Finite Group Databasebeta

← #37536290

Subgroups — #37536290

Order 23168.

588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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×#19153814C211yesmaximal, characteristic
2C2×Q8×Dic181C211yesmaximal, characteristic
3C2×#19153781C211yesmaximal, characteristic
4#26041185C211yesmaximal, characteristic
5#26041176C214yesmaximal
6#26041148C211yesmaximal, characteristic
7#26041130C214yesmaximal
8C42×Dic181C211yesmaximal, characteristic
9#391×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11#19153814C418yescomplemented
12Q8×Dic181—24no—
13#19153781—24no—
14C2×C4⋊Dic181C2211yescharacteristic
15C2×C4⋊Dic181C2212yes—
16C2×#15668428C2214yes—
17C2×#15668428C2212yes—
18C2×#15668428C2214yes—
19C2×C4×Dic181C2212yes—
20C2×C4×Dic181C2212yes—
21C2×C4×Dic181C2212yes—
22C2×C4×Dic181C2211yescharacteristic
23C2×C4×Dic181C2211yescharacteristic
24C2×C4×Dic181C2212yes—
25#19153723C2214yes—
26C22×Q8×C181C2211yescharacteristic
27C2×C4⋊C4×C181C2211yescharacteristic
28C2×C42×C181C2211yescharacteristic
29#95×C181C2214yes—
30C2×Dic362—22no—
31C2×Dic362—22no—
32C2×Dic362C2×C414yes—
33C22×Dic181C2312yes—
34C22×Dic181C2312yes—
35C22×Dic181C2311yescharacteristic
36C22×Dic181C2312yes—
37C22×Dic181C2311yescharacteristic
38C4⋊Dic181—24no—
39C4⋊Dic181—22no—
40#15668428—24no—
41#15668428—28no—
42#15668428C2×C4116yes—
43C4×Dic181—24no—
44C4×Dic181—24no—
45C4×Dic181—24no—
46C4×Dic181C2×C414yes—
47C4×Dic181D414yes—
48C4×Dic181Q818yescomplemented
49C2×Q8×C181—22no—
50C2×Q8×C181—22no—
51C2×Q8×C181C2×C414yes—
52C22×C4×C181C2312yes—
53C22×C4×C181C2312yes—
54C22×C4×C181C2312yes—
55C22×C4×C181C2311yescharacteristic
56C4⋊C4×C181—22no—
57C42×C181—22no—
58Dic362—28no—
59C2×Dic181—22no—
60C2×Dic181—22no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—24no—
64C2×Dic181C22×C414yes—
65C2×Dic181C22×C414yes—
66C2×Dic181C2×D412yes—
67C2×Dic181C2×D412yes—
68C2×Dic181C2×Q814yes—
69C2×Dic181C2×Q814yes—
70C2×Dic181#4114yes—
71C23×C181C2411yescharacteristic
72Q8×C181—28no—
73C2×C4×C181—22no—
74C2×C4×C181—22no—
75C2×C4×C181—22no—
76C2×C4×C181—22no—
77C2×C4×C181—22no—
78C2×C4×C181—24no—
79C2×C4×C181C22×C414yes—
80C2×C4×C181C22×C412yes—
81C2×C4×C181C2×D412yes—
82C2×C4×C181C2×Q814yes—
83C2×C4×C181#4112yes—
84C22×C181C23×C411yescharacteristic
85C22×C181C22×D411yescharacteristic
86C22×C181C22×Q812yes—
87C22×C181C2×#4112yes—
88C22×C181C2×#4111yescharacteristic
89C4×C181—24no—
90C4×C181—24no—
91C4×C181—42no—
92C4×C181C4×D414yes—
93Dic181—24no—
94Dic181—22no—
95Dic181—42no—
96Dic181C4×Q818yescomplemented
97Dic181C4⋊Q814yescomplemented
98C2×C181C2×C4×D411yescharacteristic
99C2×C181C2×C4×Q812yes—
