Finite Group Databasebeta

← #37532262

Subgroups — #37532262

Order 23168.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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#26041790C211yesmaximal, characteristic
2#26041243C211yesmaximal, characteristic, complemented
3#26041015C212yesmaximal, complemented
4#26041012C212yesmaximal, complemented
5#380×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C411yescharacteristic, complemented
7C2×#15668433—21no—
8#19153799—21no—
9#19153795C411yescharacteristic, complemented
10#19153795C2211yescharacteristic
11#19153790C411yescharacteristic, complemented
12#19153790C2211yescharacteristic
13#19153785—21no—
14C4×C22⋊D181—21no—
15C4×C22⋊D181—21no—
16#19153775C411yescharacteristic, complemented
17C22⋊#13089678—21no—
18C22⋊#13089678C2212yes—
19C23×C4×C181C2211yescharacteristic
20C2×C4⋊C4×C181—21no—
21C2×C22⋊C4×C181—21no—
22#95×C181C2212yes—
23#15668449—42no—
24C2×Dic362C2×C411yescharacteristic
25C2×D724C2×C411yescharacteristic
26C2×C22⋊D181—21no—
27C2×C22⋊D181—21no—
28C2×C22⋊D181—21no—
29C2×C4×D181—21no—
30#15668433—42no—
31C2×#13089678—22no—
32C4⋊Dic181C2×C411yescharacteristic
33C4⋊Dic181C2×C411yescharacteristic
34#15668428—21no—
35#15668428—21no—
36C22⋊Dic181—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39#15668426—21no—
40#15668426—21no—
41C4×Dic181—21no—
42C24×C181D411yescharacteristic
43C22×C4×C181—22no—
44C22×C4×C181—42no—
45C22×C4×C181C2×C411yescharacteristic
46C22×C4×C181C2×C411yescharacteristic
47C22×C4×C181D411yescharacteristic
48C22×C4×C181D411yescharacteristic
49C22×C4×C181D411yescharacteristic
50C22×C4×C181C2311yescharacteristic
51C4⋊C4×C181—42no—
52C22⋊C4×C181—42no—
53C22×D181—21no—
54Dic362—41no—
55D724—41no—
56C22⋊D181—42no—
57C22⋊D181—42no—
58C22⋊D181—41no—
59C22⋊D181—41no—
60C2×Dic181—21no—
61C2×Dic181—21no—
62C2×Dic181—21no—
63C4×D181—42no—
64#13089678—42no—
65C23×C181—21no—
66C23×C181—21no—
67C23×C181—41no—
68C23×C181—41no—
69C23×C181C22⋊C411yescharacteristic
70C23×C181C22⋊C411yescharacteristic
71C23×C181C2×D411yescharacteristic
72C2×C4×C181—22no—
73C2×C4×C181—22no—
74C2×C4×C181—22no—
75C2×C4×C181—41no—
76C2×C4×C181—41no—
77C2×C4×C181—41no—
78C2×C4×C181—42no—
79C2×C4×C181—42no—
80C2×C4×C181—42no—
81C2×C4×C181C22⋊C411yescharacteristic
82C2×C4×C181C22⋊C411yescharacteristic
83C2×C4×C181C22×C411yescharacteristic
84C2×C4×C181C2×D411yescharacteristic
85C22×C181—21no—
86C22×C181—21no—
87C22×C181—21no—
88C22×C181—21no—
89C22×C181—21no—
90C22×C181—21no—
91C22×C181—21no—
92C22×C181—21no—
93C22×C181—42no—
94C22×C181—42no—
95C22×C181C2×C22⋊C411yescharacteristic
96C22×C181AGL(1,ℤ/8)11yescharacteristic
97C22×C181#13711yescharacteristic
98C2×D181—41no—
99C2×D181—41no—
100C2×D181—41no—
101C4×C181—21no—
102C4×C181—21no—
103C4×C181—21no—
104C4×C181—82no—
105C4×C181#9912yes—
106Dic181—41no—
107Dic181—41no—
108Dic181—41no—
109C2×C181—21no—
110C2×C181—21no—
111C2×C181—41no—
112C2×C181—41no—
