Finite Group Databasebeta

← #37533916

Subgroups — #37533916

Order 23168.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic181C211yesmaximal, characteristic
2C2×#19153748C212yesmaximal, complemented
3#26041139C211yesmaximal, characteristic
4#26041042C212yesmaximal, complemented
5C2×C22⋊C8×C181C211yesmaximal, characteristic
6C23×Dic181—21no—
7C2×C4⋊Dic181—24no—
8C2×C4⋊Dic181—21no—
9C2×C4⋊Dic181C2211yescharacteristic
10C2×C4⋊Dic181C2211yescharacteristic
11C2×C4⋊Dic181C2212yes—
12C2×C22⋊Dic181—21no—
13#19153748—24no—
14#19153723—21no—
15C23×C4×C181C2211yescharacteristic
16C22×C8×C181C2212yes—
17C22⋊C8×C181C414yescomplemented
18C22×Dic181—24no—
19C22×Dic181—22no—
20C22×Dic181—21no—
21C22×Dic181—21no—
22C22×Dic181—42no—
23C22×Dic181—42no—
24C4⋊Dic181—24no—
25C4⋊Dic181—22no—
26C4⋊Dic181—22no—
27C4⋊Dic181—22no—
28C4⋊Dic181D414yes—
29C22⋊Dic181—42no—
30C24×C181D411yescharacteristic
31C22×C4×C181—22no—
32C22×C4×C181—22no—
33C22×C4×C181C2×C412yes—
34C22×C4×C181D411yescharacteristic
35C22×C4×C181Q812yes—
36C22×C4×C181C2311yescharacteristic
37C2×C8×C181—22no—
38C2×C8×C181—22no—
39C2×C8×C181C2×C414yescomplemented
40C2×Dic181—24no—
41C2×Dic181—48no—
42C2×Dic181—42no—
43C2×Dic181—42no—
44C2×Dic181—41no—
45C2×Dic181—41no—
46C2×Dic181—41no—
47C2×Dic181—41no—
48C2×Dic181—41no—
49C2×Dic181—41no—
50C23×C181—21no—
51C23×C181—21no—
52C23×C181—21no—
53C23×C181—21no—
54C23×C181C4⋊C412yes—
55C23×C181SD1612yes—
56C23×C181SD1612yes—
57C23×C181C2×D411yescharacteristic
58C2×C4×C181—21no—
59C2×C4×C181—21no—
60C2×C4×C181—24no—
61C2×C4×C181—24no—
62C2×C4×C181C4⋊C412yes—
63C2×C4×C181C22×C411yescharacteristic
64C2×C4×C181C2×D412yes—
65C2×C4×C181C2×Q811yescharacteristic
66C2×C4×C181#4112yes—
67C8×C181—24no—
68C22×C181—22no—
69C22×C181—22no—
70C22×C181—22no—
71C22×C181—22no—
72C22×C181—22no—
73C22×C181—22no—
74C22×C181#10614yescomplemented
75C22×C181C2×C4⋊C411yescharacteristic
76C22×C181#12011yescharacteristic
77C22×C181#12311yescharacteristic
78C22×C181C2×SD1611yescharacteristic
79C22×C181C2×SD1611yescharacteristic
80C22×C181AGL(1,ℤ/8)11yescharacteristic
81C22×C181#13711yescharacteristic
82C4×C181—22no—
83C4×C181C4×D412yes—
84C4×C181C22⋊Q812yes—
85Dic181—44no—
86Dic181—81no—
87Dic181—81no—
88C2×C181—24no—
89C2×C181#38511yescharacteristic
90C2×C181C2×#10611yescharacteristic
91C2×C181#42811yescharacteristic
92C2×C181#44811yescharacteristic
93C2×C181#45011yescharacteristic
94C2×C181#48111yescharacteristic
95C2×C181#48311yescharacteristic
96C181#185311yescharacteristic, complemented, Sylow
97#1853—1811nomaximal, Sylow
98C22×C4⋊C4—1811no—
99C2×#106—1812no—
100C2×C22⋊C8D18111yescharacteristic
101#380—1811no—
102#340—1812no—
103C23×C4—3621no—
104C23×C4C2×D18111yescharacteristic
105C22×C8C2×D18112yes—
106C2×C4⋊C4—3621no—
107C2×C4⋊C4—3624no—
108C2×C4⋊C4—1811no—
109C2×C4⋊C4—1811no—
110C2×C4⋊C4—1812no—
111C2×C22⋊C4—3621no—
112#106—3624no—
113C22⋊C8Dic18114yescomplemented
114#95—3621no—
115C24D72411yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—3624no—
119C22×C4—3622no—
120C22×C4—3621no—
121C22×C4—3621no—
122C22×C4—7242no—
123C22×C4—7242no—
124C22×C4C2×Dic18112yes—
125C22×C4D72411yescharacteristic
126C22×C4Dic36212yes—
127C22×C4C22×D18111yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic18114yescomplemented
131C4⋊C4—3624no—
132C4⋊C4—3622no—
133C4⋊C4—3622no—
134C4⋊C4—3622no—
135C4⋊C4—1814no—
136C22⋊C4—7242no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic18112yes—
142C23#1566843812yes—
143C23#1566843812yes—
144C23C2×D72411yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—3624no—
150C2×C4—7241no—
151C2×C4—7241no—
152C2×C4—7241no—
153C2×C4—7241no—
154C2×C4—7241no—
155C2×C4—7241no—
156C2×C4—7248no—
157C2×C4—7242no—
158C2×C4—7242no—
159C2×C4C4⋊Dic18112yes—
160C2×C4C22×Dic18111yescharacteristic
161C2×C4D4×D18112yes—
162C2×C4C2×Dic36211yescharacteristic
163C2×C4#1566844812yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#1915374814yescomplemented
172C22C2×C4⋊Dic18111yescharacteristic
173C22#1915378611yescharacteristic
174C22#1915380011yescharacteristic
175C22C2×#1566843811yescharacteristic
176C22C2×#1566843811yescharacteristic
177C22#1915384411yescharacteristic
178C22#1915384911yescharacteristic
179C4—22no—
180C4—14481no—
181C4—14481no—
182C4—7244no—
183C4D4×Dic18112yes—
184C4#1915379312yes—
185C2—24no—
186C2#2604115411yescharacteristic
187C2C2×#1915374811yescharacteristic
188C2#2604132111yescharacteristic
189C2#2604142011yescharacteristic
190C2#2604143211yescharacteristic
191C2#2604159811yescharacteristic
192C2#2604161011yescharacteristic
subgroupG/H
centerC23C2×D724
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
FittingC2×C22⋊C8×C181C2
socleC23×C181C2×D4
prime radicalC24×C181D4
cosocleC22×C4×C181C23
hypercenterC2×C22⋊C8D181
hypocenterC181#1853
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#19153748C2semidirect
4. #26041042C2semidirect
17. C22⋊C8×C181C4semidirect
39. C2×C8×C181C2×C4semidirect
74. C22×C181#106semidirect
96. C181#1853semidirect
113. C22⋊C8Dic181semidirect
130. C2×C8C2×Dic181semidirect
171. C22#19153748semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1812
C2×#191537482
#260411392
#260410422
C2×C22⋊C8×C1812
#1853181