Finite Group Databasebeta

← #37534845

Subgroups — #37534845

Order 23168.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19153784C212yesmaximal, complemented
2C2×#19153731C212yesmaximal, complemented
3#26041134C211yesmaximal, characteristic, complemented
4#26041046C211yesmaximal, characteristic, complemented
5#534×C181C211yesmaximal, characteristic, complemented
6C24×D181C2211yescharacteristic
7C22×C22⋊D181—22no—
8#19153784—24no—
9#19153784—24no—
10C2×C22⋊Dic181C2211yescharacteristic
11C2×C22⋊Dic181C2212yes—
12C2×#15668426—22no—
13C2×#15668426—22no—
14#19153731—24no—
15C22×D4×C181C2212yes—
16#123×C181—22no—
17#121×C181—22no—
18#120×C181—22no—
19C2×C22⋊C4×C181C2211yescharacteristic
20C23×D181—21no—
21C23×D181—22no—
22C23×D181—22no—
23C23×D181—21no—
24C23×D181—44no—
25C23×D181D412yescomplemented
26C2×C22⋊D181—28no—
27C2×C22⋊D181—48no—
28C22×Dic181—22no—
29C22×Dic181—21no—
30C22⋊Dic181—22no—
31C22⋊Dic181—22no—
32C22⋊Dic181D414yescomplemented
33#15668426—28no—
34#15668426—44no—
35C24×C181D412yescomplemented
36C24×C181C2311yescharacteristic
37C2×D4×C181—22no—
38C2×D4×C181—22no—
39C2×D4×C181—22no—
40C2×D4×C181—22no—
41C2×D4×C181—22no—
42C2×D4×C181—22no—
43C2×D4×C181—22no—
44C22×C4×C181D412yescomplemented
45C22×C4×C181D412yescomplemented
46C4⋊C4×C181—22no—
47C22⋊C4×C181—24no—
48C22⋊C4×C181—22no—
49C22×D181—24no—
50C22×D181—28no—
51C22×D181—24no—
52C22×D181—22no—
53C22×D181—44no—
54C22×D181—48no—
55C22×D181—44no—
56C22×D181—42no—
57C22×D181—44no—
58C22×D181—41no—
59C22×D181—42no—
60C22×D181—41no—
61C22⋊D181—416no—
62C2×Dic181—24no—
63C2×Dic181—44no—
64C2×Dic181—42no—
65C2×Dic181—41no—
66C23×C181—22no—
67C23×C181—22no—
68C23×C181—22no—
69C23×C181—42no—
70C23×C181—44no—
71C23×C181C2×D411yescharacteristic
72C23×C181C2×D412yes—
73C23×C181C2×D412yes—
74C23×C181C2×D411yescharacteristic
75C23×C181#4111yescharacteristic
76D4×C181—42no—
77D4×C181—42no—
78D4×C181—42no—
79D4×C181—42no—
80D4×C181—82no—
81C2×C4×C181—22no—
82C2×C4×C181—22no—
83C2×C4×C181—22no—
84C2×C4×C181—22no—
85C2×C4×C181—42no—
86C22×C181—22no—
87C22×C181—24no—
88C22×C181—41no—
89C22×C181—42no—
90C22×C181—41no—
91C22×C181—44no—
92C22×C181—42no—
93C22×C181—44no—
94C22×C181—82no—
95C22×C181#12012yes—
96C22×C181#12011yescharacteristic
97C22×C181#12111yescharacteristic
98C22×C181#12112yes—
99C22×C181#12711yescharacteristic
100C2×D181—28no—
101C2×D181—24no—
102C2×D181—44no—
103C2×D181—44no—
104C2×D181—42no—
105C2×D181—48no—
106C2×D181—42no—
107C2×D181—44no—
108C4×C181—42no—
109C4×C181—42no—
110Dic181—44no—
111Dic181—81no—
112C2×C181—22no—
113C2×C181—21no—
114C2×C181—21no—
115C2×C181—22no—
116C2×C181—82no—
117C2×C181#39211yescharacteristic
118C2×C181B(4,𝔽2)12yes—
119D181—44no—
120C181#202111yescharacteristic, complemented, Sylow
121#2021—1811nomaximal, Sylow
122#534D18111yescharacteristic, complemented
123C2×#120—1812no—
124C2×#99—1812no—
125#379—1811no—
126#342—1811no—
127C25—1811no—
128C22×D4—3622no—
129C22×D4C2×D18112yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—3624no—
134#120—3624no—
135C2×C22⋊C4—3622no—
136C2×C22⋊C4—3622no—
137C2×C22⋊C4—1811no—
138C2×C22⋊C4—1812no—
139C2×C22⋊C4C2×D18111yescharacteristic
140#99—3624no—
141C24—3621no—
142C24—3622no—
143C24—3622no—
144C24—3621no—
145C24—1812no—
146C24—7244no—
147C24C22⋊D18112yescomplemented
148C24C22×D18111yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—3628no—
157C2×D4—7248no—
158C22×C4—3622no—
159C22×C4—3621no—
160C22×C4C22⋊D18112yescomplemented
161C22×C4C22⋊D18112yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—3628no—
166C22⋊C4—3622no—
167C22⋊C4—3622no—
168C22⋊C4—1814no—
169C22⋊C4—7244no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—3624no—
176C23—3628no—
177C23—3624no—
178C23—3622no—
179C23—7242no—
180C23—7244no—
181C23—7244no—
182C23—7248no—
183C23—7244no—
184C23—7241no—
185C23—7242no—
186C23—7241no—
187C23D4×D18111yescharacteristic
188C23D4×D18112yes—
189C23C2×C22⋊D18112yes—
190C23C2×C22⋊D18111yescharacteristic
191C23#1566844811yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—72416no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—3624no—
204C2×C4—7242no—
205C2×C4—7241no—
206C2×C4—7244no—
207C22—22no—
208C22—24no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—3628no—
217C22—3624no—
218C22—7244no—
219C22—7244no—
220C22—7248no—
221C22—7242no—
222C22—7242no—
223C22—7244no—
224C22#1915378412yes—
225C22#1915378511yescharacteristic
226C22#1915378711yescharacteristic
227C22#1915378812yes—
228C22#1915381211yescharacteristic
229C4—42no—
230C4—42no—
231C4—14481no—
232C4—7244no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—7244no—
239C2#2604118711yescharacteristic
240C2#2604146612yes—
subgroupG/H
centerC22#19153785
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting#534×C181C2
socleC22×C181#120
hypercenter#534D181
hypocenterC181#2021
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#19153784C2semidirect
2. C2×#19153731C2semidirect
3. #26041134C2semidirect
4. #26041046C2semidirect
5. #534×C181C2semidirect
25. C23×D181D4semidirect
32. C22⋊Dic181D4semidirect
35. C24×C181D4semidirect
44. C22×C4×C181D4semidirect
45. C22×C4×C181D4semidirect
120. C181#2021semidirect
122. #534D181semidirect
147. C24C22⋊D181semidirect
160. C22×C4C22⋊D181semidirect
161. C22×C4C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537842
C2×#191537312
#260411342
#260410462
#534×C1812
#2021181