Finite Group Databasebeta

← #37532697

Subgroups — #37532697

Order 23168.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#26041910C211yesmaximal, characteristic
2#26041437C212yesmaximal
3#26041199C211yesmaximal, characteristic
4#26041017C212yesmaximal
5#324×C181C211yesmaximal, characteristic
6C22×Dic362C2211yescharacteristic
7C2×Dic724—22no—
8#19153815—22no—
9#19153807—22no—
10#19153795—21no—
11#19153795C2212yes—
12#19153775—21no—
13C2×#15668428—21no—
14#19153741—22no—
15#19153723—21no—
16#19153723C2211yescharacteristic
17C2×C4⋊C4×C181C2211yescharacteristic
18C22⋊C8×C181C2212yes—
19C2×Dic362—21no—
20C2×Dic362—22no—
21C2×Dic362—42no—
22C2×Dic362D412yes—
23C22×Dic181—21no—
24C22×Dic181—21no—
25Dic724—44no—
26C4⋊Dic181—21no—
27C4⋊Dic181—21no—
28C4⋊Dic181—21no—
29C4⋊Dic181D412yes—
30#15668428—22no—
31#15668428—22no—
32#15668428—42no—
33C22⋊Dic181—22no—
34C4×Dic181—22no—
35C22×C4×C181D411yescharacteristic
36C22×C4×C181D411yescharacteristic
37C22×C4×C181C2311yescharacteristic
38C2×C8×C181—22no—
39C4⋊C4×C181—22no—
40Dic362—24no—
41Dic362—42no—
42Dic362—42no—
43Dic362—81no—
44C2×Dic181—22no—
45C2×Dic181—22no—
46C2×Dic181—22no—
47C2×Dic181—22no—
48C2×Dic181—41no—
49C2×Dic181—41no—
50C2×Dic181—41no—
51C2×Dic181—41no—
52C23×C181C2×D411yescharacteristic
53C2×C4×C181—21no—
54C2×C4×C181—21no—
55C2×C4×C181—21no—
56C2×C4×C181—41no—
57C2×C4×C181—41no—
58C2×C4×C181Q1612yescomplemented
59C2×C4×C181C2×D412yes—
60C8×C181—42no—
61C22×C181—21no—
62C22×C181—21no—
63C22×C181#12011yescharacteristic
64C22×C181C2×Q1611yescharacteristic
65C22×C181#13711yescharacteristic
66C4×C181—22no—
67C4×C181—22no—
68C4×C181—41no—
69Dic181—42no—
70Dic181—42no—
71Dic181—42no—
72Dic181—42no—
73C2×C181—21no—
74C2×C181—21no—
75C2×C181#45111yescharacteristic
76C2×C181#45611yescharacteristic
77C2×C181B(4,𝔽2)11yescharacteristic
78C181#160511yescharacteristic, complemented, Sylow
79#1605—1811nomaximal, Sylow
80#544—1811no—
81#451—1812no—
82#395—1811no—
83#328—1812no—
84#324D18111yescharacteristic
85C22×Q8—1811no—
86C2×Q16—3622no—
87C4⋊Q8—3622no—
88#125—3622no—
89C22⋊Q8—3621no—
90C22⋊Q8—1812no—
91#117—3621no—
92C2×C4⋊C4—3621no—
93C2×C4⋊C4C2×D18111yescharacteristic
94#103—3622no—
95C22⋊C8C2×D18112yes—
96#95—3621no—
97#95—1811no—
98C2×Q8—3621no—
99C2×Q8—3622no—
100C2×Q8—1812no—
101C2×Q8—7242no—
102C22×C4—3621no—
103C22×C4—3621no—
104C22×C4D72411yescharacteristic
105C22×C4D72411yescharacteristic
106C22×C4C22×D18111yescharacteristic
107Q16—7244no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—3621no—
111C4⋊C4—3621no—
112C4⋊C4—3621no—
113C4⋊C4—3622no—
114C4⋊C4—3622no—
115C4⋊C4—1812no—
116C4⋊C4—7242no—
117C22⋊C4—3622no—
118C42—3622no—
119C23C2×D72411yescharacteristic
120Q8—3624no—
121Q8—14481no—
122Q8—7242no—
123Q8—7242no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—3622no—
130C2×C4—3622no—
131C2×C4—3622no—
132C2×C4—3622no—
133C2×C4—7241no—
134C2×C4—7241no—
135C2×C4—7241no—
136C2×C4—7241no—
137C2×C4Dic72412yescomplemented
138C2×C4D4×D18112yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1915378611yescharacteristic
143C22C2×Dic72411yescharacteristic
144C22#1915384911yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—7242no—
149C4—7242no—
150C4—7242no—
151C4—7242no—
152C2—21no—
153C2—21no—
154C2#2604143711yescharacteristic
155C2#2604146411yescharacteristic
156C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#324×C181C2
socleC22×C181#120
prime radicalC23×C181C2×D4
hypercenter#324D181
hypocenterC181#1605
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C181Q16semidirect
78. C181#1605semidirect
137. C2×C4Dic724semidirect

Maximal subgroups

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

subgroupindex
#260419102
#260414372
#260411992
#260410172
#324×C1812
#1605181