Finite Group Databasebeta

← #421×C373 #37558975

Subgroups — #421×C373 #37558975

Order 23872.

154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 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×#41×C373C211yesmaximal, characteristic
2C2×M16×C373C211yesmaximal, characteristic, complemented
3#117×C373C211yesmaximal, characteristic, complemented
4#104×C373C214yesmaximal, complemented
5#41×C373—21no—
6#41×C373—21no—
7#41×C373C412yescomplemented
8#41×C373C2212yes—
9C2×Q8×C373C411yescharacteristic, complemented
10C2×D4×C373—21no—
11C2×D4×C373C411yescharacteristic, complemented
12C22×C4×C373—21no—
13C22×C4×C373C2211yescharacteristic
14M16×C373—21no—
15M16×C373C2212yescomplemented
16C2×C8×C373—21no—
17C4⋊C4×C373—21no—
18C22⋊C4×C373—21no—
19C42×C373C2212yescomplemented
20C23×C373—21no—
21C23×C373D411yescharacteristic
22Q8×C373—21no—
23Q8×C373C2×C412yes—
24D4×C373—22no—
25D4×C373—22no—
26D4×C373—21no—
27D4×C373C2×C412yes—
28C2×C4×C373—21no—
29C2×C4×C373—22no—
30C2×C4×C373—22no—
31C2×C4×C373—22no—
32C2×C4×C373C2×C411yescharacteristic
33C2×C4×C373C2×C411yescharacteristic
34C2×C4×C373D411yescharacteristic
35C2×C4×C373D411yescharacteristic
36C2×C4×C373D411yescharacteristic
37C2×C4×C373C2311yescharacteristic
38C8×C373—22no—
39C22×C373—22no—
40C22×C373—41no—
41C22×C373—42no—
42C22×C373C22⋊C411yescharacteristic
43C22×C373C22⋊C411yescharacteristic
44C22×C373C2×D411yescharacteristic
45C4×C373—22no—
46C4×C373—42no—
47C4×C373C22⋊C411yescharacteristic
48C4×C373C22⋊C411yescharacteristic
49C4×C373C22×C411yescharacteristic
50C4×C373C2×D411yescharacteristic
51C2×C373—21no—
52C2×C373—21no—
53C2×C373—21no—
54C2×C373—42no—
55C2×C373C2×C22⋊C411yescharacteristic
56C373#42111yescharacteristic, direct product factor, Sylow
57#421C37311yesmaximal, characteristic, direct product factor, Sylow
58C2×#41C2×C37311yescharacteristic
59C2×M16C2×C37311yescharacteristic, complemented
60#117C2×C37311yescharacteristic, complemented
61#104C2×C37314yescomplemented
62#41—21no—
63#41—21no—
64#41C4×C37312yescomplemented
65#41C22×C37312yes—
66C2×Q8C4×C37311yescharacteristic, complemented
67C2×D4—21no—
68C2×D4C4×C37311yescharacteristic, complemented
69C22×C4—21no—
70C22×C4C22×C37311yescharacteristic
71M16—21no—
72M16C22×C37312yescomplemented
73C2×C8—21no—
74C4⋊C4—21no—
75C22⋊C4—21no—
76C42C22×C37312yescomplemented
77C23—21no—
78C23D4×C37311yescharacteristic
79Q8—21no—
80Q8C2×C4×C37312yes—
81D4—22no—
82D4—22no—
83D4—21no—
84D4C2×C4×C37312yes—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4C2×C4×C37311yescharacteristic
90C2×C4C2×C4×C37311yescharacteristic
91C2×C4D4×C37311yescharacteristic
92C2×C4D4×C37311yescharacteristic
93C2×C4D4×C37311yescharacteristic
94C2×C4C23×C37311yescharacteristic
95C8—22no—
96C22—22no—
97C22—41no—
98C22—42no—
99C22C22⋊C4×C37311yescharacteristic
100C22C22⋊C4×C37311yescharacteristic
101C22C2×D4×C37311yescharacteristic
102C4—22no—
103C4—42no—
104C4C22⋊C4×C37311yescharacteristic
105C4C22⋊C4×C37311yescharacteristic
106C4C22×C4×C37311yescharacteristic
107C4C2×D4×C37311yescharacteristic
108C2—21no—
109C2—21no—
110C2—21no—
111C2—42no—
112C2C2×C22⋊C4×C37311yescharacteristic
subgroupG/H
centerC4×C373C2×D4
commutatorC4C22×C4×C373
Frattini, cosocleC2×C4C23×C373
socleC2×C373C2×C22⋊C4
prime radicalC2×#41×C373C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C373 ◁ C22×C4×C373 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×M16×C373C2semidirect
3. #117×C373C2semidirect
4. #104×C373C2semidirect
7. #41×C373C4semidirect
9. C2×Q8×C373C4semidirect
11. C2×D4×C373C4semidirect
15. M16×C373C22semidirect
19. C42×C373C22semidirect
56. C373#421direct (N × H)
57. #421C373direct (N × H)
59. C2×M16C2×C373semidirect
60. #117C2×C373semidirect
61. #104C2×C373semidirect
64. #41C4×C373semidirect
66. C2×Q8C4×C373semidirect
68. C2×D4C4×C373semidirect
72. M16C22×C373semidirect
76. C42C22×C373semidirect

Maximal subgroups

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

subgroupindex
C2×#41×C3732
C2×M16×C3732
#117×C3732
#104×C3732
#421373