Finite Group Databasebeta

← #37549772

Subgroups — #37549772

Order 23168.

340 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 66 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#26042357C211yesmaximal, characteristic, complemented
2#26042266C211yesmaximal, characteristic, complemented
3C2×#19153876C212yesmaximal, complemented
4#26042160C212yesmaximal, complemented
5#26041844C211yesmaximal, characteristic
6#19153903—24no—
7D181⋊M16—21no—
8C2×#15668458—21no—
9C2×#15668458C2212yes—
10C2×#15668458C2212yes—
11#19153877—21no—
12#19153876—24no—
13#19153876—21no—
14C2×#15668448C2211yescharacteristic
15C22×C22⋊D181C2211yescharacteristic
16#19153812—21no—
17#19153802—21no—
18#19153789—21no—
19#19153788—21no—
20#19153786C411yescharacteristic
21#19153785C411yescharacteristic
22#19153784C411yescharacteristic
23#19153772C2211yescharacteristic
24#120×C181C411yescharacteristic
25#15668459—22no—
26#15668458—22no—
27#15668458—22no—
28#15668458—22no—
29#15668458—22no—
30#15668458D414yescomplemented
31D181⋊C8—22no—
32C2×#13089685—22no—
33C2×#13089685—22no—
34C2×#13089685—22no—
35C23×D181C2×C411yescharacteristic
36#15668448—22no—
37#15668448—22no—
38C2×Dic362D411yescharacteristic
39C2×D724—21no—
40C2×C22⋊D181—21no—
41C2×C22⋊D181—21no—
42C2×C22⋊D181—21no—
43C2×C22⋊D181—22no—
44C2×C22⋊D181—22no—
45C2×C22⋊D181—21no—
46C2×C22⋊D181C2×C411yescharacteristic
47C2×C22⋊D181D411yescharacteristic
48C22×Dic181D411yescharacteristic
49C22×Dic181C2311yescharacteristic
50C2×C4×D181D411yescharacteristic
51#15668428—21no—
52C22⋊Dic181—21no—
53C22⋊Dic181C2×C411yescharacteristic
54#15668426—21no—
55#15668426—21no—
56C4×Dic181—21no—
57C24×C181C2×C411yescharacteristic
58C2×D4×C181—21no—
59C2×D4×C181C2×C411yescharacteristic
60C22⋊C4×C181—21no—
61C22⋊C4×C181C2×C411yescharacteristic
62#13089685—24no—
63#13089685—42no—
64C22×D181—21no—
65C22×D181—22no—
66C22×D181—81no—
67C22×D181C22⋊C411yescharacteristic
68Dic362—22no—
69D724—81no—
70C22⋊D181—22no—
71C22⋊D181—22no—
72C22⋊D181—41no—
73C22⋊D181—42no—
74C22⋊D181—41no—
75C22⋊D181—81no—
76C22⋊D181—81no—
77C2×Dic181—21no—
78C2×Dic181—21no—
79C2×Dic181—21no—
80C2×Dic181C22⋊C411yescharacteristic
81C2×Dic181C2×D412yes—
82C2×Dic181C2×D411yescharacteristic
83C2×Dic181C2×D411yescharacteristic
84C2×Dic181#4112yes—
85C4×D181—21no—
86C4×D181—21no—
87C23×C181—21no—
88C23×C181—22no—
89C23×C181—81no—
90C23×C181C22⋊C411yescharacteristic
91C23×C181C22×C411yescharacteristic
92D4×C181—22no—
93D4×C181—81no—
94C2×C4×C181—21no—
95C2×C4×C181C22⋊C411yescharacteristic
96C22×C181—21no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181—41no—
100C22×C181—41no—
101C22×C181—41no—
102C22×C181—82no—
103C22×C181C2×C22⋊C411yescharacteristic
104C22×C181C4×D412yes—
105C2×D181—21no—
106C2×D181—41no—
107C2×D181—41no—
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—82no—
111C4×C181—21no—
112C4×C181—41no—
113Dic181—21no—
114Dic181—41no—
115Dic181#12011yescharacteristic
