Finite Group Databasebeta

← #37535720

Subgroups — #37535720

Order 23168.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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#26042022C211yesmaximal, characteristic, complemented
2#26041466C211yesmaximal, characteristic, complemented
3#26041457C211yesmaximal, characteristic, complemented
4#26041073C211yesmaximal, characteristic, complemented
5#26041066C211yesmaximal, characteristic, complemented
6#26041065C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C181C211yesmaximal, characteristic, complemented
8#19153846—21no—
9#19153803—21no—
10#19153803C2211yescharacteristic
11#19153802—21no—
12#19153802—21no—
13#19153802—21no—
14#19153785—21no—
15#19153784—21no—
16#19153784—21no—
17#19153784C2211yescharacteristic
18#19153743—21no—
19#19153733C2211yescharacteristic
20#19153731—21no—
21#19153731C2211yescharacteristic
22H(5,𝔽2)×C181C2211yescharacteristic
23#120×C181—21no—
24#120×C181C2211yescharacteristic
25#99×C181—21no—
26#99×C181C2211yescharacteristic
27C23×D181D411yescharacteristic, complemented
28C2×Dic362—21no—
29C2×Dic362D411yescharacteristic, complemented
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—41no—
34#15668439—41no—
35#15668437—41no—
36#15668433—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39C22⋊Dic181—21no—
40C22⋊Dic181—21no—
41C22⋊Dic181—21no—
42C22⋊Dic181D411yescharacteristic, complemented
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—41no—
47C4×Dic181—21no—
48C4×Dic181D411yescharacteristic, complemented
49C24×C181D411yescharacteristic, complemented
50#41×C181—21no—
51#41×C181—41no—
52C2×D4×C181—21no—
53C2×D4×C181—21no—
54C2×D4×C181—21no—
55C2×D4×C181—41no—
56C2×D4×C181—41no—
57C2×D4×C181C2311yescharacteristic
58C22⋊C4×C181—21no—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—41no—
61C22⋊C4×C181D411yescharacteristic, complemented
62C22×D181—21no—
63C22×D181—41no—
64C22×D181—81no—
65Dic362—41no—
66Dic362—81no—
67C22⋊D181—81no—
68C22⋊D181—81no—
69C22⋊D181—81no—
70C22⋊D181—82no—
71C2×Dic181—21no—
72C2×Dic181—21no—
73C2×Dic181—21no—
74C2×Dic181—41no—
75#13089678—41no—
76C23×C181—21no—
77C23×C181—21no—
78C23×C181—21no—
79C23×C181—41no—
80C23×C181—81no—
81C23×C181C2×D411yescharacteristic
82C23×C181C2×D411yescharacteristic
83Q8×C181—21no—
84D4×C181—21no—
85D4×C181—41no—
86D4×C181—41no—
87D4×C181—41no—
88D4×C181—41no—
89D4×C181—81no—
90D4×C181—82no—
91C2×C4×C181—21no—
92C2×C4×C181—21no—
93C2×C4×C181—21no—
94C2×C4×C181—41no—
95C2×C4×C181—41no—
96C2×C4×C181C2×D411yescharacteristic
97C22×C181—21no—
98C22×C181—21no—
99C22×C181—41no—
100C22×C181—41no—
101C22×C181—41no—
102C22×C181—41no—
103C22×C181—42no—
104C22×C181—41no—
105C22×C181—41no—
106C22×C181—81no—
107C22×C181—81no—
108C22×C181—81no—
109C22×C181#12011yescharacteristic
110C2×D181—41no—
111C2×D181—41no—
112C2×D181—42no—
113C2×D181—41no—
114C2×D181—81no—
115C4×C181—21no—
116C4×C181—41no—
117C4×C181—41no—
118C4×C181—81no—
119Dic181—41no—
120Dic181—41no—
121Dic181—41no—
122Dic181—81no—
123C2×C181—21no—
124C2×C181—41no—
125C2×C181—41no—
126C2×C181—81no—
127C2×C181—81no—
128C2×C181B(4,𝔽2)11yescharacteristic
129D181—81no—
130C181#219911yescharacteristic, complemented, Sylow
131#2199—1811nomaximal, Sylow
132H(3,𝔽4)—1811no—
133B(4,𝔽2)—1811no—
134B(4,𝔽2)D18111yescharacteristic, complemented
135#455—1811no—
136#354—1811no—
137AΣL(1,𝔽16)—1811no—
138AΣL(1,𝔽16)—1811no—
139H(5,𝔽2)C2×D18111yescharacteristic
140#137—3621no—
141#124—3621no—
142#124—3621no—
143#124—3621no—
144#124—3621no—
145#124—1811no—
146#120—21no—
147#120—3621no—
148#120—3621no—
149#120—3621no—
150#120—1811no—
151#120C2×D18111yescharacteristic
152#104—3621no—
153#100—1811no—
154#99—21no—
155#99—3621no—
156#99—1811no—
157#99C2×D18111yescharacteristic
158C24—1811no—
159C24C22⋊D18111yescharacteristic, complemented
160#41—21no—
161#41—41no—
162C2×Q8—3621no—
163C2×Q8—1811no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—3621no—
170C2×D4—3621no—
171C2×D4—3621no—
172C2×D4—7241no—
173C2×D4C22×D18111yescharacteristic
174Q16—7241no—
175SD16—7241no—
176M16—3621no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—41no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—3621no—
183C22⋊C4—3621no—
184C22⋊C4—3621no—
185C22⋊C4—3621no—
186C22⋊C4—3621no—
187C22⋊C4—3621no—
188C22⋊C4—1811no—
189C22⋊C4—7241no—
190C22⋊C4C22⋊D18111yescharacteristic, complemented
191C42—3621no—
192C42—1811no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23—3621no—
199C23—14481no—
200C23—7241no—
201C23D4×D18111yescharacteristic
202C23C2×C22⋊D18111yescharacteristic
203Q8—21no—
204Q8—14481no—
205Q8—7241no—
206D4—21no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—81no—
212D4—82no—
213D4—14482no—
214D4—14481no—
215D4—14481no—
216D4—14481no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3621no—
223C2×C4—3621no—
224C2×C4—3621no—
225C2×C4—7241no—
226C2×C4D4×D18111yescharacteristic
227C8—7241no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—14481no—
241C22—7241no—
242C22—7241no—
243C22—7242no—
244C22—7241no—
245C22#1915378411yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—14481no—
251C4—7241no—
252C4—7241no—
253C4—7241no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—81no—
259C2—14481no—
260C2#2604146511yescharacteristic
subgroupG/H
centerC2#26041465
commutator, cosocleC2×D4×C181C23
FrattiniC2×D4C22×D181
FittingB(4,𝔽2)×C181C2
socleC2×C181B(4,𝔽2)
hypercenterB(4,𝔽2)D181
hypocenterC181#2199
LCSG ▷ C2×D4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26042022C2semidirect
2. #26041466C2semidirect
3. #26041457C2semidirect
4. #26041073C2semidirect
5. #26041066C2semidirect
6. #26041065C2semidirect
7. B(4,𝔽2)×C181C2semidirect
27. C23×D181D4semidirect
29. C2×Dic362D4semidirect
42. C22⋊Dic181D4semidirect
48. C4×Dic181D4semidirect
49. C24×C181D4semidirect
61. C22⋊C4×C181D4semidirect
130. C181#2199semidirect
134. B(4,𝔽2)D181semidirect
159. C24C22⋊D181semidirect
190. C22⋊C4C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
#260420222
#260414662
#260414572
#260410732
#260410662
#260410652
B(4,𝔽2)×C1812
#2199181