Finite Group Databasebeta

← #36374357

Subgroups — #36374357

Order 22336.

124 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C2×#19150221C211yesmaximal, characteristic, complemented
2#26013626C211yesmaximal, characteristic, complemented
3D4×Dic349C211yesmaximal, characteristic, complemented
4#26013568C211yesmaximal, characteristic, complemented
5#26013564C211yesmaximal, characteristic, complemented
6#26013560C211yesmaximal, characteristic, complemented
7C2×D8×C349C211yesmaximal, characteristic
8C2×Dic698C2211yescharacteristic
9C22×Dic349—21no—
10#19150221—22no—
11C2×#14584095C2211yescharacteristic
12C4⋊Dic349C2211yescharacteristic
13C22⋊Dic349—21no—
14C22⋊Dic349—21no—
15C22⋊Dic349—21no—
16C4×Dic349C2211yescharacteristic
17C2×D4×C349C2211yescharacteristic
18C2×D4×C349C2211yescharacteristic
19D8×C349—22no—
20C2×C8×C349C2211yescharacteristic
21Dic698—21no—
22Dic698—21no—
23C2×Dic349—21no—
24C2×Dic349—21no—
25C2×Dic349—42no—
26C2×Dic349D411yescharacteristic
27C2×Dic349D411yescharacteristic
28#14584095—21no—
29C23×C349—21no—
30C23×C349—21no—
31D4×C349—21no—
32D4×C349—21no—
33D4×C349—21no—
34D4×C349D412yes—
35C2×C4×C349C2311yescharacteristic
36C8×C349—21no—
37C22×C349—22no—
38C22×C349—41no—
39C22×C349—41no—
40C22×C349—41no—
41C22×C349—41no—
42C22×C349—41no—
43C22×C349C2×D411yescharacteristic
44C4×C349C2×D411yescharacteristic
45C4×C349#4111yescharacteristic
46Dic349—21no—
47Dic349—21no—
48Dic349—41no—
49Dic349—41no—
50C2×C349—42no—
51C2×C349—81no—
52C2×C349#12111yescharacteristic
53C2×C349#13511yescharacteristic
54C2×C349AGL(1,ℤ/8)11yescharacteristic
55C349#46311yescharacteristic, complemented, Sylow
56#463—3491nomaximal, Sylow
57C2×SD16—3491no—
58C2×D8D34911yescharacteristic
59#124—3491no—
60C4×D4—3491no—
61C4⋊C8—3491no—
62#103—3491no—
63#102—3491no—
64C2×Q8—3491no—
65C2×D4C2×D34911yescharacteristic
66C2×D4C2×D34911yescharacteristic
67C22×C4—6981no—
68SD16—6982no—
69D8—22no—
70C2×C8—3491no—
71C2×C8C2×D34911yescharacteristic
72C4⋊C4—3491no—
73C22⋊C4—6981no—
74C22⋊C4—6981no—
75C22⋊C4—6981no—
76C42—3491no—
77C23—21no—
78C23—21no—
79Q8—6981no—
80Q8—6981no—
81D4—21no—
82D4—21no—
83D4—21no—
84D4C22⋊D34912yes—
85C2×C4—3491no—
86C2×C4—3491no—
87C2×C4—13962no—
88C2×C4—6981no—
89C2×C4—6981no—
90C2×C4C22×D34911yescharacteristic
91C8—21no—
92C8—6981no—
93C22—22no—
94C22—41no—
95C22—41no—
96C22—41no—
97C22—41no—
98C22—41no—
99C22D4×D34911yescharacteristic
100C4—13961no—
101C4—13961no—
102C4—6981no—
103C4—6981no—
104C4C2×C22⋊D34911yescharacteristic
105C4#1915023211yescharacteristic
106C2—42no—
107C2—81no—
108C2#2601361111yescharacteristic
109C2#2601365811yescharacteristic
110C2#2601366311yescharacteristic
subgroupG/H
centerC22D4×D349
commutator, cosocleC2×C4×C349C23
FrattiniC2×C4C22×D349
Fitting, prime radicalC2×D8×C349C2
socleC22×C349C2×D4
hypercenterC2×D8D349
hypocenterC349#463
LCSG ▷ C2×C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19150221C2semidirect
2. #26013626C2semidirect
3. D4×Dic349C2semidirect
4. #26013568C2semidirect
5. #26013564C2semidirect
6. #26013560C2semidirect
55. C349#463semidirect

Maximal subgroups

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

subgroupindex
C2×#191502212
#260136262
D4×Dic3492
#260135682
#260135642
#260135602
C2×D8×C3492
#463349