Finite Group Databasebeta

← #37532671

Subgroups — #37532671

Order 23168.

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

normal subgroups only

This group has 308 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#26041418C211yesmaximal, characteristic, complemented
3#26041190C211yesmaximal, characteristic, complemented
4#26041014C211yesmaximal, characteristic, complemented
5#26041013C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic, complemented
7#448×C181C211yesmaximal, characteristic, complemented
8C2×D4×D181—21no—
9C2×#15668436—21no—
10#19153801—21no—
11#19153798—21no—
12#19153790C2211yescharacteristic
13#19153788—21no—
14#19153786—21no—
15#19153784—21no—
16C2×#15668426—21no—
17C2×#15668426C2211yescharacteristic
18#19153740—21no—
19C22⋊#13089678C2211yescharacteristic, complemented
20#19153723—21no—
21#19153723C2211yescharacteristic, complemented
22C22×Q8×C181C2211yescharacteristic, complemented
23C2×SD16×C181—21no—
24#121×C181C2211yescharacteristic
25#103×C181—21no—
26C22⋊C8×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—
39#15668436—42no—
40C2×#13089678—21no—
41C4⋊Dic181D411yescharacteristic
42#15668428—21no—
43C22⋊Dic181—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47#15668426—21no—
48#15668426—42no—
49C2×Q8×C181—21no—
50C2×Q8×C181—21no—
51C2×Q8×C181—42no—
52C2×D4×C181—21no—
53C2×D4×C181D411yescharacteristic
54C22×C4×C181—21no—
55C22×C4×C181C2311yescharacteristic
56SD16×C181—42no—
57C2×C8×C181—21no—
58C4⋊C4×C181D411yescharacteristic
59C22⋊C4×C181—21no—
60C22×D181—21no—
61C22×D181—21no—
62C22×D181—21no—
63C22×D181—42no—
64C22×D181—41no—
65C22×D181—41no—
66C22×D181—81no—
67C22×D181—81no—
68C22×D181SD1612yescomplemented
69D724—41no—
70D724—42no—
71C22⋊D181—42no—
72C22⋊D181—41no—
73C22⋊D181—41no—
74C22⋊D181—81no—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—41no—
78C2×Dic181—41no—
79C2×Dic181—41no—
80C2×Dic181—41no—
81C4×D181—42no—
82#13089678—41no—
83C23×C181—21no—
84C23×C181C2×D411yescharacteristic
85Q8×C181—22no—
86Q8×C181—22no—
87Q8×C181—41no—
88Q8×C181—41no—
89D4×C181—41no—
90D4×C181—41no—
91D4×C181—41no—
92C2×C4×C181—22no—
93C2×C4×C181—21no—
94C2×C4×C181—41no—
95C2×C4×C181—41no—
96C2×C4×C181—41no—
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181C2×D411yescharacteristic
99C8×C181—41no—
100C22×C181—21no—
101C22×C181—21no—
102C22×C181—41no—
103C22×C181—41no—
104C22×C181—41no—
105C22×C181#12011yescharacteristic
106C22×C181C2×SD1611yescharacteristic
107C22×C181AGL(1,ℤ/8)11yescharacteristic
108C2×D181—22no—
109C2×D181—41no—
110C2×D181—41no—
111C2×D181—42no—
112C2×D181—41no—
113C2×D181—41no—
114C2×D181—41no—
115C2×D181—81no—
116C2×D181—84no—
117C4×C181—21no—
118C4×C181—21no—
119C4×C181—41no—
120C4×C181—42no—
121Dic181—41no—
122Dic181—41no—
123Dic181—81no—
124C2×C181—21no—
125C2×C181—21no—
126C2×C181—81no—
127C2×C181#45011yescharacteristic
128C2×C181#45511yescharacteristic
129C2×C181B(4,𝔽2)11yescharacteristic
130D181—42no—
131D181—81no—
132C181#160111yescharacteristic, complemented, Sylow
133#1601—1811nomaximal, Sylow
134#534—1811no—
135#448—1811no—
136#448D18111yescharacteristic, complemented
137#393—1811no—
138#327—1811no—
139#327—1811no—
140#323—1811no—
141C22×Q8C2×D18111yescharacteristic, complemented
142C22×D4—3621no—
143C2×SD16—21no—
144C2×SD16—3621no—
145#123—3621no—
146#123—3621no—
147#121—3621no—
148#121—1811no—
149#121C2×D18111yescharacteristic
150#120—3621no—
151#120—3621no—
152C2×C22⋊C4—3621no—
153C2×C22⋊C4—1811no—
154#103—21no—
155#103—3621no—
156C22⋊C8—1811no—
157C22⋊C8C2×D18111yescharacteristic, complemented
158#95—3621no—
159#95—1811no—
160C24—3621no—
161C24—1811no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×D4—21no—
166C2×D4—3621no—
167C2×D4—3621no—
168C2×D4—3621no—
169C2×D4—3621no—
170C2×D4—1811no—
171C2×D4—7242no—
172C2×D4—7242no—
173C2×D4C22⋊D18111yescharacteristic
174C22×C4—21no—
175C22×C4—3621no—
176C22×C4—3621no—
177C22×C4—1811no—
178C22×C4C22×D18111yescharacteristic
179SD16—42no—
180SD16—7242no—
181C2×C8—21no—
182C2×C8—3621no—
183C4⋊C4—3621no—
184C4⋊C4—1811no—
185C4⋊C4C22⋊D18111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3621no—
188C22⋊C4—3621no—
189C22⋊C4—3621no—
190C22⋊C4—3621no—
191C22⋊C4—3621no—
192C22⋊C4—7242no—
193C23—21no—
194C23—3621no—
195C23—3621no—
196C23—3621no—
197C23—14481no—
198C23—14481no—
199C23—1812no—
200C23—7241no—
201C23—7241no—
202C23—7242no—
203C23D4×D18111yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—14481no—
212D4—7241no—
213D4—7242no—
214D4—7242no—
215D4—7241no—
216D4—7241no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3621no—
223C2×C4—3621no—
224C2×C4—7242no—
225C2×C4—7241no—
226C2×C4—7241no—
227C2×C4—7241no—
228C2×C4—7241no—
229C2×C4D4×D18111yescharacteristic
230C2×C4C2×C22⋊D18111yescharacteristic
231C8—41no—
232C8—7241no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—3622no—
239C22—14481no—
240C22—14484no—
241C22—7242no—
242C22—7241no—
243C22—7241no—
244C22—7241no—
245C22—7241no—
246C22—7241no—
247C22#1915378411yescharacteristic
248C22SD16×D18111yescharacteristic
249C22#1915384311yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—42no—
254C4—14481no—
255C4—7241no—
256C4—7241no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—14481no—
261C2—7242no—
262C2#2604142811yescharacteristic
263C2#2604145511yescharacteristic
264C2#2604146511yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#448×C181C2
socleC22×C181#120
prime radical#26041837C2
hypercenter#448D181
hypocenterC181#1601
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #26041418C2semidirect
3. #26041190C2semidirect
4. #26041014C2semidirect
5. #26041013C2semidirect
6. #26041002C2semidirect
7. #448×C181C2semidirect
19. C22⋊#13089678C22semidirect
21. #19153723C22semidirect
22. C22×Q8×C181C22semidirect
26. C22⋊C8×C181C22semidirect
68. C22×D181SD16semidirect
132. C181#1601semidirect
136. #448D181semidirect
141. C22×Q8C2×D181semidirect
157. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418372
#260414182
#260411902
#260410142
#260410132
#260410022
#448×C1812
#1601181