113C2×C181C2×#9911yescharacteristic
114C2×C181#41711yescharacteristic
115C2×C181#42111yescharacteristic
116D181—81no—
117C181#151311yescharacteristic, complemented, Sylow
118#1513—1811nomaximal, Sylow
119#525—1811no—
120#407—1811no—
121#380D18111yescharacteristic, complemented
122#328—1812no—
123#327—1812no—
124C2×#41—1811no—
125C23×C4C2×D18111yescharacteristic
126C2×M16—3621no—
127#123—3621no—
128C22⋊Q8—1811no—
129C22⋊Q8—1811no—
130#121—1811no—
131#121—1811no—
132#120—3621no—
133C4×D4—3621no—
134C4×D4—3621no—
135#117—1811no—
136C2×C4⋊C4—21no—
137C2×C22⋊C4—21no—
138C22⋊C8—3621no—
139C22⋊C8—1812no—
140#95C2×D18112yes—
141C24C22⋊D18111yescharacteristic
142#41—7242no—
143C2×Q8—1811no—
144C2×D4—3621no—
145C2×D4—3621no—
146C2×D4—3621no—
147C2×D4—1811no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—3621no—
151C22×C4C4×D18111yescharacteristic
152C22×C4C4×D18111yescharacteristic
153C22×C4C22⋊D18111yescharacteristic
154C22×C4D72411yescharacteristic
155C22×C4D72411yescharacteristic
156C22×C4C22×D18111yescharacteristic
157M16—7242no—
158C2×C8—3622no—
159C4⋊C4—42no—
160C4⋊C4—3621no—
161C4⋊C4—3621no—
162C4⋊C4—1811no—
163C4⋊C4—1811no—
164C22⋊C4—42no—
165C22⋊C4—3621no—
166C22⋊C4—3621no—
167C22⋊C4—3621no—
168C22⋊C4—3621no—
169C22⋊C4—3621no—
170C42—3621no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—3621no—
176C23#1566842611yescharacteristic
177C23#1566842611yescharacteristic
178C23C2×C22⋊D18111yescharacteristic
179Q8—7241no—
180D4—7241no—
181D4—7242no—
182D4—7242no—
183D4—7241no—
184D4—7241no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3621no—
195C2×C4—3621no—
196C2×C4—3621no—
197C2×C4—7242no—
198C2×C4#1566842611yescharacteristic
199C2×C4#1566842611yescharacteristic
200C2×C4C2×C4×D18111yescharacteristic
201C2×C4C2×D72411yescharacteristic
202C8—7242no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—7241no—
214C22—7241no—
215C22—7241no—
216C22C2×#1566842611yescharacteristic
217C22#1915384211yescharacteristic
218C22#1915384711yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—82no—
223C4—7241no—
224C4—7241no—
225C4—7241no—
226C4#1915373012yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—14481no—
232C2C2×#1915373011yescharacteristic
233C2#2604127811yescharacteristic
234C2#2604129711yescharacteristic
subgroupG/H
centerC22C2×#15668426
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#380×C181C2
socleC22×C181C2×C22⋊C4
prime radical#26041790C2
cosocleC22×C4×C181C23
hypercenter#380D181
hypocenterC181#1513
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #26041243C2semidirect
3. #26041015C2semidirect
4. #26041012C2semidirect
5. #380×C181C2semidirect
6. C2×#15668449C4semidirect
9. #19153795C4semidirect
11. #19153790C4semidirect
16. #19153775C4semidirect
117. C181#1513semidirect
121. #380D181semidirect

Maximal subgroups

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

subgroupindex
#260417902
#260412432
#260410152
#260410122
#380×C1812
#1513181