Finite Group Databasebeta

← #37544105

Subgroups — #37544105

Order 23168.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#26041832C211yesmaximal, characteristic, complemented
2C2×#19153803C211yesmaximal, characteristic
3#26041637C214yesmaximal, complemented
4#26041632C214yesmaximal, complemented
5#26041289C211yesmaximal, characteristic, complemented
6#26041287C211yesmaximal, characteristic, complemented
7#26041284C211yesmaximal, characteristic
8M16×Dic181C211yesmaximal, characteristic, complemented
9#418×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11C2×#15668433C2211yescharacteristic
12#19153815C2212yescomplemented
13#19153807C2212yescomplemented
14#19153803—22no—
15#19153803C2212yes—
16#19153803C2212yes—
17#19153793—22no—
18C4×Dic362—22no—
19#19153775C2211yescharacteristic
20C2×C22⋊Dic181—22no—
21C2×C4×Dic181C2211yescharacteristic
22#19153741C2214yescomplemented
23#19153739C2214yescomplemented
24#19153737C2214yes—
25#19153726C2212yescomplemented
26C8×Dic181C2212yescomplemented
27C22×D4×C181C2211yescharacteristic
28C2×M16×C181C2211yescharacteristic
29#117×C181C2211yescharacteristic
30#102×C181C2214yes—
31C2×Dic362—22no—
32C2×Dic362—21no—
33C2×Dic362—22no—
34C2×Dic362—22no—
35C2×Dic362C2312yes—
36C22×Dic181—21no—
37C22×Dic181D412yes—
38#15668433—22no—
39C2×#13089678C2312yes—
40C4⋊Dic181—22no—
41C4⋊Dic181C2312yes—
42#15668428—22no—
43#15668428—22no—
44C22⋊Dic181—24no—
45C22⋊Dic181—24no—
46C22⋊Dic181—21no—
47C4×Dic181—21no—
48C4×Dic181D412yes—
49C4×Dic181C2311yescharacteristic
50C4×Dic181C2311yescharacteristic
51C24×C181—21no—
52C2×D4×C181—21no—
53C2×D4×C181—22no—
54C2×D4×C181—22no—
55C2×D4×C181C2312yes—
56C22×C4×C181C2311yescharacteristic
57M16×C181—22no—
58C2×C8×C181C2312yes—
59C4⋊C4×C181C2312yes—
60C22⋊C4×C181—21no—
61C42×C181—21no—
62Dic362—22no—
63Dic362—22no—
64Dic362—44no—
65Dic362—42no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181—42no—
69C2×Dic181—42no—
70C2×Dic181C2×D412yes—
71C2×Dic181C2×D412yes—
72#13089678—22no—
73C23×C181—22no—
74C23×C181—44no—
75C23×C181—42no—
76C23×C181C2×D411yescharacteristic
77D4×C181—22no—
78D4×C181—22no—
79D4×C181—42no—
80C2×C4×C181—22no—
81C2×C4×C181C2×D411yescharacteristic
82C2×C4×C181#4112yes—
83C2×C4×C181#4112yes—
84C2×C4×C181C2411yescharacteristic
85C8×C181—22no—
86C22×C181—22no—
87C22×C181—42no—
88C22×C181—44no—
89C22×C181—84no—
90C22×C181#12412yes—
91C22×C181C22×D411yescharacteristic
92C4×C181—42no—
93C4×C181#12412yes—
94C4×C181C2×#4111yescharacteristic
95C4×C181C2×#4111yescharacteristic
96Dic181—22no—
97Dic181—22no—
98Dic181—42no—
99Dic181—42no—
100C2×C181—22no—
101C2×C181—82no—
102C2×C181C2×#12411yescharacteristic
103C2×C181#57512yes—
104C181#314011yescharacteristic, complemented, Sylow
105#3140—1811nomaximal, Sylow
106#533—1811no—
107C2×#124—1811no—
108#489—1814no—
109#488—1814no—
110#419—1811no—
111#419—1811no—
112#418—1811no—
113#418D18111yescharacteristic
114C4×M16—1811no—
115C22×Q8—1811no—
116C22×D4C2×D18111yescharacteristic
117C2×M16—1811no—
118C2×M16C2×D18111yescharacteristic
119C4⋊Q8—1812no—
120#125—1812no—
121#124—3622no—
122#124—1812no—
123#124—1812no—
124C22⋊Q8—3622no—
125C4×Q8—3622no—
126#117—1811no—
127#117C2×D18111yescharacteristic
128C2×C22⋊C4—3622no—
129C2×C42—1811no—
130#103—1814no—
131#103—1814no—
132#102—1814no—
133#102C2×D18114yes—
134#97—1812no—
135C4×C8—1812no—
136C24—21no—
137C2×Q8—3622no—
138C2×Q8—3621no—
139C2×Q8—3622no—
140C2×Q8—3622no—
141C2×Q8—1812no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22×D18112yes—
146C22×C4—3621no—
147C22×C4—1812no—
148C22×C4C22×D18111yescharacteristic
149M16—22no—
150M16—3622no—
151C2×C8—1812no—
152C2×C8C22×D18112yes—
153C4⋊C4—3622no—
154C4⋊C4—3622no—
155C4⋊C4—3622no—
156C4⋊C4—1812no—
157C4⋊C4C22×D18112yes—
158C22⋊C4—21no—
159C22⋊C4—3621no—
160C22⋊C4—3624no—
161C22⋊C4—3624no—
162C42—21no—
163C42—3621no—
164C42—1812no—
165C42—1811no—
166C42—1811no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D18111yescharacteristic
171Q8—3622no—
172Q8—3622no—
173Q8—7244no—
174Q8—7242no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—1812no—
182C2×C4—1812no—
183C2×C4—7242no—
184C2×C4—7242no—
185C2×C4D4×D18111yescharacteristic
186C2×C4#1566844812yes—
187C2×C4#1566844912yes—
188C2×C4C23×D18111yescharacteristic
189C8—22no—
190C8—3622no—
191C22—22no—
192C22—42no—
193C22—44no—
194C22—84no—
195C22#1915380212yes—
196C22C2×D4×D18111yescharacteristic
197C4—42no—
198C4—3622no—
199C4—3622no—
200C4—7242no—
201C4—7242no—
202C4#1915380212yes—
203C4C2×#1566844811yescharacteristic
204C4C2×#1566844911yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#1915380211yescharacteristic
208C2#2604207512yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#418×C181C2
socleC22×C181C22×D4
prime radicalC22×D4×C181C22
hypercenter#418D181
hypocenterC181#3140
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #26041832C2semidirect
3. #26041637C2semidirect
4. #26041632C2semidirect
5. #26041289C2semidirect
6. #26041287C2semidirect
8. M16×Dic181C2semidirect
12. #19153815C22semidirect
13. #19153807C22semidirect
22. #19153741C22semidirect
23. #19153739C22semidirect
25. #19153726C22semidirect
26. C8×Dic181C22semidirect
104. C181#3140semidirect

Maximal subgroups

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

subgroupindex
#260418322
C2×#191538032
#260416372
#260416322
#260412892
#260412872
#260412842
M16×Dic1812
#418×C1812
#3140181