Finite Group Databasebeta

← #36352874

Subgroups — #36352874

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#26011901C212yesmaximal, complemented
2#26011886C211yesmaximal, characteristic
3#26011233C212yesmaximal, complemented
4#26011231C211yesmaximal, characteristic
5#563×C173C211yesmaximal, characteristic
6C2×#14583816—21no—
7C2×#14583814—21no—
8#19149679C2211yescharacteristic
9#19149612—22no—
10#19149610—22no—
11#19149590C2212yes—
12#19149590C2211yescharacteristic
13C4⋊Q8×C173—21no—
14#126×C173—22no—
15#125×C173C2212yes—
16#124×C173C2211yescharacteristic
17#123×C173—21no—
18C22⋊Q8×C173—22no—
19C2×Dic346—21no—
20#14583816—42no—
21#14583814—42no—
22C2×#13063436—22no—
23C2×#13063436—21no—
24C4⋊Dic173—22no—
25C2×Q8×C173—21no—
26C2×Q8×C173D411yescharacteristic
27C2×D4×C173D411yescharacteristic
28C22×C4×C173—21no—
29C4⋊C4×C173—22no—
30C4⋊C4×C173—22no—
31C4⋊C4×C173—22no—
32C4⋊C4×C173D412yes—
33C4⋊C4×C173D412yes—
34C22⋊C4×C173—22no—
35C22⋊C4×C173—21no—
36C22⋊C4×C173—22no—
37C42×C173—21no—
38C42×C173C2311yescharacteristic
39Dic346—41no—
40Dic346—41no—
41C2×Dic173—41no—
42#13063436—24no—
43#13063436—22no—
44C23×C173—21no—
45Q8×C173—41no—
46Q8×C173—81no—
47D4×C173—41no—
48C2×C4×C173—22no—
49C2×C4×C173—21no—
50C2×C4×C173—22no—
51C2×C4×C173—81no—
52C2×C4×C173C2×D411yescharacteristic
53C2×C4×C173C2×D412yes—
54C22×C173—41no—
55C22×C173—42no—
56C22×C173#12011yescharacteristic
57C4×C173—21no—
58C4×C173—22no—
59C4×C173—42no—
60C4×C173—41no—
61C4×C173—42no—
62Dic173—81no—
63C2×C173—81no—
64C2×C173#45311yescharacteristic
65C2×C173#45612yes—
66C173#166011yescharacteristic, complemented, Sylow
67#1660—1731nomaximal, Sylow
68#563D17311yescharacteristic
69#501—1732no—
70#497—1731no—
71#330—1732no—
72#329—1731no—
73C2×Q16—3461no—
74C2×SD16—3461no—
75C4⋊Q8—21no—
76C4⋊Q8—1731no—
77#126—22no—
78#125C2×D17312yes—
79#124C2×D17311yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—3462no—
83#106—3462no—
84#97—1732no—
85#97—1731no—
86C2×Q8—21no—
87C2×Q8—3461no—
88C2×Q8C22⋊D17311yescharacteristic
89C2×D4C22⋊D17311yescharacteristic
90C22×C4—21no—
91Q16—6922no—
92SD16—6922no—
93C2×C8—3462no—
94C2×C8—3461no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—3462no—
99C4⋊C4C22⋊D17312yes—
100C4⋊C4C22⋊D17312yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D17311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—6921no—
110Q8—6921no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—6921no—
117C2×C4C2×C22⋊D17312yes—
118C2×C4C2×C22⋊D17311yescharacteristic
119C8—3464no—
120C8—3462no—
121C22—41no—
122C22—42no—
123C22#1914964811yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—13841no—
130C2—81no—
131C2#2601165911yescharacteristic
132C2#2601167412yes—
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#563×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenter#563D173
hypocenterC173#1660
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26011901C2semidirect
3. #26011233C2semidirect
66. C173#1660semidirect

Maximal subgroups

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

subgroupindex
#260119012
#260118862
#260112332
#260112312
#563×C1732
#1660173