Finite Group Databasebeta

← #36374333

Subgroups — #36374333

Order 22336.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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#26013684C211yesmaximal, characteristic
2#26013622C212yesmaximal, complemented
3#26013554C211yesmaximal, characteristic, complemented
4#26013553C212yesmaximal, complemented
5#123×C349C211yesmaximal, characteristic, complemented
6#19150233—21no—
7#19150232—22no—
8C2×C22⋊D349—21no—
9C2×C22⋊D349C2212yes—
10D4×D349—22no—
11#19150212—22no—
12C22⋊Dic349C2212yescomplemented
13#19150210—22no—
14C2×D4×C349C2211yescharacteristic
15C22×C4×C349C2211yescharacteristic, complemented
16C4⋊C4×C349—21no—
17C22⋊C4×C349—21no—
18C22⋊C4×C349C2211yescharacteristic, complemented
19C22×D349—21no—
20C22×D349D412yes—
21Dic698—21no—
22D1396—21no—
23C22⋊D349—22no—
24C22⋊D349—22no—
25C22⋊D349—22no—
26C2×Dic349—21no—
27C2×Dic349—22no—
28C2×Dic349D412yes—
29C4×D349—22no—
30C23×C349D411yescharacteristic
31C23×C349C2311yescharacteristic
32D4×C349—22no—
33C2×C4×C349—21no—
34C2×C4×C349—21no—
35C2×C4×C349—22no—
36C2×C4×C349D411yescharacteristic
37C22×C349—21no—
38C22×C349—41no—
39C22×C349—41no—
40C22×C349C2×D412yes—
41C22×C349C2×D411yescharacteristic
42C2×D349—22no—
43C2×D349—42no—
44C2×D349—81no—
45C4×C349—21no—
46C4×C349—41no—
47C4×C349—41no—
48Dic349—22no—
49Dic349—42no—
50C2×C349—22no—
51C2×C349—21no—
52C2×C349—41no—
53C2×C349#12011yescharacteristic
54D349—42no—
55C349#45811yescharacteristic, complemented, Sylow
56#458—3491nomaximal, Sylow
57H(5,𝔽2)—3491no—
58#123—3492no—
59#123D34911yescharacteristic, complemented
60#99—3491no—
61#99—3492no—
62#41—6982no—
63#41—6981no—
64C2×D4—3492no—
65C2×D4—6982no—
66C2×D4—6981no—
67C2×D4C2×D34911yescharacteristic
68C22×C4C2×D34911yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—6982no—
71C22⋊C4—21no—
72C22⋊C4—3492no—
73C22⋊C4—6982no—
74C22⋊C4C2×D34911yescharacteristic, complemented
75C23—3492no—
76C23—6981no—
77C23C22⋊D34911yescharacteristic
78C23C22×D34911yescharacteristic
79Q8—6981no—
80D4—22no—
81D4—6981no—
82D4—6982no—
83D4—6982no—
84D4—6982no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3492no—
89C2×C4—6982no—
90C2×C4—6982no—
91C2×C4—6981no—
92C2×C4C22⋊D34911yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—13962no—
97C22—6982no—
98C22—27921no—
99C22D4×D34912yes—
100C22C2×C22⋊D34911yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—13962no—
105C4—6982no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—13962no—
110C2#2601360711yescharacteristic
subgroupG/H
centerC2#26013607
commutator, cosocleC23×C349C23
FrattiniC23C22×D349
Fitting#123×C349C2
socleC2×C349#120
prime radical#26013684C2
hypercenter#123D349
hypocenterC349#458
LCSG ▷ C23×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #26013622C2semidirect
3. #26013554C2semidirect
4. #26013553C2semidirect
5. #123×C349C2semidirect
12. C22⋊Dic349C22semidirect
15. C22×C4×C349C22semidirect
18. C22⋊C4×C349C22semidirect
55. C349#458semidirect
59. #123D349semidirect
68. C22×C4C2×D349semidirect
74. C22⋊C4C2×D349semidirect

Maximal subgroups

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

subgroupindex
#260136842
#260136222
#260135542
#260135532
#123×C3492
#458349