Finite Group Databasebeta

← #37534848

Subgroups — #37534848

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
1#26041841C211yesmaximal, characteristic, complemented
2C2×#19153786C212yesmaximal, complemented
3C2×#19153730C212yesmaximal, complemented
4#26041134C211yesmaximal, characteristic, complemented
5#342×C181C211yesmaximal, characteristic, complemented
6C24×D181C2211yescharacteristic
7C22×D724—22no—
8C22×C22⋊D181C2212yes—
9#19153800—22no—
10#19153790—22no—
11#19153786—24no—
12#19153786—22no—
13#19153786—24no—
14C2×C22⋊Dic181C2211yescharacteristic
15C2×#15668426—22no—
16C2×#15668426—22no—
17#19153730—24no—
18C2×C22⋊C4×C181C2211yescharacteristic
19C2×C22⋊C4×C181C2212yes—
20C23×D181—21no—
21C23×D181—22no—
22C23×D181—22no—
23C23×D181—21no—
24C23×D181—44no—
25C23×D181D412yescomplemented
26C23×D181D412yescomplemented
27C2×D724—28no—
28C2×D724—22no—
29C2×D724—48no—
30C2×C22⋊D181—22no—
31C2×C22⋊D181—22no—
32C2×C22⋊D181—22no—
33C2×C22⋊D181—22no—
34C2×C22⋊D181—22no—
35C2×C22⋊D181—22no—
36C22×Dic181—21no—
37C22×Dic181D412yescomplemented
38C4⋊Dic181—22no—
39C22⋊Dic181—22no—
40#15668426—28no—
41#15668426—24no—
42#15668426—44no—
43C24×C181C2311yescharacteristic
44C22×C4×C181—22no—
45C22×C4×C181D412yescomplemented
46C22⋊C4×C181—22no—
47C22⋊C4×C181—22no—
48C22⋊C4×C181D414yescomplemented
49C22×D181—24no—
50C22×D181—28no—
51C22×D181—24no—
52C22×D181—22no—
53C22×D181—22no—
54C22×D181—22no—
55C22×D181—22no—
56C22×D181—44no—
57C22×D181—48no—
58C22×D181—44no—
59C22×D181—42no—
60C22×D181—44no—
61C22×D181—41no—
62C22×D181—42no—
63C22×D181—41no—
64C22×D181—44no—
65D724—416no—
66D724—82no—
67C22⋊D181—42no—
68C22⋊D181—42no—
69C22⋊D181—42no—
70C22⋊D181—42no—
71C2×Dic181—22no—
72C2×Dic181—22no—
73C2×Dic181—22no—
74C2×Dic181—42no—
75C2×Dic181—41no—
76C23×C181—42no—
77C23×C181C2×D412yes—
78C23×C181C2×D411yescharacteristic
79C23×C181C2×D411yescharacteristic
80C23×C181C2×D412yes—
81C23×C181#4111yescharacteristic
82C2×C4×C181—22no—
83C2×C4×C181—24no—
84C2×C4×C181—44no—
85C2×C4×C181—42no—
86C22×C181—22no—
87C22×C181—24no—
88C22×C181—41no—
89C22×C181—42no—
90C22×C181—41no—
91C22×C181#12012yes—
92C22×C181#12011yescharacteristic
93C22×C181#12112yes—
94C22×C181#12111yescharacteristic
95C22×C181#12711yescharacteristic
96C2×D181—28no—
97C2×D181—24no—
98C2×D181—44no—
99C2×D181—44no—
100C2×D181—42no—
101C2×D181—48no—
102C2×D181—42no—
103C2×D181—44no—
104C2×D181—44no—
105C2×D181—42no—
106C2×D181—44no—
107C2×D181—82no—
108C4×C181—44no—
109C4×C181—42no—
110Dic181—42no—
111Dic181—81no—
112C2×C181—21no—
113C2×C181—22no—
114C2×C181—21no—
115C2×C181—22no—
116C2×C181#39211yescharacteristic
117C2×C181B(4,𝔽2)12yes—
118D181—44no—
119D181—82no—
120C181#202111yescharacteristic, complemented, Sylow
121#2021—1811nomaximal, Sylow
122#534—1811no—
123C2×#120—1812no—
124C2×#99—1812no—
