Finite Group Databasebeta

← #36374332

Subgroups — #36374332

Order 22336.

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

normal subgroups only

This group has 198 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, complemented
2#26013609C212yesmaximal, complemented
3#26013607C211yesmaximal, characteristic, complemented
4#26013553C212yesmaximal, complemented
5#99×C349C211yesmaximal, characteristic, complemented
6C23×D349C2211yescharacteristic, complemented
7#19150233—21no—
8#19150232—22no—
9C2×D1396—22no—
10C2×C22⋊D349—21no—
11C2×C22⋊D349—21no—
12C2×C22⋊D349C2212yes—
13D4×D349—22no—
14C22⋊Dic349C2211yescharacteristic, complemented
15#19150210—22no—
16#19150210—21no—
17C2×D4×C349C2211yescharacteristic
18C22⋊C4×C349C2212yescomplemented
19C22×D349—21no—
20C22×D349—22no—
21C22×D349—21no—
22C22×D349—42no—
23C22×D349D412yescomplemented
24Dic698—21no—
25D1396—21no—
26D1396—44no—
27C22⋊D349—22no—
28C22⋊D349—22no—
29C22⋊D349—22no—
30C22⋊D349—42no—
31C2×Dic349—21no—
32C2×Dic349—21no—
33C2×Dic349D412yescomplemented
34C4×D349—22no—
35C23×C349D411yescharacteristic, complemented
36C23×C349C2311yescharacteristic
37D4×C349—22no—
38C2×C4×C349—22no—
39C2×C4×C349D411yescharacteristic, complemented
40C22×C349—21no—
41C22×C349—41no—
42C22×C349—41no—
43C22×C349C2×D412yes—
44C22×C349C2×D411yescharacteristic
45C2×D349—22no—
46C2×D349—24no—
47C2×D349—22no—
48C2×D349—41no—
49C2×D349—42no—
50C2×D349—41no—
51C2×D349—42no—
52C2×D349—81no—
53C4×C349—21no—
54C4×C349—42no—
55Dic349—22no—
56Dic349—41no—
57C2×C349—21no—
58C2×C349—22no—
59C2×C349—41no—
60C2×C349#12011yescharacteristic
61D349—42no—
62D349—42no—
63C349B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—3491nomaximal, Sylow
65H(5,𝔽2)—3491no—
66#120—3491no—
67#120—3492no—
68#99—3492no—
69#99D34911yescharacteristic, complemented
70C24—3491no—
71#41—6982no—
72#41—6981no—
73C2×D4—3492no—
74C2×D4—6982no—
75C2×D4—6982no—
76C2×D4—6981no—
77C2×D4—6981no—
78C2×D4C2×D34911yescharacteristic
79C22⋊C4—3491no—
80C22⋊C4—6982no—
81C22⋊C4—6981no—
82C22⋊C4C2×D34912yescomplemented
83C23—3492no—
84C23—13962no—
85C23—6981no—
86C23—6982no—
87C23—6981no—
88C23D139611yescharacteristic, complemented
89C23C22×D34911yescharacteristic
90Q8—6981no—
91D4—22no—
92D4—13964no—
93D4—13962no—
94D4—6981no—
95D4—6982no—
96D4—6982no—
97D4—6982no—
98C2×C4—22no—
99C2×C4—3492no—
100C2×C4—6982no—
101C2×C4—6981no—
102C2×C4—6981no—
103C2×C4D139611yescharacteristic, complemented
104C22—21no—
105C22—41no—
106C22—41no—
107C22—13962no—
108C22—13962no—
109C22—13961no—
110C22—13961no—
111C22—6984no—
112C22—6982no—
113C22—6982no—
114C22—27921no—
115C22D4×D34912yes—
116C22C2×D139611yescharacteristic
117C4—21no—
118C4—42no—
119C4—13961no—
120C4—6982no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—13962no—
125C2—13962no—
126C2#2601360911yescharacteristic
subgroupG/H
centerC2#26013609
commutator, cosocleC23×C349C23
FrattiniC23C22×D349
Fitting#99×C349C2
socleC2×C349#120
hypercenter#99D349
hypocenterC349B(4,𝔽2)
LCSG ▷ C23×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #26013684C2semidirect
2. #26013609C2semidirect
3. #26013607C2semidirect
4. #26013553C2semidirect
5. #99×C349C2semidirect
6. C23×D349C22semidirect
14. C22⋊Dic349C22semidirect
18. C22⋊C4×C349C22semidirect
23. C22×D349D4semidirect
33. C2×Dic349D4semidirect
35. C23×C349D4semidirect
39. C2×C4×C349D4semidirect
63. C349B(4,𝔽2)semidirect
69. #99D349semidirect
82. C22⋊C4C2×D349semidirect
88. C23D1396semidirect
103. C2×C4D1396semidirect

Maximal subgroups

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

subgroupindex
#260136842
#260136092
#260136072
#260135532
#99×C3492
B(4,𝔽2)349