Finite Group Databasebeta

← #27213372

Subgroups — #27213372

Order 13504.

124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C2×C8×D211C211yesmaximal, characteristic, complemented
2#20269953C211yesmaximal, characteristic, complemented
3#20269948C211yesmaximal, characteristic
4#20269909C211yesmaximal, characteristic, complemented
5#20269901C211yesmaximal, characteristic, complemented
6#20269898C211yesmaximal, characteristic, complemented
7C2×SD16×C211C211yesmaximal, characteristic, complemented
8C2×C22⋊D211—21no—
9C2×C4×D211C2211yescharacteristic
10C2×C211⋊C8C2211yescharacteristic
11C8×D211—22no—
12C4⋊Dic211C2211yescharacteristic
13C4⋊Dic211C2211yescharacteristic
14Dic211⋊C4—21no—
15C22⋊Dic211—21no—
16(C2×D211)⋊C4—21no—
17C2×Q8×C211C2211yescharacteristic
18C2×D4×C211C2211yescharacteristic
19SD16×C211—22no—
20C2×C8×C211C2211yescharacteristic
21C22×D211D411yescharacteristic
22C22⋊D211—42no—
23C2×Dic211—21no—
24C2×Dic211—21no—
25C2×Dic211D411yescharacteristic
26C4×D211—21no—
27C4×D211—21no—
28C211⋊C8—21no—
29C23×C211—21no—
30Q8×C211—21no—
31Q8×C211—21no—
32D4×C211—21no—
33D4×C211—21no—
34C2×C4×C211—21no—
35C2×C4×C211C2311yescharacteristic
36C8×C211D412yescomplemented
37C22×C211—41no—
38C22×C211—41no—
39C22×C211—41no—
40C22×C211C2×D411yescharacteristic
41C2×D211—21no—
42C2×D211—21no—
43C2×D211SD1612yescomplemented
44C4×C211—41no—
45C4×C211C2×D411yescharacteristic
46C4×C211#4111yescharacteristic
47Dic211—21no—
48Dic211—41no—
49Dic211—41no—
50C2×C211—81no—
51C2×C211#12111yescharacteristic
52C2×C211C2×SD1611yescharacteristic
53C2×C211#13511yescharacteristic
54D211—22no—
55C211#46511yescharacteristic, complemented, Sylow
56#465—2111nomaximal, Sylow
57C2×SD16D21111yescharacteristic, complemented
58C22×C8—2111no—
59C22⋊Q8—2111no—
60#121—2111no—
61#106—2111no—
62#103—2111no—
63#102—2111no—
64C2×Q8C2×D21111yescharacteristic
65C2×D4—4221no—
66C2×D4C2×D21111yescharacteristic
67C22×C4—2111no—
68SD16—22no—
69C2×C8—4222no—
70C2×C8—2111no—
71C2×C8C2×D21111yescharacteristic
72C4⋊C4—4221no—
73C4⋊C4—2111no—
74C4⋊C4—2111no—
75C22⋊C4—4221no—
76C22⋊C4—4221no—
77C23—21no—
78C23—2111no—
79Q8—21no—
80Q8—21no—
81D4—21no—
82D4—21no—
83D4—8442no—
84C2×C4—21no—
85C2×C4—4221no—
86C2×C4—4221no—
87C2×C4—4221no—
88C2×C4—4221no—
89C2×C4—2111no—
90C2×C4C22×D21111yescharacteristic
91C8—4221no—
92C8C22⋊D21112yescomplemented
93C22—41no—
94C22—41no—
95C22—41no—
96C22—4221no—
97C22—4221no—
98C22—2112no—
99C22D4×D21111yescharacteristic
100C4—41no—
101C4—8441no—
102C4—8441no—
103C4—4221no—
104C4C2×C22⋊D21111yescharacteristic
105C4#1680996111yescharacteristic
106C2—81no—
107C2—4222no—
108C2#2026994811yescharacteristic
109C2SD16×D21111yescharacteristic
110C2#2026999811yescharacteristic
subgroupG/H
centerC22D4×D211
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
FittingC2×SD16×C211C2
socleC22×C211C2×D4
prime radical#20269948C2
hypercenterC2×SD16D211
hypocenterC211#465
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C8×D211C2semidirect
2. #20269953C2semidirect
4. #20269909C2semidirect
5. #20269901C2semidirect
6. #20269898C2semidirect
7. C2×SD16×C211C2semidirect
36. C8×C211D4semidirect
43. C2×D211SD16semidirect
55. C211#465semidirect
57. C2×SD16D211semidirect
92. C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×C8×D2112
#202699532
#202699482
#202699092
#202699012
#202698982
C2×SD16×C2112
#465211