Finite Group Databasebeta

← #37533017

Subgroups — #37533017

Order 23168.

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#26042029C211yesmaximal, characteristic, complemented
2#26041643C212yesmaximal, complemented
3#26041020C212yesmaximal, complemented
4#26041019C211yesmaximal, characteristic, complemented
5#496×C181C211yesmaximal, characteristic
6#19153814—21no—
7#19153809—22no—
8#19153808C2212yes—
9#19153797—21no—
10#19153793—22no—
11#19153737—22no—
12#19153737—22no—
13#19153727C2212yes—
14C2×SD16×C181—21no—
15C2×D8×C181—21no—
16#127×C181C2211yescharacteristic
17#124×C181C2211yescharacteristic
18#97×C181C2211yescharacteristic
19C2×Dic362—21no—
20C22×Dic181—21no—
21C2×#13089678—22no—
22C4⋊Dic181D412yes—
23C4⋊Dic181D412yes—
24#15668428—22no—
25#15668428—22no—
26#15668428—22no—
27C22⋊Dic181—22no—
28C22⋊Dic181—21no—
29C4×Dic181—21no—
30C2×Q8×C181D411yescharacteristic
31C2×D4×C181—21no—
32C2×D4×C181—22no—
33C2×D4×C181D411yescharacteristic
34SD16×C181—42no—
35D8×C181—42no—
36C2×C8×C181—21no—
37C22⋊C4×C181—22no—
38C42×C181C2311yescharacteristic
39Dic362—81no—
40C2×Dic181—22no—
41C2×Dic181—22no—
42C2×Dic181—81no—
43#13089678—42no—
44C23×C181—21no—
45C23×C181—41no—
46Q8×C181—41no—
47D4×C181—41no—
48D4×C181—41no—
49D4×C181—41no—
50D4×C181—44no—
51C2×C4×C181—21no—
52C2×C4×C181C2×D412yes—
53C2×C4×C181C2×D411yescharacteristic
54C8×C181—22no—
55C22×C181—41no—
56C22×C181—42no—
57C22×C181—81no—
58C22×C181—82no—
59C22×C181#12011yescharacteristic
60C4×C181—21no—
61C4×C181—22no—
62C4×C181—41no—
63Dic181—42no—
64Dic181—42no—
65C2×C181—81no—
66C2×C181—161no—
67C2×C181#45311yescharacteristic
68C2×C181#45412yes—
69C181#165711yescharacteristic, complemented, Sylow
70#1657—1811nomaximal, Sylow
71#563—1811no—
72#496D18111yescharacteristic
73#490—1812no—
74#330—1812no—
75#329—1811no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—3621no—
79#127C2×D18111yescharacteristic
80#126—3622no—
81#125—1812no—
82#124C2×D18111yescharacteristic
83#123—3621no—
84C22⋊Q8—3622no—
85#102—3622no—
86#102—3622no—
87#97—1812no—
88#97C2×D18111yescharacteristic
89C2×Q8—3621no—
90C2×Q8C22⋊D18111yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D18111yescharacteristic
94C22×C4—3621no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—3622no—
99C4⋊C4—3622no—
100C4⋊C4—3622no—
101C4⋊C4—3622no—
102C4⋊C4—1812no—
103C4⋊C4—1812no—
104C22⋊C4—22no—
105C22⋊C4—3622no—
106C22⋊C4—3621no—
107C42—3621no—
108C42C22×D18111yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—14481no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—3622no—
119C2×C4—3622no—
120C2×C4—14481no—
121C2×C4D4×D18112yes—
122C2×C4C2×C22⋊D18111yescharacteristic
123C8—22no—
124C8—7242no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1915378411yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—7242no—
134C4—7242no—
135C2—81no—
136C2—161no—
137C2#2604144611yescharacteristic
138C2#2604145012yes—
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting, prime radical#496×C181C2
socleC22×C181#120
hypercenter#496D181
hypocenterC181#1657
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26042029C2semidirect
2. #26041643C2semidirect
3. #26041020C2semidirect
4. #26041019C2semidirect
69. C181#1657semidirect

Maximal subgroups

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

subgroupindex
#260420292
#260416432
#260410202
#260410192
#496×C1812
#1657181