Finite Group Databasebeta

← #36352872

Subgroups — #36352872

Order 22144.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#26012241C211yesmaximal, characteristic
2#26011897C212yesmaximal, complemented
3#26011232C212yesmaximal, complemented
4#26011231C211yesmaximal, characteristic
5#497×C173C211yesmaximal, characteristic
6#19149677—21no—
7#19149672—22no—
8#19149671C2212yes—
9#19149660—21no—
10#19149656—22no—
11#19149612—22no—
12#19149610—22no—
13#19149590C2212yes—
14C2×Q16×C173—21no—
15C2×SD16×C173—21no—
16C4⋊Q8×C173C2211yescharacteristic
17#124×C173C2211yescharacteristic
18#97×C173C2211yescharacteristic
19C2×Dic346—21no—
20C22×Dic173—21no—
21C2×#13063436—22no—
22C4⋊Dic173D412yes—
23C4⋊Dic173D412yes—
24#14583805—22no—
25#14583805—22no—
26#14583805—22no—
27C22⋊Dic173—22no—
28C22⋊Dic173—21no—
29C4×Dic173—21no—
30C2×Q8×C173—21no—
31C2×Q8×C173D411yescharacteristic
32C2×D4×C173D411yescharacteristic
33Q16×C173—42no—
34SD16×C173—42no—
35C2×C8×C173—21no—
36C4⋊C4×C173—22no—
37C22⋊C4×C173—22no—
38C42×C173C2311yescharacteristic
39Dic346—81no—
40C2×Dic173—22no—
41C2×Dic173—22no—
42C2×Dic173—81no—
43#13063436—24no—
44C23×C173—21no—
45Q8×C173—41no—
46Q8×C173—41no—
47Q8×C173—41no—
48D4×C173—41no—
49C2×C4×C173—21no—
50C2×C4×C173—41no—
51C2×C4×C173C2×D412yes—
52C2×C4×C173C2×D411yescharacteristic
53C8×C173—22no—
54C22×C173—41no—
55C22×C173—42no—
56C22×C173#12011yescharacteristic
57C4×C173—21no—
58C4×C173—22no—
59C4×C173—41no—
60C4×C173—81no—
61Dic173—42no—
62Dic173—42no—
63C2×C173—81no—
64C2×C173#45311yescharacteristic
65C2×C173#45612yes—
66C173#166011yescharacteristic, complemented, Sylow
67#1660—1731nomaximal, Sylow
68#563—1731no—
69#501—1732no—
70#497D17311yescharacteristic
71#330—1732no—
72#329—1731no—
73C2×Q16—21no—
74C2×SD16—21no—
75C4⋊Q8—3461no—
76C4⋊Q8C2×D17311yescharacteristic
77#126—3462no—
78#125—1732no—
79#124C2×D17311yescharacteristic
80#123—3461no—
81C22⋊Q8—3462no—
82#107—3462no—
83#106—3462no—
84#97—1732no—
85#97C2×D17311yescharacteristic
86C2×Q8—21no—
87C2×Q8—3461no—
88C2×Q8C22⋊D17311yescharacteristic
89C2×D4C22⋊D17311yescharacteristic
90C22×C4—3461no—
91Q16—42no—
92SD16—42no—
93C2×C8—21no—
94C2×C8—3462no—
95C4⋊C4—22no—
96C4⋊C4—3462no—
97C4⋊C4—3462no—
98C4⋊C4—3462no—
99C4⋊C4—1732no—
100C4⋊C4—1732no—
101C22⋊C4—22no—
102C22⋊C4—3462no—
103C22⋊C4—3461no—
104C42—3461no—
105C42C22×D17311yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—13841no—
111D4—41no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—3462no—
115C2×C4—3462no—
116C2×C4—13841no—
117C2×C4D4×D17312yes—
118C2×C4C2×C22⋊D17311yescharacteristic
119C8—22no—
120C8—3464no—
121C22—41no—
122C22—42no—
123C22#1914964711yescharacteristic
124C4—21no—
125C4—22no—
126C4—41no—
127C4—81no—
128C4—6922no—
129C4—6922no—
130C2—81no—
131C2#2601165811yescharacteristic
132C2#2601167212yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#497×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenter#497D173
hypocenterC173#1660
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #26011897C2semidirect
3. #26011232C2semidirect
66. C173#1660semidirect

Maximal subgroups

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

subgroupindex
#260122412
#260118972
#260112322
#260112312
#497×C1732
#1660173