Finite Group Databasebeta

← #37532698

Subgroups — #37532698

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#26041866C211yesmaximal, characteristic
2#26041435C211yesmaximal, characteristic, complemented
3#26041190C211yesmaximal, characteristic, complemented
4#26041017C211yesmaximal, characteristic, complemented
5#26041013C211yesmaximal, characteristic, complemented
6#26041002C211yesmaximal, characteristic, complemented
7#448×C181C211yesmaximal, characteristic, complemented
8C2×#15668439—21no—
9#19153811—21no—
10#19153802—21no—
11#19153800—21no—
12#19153795C2211yescharacteristic
13#19153788—21no—
14#19153784—21no—
15D4×Dic181—21no—
16C2×#15668426—21no—
17C2×#15668426C2211yescharacteristic
18#19153740—21no—
19C22⋊#13089678C2211yescharacteristic
20#19153723—21no—
21#19153723C2211yescharacteristic
22C22×Q8×C181C2211yescharacteristic
23C2×SD16×C181—21no—
24#121×C181C2211yescharacteristic
25#103×C181—21no—
26C22⋊C8×C181C2211yescharacteristic
27C23×D181D411yescharacteristic
28C2×Dic362D411yescharacteristic
29C2×C22⋊D181—21no—
30C2×C22⋊D181—21no—
31C22×Dic181—21no—
32C22×Dic181—21no—
33C22×Dic181D411yescharacteristic
34#15668439—42no—
35C2×#13089678—21no—
36C4⋊Dic181—21no—
37C4⋊Dic181D411yescharacteristic
38#15668428—21no—
39C22⋊Dic181—21no—
40C22⋊Dic181—21no—
41#15668426—21no—
42#15668426—21no—
43#15668426—21no—
44#15668426—21no—
45#15668426—42no—
46C4×Dic181—21no—
47C2×Q8×C181—21no—
48C2×Q8×C181—21no—
49C2×Q8×C181—42no—
50C2×D4×C181—21no—
51C2×D4×C181D411yescharacteristic
52C22×C4×C181—21no—
53C22×C4×C181C2311yescharacteristic
54SD16×C181—42no—
55C2×C8×C181—21no—
56C4⋊C4×C181D411yescharacteristic
57C22⋊C4×C181—21no—
58C22×D181—21no—
59C22×D181—21no—
60C22×D181—21no—
61C22×D181—41no—
62C22×D181—41no—
63Dic362—41no—
64C22⋊D181—81no—
65C22⋊D181—81no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181—21no—
69C2×Dic181—41no—
70C2×Dic181—41no—
71C2×Dic181—41no—
72C2×Dic181—41no—
73C2×Dic181—81no—
74#13089678—41no—
75C23×C181—21no—
76C23×C181C2×D411yescharacteristic
77Q8×C181—22no—
78Q8×C181—22no—
79Q8×C181—41no—
80Q8×C181—41no—
81D4×C181—41no—
82D4×C181—41no—
83D4×C181—41no—
84C2×C4×C181—22no—
85C2×C4×C181—21no—
86C2×C4×C181—41no—
87C2×C4×C181—41no—
88C2×C4×C181—41no—
89C2×C4×C181C2×D411yescharacteristic
90C2×C4×C181C2×D411yescharacteristic
91C8×C181—41no—
92C22×C181—21no—
93C22×C181—21no—
94C22×C181—41no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181#12011yescharacteristic
98C22×C181#13511yescharacteristic
99C22×C181#13711yescharacteristic
100C2×D181—42no—
101C2×D181—41no—
102C2×D181—41no—
103C2×D181—41no—
104C2×D181—81no—
105C4×C181—21no—
106C4×C181—21no—
107C4×C181—41no—
108C4×C181—42no—
109Dic181—41no—
110Dic181—41no—
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#448D18111yescharacteristic, complemented
125#393—1811no—
126#328—1811no—
127#327—1811no—
128#323—1811no—
129C22×Q8C2×D18111yescharacteristic
130C2×Q16—3621no—
131C2×SD16—21no—
132#126—3621no—
133#124—3621no—
134#123—3621no—
135C22⋊Q8—1811no—
136#121—3621no—
137#121C2×D18111yescharacteristic
138#120—3621no—
139C4×D4—3621no—
140C2×C22⋊C4—3621no—
141C2×C22⋊C4—1811no—
142#103—21no—
143#103—3621no—
144C22⋊C8—1811no—
145C22⋊C8C2×D18111yescharacteristic
146#95—3621no—
147#95—1811no—
148C24—1811no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—1811no—
153C2×D4—21no—
154C2×D4—3621no—
155C2×D4—3621no—
156C2×D4C22⋊D18111yescharacteristic
157C22×C4—21no—
158C22×C4—3621no—
159C22×C4—3621no—
160C22×C4—1811no—
161C22×C4C22×D18111yescharacteristic
162Q16—7242no—
163SD16—42no—
164C2×C8—21no—
165C2×C8—3621no—
166C4⋊C4—3621no—
167C4⋊C4—3621no—
168C4⋊C4—1811no—
169C4⋊C4C22⋊D18111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3621no—
172C22⋊C4—3621no—
173C22⋊C4—3621no—
174C22⋊C4—3621no—
175C22⋊C4—3621no—
176C22⋊C4—3621no—
177C22⋊C4—7242no—
178C42—3621no—
179C23—21no—
180C23—3621no—
181C23—3621no—
182C23—3621no—
183C23—7241no—
184C23—7241no—
185C23D4×D18111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—7241no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—14481no—
195D4—14481no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—3621no—
202C2×C4—3621no—
203C2×C4—3621no—
204C2×C4—14481no—
205C2×C4—7241no—
206C2×C4—7241no—
207C2×C4—7241no—
208C2×C4—7241no—
209C2×C4D4×D18111yescharacteristic
210C2×C4C2×C22⋊D18111yescharacteristic
211C8—41no—
212C8—7241no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—14481no—
219C22—7242no—
220C22—7241no—
221C22—7241no—
222C22—7241no—
223C22#1915378411yescharacteristic
224C22#1915383711yescharacteristic
225C22#1915384511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—14481no—
231C4—7241no—
232C4—7241no—
233C4—7241no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—14481no—
238C2#2604143811yescharacteristic
239C2#2604145511yescharacteristic
240C2#2604146511yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#448×C181C2
socleC22×C181#120
prime radical#26041866C2
hypercenter#448D181
hypocenterC181#1606
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. #26041435C2semidirect
3. #26041190C2semidirect
4. #26041017C2semidirect
5. #26041013C2semidirect
6. #26041002C2semidirect
7. #448×C181C2semidirect
120. C181#1606semidirect
124. #448D181semidirect

Maximal subgroups

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

subgroupindex
#260418662
#260414352
#260411902
#260410172
#260410132
#260410022
#448×C1812
#1606181