Finite Group Databasebeta

← #36374310

Subgroups — #36374310

Order 22336.

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

normal subgroups only

This group has 168 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, complemented
2#26013665C211yesmaximal, characteristic, complemented
3#26013635C211yesmaximal, characteristic, complemented
4#26013566C211yesmaximal, characteristic, complemented
5#26013565C211yesmaximal, characteristic, complemented
6#26013555C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C349C211yesmaximal, characteristic, complemented
8#19150234—21no—
9#19150233—21no—
10#19150233C2211yescharacteristic
11C2×D1396—21no—
12C2×D1396C2211yescharacteristic
13C2×C22⋊D349—22no—
14D4×D349—22no—
15D4×D349—21no—
16#19150221—21no—
17#19150218—21no—
18#19150217C2211yescharacteristic, complemented
19C4×Dic349C2211yescharacteristic, complemented
20#41×C349C2211yescharacteristic
21C2×D4×C349C2211yescharacteristic, complemented
22SD16×C349—21no—
23D8×C349—21no—
24M16×C349C2211yescharacteristic, complemented
25C22×D349—22no—
26C22×D349D412yes—
27Dic698D411yescharacteristic, complemented
28D1396—22no—
29D1396—21no—
30D1396—21no—
31D1396D411yescharacteristic, complemented
32C22⋊D349—22no—
33C22⋊D349—21no—
34C22⋊D349—44no—
35C2×Dic349—21no—
36C4×D349—22no—
37C4×D349—21no—
38#14584095—21no—
39C23×C349—21no—
40Q8×C349D411yescharacteristic, complemented
41D4×C349—21no—
42D4×C349—21no—
43D4×C349—21no—
44D4×C349D411yescharacteristic, complemented
45C2×C4×C349—21no—
46C2×C4×C349C2311yescharacteristic
47C8×C349—21no—
48C22×C349—21no—
49C22×C349—42no—
50C22×C349—41no—
51C22×C349C2×D411yescharacteristic
52C2×D349—22no—
53C2×D349—21no—
54C2×D349—42no—
55C2×D349—82no—
56C4×C349—21no—
57C4×C349C2×D411yescharacteristic
58C4×C349C2×D411yescharacteristic
59Dic349—22no—
60Dic349—21no—
61C2×C349—21no—
62C2×C349—41no—
63C2×C349—81no—
64C2×C349#12011yescharacteristic
65D349—42no—
66D349—41no—
67C349#45311yescharacteristic, complemented, Sylow
68#453—3491nomaximal, Sylow
69H(5,𝔽2)—3491no—
70AGL(1,ℤ/8)—3491no—
71AGL(1,ℤ/8)D34911yescharacteristic, complemented
72#127—3491no—
73#104—3491no—
74#104—3491no—
75#100—3491no—
76#41—3491no—
77#41—6981no—
78#41—6981no—
79#41C2×D34911yescharacteristic
80C2×D4—3491no—
81C2×D4—6981no—
82C2×D4—6982no—
83C2×D4—6982no—
84C2×D4—6981no—
85C2×D4C2×D34911yescharacteristic, complemented
86SD16—21no—
87SD16—6981no—
88D8—21no—
89D8—6981no—
90M16—3491no—
91M16C2×D34911yescharacteristic, complemented
92C42—3491no—
93C23—21no—
94C23—3492no—
95C23—6982no—
96Q8—3491no—
97Q8C22⋊D34911yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—3491no—
102D4—13964no—
103D4—6982no—
104D4—6982no—
105D4—6981no—
106D4—6981no—
107D4—6981no—
108D4C22⋊D34911yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—6982no—
111C2×C4—6981no—
112C2×C4—6981no—
113C2×C4C22×D34911yescharacteristic
114C8—21no—
115C8—6981no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—13962no—
120C22—6981no—
121C22—6982no—
122C22—27922no—
123C22D4×D34911yescharacteristic
124C4—21no—
125C4—6982no—
126C4—6981no—
127C4D4×D34911yescharacteristic
128C4C2×C22⋊D34911yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—13962no—
133C2—13961no—
134C2#2601360711yescharacteristic
subgroupG/H
centerC2#26013607
commutator, cosocleC2×C4×C349C23
FrattiniC2×C4C22×D349
FittingAGL(1,ℤ/8)×C349C2
socleC2×C349#120
hypercenterAGL(1,ℤ/8)D349
hypocenterC349#453
LCSG ▷ C2×C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26013685C2semidirect
2. #26013665C2semidirect
3. #26013635C2semidirect
4. #26013566C2semidirect
5. #26013565C2semidirect
6. #26013555C2semidirect
7. AGL(1,ℤ/8)×C349C2semidirect
18. #19150217C22semidirect
19. C4×Dic349C22semidirect
21. C2×D4×C349C22semidirect
24. M16×C349C22semidirect
27. Dic698D4semidirect
31. D1396D4semidirect
40. Q8×C349D4semidirect
44. D4×C349D4semidirect
67. C349#453semidirect
71. AGL(1,ℤ/8)D349semidirect
85. C2×D4C2×D349semidirect
91. M16C2×D349semidirect
97. Q8C22⋊D349semidirect
108. D4C22⋊D349semidirect

Maximal subgroups

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

subgroupindex
#260136852
#260136652
#260136352
#260135662
#260135652
#260135552
AGL(1,ℤ/8)×C3492
#453349