Finite Group Databasebeta

← #37536809

Subgroups — #37536809

Order 23168.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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
2#26041186C211yesmaximal, characteristic
3#26041183C212yesmaximal
4#26041176C212yesmaximal
5#26041174C212yesmaximal
6C4×#15668428C212yesmaximal
7#26041130C212yesmaximal
8#26041130C212yesmaximal
9#391×C181C211yesmaximal, characteristic
10C22×Dic362C2211yescharacteristic
11#19153814C418yescomplemented
12C2×C4⋊Dic181C2211yescharacteristic
13C2×#15668428C2212yes—
14C2×#15668428C2212yes—
15C2×#15668428C2212yes—
16C2×#15668428C2212yes—
17C2×#15668428C2212yes—
18C2×C4×Dic181C2212yes—
19C2×C4×Dic181C2212yes—
20C2×C4×Dic181C2211yescharacteristic
21C2×C4×Dic181C2212yes—
22C2×C4×Dic181C2211yescharacteristic
23#19153723C2212yes—
24#19153723C2212yes—
25#19153723C2212yes—
26#19153723C2212yes—
27C22×Q8×C181C2211yescharacteristic
28C2×C4⋊C4×C181C2211yescharacteristic
29C2×C42×C181C2211yescharacteristic
30#95×C181C2212yes—
31#95×C181C2212yes—
32C2×Dic362—41no—
33C2×Dic362—41no—
34C2×Dic362C2×C414yes—
35C22×Dic181C2312yes—
36C22×Dic181C2312yes—
37C22×Dic181C2312yes—
38C22×Dic181C2312yes—
39C4⋊Dic181—41no—
40#15668428—24no—
41#15668428—42no—
42#15668428C2×C418yes—
43#15668428C2×C418yes—
44#15668428Q818yescomplemented
45C4×Dic181—24no—
46C4×Dic181—24no—
47C4×Dic181—42no—
48C4×Dic181—41no—
49C4×Dic181C2×C414yes—
50C2×Q8×C181—41no—
51C2×Q8×C181—41no—
52C2×Q8×C181C2×C414yes—
53C22×C4×C181C2312yes—
54C22×C4×C181C2311yescharacteristic
55C22×C4×C181C2311yescharacteristic
56C22×C4×C181C2311yescharacteristic
57C22×C4×C181C2312yes—
58C4⋊C4×C181—41no—
59C42×C181—41no—
60Dic362—44no—
61C2×Dic181—22no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—22no—
65C2×Dic181—22no—
66C2×Dic181—22no—
67C2×Dic181—22no—
68C2×Dic181C22×C414yes—
69C2×Dic181C22×C414yes—
70C2×Dic181C2×Q814yes—
71C2×Dic181C2×Q814yes—
72C2×Dic181#4114yes—
73C23×C181C2411yescharacteristic
74Q8×C181—44no—
75C2×C4×C181—22no—
76C2×C4×C181—21no—
77C2×C4×C181—22no—
78C2×C4×C181—21no—
79C2×C4×C181—21no—
80C2×C4×C181—21no—
81C2×C4×C181—21no—
82C2×C4×C181—21no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181—22no—
86C2×C4×C181C22×C412yes—
87C2×C4×C181C22×C412yes—
88C2×C4×C181C22×C412yes—
89C2×C4×C181C2×Q814yes—
90C22×C181C23×C411yescharacteristic
91C22×C181C22×Q812yes—
92C22×C181C2×#4112yes—
93C22×C181H(5,𝔽2)11yescharacteristic
94C22×C181H(5,𝔽2)11yescharacteristic
95C4×C181—22no—
96C4×C181—22no—
97C4×C181—22no—
98C4×C181—24no—
99C4×C181—42no—
100Dic181—24no—
101Dic181—24no—
102Dic181—42no—
103Dic181C4×Q818yes—
104C2×C181C2×C4×Q812yes—
105C2×C181#51811yescharacteristic
106C2×C181#55412yes—
107C2×C181#55511yescharacteristic
108C2×C181Q8211yescharacteristic
109C181#236911yescharacteristic, complemented, Sylow
110#2369—1811nomaximal, Sylow
111C2×C4⋊Q8—1811no—
112#391—1811no—
113#391—1812no—
114#391D18111yescharacteristic
115#389—1812no—
116#389—1812no—
117C4×C4⋊C4—1812no—
118C4×C4⋊C4—1812no—
119C4×C4⋊C4—1812no—
120C22×Q8—1811no—
121C22×Q8C2×D18111yescharacteristic
122C4⋊Q8—1818no—
123C2×C4⋊C4—1812no—
124C2×C4⋊C4—1812no—
125C2×C4⋊C4—1812no—
126C2×C4⋊C4—1812no—
127C2×C4⋊C4—1812no—
128C2×C4⋊C4—1811no—
129C2×C4⋊C4C2×D18111yescharacteristic
130C2×C42—1812no—
131C2×C42—1812no—
132C2×C42—1811no—
133C2×C42—1811no—
134C2×C42—1812no—
135C2×C42C2×D18111yescharacteristic
136#95—1812no—
137#95—1812no—
138#95—1812no—
139#95—1812no—
140#95C2×D18112yes—
141#95C2×D18112yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—1814no—
145C2×Q8—7241no—
146C2×Q8—7241no—
147C2×Q8C4×D18114yes—
148C22×C4—1812no—
149C22×C4—1812no—
150C22×C4—1812no—
151C22×C4—1812no—
152C22×C4C22×D18112yes—
153C22×C4C22×D18111yescharacteristic
154C22×C4C22×D18111yescharacteristic
155C22×C4C22×D18111yescharacteristic
156C22×C4C22×D18112yes—
157C4⋊C4—41no—
158C4⋊C4—3624no—
159C4⋊C4—1818no—
160C4⋊C4—1818no—
161C4⋊C4—1818no—
162C4⋊C4—7242no—
163C4⋊C4—7241no—
164C42—41no—
165C42—3624no—
166C42—3624no—
167C42—1814no—
168C42—7242no—
169C42—7241no—
170C23C23×D18111yescharacteristic
171Q8—44no—
172Q8—7244no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—3622no—
185C2×C4—3622no—
186C2×C4—3622no—
187C2×C4—3622no—
188C2×C4—3622no—
189C2×C4—3622no—
190C2×C4—3622no—
191C2×C4—1814no—
192C2×C4—1814no—
193C2×C4—1814no—
194C2×C4—1814no—
195C2×C4—1814no—
196C2×C4C2×C4×D18112yes—
197C2×C4C2×C4×D18112yes—
198C2×C4C2×C4×D18112yes—
199C2×C4Q8×D18114yes—
200C22C22×C4×D18111yescharacteristic
201C22C2×Q8×D18112yes—
202C22#41×D18112yes—
203C22#1915386111yescharacteristic
204C22#1915386111yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—3624no—
211C4—3624no—
212C4—1818no—
213C4—7242no—
214C2C4×Q8×D18112yes—
215C2#2604175211yescharacteristic
216C2#2604198312yes—
217C2#2604199111yescharacteristic
218C2#2604200911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#391×C181C2
socle, prime radical, cosocleC23×C181C24
hypercenter#391D181
hypocenterC181#2369
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #19153814C4semidirect
44. #15668428Q8semidirect
109. C181#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#191538142
#260411862
#260411832
#260411762
#260411742
C4×#156684282
#260411302
#391×C1812
#2369181