Finite Group Databasebeta

← #36352680

Subgroups — #36352680

Order 22144.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#26012126C211yesmaximal, characteristic
2#26011863C211yesmaximal, characteristic, complemented
3#26011782C212yesmaximal, complemented
4#26011218C212yesmaximal, complemented
5#331×C173C211yesmaximal, characteristic, complemented
6C2×D4×D173—22no—
7C2×#14583814—22no—
8#19149679C2211yescharacteristic, complemented
9#19149650—22no—
10#19149647—22no—
11C4×D692C2212yes—
12#19149611—22no—
13#19149601—22no—
14C4×#13063436C2211yescharacteristic, complemented
15#127×C173C2211yescharacteristic
16C4⋊C8×C173C2212yescomplemented
17C23×D173—22no—
18C2×Dic346—21no—
19C2×D692D412yes—
20C2×C22⋊D173—24no—
21D4×D173—44no—
22D4×D173—44no—
23C2×C4×D173—22no—
24#14583814—44no—
25C2×#13063436—21no—
26C4⋊Dic173—22no—
27C4⋊Dic173D412yes—
28C22⋊Dic173—22no—
29#14583803—22no—
30C2×D4×C173—22no—
31C2×D4×C173D412yes—
32C2×C8×C173—22no—
33C42×C173C2311yescharacteristic
34C22×D173—22no—
35C22×D173—48no—
36C22×D173—82no—
37Dic346—42no—
38D692—22no—
39D692SD1614yescomplemented
40C22⋊D173—48no—
41C2×Dic173—22no—
42C2×Dic173—41no—
43C4×D173—42no—
44C4×D173—42no—
45#13063436—22no—
46C23×C173—22no—
47D4×C173—42no—
48D4×C173—42no—
49D4×C173—42no—
50C2×C4×C173C2×D412yes—
51C2×C4×C173C2×D411yescharacteristic
52C8×C173—42no—
53C22×C173—44no—
54C22×C173—42no—
55C22×C173#12011yescharacteristic
56C2×D173—24no—
57C2×D173—42no—
58C2×D173—42no—
59C2×D173—88no—
60C4×C173—21no—
61C4×C173C2×SD1612yes—
62C4×C173AGL(1,ℤ/8)12yes—
63Dic173—42no—
64Dic173—81no—
65C2×C173—82no—
66C2×C173#45012yes—
67C2×C173#45311yescharacteristic
68D173—44no—
69C173#162911yescharacteristic, complemented, Sylow
70#1629—1731nomaximal, Sylow
71D42—1731no—
72#492—1731no—
73#476—1732no—
74#331D17311yescharacteristic, complemented
75D4⋊C8—1732no—
76C22×D4—3462no—
77C2×SD16—3462no—
78C4⋊Q8—1731no—
79#127C2×D17311yescharacteristic
80#121—3462no—
81#120—3462no—
82C4×D4—1732no—
83#106—3462no—
84C4⋊C8C2×D17312yescomplemented
85#102—3462no—
86C4×C8—1731no—
87C24—3462no—
88C2×Q8—3461no—
89C2×D4—22no—
90C2×D4—3464no—
91C2×D4—1732no—
92C2×D4—6924no—
93C2×D4—6924no—
94C2×D4D69212yes—
95C22×C4—3462no—
96SD16—6924no—
97C2×C8—22no—
98C2×C8—3461no—
99C4⋊C4—3462no—
100C4⋊C4—1732no—
101C22⋊C4—3462no—
102C22⋊C4—3462no—
103C42C22×D17311yescharacteristic
104C23—22no—
105C23—3462no—
106C23—13842no—
107C23—6928no—
108Q8—6922no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—3462no—
113D4—1734no—
114D4—6928no—
115C2×C4—3462no—
116C2×C4—6921no—
117C2×C4—6922no—
118C2×C4—6922no—
119C2×C4D4×D17312yes—
120C2×C4C2×D69211yescharacteristic
121C8—42no—
122C8—3462no—
123C22—42no—
124C22—44no—
125C22—3464no—
126C22—13848no—
127C22—6922no—
128C22—6922no—
129C22#1914964911yescharacteristic
130C4—21no—
131C4—13841no—
132C4—6922no—
133C4SD16×D17312yes—
134C4#1914970312yes—
135C2—82no—
136C2—6924no—
137C2#2601164312yes—
138C2#2601166111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#331×C173C2
socleC22×C173#120
prime radical#26012126C2
hypercenter#331D173
hypocenterC173#1629
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #26011863C2semidirect
3. #26011782C2semidirect
4. #26011218C2semidirect
5. #331×C173C2semidirect
8. #19149679C22semidirect
14. C4×#13063436C22semidirect
16. C4⋊C8×C173C22semidirect
39. D692SD16semidirect
69. C173#1629semidirect
74. #331D173semidirect
84. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121262
#260118632
#260117822
#260112182
#331×C1732
#1629173