Finite Group Databasebeta

← #37549493

Subgroups — #37549493

Order 23168.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#19153879C212yesmaximal
2#26042238C211yesmaximal, characteristic
3C4×C4⋊#10624382C211yesmaximal, characteristic
4#26042152C212yesmaximal
5C4×Q8×D181C211yesmaximal, characteristic
6C2×D181⋊C8C2212yes—
7C2×C4⋊#10624382C2212yes—
8C2×C4×#10624382—21no—
9#19153879C418yescomplemented
10#19153871—21no—
11#19153871—21no—
12(C2×#10624382)⋊C4—21no—
13C2×Q8×D181C2211yescharacteristic
14Q8×Dic181C411yescharacteristic, complemented
15C4×Dic362—21no—
16C4×Dic362C411yescharacteristic, complemented
17#19153781—21no—
18#19153781C411yescharacteristic, complemented
19C4⋊C4×D181—21no—
20C4⋊C4×D181C2211yescharacteristic
21C42×D181—21no—
22C42×D181C2211yescharacteristic
23C4×Q8×C181C411yescharacteristic, complemented
24C22×#10624382—22no—
25D181⋊C8—22no—
26D181⋊C8C2×C414yes—
27C2×#13089685—22no—
28C4⋊#10624382—22no—
29C4⋊#10624382C2×C414yes—
30C4×#10624382—42no—
31C2×Dic362—21no—
32C2×Dic362C2×C411yescharacteristic
33Q8×D181—22no—
34Q8×D181C2×C414yes—
35C2×C4×D181—21no—
36C2×C4×D181—21no—
37C2×C4×D181D411yescharacteristic
38C2×C4×D181D411yescharacteristic
39C2×C4×D181C2311yescharacteristic
40C4⋊Dic181—21no—
41C4⋊Dic181C2×C411yescharacteristic
42#15668428—21no—
43#15668428—21no—
44#15668428—21no—
45C4×Dic181—21no—
46C4×Dic181—21no—
47C4×Dic181—21no—
48C4×Dic181C2×C411yescharacteristic
49C4×Dic181D411yescharacteristic
50C4×Dic181D411yescharacteristic
51C2×Q8×C181C2×C411yescharacteristic
52C4⋊C4×C181—21no—
53C4⋊C4×C181C2×C411yescharacteristic
54C42×C181—21no—
55C42×C181C2×C411yescharacteristic
56C2×#10624382—24no—
57C2×#10624382—42no—
58C2×#10624382—42no—
59#13089685—24no—
60C22×D181C2×D411yescharacteristic
61Dic362—21no—
62Dic362—24no—
63Dic362C4212yescomplemented
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181C22⋊C411yescharacteristic
67C2×Dic181C22⋊C411yescharacteristic
68C2×Dic181C2×D411yescharacteristic
69C4×D181—24no—
70C4×D181—21no—
71C4×D181—21no—
72C4×D181—22no—
73C4×D181—41no—
74C4×D181—41no—
75C4×D181C22×C412yes—
76C4×D181#4112yes—
77Q8×C181—21no—
78Q8×C181C4212yescomplemented
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181C22⋊C411yescharacteristic
82C2×C4×C181C22⋊C411yescharacteristic
83C2×C4×C181C22×C411yescharacteristic
84C22×C181C2×C22⋊C411yescharacteristic
85C2×D181C4×D412yes—
86C2×D181AGL(1,ℤ/8)11yescharacteristic
87C2×D181#13712yes—
88C2×D181#13711yescharacteristic
89#10624382—44no—
90C4×C181—22no—
91C4×C181—21no—
92C4×C181—21no—
93C4×C181—41no—
94C4×C181C2×C4211yescharacteristic
95C4×C181#11711yescharacteristic
96Dic181—22no—
97Dic181—21no—
98Dic181—21no—
99Dic181—41no—
100Dic181C4×D412yes—
101C2×C181C4×C22⋊C411yescharacteristic
102C2×C181#41711yescharacteristic
103C2×C181#41911yescharacteristic
