Finite Group Databasebeta

← #36352861

Subgroups — #36352861

Order 22144.

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

normal subgroups only

This group has 206 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#26011881C211yesmaximal, characteristic
2#26011857C212yesmaximal, complemented
3#26011233C212yesmaximal, complemented
4#26011231C211yesmaximal, characteristic, complemented
5#563×C173C211yesmaximal, characteristic, complemented
6C2×#14583813—21no—
7C2×#14583811—21no—
8#19149676C2211yescharacteristic
9#19149599—22no—
10#19149599—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×D692—22no—
20C2×D692—21no—
21#14583813—42no—
22#14583811—42no—
23C2×#13063436—21no—
24C2×#13063436—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
39C22×D173—41no—
40D692—41no—
41D692—41no—
42D692—44no—
43#13063436—22no—
44#13063436—42no—
45C23×C173—21no—
46Q8×C173—41no—
47Q8×C173—81no—
48D4×C173—41no—
49C2×C4×C173—22no—
50C2×C4×C173—21no—
51C2×C4×C173—22no—
52C2×C4×C173—81no—
53C2×C4×C173C2×D412yes—
54C2×C4×C173C2×D411yescharacteristic
55C22×C173—41no—
56C22×C173—42no—
57C22×C173#12011yescharacteristic
58C2×D173—81no—
59C2×D173—82no—
60C4×C173—21no—
61C4×C173—22no—
62C4×C173—42no—
63C4×C173—41no—
64C4×C173—42no—
65C2×C173—81no—
66C2×C173#45311yescharacteristic
67C2×C173#45412yes—
68D173—161no—
69C173#165711yescharacteristic, complemented, Sylow
70#1657—1731nomaximal, Sylow
71#563D17311yescharacteristic, complemented
72#496—1731no—
73#490—1732no—
74#330—1732no—
75#329—1731no—
76C2×SD16—3461no—
77C2×D8—3461no—
78C4⋊Q8—21no—
79#127—1731no—
80#126—22no—
81#125C2×D17312yes—
82#124C2×D17311yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—3462no—
86#102—3462no—
87#97—1732no—
88#97—1731no—
89C2×Q8—21no—
90C2×Q8C22⋊D17311yescharacteristic
91C2×D4—3462no—
92C2×D4—3461no—
93C2×D4C22⋊D17311yescharacteristic
94C22×C4—21no—
95SD16—6922no—
96D8—6922no—
97C2×C8—3461no—
98C2×C8—3462no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D17312yes—
103C4⋊C4C22⋊D17312yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D17311yescharacteristic
109C23—21no—
110C23—6921no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—6921no—
115D4—6921no—
116D4—6924no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D17312yes—
122C2×C4C2×C22⋊D17311yescharacteristic
123C8—3462no—
124C8—6922no—
125C22—41no—
126C22—42no—
127C22—13841no—
128C22—13842no—
129C22#1914964811yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—27681no—
137C2#2601165911yescharacteristic
138C2#2601166412yes—
subgroupG/H
centerC22#19149648
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#563×C173C2
socleC22×C173#120
prime radical#26011881C2
hypercenter#563D173
hypocenterC173#1657
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #26011857C2semidirect
3. #26011233C2semidirect
4. #26011231C2semidirect
5. #563×C173C2semidirect
69. C173#1657semidirect
71. #563D173semidirect

Maximal subgroups

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

subgroupindex
#260118812
#260118572
#260112332
#260112312
#563×C1732
#1657173