Finite Group Databasebeta

← #36374320

Subgroups — #36374320

Order 22336.

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

normal subgroups only

This group has 152 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#26013685C211yesmaximal, characteristic
2#26013670C211yesmaximal, characteristic, complemented
3#26013628C211yesmaximal, characteristic, complemented
4#26013566C211yesmaximal, characteristic, complemented
5#26013565C211yesmaximal, characteristic, complemented
6#26013555C211yesmaximal, characteristic, complemented
7#137×C349C211yesmaximal, characteristic, complemented
8#19150234—21no—
9#19150233—21no—
10#19150233C2211yescharacteristic
11C2×D1396—21no—
12C2×D1396C2211yescharacteristic
13D4×D349—22no—
14D4×D349—21no—
15#19150223—21no—
16#19150220—21no—
17#19150217C2211yescharacteristic, complemented
18#19150210—22no—
19C4×Dic349C2211yescharacteristic, complemented
20#41×C349C2211yescharacteristic
21C2×Q8×C349C2211yescharacteristic, complemented
22Q16×C349—21no—
23SD16×C349—21no—
24M16×C349C2211yescharacteristic, complemented
25C22×D349—22no—
26C22×D349D412yes—
27Dic698D411yescharacteristic
28D1396—22no—
29D1396—21no—
30D1396—21no—
31D1396D411yescharacteristic
32C22⋊D349—22no—
33C22⋊D349—21no—
34C2×Dic349—21no—
35C4×D349—22no—
36C4×D349—21no—
37#14584095—21no—
38Q8×C349—21no—
39Q8×C349—21no—
40Q8×C349D411yescharacteristic
41D4×C349—21no—
42D4×C349D411yescharacteristic
43C2×C4×C349—21no—
44C2×C4×C349—21no—
45C2×C4×C349C2311yescharacteristic
46C8×C349—21no—
47C22×C349—21no—
48C22×C349C2×D411yescharacteristic
49C2×D349—22no—
50C2×D349—21no—
51C2×D349—42no—
52C2×D349—82no—
53C4×C349—21no—
54C4×C349—41no—
55C4×C349C2×D411yescharacteristic
56C4×C349C2×D411yescharacteristic
57Dic349—21no—
58Dic349—41no—
59C2×C349—21no—
60C2×C349—41no—
61C2×C349#12011yescharacteristic
62D349—42no—
63D349—41no—
64C349#45511yescharacteristic, complemented, Sylow
65#455—3491nomaximal, Sylow
66H(5,𝔽2)—3491no—
67#137—3491no—
68#137D34911yescharacteristic, complemented
69#124—3491no—
70#104—3491no—
71#104—3491no—
72#100—3491no—
73#41—3491no—
74#41—6981no—
75#41—6981no—
76#41C2×D34911yescharacteristic
77C2×Q8C2×D34911yescharacteristic, complemented
78C2×D4—3491no—
79C2×D4—6981no—
80C2×D4—6982no—
81C2×D4—6981no—
82Q16—21no—
83Q16—6981no—
84SD16—21no—
85SD16—6981no—
86M16—3491no—
87M16C2×D34911yescharacteristic, complemented
88C22⋊C4—6982no—
89C42—3491no—
90C23—3492no—
91C23—6982no—
92Q8—21no—
93Q8—21no—
94Q8—3491no—
95Q8C22⋊D34911yescharacteristic
96D4—21no—
97D4—3491no—
98D4—6982no—
99D4—6982no—
100D4—6981no—
101D4—6981no—
102D4—6981no—
103D4C22⋊D34911yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—6982no—
107C2×C4—6981no—
108C2×C4—6981no—
109C2×C4C22×D34911yescharacteristic
110C8—21no—
111C8—6981no—
112C22—21no—
113C22—13962no—
114C22—6981no—
115C22—6982no—
116C22—27922no—
117C22D4×D34911yescharacteristic
118C4—21no—
119C4—41no—
120C4—13961no—
121C4—6981no—
122C4D4×D34911yescharacteristic
123C4C2×C22⋊D34911yescharacteristic
124C2—21no—
125C2—41no—
126C2—13962no—
127C2—13961no—
128C2#2601360711yescharacteristic
subgroupG/H
centerC2#26013607
commutator, cosocleC2×C4×C349C23
FrattiniC2×C4C22×D349
Fitting#137×C349C2
socleC2×C349#120
prime radical#26013685C2
hypercenter#137D349
hypocenterC349#455
LCSG ▷ C2×C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #26013670C2semidirect
3. #26013628C2semidirect
4. #26013566C2semidirect
5. #26013565C2semidirect
6. #26013555C2semidirect
7. #137×C349C2semidirect
17. #19150217C22semidirect
19. C4×Dic349C22semidirect
21. C2×Q8×C349C22semidirect
24. M16×C349C22semidirect
64. C349#455semidirect
68. #137D349semidirect
77. C2×Q8C2×D349semidirect
87. M16C2×D349semidirect

Maximal subgroups

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

subgroupindex
#260136852
#260136702
#260136282
#260135662
#260135652
#260135552
#137×C3492
#455349