Finite Group Databasebeta

← #36352859

Subgroups — #36352859

Order 22144.

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

normal subgroups only

This group has 206 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, complemented
2#26011855C212yesmaximal, complemented
3#26011232C212yesmaximal, complemented
4#26011231C211yesmaximal, characteristic, complemented
5#496×C173C211yesmaximal, characteristic
6#19149677—21no—
7#19149672—22no—
8#19149671C2212yes—
9#19149660—21no—
10#19149656—22no—
11#19149600—22no—
12#19149600—22no—
13#19149590C2212yes—
14C2×SD16×C173—21no—
15C2×D8×C173—21no—
16#127×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×C173D411yescharacteristic
31C2×D4×C173—21no—
32C2×D4×C173—22no—
33C2×D4×C173D411yescharacteristic
34SD16×C173—42no—
35D8×C173—42no—
36C2×C8×C173—21no—
37C22⋊C4×C173—22no—
38C42×C173C2311yescharacteristic
39Dic346—81no—
40C2×Dic173—22no—
41C2×Dic173—22no—
42C2×Dic173—81no—
43#13063436—42no—
44C23×C173—21no—
45C23×C173—41no—
46Q8×C173—41no—
47D4×C173—41no—
48D4×C173—41no—
49D4×C173—41no—
50D4×C173—44no—
51C2×C4×C173—21no—
52C2×C4×C173C2×D412yes—
53C2×C4×C173C2×D411yescharacteristic
54C8×C173—22no—
55C22×C173—41no—
56C22×C173—42no—
57C22×C173—81no—
58C22×C173—82no—
59C22×C173#12011yescharacteristic
60C4×C173—21no—
61C4×C173—22no—
62C4×C173—41no—
63Dic173—42no—
64Dic173—42no—
65C2×C173—81no—
66C2×C173—161no—
67C2×C173#45311yescharacteristic
68C2×C173#45412yes—
69C173#165711yescharacteristic, complemented, Sylow
70#1657—1731nomaximal, Sylow
71#563—1731no—
72#496D17311yescharacteristic
73#490—1732no—
74#330—1732no—
75#329—1731no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—3461no—
79#127C2×D17311yescharacteristic
80#126—3462no—
81#125—1732no—
82#124C2×D17311yescharacteristic
83#123—3461no—
84C22⋊Q8—3462no—
85#102—3462no—
86#102—3462no—
87#97—1732no—
88#97C2×D17311yescharacteristic
89C2×Q8—3461no—
90C2×Q8C22⋊D17311yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D17311yescharacteristic
94C22×C4—3461no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—3462no—
99C4⋊C4—3462no—
100C4⋊C4—3462no—
101C4⋊C4—3462no—
102C4⋊C4—1732no—
103C4⋊C4—1732no—
104C22⋊C4—22no—
105C22⋊C4—3462no—
106C22⋊C4—3461no—
107C42—3461no—
108C42C22×D17311yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—13841no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—3462no—
119C2×C4—3462no—
120C2×C4—13841no—
121C2×C4D4×D17312yes—
122C2×C4C2×C22⋊D17311yescharacteristic
123C8—22no—
124C8—6922no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1914964711yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—6922no—
134C4—6922no—
135C2—81no—
136C2—161no—
137C2#2601165811yescharacteristic
138C2#2601166212yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting, prime radical#496×C173C2
socleC22×C173#120
hypercenter#496D173
hypocenterC173#1657
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26012241C2semidirect
2. #26011855C2semidirect
3. #26011232C2semidirect
4. #26011231C2semidirect
69. C173#1657semidirect

Maximal subgroups

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

subgroupindex
#260122412
#260118552
#260112322
#260112312
#496×C1732
#1657173