116Dic181#12112yes—
117Dic181#12311yescharacteristic
118C2×C181—21no—
119C2×C181—22no—
120C2×C181—41no—
121C2×C181—81no—
122C2×C181#38611yescharacteristic
123D181—41no—
124D181—81no—
125C181#191411yescharacteristic, complemented, Sylow
126#1914—1811nomaximal, Sylow
127#568—1811no—
128#535—1811no—
129#413—1811no—
130C2×#100—1812no—
131#344—1812no—
132C2×#41—1811no—
133C22×D4—1811no—
134#131—3624no—
135C2×M16—3621no—
136C2×M16—3621no—
137C2×M16—1812no—
138C2×M16—1812no—
139#127—3621no—
140#124—3621no—
141#121—3621no—
142#121—3621no—
143#120—1811no—
144#120—1811no—
145#120—1811no—
146#120#1062438211yescharacteristic
147#117—1811no—
148#101—3621no—
149#100—3621no—
150#100—3624no—
151C24—1811no—
152C24C2×#1062438211yescharacteristic
153#41—3622no—
154#41—3622no—
155C2×Q8—1811no—
156C2×D4—21no—
157C2×D4—3621no—
158C2×D4—3621no—
159C2×D4—3621no—
160C2×D4—3622no—
161C2×D4—3621no—
162C2×D4—3621no—
163C2×D4—3622no—
164C2×D4—1811no—
165C2×D4—1811no—
166C2×D4C2×#1062438211yescharacteristic
167C22×C4—1811no—
168C22×C4—1811no—
169C22×C4—1811no—
170M16—3622no—
171M16—3622no—
172M16—3622no—
173M16—3622no—
174M16—3622no—
175M16—1814no—
176C2×C8—3622no—
177C2×C8—3622no—
178C2×C8—3622no—
179C2×C8—3622no—
180C4⋊C4—3621no—
181C22⋊C4—21no—
182C22⋊C4—3621no—
183C22⋊C4—3621no—
184C22⋊C4—3621no—
185C22⋊C4—1811no—
186C22⋊C4C2×#1062438211yescharacteristic
187C42—3621no—
188C23—21no—
189C23—22no—
190C23—81no—
191C23—3622no—
192C23—3621no—
193C23—14481no—
194C23—1811no—
195C23C22⋊#1062438211yescharacteristic
196C23C22×#1062438211yescharacteristic
197Q8—3622no—
198D4—22no—
199D4—81no—
200D4—3622no—
201D4—3622no—
202D4—14481no—
203D4—14481no—
204D4—14481no—
205D4—7241no—
206D4—7242no—
207D4—7241no—
208C2×C4—21no—
209C2×C4—3621no—
210C2×C4—3621no—
211C2×C4—3621no—
212C2×C4—3621no—
213C2×C4—3621no—
214C2×C4—1811no—
215C2×C4—1811no—
216C2×C4—1811no—
217C2×C4—1812no—
218C2×C4—1812no—
219C2×C4C22⋊#1062438211yescharacteristic
220C8—3624no—
221C8—7242no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—82no—
229C22—3621no—
230C22—14482no—
231C22—7241no—
232C22—7241no—
233C22—7241no—
234C22—7241no—
235C22C2×C22⋊#1062438211yescharacteristic
236C22D4×#1062438212yes—
237C4—21no—
238C4—41no—
239C4—3621no—
240C4—1811no—
241C4—1812no—
242C4—1811no—
243C4—7241no—
244C2—21no—
245C2—22no—
246C2—41no—
247C2—81no—
248C2—14481no—
249C2—7241no—
250C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
Fitting#120×C181C4
socleC2×C181#386
prime radical#26041844C2
cosocleC22×Dic181C23
hypercenter#120#10624382
hypocenterC181#1914
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #26042357C2semidirect
2. #26042266C2semidirect
3. C2×#19153876C2semidirect
4. #26042160C2semidirect
30. #15668458D4semidirect
125. C181#1914semidirect

Maximal subgroups

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

subgroupindex
#260423572
#260422662
C2×#191538762
#260421602
#260418442
#1914181