Finite Group Databasebeta

← #37532708

Subgroups — #37532708

Order 23168.

280 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 51 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#26041837C211yesmaximal, characteristic
2#26041593C211yesmaximal, characteristic, complemented
3#26041590C211yesmaximal, characteristic, complemented
4#26041206C211yesmaximal, characteristic, complemented
5#26041012C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic, complemented
7#327×C181C211yesmaximal, characteristic, complemented
8C2×D4×D181—21no—
9#19153801—21no—
10#19153798—21no—
11#19153790C2211yescharacteristic
12#19153788—21no—
13#19153786—21no—
14#19153784—21no—
15C2×C4⋊Dic181C2211yescharacteristic, complemented
16C2×#15668426—21no—
17C2×#15668426C2211yescharacteristic
18#19153750—21no—
19#19153749—21no—
20#19153738—21no—
21#19153737—21no—
22C22⋊#13089678C2211yescharacteristic, complemented
23#19153723—21no—
24#121×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic, complemented
26#95×C181C2211yescharacteristic, complemented
27C23×D181—21no—
28C23×D181D411yescharacteristic
29C2×D724—21no—
30C2×D724D411yescharacteristic
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34D4×D181—42no—
35D4×D181—42no—
36C22×Dic181—21no—
37C22×Dic181D411yescharacteristic
38C2×C4×D181—21no—
39C2×#13089678—21no—
40C4⋊Dic181—21no—
41C4⋊Dic181—21no—
42C4⋊Dic181D411yescharacteristic
43#15668428—21no—
44C22⋊Dic181—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—21no—
49#15668426—42no—
50C2×D4×C181—21no—
51C2×D4×C181D411yescharacteristic
52C22×C4×C181—21no—
53C22×C4×C181C2311yescharacteristic
54C2×C8×C181—21no—
55C4⋊C4×C181D411yescharacteristic
56C22⋊C4×C181—21no—
57C22×D181—21no—
58C22×D181—21no—
59C22×D181—21no—
60C22×D181—42no—
61C22×D181—41no—
62C22×D181—41no—
63C22×D181—81no—
64C22×D181—81no—
65C22×D181D812yes—
66D724—41no—
67D724—42no—
68C22⋊D181—42no—
69C22⋊D181—41no—
70C22⋊D181—41no—
71C22⋊D181—81no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—41no—
75C2×Dic181—41no—
76C2×Dic181—41no—
77C2×Dic181—41no—
78C4×D181—42no—
79#13089678—41no—
80C23×C181—21no—
81C23×C181C2×D411yescharacteristic
82D4×C181—41no—
83D4×C181—41no—
84D4×C181—41no—
85C2×C4×C181—21no—
86C2×C4×C181—41no—
87C2×C4×C181—41no—
88C2×C4×C181—41no—
89C2×C4×C181—41no—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D411yescharacteristic
92C8×C181—41no—
93C22×C181—21no—
94C22×C181—21no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181#12011yescharacteristic
99C22×C181C2×D811yescharacteristic
100C22×C181AGL(1,ℤ/8)11yescharacteristic
101C2×D181—22no—
102C2×D181—41no—
103C2×D181—41no—
104C2×D181—42no—
105C2×D181—41no—
106C2×D181—41no—
107C2×D181—41no—
108C2×D181—81no—
109C2×D181—84no—
110C4×C181—21no—
111C4×C181—21no—
112C4×C181—41no—
113C4×C181—81no—
114Dic181—41no—
115Dic181—41no—
116Dic181—81no—
117C2×C181—21no—
118C2×C181—21no—
119C2×C181—81no—
120C2×C181#44711yescharacteristic
121C2×C181#45511yescharacteristic
122C2×C181#45811yescharacteristic
123D181—42no—
124D181—81no—
125C181#160711yescharacteristic, complemented, Sylow
126#1607—1811nomaximal, Sylow
127#534—1811no—
128#480—1811no—
129#480—1811no—
130#397—1811no—
131#327—1811no—
132#327D18111yescharacteristic, complemented
133#323—1811no—
134C22×D4—3621no—
135#123—3621no—
136#123—3621no—
137#121—3621no—
138#121—1811no—
139#121C2×D18111yescharacteristic
140#120—3621no—
141#120—3621no—
142C2×C4⋊C4—1811no—
143C2×C22⋊C4—3621no—
144C2×C22⋊C4—1811no—
145#107—3621no—
146#107—3621no—
147#102—3621no—
148#102—3621no—
149C22⋊C8—1811no—
150C22⋊C8C2×D18111yescharacteristic, complemented
151#95—3621no—
152#95C2×D18111yescharacteristic, complemented
153C24—3621no—
154C24—1811no—
155C2×D4—21no—
156C2×D4—3621no—
157C2×D4—3621no—
158C2×D4—3621no—
159C2×D4—3621no—
160C2×D4—1811no—
161C2×D4—7242no—
162C2×D4—7242no—
163C2×D4D72411yescharacteristic
164C22×C4—21no—
165C22×C4—3621no—
166C22×C4—3621no—
167C22×C4—1811no—
168C22×C4C22×D18111yescharacteristic
169C2×C8—21no—
170C2×C8—3621no—
171C4⋊C4—3621no—
172C4⋊C4—3621no—
173C4⋊C4—3621no—
174C4⋊C4—1811no—
175C4⋊C4D72411yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—7242no—
183C23—21no—
184C23—3621no—
185C23—3621no—
186C23—3621no—
187C23—14481no—
188C23—14481no—
189C23—1812no—
190C23—7241no—
191C23—7241no—
192C23—7242no—
193C23D4×D18111yescharacteristic
194D4—41no—
195D4—41no—
196D4—41no—
197D4—14481no—
198D4—7241no—
199D4—7242no—
200D4—7242no—
201D4—7241no—
202D4—7241no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—3621no—
209C2×C4—3621no—
210C2×C4—7242no—
211C2×C4—7241no—
212C2×C4—7241no—
213C2×C4—7241no—
214C2×C4—7241no—
215C2×C4D4×D18111yescharacteristic
216C2×C4C2×D72411yescharacteristic
217C8—41no—
218C8—7241no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—3622no—
225C22—14481no—
226C22—14484no—
227C22—7242no—
228C22—7241no—
229C22—7241no—
230C22—7241no—
231C22—7241no—
232C22—7241no—
233C22#1915378611yescharacteristic
234C22D8×D18111yescharacteristic
235C22#1915384311yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—81no—
240C4—14481no—
241C4—7241no—
242C4—7241no—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—14481no—
247C2—7242no—
248C2#2604141411yescharacteristic
249C2#2604145811yescharacteristic
250C2#2604147011yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#327×C181C2
socleC22×C181#120
prime radical#26041837C2
hypercenter#327D181
hypocenterC181#1607
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #26041593C2semidirect
3. #26041590C2semidirect
4. #26041206C2semidirect
5. #26041012C2semidirect
6. #26041002C2semidirect
7. #327×C181C2semidirect
15. C2×C4⋊Dic181C22semidirect
22. C22⋊#13089678C22semidirect
25. C22⋊C8×C181C22semidirect
26. #95×C181C22semidirect
125. C181#1607semidirect
132. #327D181semidirect
150. C22⋊C8C2×D181semidirect
152. #95C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418372
#260415932
#260415902
#260412062
#260410122
#260410022
#327×C1812
#1607181