Finite Group Databasebeta

← #36374325

Subgroups — #36374325

Order 22336.

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

normal subgroups only

This group has 142 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#26013686C211yesmaximal, characteristic
2#26013670C211yesmaximal, characteristic, complemented
3#26013637C211yesmaximal, characteristic, complemented
4#26013566C211yesmaximal, characteristic, complemented
5#26013565C211yesmaximal, characteristic, complemented
6#26013557C211yesmaximal, characteristic, complemented
7#137×C349C211yesmaximal, characteristic, complemented
8#19150233—21no—
9#19150233C2211yescharacteristic
10#19150232—22no—
11#19150232—21no—
12C2×Dic698—21no—
13C2×Dic698C2211yescharacteristic
14Q8×D349—21no—
15#19150223—21no—
16#19150220—21no—
17#19150217C2211yescharacteristic
18#19150212—22no—
19C4×Dic349C2211yescharacteristic
20#41×C349C2211yescharacteristic
21C2×Q8×C349C2211yescharacteristic
22Q16×C349—21no—
23SD16×C349—21no—
24M16×C349C2211yescharacteristic
25Dic698—22no—
26Dic698—21no—
27Dic698—21no—
28Dic698D411yescharacteristic
29D1396D411yescharacteristic
30C22⋊D349—22no—
31C22⋊D349—21no—
32C2×Dic349—22no—
33C2×Dic349—21no—
34C2×Dic349D412yes—
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—21no—
50C4×C349—21no—
51C4×C349—41no—
52C4×C349C2×D411yescharacteristic
53C4×C349C2×D411yescharacteristic
54Dic349—22no—
55Dic349—21no—
56Dic349—22no—
57C2×C349—21no—
58C2×C349—41no—
59C2×C349#12011yescharacteristic
60D349—41no—
61C349#45611yescharacteristic, complemented, Sylow
62#456—3491nomaximal, Sylow
632−1+4—3491no—
64#137—3491no—
65#137D34911yescharacteristic, complemented
66C4⋊Q8—3491no—
67#104—3491no—
68#104—3491no—
69#101—3491no—
70#41—3491no—
71#41—6981no—
72#41—6982no—
73#41—6981no—
74#41C2×D34911yescharacteristic
75C2×Q8—3491no—
76C2×Q8—6981no—
77C2×Q8—6981no—
78C2×Q8C2×D34911yescharacteristic
79Q16—21no—
80Q16—6981no—
81SD16—21no—
82SD16—6981no—
83M16—3491no—
84M16C2×D34911yescharacteristic
85C4⋊C4—6982no—
86C42—3491no—
87Q8—21no—
88Q8—21no—
89Q8—3491no—
90Q8—6982no—
91Q8—6981no—
92Q8—6981no—
93Q8C22⋊D34911yescharacteristic
94D4—21no—
95D4—3491no—
96D4—6982no—
97D4—6981no—
98D4C22⋊D34911yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—3492no—
102C2×C4—6982no—
103C2×C4—6982no—
104C2×C4—6981no—
105C2×C4—6981no—
106C2×C4C22×D34911yescharacteristic
107C8—21no—
108C8—6981no—
109C22—21no—
110C22—6981no—
111C22D4×D34911yescharacteristic
112C4—21no—
113C4—41no—
114C4—6982no—
115C4—6982no—
116C4—6981no—
117C4D4×D34911yescharacteristic
118C4C2×C22⋊D34911yescharacteristic
119C2—21no—
120C2—41no—
121C2—13961no—
122C2#2601360711yescharacteristic
subgroupG/H
centerC2#26013607
commutator, cosocleC2×C4×C349C23
FrattiniC2×C4C22×D349
Fitting#137×C349C2
socleC2×C349#120
prime radical#26013686C2
hypercenter#137D349
hypocenterC349#456
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. #26013637C2semidirect
4. #26013566C2semidirect
5. #26013565C2semidirect
6. #26013557C2semidirect
7. #137×C349C2semidirect
61. C349#456semidirect
65. #137D349semidirect

Maximal subgroups

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

subgroupindex
#260136862
#260136702
#260136372
#260135662
#260135652
#260135572
#137×C3492
#456349