Finite Group Databasebeta

← #37532680

Subgroups — #37532680

Order 23168.

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

normal subgroups only

This group has 256 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#26041905C211yesmaximal, characteristic
2#26041437C212yesmaximal, complemented
3#26041191C211yesmaximal, characteristic
4#26041017C212yesmaximal, complemented
5#323×C181C211yesmaximal, characteristic
6C22×Dic362C2211yescharacteristic
7C2×Dic724—22no—
8#19153795—21no—
9#19153795C2212yes—
10#19153793—22no—
11#19153793—22no—
12C2×C22⋊Dic181—21no—
13C2×C22⋊Dic181—21no—
14#19153741—22no—
15#19153723—21no—
16#19153723C2211yescharacteristic
17C2×C22⋊C4×C181C2211yescharacteristic
18C22⋊C8×C181C2212yes—
19C2×Dic362—21no—
20C2×Dic362—22no—
21C2×Dic362—42no—
22C2×Dic362D412yes—
23C22×Dic181—21no—
24C22×Dic181—21no—
25C22×Dic181—21no—
26C22×Dic181—21no—
27Dic724—44no—
28C4⋊Dic181—21no—
29C4⋊Dic181D412yes—
30#15668428—22no—
31#15668428—22no—
32C22⋊Dic181—22no—
33C22⋊Dic181—22no—
34C22⋊Dic181—42no—
35C24×C181D411yescharacteristic
36C22×C4×C181D411yescharacteristic
37C22×C4×C181C2311yescharacteristic
38C2×C8×C181—22no—
39C22⋊C4×C181—22no—
40Dic362—24no—
41Dic362—42no—
42Dic362—42no—
43Dic362—81no—
44C2×Dic181—22no—
45C2×Dic181—22no—
46C2×Dic181—22no—
47C2×Dic181—41no—
48C2×Dic181—41no—
49C2×Dic181—41no—
50C2×Dic181—41no—
51C2×Dic181—41no—
52C2×Dic181—81no—
53C2×Dic181—81no—
54C23×C181—21no—
55C23×C181—21no—
56C23×C181—41no—
57C23×C181—41no—
58C23×C181Q1612yescomplemented
59C23×C181C2×D411yescharacteristic
60C2×C4×C181—21no—
61C2×C4×C181—41no—
62C2×C4×C181—41no—
63C2×C4×C181C2×D412yes—
64C8×C181—42no—
65C22×C181—21no—
66C22×C181—21no—
67C22×C181—22no—
68C22×C181—42no—
69C22×C181—41no—
70C22×C181—41no—
71C22×C181—81no—
72C22×C181#12011yescharacteristic
73C22×C181C2×Q1611yescharacteristic
74C22×C181#13711yescharacteristic
75C4×C181—22no—
76C4×C181—41no—
77Dic181—42no—
78Dic181—42no—
79Dic181—42no—
80Dic181—81no—
81C2×C181—21no—
82C2×C181—21no—
83C2×C181—42no—
84C2×C181#45111yescharacteristic
85C2×C181#45511yescharacteristic
86C2×C181B(4,𝔽2)11yescharacteristic
87C181#160211yescharacteristic, complemented, Sylow
88#1602—1811nomaximal, Sylow
89#543—1811no—
90#451—1812no—
91#393—1811no—
92#328—1812no—
93#323D18111yescharacteristic
94C22×Q8—1811no—
95C2×Q16—3622no—
96C22⋊Q8—3622no—
97C22⋊Q8—3622no—
98C22⋊Q8—3621no—
99C22⋊Q8—1812no—
100C2×C22⋊C4—3621no—
101C2×C22⋊C4—3621no—
102C2×C22⋊C4C2×D18111yescharacteristic
103#103—3622no—
104C22⋊C8C2×D18112yes—
105#95—3621no—
106#95—1811no—
107C24D72411yescharacteristic
108C2×Q8—3621no—
109C2×Q8—3622no—
110C2×Q8—1812no—
111C2×Q8—7242no—
112C22×C4—3621no—
113C22×C4—3621no—
114C22×C4—3621no—
115C22×C4—3621no—
116C22×C4D72411yescharacteristic
117C22×C4C22×D18111yescharacteristic
118Q16—7244no—
119C2×C8—22no—
120C4⋊C4—3622no—
121C4⋊C4—3621no—
122C4⋊C4—3622no—
123C4⋊C4—1812no—
124C22⋊C4—22no—
125C22⋊C4—3622no—
126C22⋊C4—3622no—
127C22⋊C4—7242no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic72412yescomplemented
133C23C2×D72411yescharacteristic
134Q8—3624no—
135Q8—14481no—
136Q8—7242no—
137Q8—7242no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—3622no—
142C2×C4—3622no—
143C2×C4—3622no—
144C2×C4—14481no—
145C2×C4—14481no—
146C2×C4—7241no—
147C2×C4—7241no—
148C2×C4—7241no—
149C2×C4—7241no—
150C2×C4—7241no—
151C2×C4D4×D18112yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1915378611yescharacteristic
161C22C2×Dic72411yescharacteristic
162C22#1915384911yescharacteristic
163C4—22no—
164C4—41no—
165C4—14481no—
166C4—7242no—
167C4—7242no—
168C4—7242no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2604143711yescharacteristic
173C2#2604145911yescharacteristic
174C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#323×C181C2
socleC22×C181#120
prime radicalC24×C181D4
hypercenter#323D181
hypocenterC181#1602
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26041437C2semidirect
4. #26041017C2semidirect
58. C23×C181Q16semidirect
87. C181#1602semidirect
132. C23Dic724semidirect

Maximal subgroups

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

subgroupindex
#260419052
#260414372
#260411912
#260410172
#323×C1812
#1602181