100C2×C181C2×C4⋊Q811yescharacteristic
101C2×C181#53211yescharacteristic
102C2×C181#53312yes—
103C181C4×C4⋊Q811yescharacteristic, complemented, Sylow
104C4×C4⋊Q8—1811nomaximal, Sylow
105C2×C4⋊Q8—1811no—
106C2×C4×Q8—1811no—
107C2×C4×Q8—1811no—
108#391—1811no—
109#391D18111yescharacteristic
110#389—1814no—
111#384—1811no—
112C4×C4⋊C4—1814no—
113C43—1811no—
114C22×Q8—1811no—
115C22×Q8C2×D18111yescharacteristic
116C4⋊Q8—1818no—
117C4×Q8—3624no—
118C4×Q8—3624no—
119C2×C4⋊C4—1812no—
120C2×C4⋊C4—1811no—
121C2×C4⋊C4—1812no—
122C2×C4⋊C4—1814no—
123C2×C4⋊C4—1814no—
124C2×C4⋊C4C2×D18111yescharacteristic
125C2×C42—1812no—
126C2×C42—1812no—
127C2×C42—1812no—
128C2×C42—1811no—
129C2×C42—1811no—
130C2×C42—1812no—
131C2×C42C2×D18111yescharacteristic
132#95—1814no—
133#95C2×D18114yes—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—3622no—
137C2×Q8—3622no—
138C2×Q8—1814no—
139C2×Q8C4×D18114yes—
140C22×C4—1812no—
141C22×C4—1812no—
142C22×C4—1811no—
143C22×C4—1811no—
144C22×C4—1812no—
145C22×C4C22×D18112yes—
146C22×C4C22×D18112yes—
147C22×C4C22×D18112yes—
148C22×C4C22×D18111yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—3624no—
151C4⋊C4—3624no—
152C4⋊C4—3628no—
153C4⋊C4—3622no—
154C4⋊C4—18116no—
155C42—22no—
156C42—3624no—
157C42—3624no—
158C42—3624no—
159C42—1814no—
160C42—1814no—
161C42—1818no—
162C23C23×D18111yescharacteristic
163Q8—28no—
164Q8—3628no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—3622no—
172C2×C4—3622no—
173C2×C4—3622no—
174C2×C4—3622no—
175C2×C4—3624no—
176C2×C4—1814no—
177C2×C4—1812no—
178C2×C4—1814no—
179C2×C4—1812no—
180C2×C4—1814no—
181C2×C4—1814no—
182C2×C4—1814no—
183C2×C4C2×C4×D18114yes—
184C2×C4C2×C4×D18112yes—
185C2×C4D4×D18112yes—
186C2×C4Q8×D18114yes—
187C2×C4#1566844812yes—
188C22C22×C4×D18111yescharacteristic
189C22C2×D4×D18111yescharacteristic
190C22C2×Q8×D18112yes—
191C22#41×D18112yes—
192C22C2×#1566844811yescharacteristic
193C4—24no—
194C4—24no—
195C4—42no—
196C4—3624no—
197C4—3622no—
198C4—1818no—
199C4—1814no—
200C4—7242no—
201C4#1915377614yes—
202C2C2×#1915377611yescharacteristic
203C2C4×Q8×D18112yes—
204C2C4⋊Q8×D18111yescharacteristic
205C2#2604182811yescharacteristic
206C2#2604183112yes—
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#391×C181C2
socle, prime radical, cosocleC23×C181C24
hypercenter#391D181
hypocenterC181C4×C4⋊Q8
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #19153814C4semidirect
48. C4×Dic181Q8semidirect
96. Dic181C4×Q8semidirect
97. Dic181C4⋊Q8semidirect
103. C181C4×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#191538142
C2×Q8×Dic1812
C2×#191537812
#260411852
#260411762
#260411482
#260411302
C42×Dic1812
#391×C1812
C4×C4⋊Q8181