
27C4⋊C4×C137—42no—
28C4⋊C4×C137—82no—
29C4⋊C4×C137—82no—
30C42×C137—21no—
31C42×C137C2×C811yescharacteristic
32C42×C137M1611yescharacteristic
33C23×C137#9911yescharacteristic
34C2×C4×C137—22no—
35C2×C4×C137—22no—
36C2×C4×C137—21no—
37C2×C4×C137—21no—
38C2×C4×C137—21no—
39C2×C4×C137—21no—
40C2×C4×C137—42no—
41C2×C4×C137—82no—
42C2×C4×C137—82no—
43C2×C4×C137—81no—
44C2×C4×C137C22⋊C811yescharacteristic
45C2×C4×C137#10111yescharacteristic
46C8×C137—24no—
47C8×C137—162no—
48C22×C137—21no—
49C22×C137—21no—
50C22×C137#32411yescharacteristic
51C22×C137#35311yescharacteristic
52C22×C137#35611yescharacteristic
53C4×C137—22no—
54C4×C137—22no—
55C4×C137—21no—
56C4×C137—21no—
57C4×C137—82no—
58C4×C137—161no—
59C2×C137—21no—
60C2×C137—21no—
61C2×C137#132511yescharacteristic
62C2×C137#140911yescharacteristic
63C2×C137#141011yescharacteristic
64#7371C13711yesmaximal, characteristic, direct product factor, Sylow
65C137#737111yescharacteristic, direct product factor, Sylow
66#1844C2×C13711yescharacteristic
67#1325C2×C13712yes—
68C2×#106—22no—
69C2×C4×C8C4×C13711yescharacteristic
70#384C4×C13711yescharacteristic
71#384C22×C13711yescharacteristic
72#324—22no—
73C22×C8—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4D4×C13712yes—
78C2×C42C2×C4×C13711yescharacteristic
79#106—44no—
80C22⋊C8—42no—
81C4×C8—21no—
82C4×C8C8×C13712yescomplemented
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—41no—
86C22×C4C22⋊C4×C13711yescharacteristic
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—82no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—82no—
94C4⋊C4—82no—
95C42—21no—
96C42C2×C8×C13711yescharacteristic
97C42M16×C13711yescharacteristic
98C23#99×C13711yescharacteristic
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—42no—
106C2×C4—82no—
107C2×C4—82no—
108C2×C4—81no—
109C2×C4C22⋊C8×C13711yescharacteristic
110C2×C4#101×C13711yescharacteristic
111C8—24no—
112C8—162no—
113C22—21no—
114C22—21no—
115C22#324×C13711yescharacteristic
116C22#353×C13711yescharacteristic
117C22#356×C13711yescharacteristic
118C4—22no—
119C4—22no—
120C4—21no—
121C4—21no—
122C4—82no—
123C4—161no—
124C2—21no—
125C2—21no—
126C2#1325×C13711yescharacteristic
127C2#1409×C13711yescharacteristic
128C2#1410×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#353
commutatorC42C2×C8×C137
Frattini, cosocle#384C22×C137
prime radicalC23×C137#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C22×C4×C137 ◁ #384×C137 ◁ G

Complements

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

NH ≅ G/Ntype
17. C4×C8×C137C8semidirect
64. #7371C137direct (N × H)
65. C137#7371direct (N × H)
82. C4×C8C8×C137semidirect

Maximal subgroups

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

subgroupindex
#1844×C1372
#1325×C1372
#7371137