Finite Group Databasebeta

← #36352886

Subgroups — #36352886

Order 22144.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#26011861C211yesmaximal, characteristic, complemented
2#26011850C212yesmaximal, complemented
3#26011233C211yesmaximal, characteristic, complemented
4#26011231C212yesmaximal, complemented
5#562×C173C211yesmaximal, characteristic
6#19149679C2211yescharacteristic
7#19149603—22no—
8#19149602—22no—
9#19149600—22no—
10#19149590C2211yescharacteristic
11#19149590C2212yes—
12#125×C173C2211yescharacteristic
13#124×C173C2212yes—
14#123×C173—22no—
15C22⋊Q8×C173—22no—
16#121×C173—22no—
17C2×Dic346—21no—
18C2×#13063436—22no—
19C2×#13063436—21no—
20C4⋊Dic173—22no—
21C2×Q8×C173D412yes—
22C2×D4×C173—22no—
23C2×D4×C173D412yes—
24C22×C4×C173—22no—
25C4⋊C4×C173—22no—
26C4⋊C4×C173—22no—
27C4⋊C4×C173D412yes—
28C22⋊C4×C173—22no—
29C22⋊C4×C173—22no—
30C22⋊C4×C173—22no—
31C42×C173C2311yescharacteristic
32Dic346—42no—
33C2×Dic173—41no—
34#13063436—41no—
35#13063436—42no—
36C23×C173—22no—
37Q8×C173—42no—
38D4×C173—42no—
39D4×C173—82no—
40C2×C4×C173—22no—
41C2×C4×C173—22no—
42C2×C4×C173—82no—
43C2×C4×C173C2×D411yescharacteristic
44C2×C4×C173C2×D412yes—
45C22×C173—42no—
46C22×C173—42no—
47C22×C173—42no—
48C22×C173#12011yescharacteristic
49C4×C173—21no—
50C4×C173—22no—
51C4×C173—42no—
52C4×C173—42no—
53Dic173—81no—
54C2×C173—82no—
55C2×C173#45411yescharacteristic
56C2×C173#45512yes—
57C173#166211yescharacteristic, complemented, Sylow
58#1662—1731nomaximal, Sylow
59#562D17311yescharacteristic
60#491—1731no—
61#489—1732no—
62#330—1731no—
63#329—1732no—
64C4⋊Q8—1731no—
65#125C2×D17311yescharacteristic
66#124C2×D17312yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—3462no—
71#103—3462no—
72#102—3462no—
73#97—1731no—
74#97—1732no—
75C2×Q8—3461no—
76C2×Q8C22⋊D17312yes—
77C2×D4—22no—
78C2×D4C22⋊D17312yes—
79C22×C4—22no—
80C2×C8—3462no—
81C2×C8—3461no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—3462no—
85C4⋊C4C22⋊D17312yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D17311yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—6922no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—6921no—
99C2×C4C2×C22⋊D17311yescharacteristic
100C2×C4C2×C22⋊D17312yes—
101C8—6921no—
102C8—6922no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1914964811yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—13841no—
112C2—82no—
113C2#2601166411yescharacteristic
114C2#2601166912yes—
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting, prime radical#562×C173C2
socleC22×C173#120
hypercenter#562D173
hypocenterC173#1662
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #26011861C2semidirect
2. #26011850C2semidirect
3. #26011233C2semidirect
4. #26011231C2semidirect
57. C173#1662semidirect

Maximal subgroups

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

subgroupindex
#260118612
#260118502
#260112332
#260112312
#562×C1732
#1662173