104D181#44012yes—
105D181#44112yes—
106C181#176311yescharacteristic, complemented, Sylow
107#1763—1811nomaximal, Sylow
108C2×C4×Q8—1811no—
109C2×#103—1812no—
110C2×#97—1811no—
111C4×C4⋊C4—1811no—
112#340—1812no—
113C22×Q8—1811no—
114C22×C8—1812no—
115C4×Q8—3621no—
116C4×Q8—3621no—
117C4×Q8—1811no—
118C4×Q8—1811no—
119C4×Q8—1811no—
120C4×Q8#1062438211yescharacteristic, complemented
121C2×C4⋊C4—3621no—
122C2×C4⋊C4—1812no—
123C2×C4⋊C4—1811no—
124C2×C42—3621no—
125C2×C42—3621no—
126C2×C42—1811no—
127#103—1818no—
128#97—3621no—
129#97—3621no—
130#95—3621no—
131C2×Q8—3621no—
132C2×Q8—3622no—
133C2×Q8—1814no—
134C2×Q8—1811no—
135C2×Q8C2×#1062438211yescharacteristic
136C22×C4—3621no—
137C22×C4—3621no—
138C22×C4—3622no—
139C22×C4—1811no—
140C22×C4—1811no—
141C22×C4—1811no—
142C2×C8—3622no—
143C2×C8—3622no—
144C2×C8—1814no—
145C4⋊C4—21no—
146C4⋊C4—3621no—
147C4⋊C4—3621no—
148C4⋊C4—3621no—
149C4⋊C4—3622no—
150C4⋊C4—3621no—
151C4⋊C4—1814no—
152C4⋊C4—1811no—
153C4⋊C4C2×#1062438211yescharacteristic
154C42—21no—
155C42—3621no—
156C42—3621no—
157C42—3621no—
158C42—1811no—
159C42—1811no—
160C42—1811no—
161C42—7242no—
162C42C2×#1062438211yescharacteristic
163C23—1811no—
164Q8—21no—
165Q8—3624no—
166Q8—3621no—
167Q8—1812no—
168Q8C4×#1062438212yescomplemented
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—3621no—
172C2×C4—3624no—
173C2×C4—3621no—
174C2×C4—3622no—
175C2×C4—3621no—
176C2×C4—3624no—
177C2×C4—3621no—
178C2×C4—1811no—
179C2×C4—1811no—
180C2×C4—1811no—
181C2×C4—1812no—
182C2×C4—1812no—
183C2×C4—7241no—
184C2×C4—7241no—
185C2×C4—7242no—
186C2×C4—7242no—
187C2×C4C22⋊#1062438211yescharacteristic
188C2×C4C22⋊#1062438211yescharacteristic
189C2×C4C22×#1062438211yescharacteristic
190C8—3624no—
191C22—1812no—
192C22—1811no—
193C22—1811no—
194C22—1812no—
195C22C2×C22⋊#1062438211yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—3622no—
201C4—3621no—
202C4—3621no—
203C4—1812no—
204C4—7244no—
205C4—7241no—
206C4C2×C4×#1062438211yescharacteristic
207C4#1915389511yescharacteristic
208C2—1812no—
209C2—1812no—
210C2C4×C22⋊#1062438211yescharacteristic
211C2#2604227011yescharacteristic
212C2#2604227211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC4×C181C2×C42
FrattiniC2×C4C22×#10624382
FittingC4×Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×D181C2×D4
cosocleC2×C4×D181C23
hypercenterC4×Q8#10624382
hypocenterC181#1763
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
9. #19153879C4semidirect
14. Q8×Dic181C4semidirect
16. C4×Dic362C4semidirect
18. #19153781C4semidirect
23. C4×Q8×C181C4semidirect
63. Dic362C42semidirect
78. Q8×C181C42semidirect
106. C181#1763semidirect
120. C4×Q8#10624382semidirect
168. Q8C4×#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538792
#260422382
C4×C4⋊#106243822
#260421522
C4×Q8×D1812
#1763181