Finite Group Databasebeta

← #1392×C211 #41009012

Subgroups — #1392×C211 #41009012

Order 27008.

158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 60 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#413×C211C212yesmaximal
2#408×C211C211yesmaximal, characteristic
3C2×#41×C211C2211yescharacteristic
4C2×M16×C211—22no—
5C2×M16×C211C411yescharacteristic
6C22×C8×C211C411yescharacteristic
7#101×C211—22no—
8#100×C211C414yes—
9C22⋊C8×C211—22no—
10#41×C211—41no—
11#41×C211—41no—
12C2×Q8×C211D411yescharacteristic
13C2×D4×C211C2×C412yes—
14C2×D4×C211Q811yescharacteristic
15C22×C4×C211C2×C411yescharacteristic
16C22×C4×C211D412yes—
17M16×C211—24no—
18M16×C211—42no—
19M16×C211—41no—
20C2×C8×C211—21no—
21C2×C8×C211—21no—
22C2×C8×C211—42no—
23C2×C8×C211—41no—
24C23×C211C4211yescharacteristic
25C23×C211C4⋊C412yes—
26Q8×C211—41no—
27D4×C211—42no—
28D4×C211—41no—
29C2×C4×C211—21no—
30C2×C4×C211—22no—
31C2×C4×C211—21no—
32C2×C4×C211—22no—
33C2×C4×C211C22⋊C412yes—
34C2×C4×C211C22⋊C411yescharacteristic
35C2×C4×C211C4⋊C411yescharacteristic
36C8×C211—44no—
37C8×C211—41no—
38C8×C211—41no—
39C22×C211—21no—
40C22×C211—22no—
41C22×C211—41no—
42C22×C211—42no—
43C22×C211#9511yescharacteristic
44C4×C211—22no—
45C4×C211—21no—
46C4×C211#9911yescharacteristic
47C4×C211#9911yescharacteristic
48C2×C211—21no—
49C2×C211—41no—
50C2×C211—42no—
51C2×C211#34211yescharacteristic
52C211#139211yescharacteristic, direct product factor, Sylow
53#1392C21111yesmaximal, characteristic, direct product factor, Sylow
54#413C2×C21112yes—
55#408C2×C21111yescharacteristic
56C2×#41C22×C21111yescharacteristic
57C2×M16—22no—
58C2×M16C4×C21111yescharacteristic
59C22×C8C4×C21111yescharacteristic
60#101—22no—
61#100C4×C21114yes—
62C22⋊C8—22no—
63#41—41no—
64#41—41no—
65C2×Q8D4×C21111yescharacteristic
66C2×D4C2×C4×C21112yes—
67C2×D4Q8×C21111yescharacteristic
68C22×C4C2×C4×C21111yescharacteristic
69C22×C4D4×C21112yes—
70M16—24no—
71M16—42no—
72M16—41no—
73C2×C8—21no—
74C2×C8—21no—
75C2×C8—42no—
76C2×C8—41no—
77C23C42×C21111yescharacteristic
78C23C4⋊C4×C21112yes—
79Q8—41no—
80D4—42no—
81D4—41no—
82C2×C4—21no—
83C2×C4—22no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4C22⋊C4×C21112yes—
87C2×C4C22⋊C4×C21111yescharacteristic
88C2×C4C4⋊C4×C21111yescharacteristic
89C8—44no—
90C8—41no—
91C8—41no—
92C22—21no—
93C22—22no—
94C22—41no—
95C22—42no—
96C22#95×C21111yescharacteristic
97C4—22no—
98C4—21no—
99C4#99×C21111yescharacteristic
100C4#99×C21111yescharacteristic
101C2—21no—
102C2—41no—
103C2—42no—
104C2#342×C21111yescharacteristic
subgroupG/H
centerC4×C211#99
commutatorC23C42×C211
Frattini, cosocleC2×#41C22×C211
socleC2×C211#342
prime radicalC2×#41×C211C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C211 ◁ C2×C4×C211 ◁ C2×#41×C211 ◁ G

Complements

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

NH ≅ G/Ntype
52. C211#1392direct (N × H)
53. #1392C211direct (N × H)

Maximal subgroups

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

subgroupindex
#413×C2112
#408×C2112
#1392211