125#379—1811no—
126#342D18111yescharacteristic, complemented
127C25—1811no—
128C22×D4—3622no—
129C22×D4—1812no—
130#123—3622no—
131#121—3622no—
132#120—3624no—
133#120—3622no—
134#120—3624no—
135C2×C22⋊C4—3622no—
136C2×C22⋊C4—3622no—
137C2×C22⋊C4—1811no—
138C2×C22⋊C4C2×D18111yescharacteristic
139C2×C22⋊C4C2×D18112yes—
140#99—3624no—
141C24—3622no—
142C24—3621no—
143C24—3622no—
144C24—3621no—
145C24—1812no—
146C24—1812no—
147C24—7244no—
148C24C22×D18111yescharacteristic
149C2×D4—3622no—
150C2×D4—3628no—
151C2×D4—3622no—
152C2×D4—3622no—
153C2×D4—3622no—
154C2×D4—3622no—
155C2×D4—3622no—
156C2×D4—3622no—
157C2×D4—7248no—
158C22×C4—22no—
159C22×C4—3621no—
160C22×C4—1812no—
161C22×C4D72412yescomplemented
162C4⋊C4—3622no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—3628no—
166C22⋊C4—3624no—
167C22⋊C4—3622no—
168C22⋊C4—7244no—
169C22⋊C4D72414yescomplemented
170C23—42no—
171C23—3624no—
172C23—3628no—
173C23—3624no—
174C23—3622no—
175C23—3622no—
176C23—3622no—
177C23—3622no—
178C23—7244no—
179C23—7242no—
180C23—7244no—
181C23—7244no—
182C23—7248no—
183C23—7244no—
184C23—7241no—
185C23—7242no—
186C23—7241no—
187C23D4×D18112yes—
188C23D4×D18111yescharacteristic
189C23C2×D72411yescharacteristic
190C23C2×D72412yes—
191C23#1566845011yescharacteristic
192D4—14482no—
193D4—72416no—
194D4—7242no—
195D4—7242no—
196D4—7242no—
197D4—7242no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—3622no—
203C2×C4—3622no—
204C2×C4—3622no—
205C2×C4—7242no—
206C2×C4—7241no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—3628no—
213C22—3624no—
214C22—14482no—
215C22—7244no—
216C22—7244no—
217C22—7244no—
218C22—7242no—
219C22—7248no—
220C22—7242no—
221C22—7244no—
222C22—7242no—
223C22—7244no—
224C22#1915378612yes—
225C22#1915378611yescharacteristic
226C22#1915379112yes—
227C22#1915379111yescharacteristic
228C22#1915381311yescharacteristic
229C4—44no—
230C4—42no—
231C4—14481no—
232C4—7242no—
233C2—22no—
234C2—21no—
235C2—21no—
236C2—22no—
237C2—14482no—
238C2—7244no—
239C2#2604118911yescharacteristic
240C2#2604146712yes—
subgroupG/H
centerC22#19153786
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
Fitting#342×C181C2
socleC22×C181#120
hypercenter#342D181
hypocenterC181#2021
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #26041841C2semidirect
2. C2×#19153786C2semidirect
3. C2×#19153730C2semidirect
4. #26041134C2semidirect
5. #342×C181C2semidirect
25. C23×D181D4semidirect
26. C23×D181D4semidirect
37. C22×Dic181D4semidirect
45. C22×C4×C181D4semidirect
48. C22⋊C4×C181D4semidirect
120. C181#2021semidirect
126. #342D181semidirect
161. C22×C4D724semidirect
169. C22⋊C4D724semidirect

Maximal subgroups

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

subgroupindex
#260418412
C2×#191537862
C2×#191537302
#260411342
#342×C1812
#2021181