Finite Group Databasebeta

← #37532703

Subgroups — #37532703

Order 23168.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#26041867C211yesmaximal, characteristic
2#26041437C211yesmaximal, characteristic, complemented
3#26041420C211yesmaximal, characteristic, complemented
4#26041191C211yesmaximal, characteristic, complemented
5#26041017C211yesmaximal, characteristic, complemented
6#26041014C211yesmaximal, characteristic, complemented
7#323×C181C211yesmaximal, characteristic, complemented
8C22×Dic362C2211yescharacteristic
9C2×Dic724—21no—
10C2×#15668438—21no—
11#19153809—21no—
12#19153802—21no—
13#19153798—21no—
14#19153795C2211yescharacteristic
15#19153790C2211yescharacteristic
16#19153789—21no—
17#19153785—21no—
18C4×C22⋊D181—21no—
19C2×C22⋊Dic181—21no—
20#19153741—21no—
21#19153741—21no—
22#19153723—21no—
23#19153723C2211yescharacteristic
24C2×C22⋊C4×C181C2211yescharacteristic
25C22⋊C8×C181C2211yescharacteristic
26C22⋊C8×C181C2211yescharacteristic
27C2×Dic362—21no—
28C2×Dic362—21no—
29C2×Dic362—42no—
30C2×Dic362D411yescharacteristic
31C2×D724D411yescharacteristic
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34C2×C22⋊D181—21no—
35C22×Dic181—21no—
36C22×Dic181—21no—
37C2×C4×D181—21no—
38Dic724—42no—
39#15668438—42no—
40C4⋊Dic181D411yescharacteristic
41C4⋊Dic181D411yescharacteristic
42#15668428—21no—
43#15668428—21no—
44C22⋊Dic181—21no—
45C22⋊Dic181—21no—
46C22⋊Dic181—21no—
47C22⋊Dic181—42no—
48#15668426—21no—
49#15668426—21no—
50C4×Dic181—21no—
51C24×C181D411yescharacteristic
52C22×C4×C181D411yescharacteristic
53C22×C4×C181C2311yescharacteristic
54C2×C8×C181—21no—
55C2×C8×C181—21no—
56C22⋊C4×C181—21no—
57C22⋊C4×C181—21no—
58C22×D181—21no—
59Dic362—22no—
60Dic362—22no—
61Dic362—41no—
62Dic362—41no—
63Dic362—41no—
64D724—41no—
65C22⋊D181—41no—
66C22⋊D181—41no—
67C22⋊D181—81no—
68C22⋊D181—81no—
69C2×Dic181—22no—
70C2×Dic181—21no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—41no—
74C2×Dic181—41no—
75C2×Dic181—41no—
76C2×Dic181—41no—
77C2×Dic181—41no—
78C4×D181—81no—
79C23×C181—21no—
80C23×C181—21no—
81C23×C181—21no—
82C23×C181—41no—
83C23×C181—41no—
84C23×C181C2×D411yescharacteristic
85C2×C4×C181—21no—
86C2×C4×C181—41no—
87C2×C4×C181—41no—
88C2×C4×C181C2×D411yescharacteristic
89C2×C4×C181C2×D411yescharacteristic
90C8×C181—41no—
91C8×C181—41no—
92C22×C181—21no—
93C22×C181—21no—
94C22×C181—42no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181—81no—
99C22×C181#12011yescharacteristic
100C22×C181#13511yescharacteristic
101C22×C181#13711yescharacteristic
102C2×D181—41no—
103C2×D181—41no—
104C2×D181—41no—
105C4×C181—21no—
106C4×C181—21no—
107C4×C181—41no—
108Dic181—41no—
109Dic181—41no—
110Dic181—42no—
111Dic181—41no—
112Dic181—81no—
113C2×C181—21no—
114C2×C181—21no—
115C2×C181—81no—
116C2×C181#45211yescharacteristic
117C2×C181#45511yescharacteristic
118C2×C181B(4,𝔽2)11yescharacteristic
119D181—81no—
120C181#160611yescharacteristic, complemented, Sylow
121#1606—1811nomaximal, Sylow
122#538—1811no—
123#451—1811no—
124#448—1811no—
125#393—1811no—
126#328—1811no—
127#327—1811no—
128#323D18111yescharacteristic, complemented
129C22×Q8—1811no—
130C2×Q16—3621no—
131C2×SD16—3621no—
132#126—3621no—
133#124—3621no—
134#123—3621no—
135C22⋊Q8—1811no—
136#121—3621no—
137#121—1811no—
138#120—3621no—
139C4×D4—3621no—
140C2×C22⋊C4—3621no—
141C2×C22⋊C4C2×D18111yescharacteristic
142#103—3621no—
143#103—3621no—
144C22⋊C8C2×D18111yescharacteristic
145C22⋊C8C2×D18111yescharacteristic
146#95—3621no—
147#95—1811no—
148C24D72411yescharacteristic
149C2×Q8—3621no—
150C2×Q8—3621no—
151C2×Q8—1811no—
152C2×Q8—7242no—
153C2×D4—3621no—
154C2×D4—3621no—
155C2×D4—3621no—
156C2×D4—1811no—
157C22×C4—3621no—
158C22×C4—3621no—
159C22×C4—3621no—
160C22×C4D72411yescharacteristic
161C22×C4C22×D18111yescharacteristic
162Q16—7242no—
163SD16—7242no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—3621no—
167C4⋊C4—3621no—
168C4⋊C4—1811no—
169C4⋊C4—1811no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3621no—
173C22⋊C4—3621no—
174C22⋊C4—3621no—
175C22⋊C4—3621no—
176C22⋊C4—3621no—
177C22⋊C4—7242no—
178C42—3621no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3621no—
185C23C2×D72411yescharacteristic
186Q8—3622no—
187Q8—3622no—
188Q8—7241no—
189Q8—7241no—
190Q8—7241no—
191D4—14481no—
192D4—14481no—
193D4—7241no—
194D4—7241no—
195D4—7241no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3622no—
200C2×C4—3621no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—14481no—
204C2×C4—7241no—
205C2×C4—7241no—
206C2×C4—7241no—
207C2×C4—7241no—
208C2×C4—7241no—
209C2×C4D4×D18111yescharacteristic
210C2×C4D4×D18111yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—7241no—
221C22—7241no—
222C22—7241no—
223C22#1915378611yescharacteristic
224C22#1915383811yescharacteristic
225C22#1915384911yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—14481no—
230C4—7241no—
231C4—7241no—
232C4—7242no—
233C4—7241no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—14481no—
238C2#2604144311yescharacteristic
239C2#2604145911yescharacteristic
240C2#2604146711yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#323×C181C2
socleC22×C181#120
prime radical#26041867C2
hypercenter#323D181
hypocenterC181#1606
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26041437C2semidirect
3. #26041420C2semidirect
4. #26041191C2semidirect
5. #26041017C2semidirect
6. #26041014C2semidirect
7. #323×C181C2semidirect
120. C181#1606semidirect
128. #323D181semidirect

Maximal subgroups

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

subgroupindex
#260418672
#260414372
#260414202
#260411912
#260410172
#260410142
#323×C1812
#1606181