Finite Group Databasebeta

← #36374330

Subgroups — #36374330

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#26013608C212yesmaximal, complemented
3#26013554C211yesmaximal, characteristic, complemented
4#26013553C212yesmaximal, complemented
5#120×C349C211yesmaximal, characteristic, complemented
6#19150233—21no—
7#19150232—22no—
8C2×C22⋊D349—22no—
9C2×C22⋊D349—21no—
10C2×C22⋊D349C2212yes—
11D4×D349—22no—
12C22⋊Dic349—22no—
13C22⋊Dic349C2212yescomplemented
14C24×C349C2211yescharacteristic, complemented
15C2×D4×C349—21no—
16C2×D4×C349C2211yescharacteristic
17C22⋊C4×C349—21no—
18C22⋊C4×C349C2211yescharacteristic, complemented
19C22×D349—21no—
20C22×D349D412yescomplemented
21Dic698—21no—
22D1396—21no—
23C22⋊D349—22no—
24C22⋊D349—22no—
25C22⋊D349—22no—
26C22⋊D349—44no—
27C2×Dic349—21no—
28C2×Dic349—22no—
29C2×Dic349D412yescomplemented
30C4×D349—22no—
31C23×C349—21no—
32C23×C349—22no—
33C23×C349—42no—
34C23×C349D411yescharacteristic, complemented
35C23×C349C2311yescharacteristic
36D4×C349—22no—
37D4×C349—42no—
38C2×C4×C349—21no—
39C2×C4×C349D411yescharacteristic, complemented
40C22×C349—22no—
41C22×C349—24no—
42C22×C349—21no—
43C22×C349—41no—
44C22×C349—41no—
45C22×C349—42no—
46C22×C349—41no—
47C22×C349—41no—
48C22×C349C2×D412yes—
49C22×C349C2×D411yescharacteristic
50C2×D349—22no—
51C2×D349—42no—
52C2×D349—81no—
53C4×C349—21no—
54C4×C349—41no—
55Dic349—22no—
56Dic349—42no—
57C2×C349—22no—
58C2×C349—21no—
59C2×C349—41no—
60C2×C349—42no—
61C2×C349#12011yescharacteristic
62D349—42no—
63C349B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—3491nomaximal, Sylow
65H(5,𝔽2)—3491no—
66#120—3492no—
67#120D34911yescharacteristic, complemented
68#99—3491no—
69#99—3492no—
70C24C2×D34911yescharacteristic, complemented
71#41—6982no—
72#41—6981no—
73C2×D4—21no—
74C2×D4—3492no—
75C2×D4—6982no—
76C2×D4—6982no—
77C2×D4—6981no—
78C2×D4C2×D34911yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—3492no—
81C22⋊C4—6982no—
82C22⋊C4C2×D34911yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—3492no—
87C23—6981no—
88C23C22⋊D34911yescharacteristic, complemented
89C23C22×D34911yescharacteristic
90Q8—6981no—
91D4—22no—
92D4—42no—
93D4—13964no—
94D4—6981no—
95D4—6982no—
96D4—6982no—
97D4—6982no—
98C2×C4—21no—
99C2×C4—3492no—
100C2×C4—6982no—
101C2×C4—6982no—
102C2×C4—6981no—
103C2×C4C22⋊D34911yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—13962no—
113C22—6982no—
114C22—27921no—
115C22D4×D34912yes—
116C22C2×C22⋊D34911yescharacteristic
117C4—21no—
118C4—41no—
119C4—13962no—
120C4—6982no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—13962no—
126C2#2601360711yescharacteristic
subgroupG/H
centerC2#26013607
commutator, cosocleC23×C349C23
FrattiniC23C22×D349
Fitting#120×C349C2
socleC2×C349#120
hypercenter#120D349
hypocenterC349B(4,𝔽2)
LCSG ▷ C23×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26013684C2semidirect
2. #26013608C2semidirect
3. #26013554C2semidirect
4. #26013553C2semidirect
5. #120×C349C2semidirect
13. C22⋊Dic349C22semidirect
14. C24×C349C22semidirect
18. C22⋊C4×C349C22semidirect
20. C22×D349D4semidirect
29. C2×Dic349D4semidirect
34. C23×C349D4semidirect
39. C2×C4×C349D4semidirect
63. C349B(4,𝔽2)semidirect
67. #120D349semidirect
70. C24C2×D349semidirect
82. C22⋊C4C2×D349semidirect
88. C23C22⋊D349semidirect
103. C2×C4C22⋊D349semidirect

Maximal subgroups

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

subgroupindex
#260136842
#260136082
#260135542
#260135532
#120×C3492
B(4,𝔽2)349