Finite Group Databasebeta

← #37532650

Subgroups — #37532650

Order 23168.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#26041429C211yesmaximal, characteristic, complemented
2#26041412C211yesmaximal, characteristic
3#26041196C211yesmaximal, characteristic, complemented
4#26041016C211yesmaximal, characteristic, complemented
5#26041013C211yesmaximal, characteristic, complemented
6#26041005C211yesmaximal, characteristic, complemented
7#539×C181C211yesmaximal, characteristic, complemented
8C22×D724C2211yescharacteristic
9C2×#15668436—21no—
10C2×#15668434—21no—
11C2×#15668426—21no—
12C2×#15668426—21no—
13#19153736—21no—
14#19153736—21no—
15C22⋊#13089678C2211yescharacteristic
16C22⋊#13089678C2211yescharacteristic
17#19153723C2211yescharacteristic
18#126×C181—21no—
19#125×C181—21no—
20#123×C181—21no—
21#123×C181—21no—
22C22⋊Q8×C181—21no—
23C22⋊Q8×C181C2211yescharacteristic
24#121×C181C2211yescharacteristic
25C4×D4×C181—21no—
26C2×C4⋊C4×C181C2211yescharacteristic
27C23×D181—21no—
28C2×D724—21no—
29C2×D724—21no—
30C2×D724—42no—
31C22×Dic181—21no—
32#15668436—42no—
33#15668434—42no—
34C2×#13089678—21no—
35C2×#13089678—21no—
36#15668426—42no—
37#15668426—42no—
38C2×Q8×C181D411yescharacteristic
39C2×D4×C181—21no—
40C2×D4×C181D411yescharacteristic
41C22×C4×C181—21no—
42C22×C4×C181D411yescharacteristic
43C22×C4×C181D411yescharacteristic
44C22×C4×C181C2311yescharacteristic
45C4⋊C4×C181—21no—
46C4⋊C4×C181—21no—
47C4⋊C4×C181—21no—
48C4⋊C4×C181—21no—
49C4⋊C4×C181—21no—
50C4⋊C4×C181—21no—
51C4⋊C4×C181D411yescharacteristic
52C4⋊C4×C181D411yescharacteristic
53C22⋊C4×C181—21no—
54C22⋊C4×C181—21no—
55C22⋊C4×C181—21no—
56C22⋊C4×C181—21no—
57C22⋊C4×C181—21no—
58C42×C181—21no—
59C22×D181—22no—
60C22×D181—41no—
61C22×D181—41no—
62C22×D181—41no—
63C22×D181—41no—
64C22×D181—41no—
65C22×D181—41no—
66D724—22no—
67D724—22no—
68D724—41no—
69D724—41no—
70C2×Dic181—41no—
71C2×Dic181—41no—
72C2×Dic181—41no—
73#13089678—41no—
74#13089678—41no—
75C23×C181—21no—
76C23×C181C2×D411yescharacteristic
77Q8×C181—41no—
78D4×C181—41no—
79D4×C181—41no—
80D4×C181—41no—
81C2×C4×C181—21no—
82C2×C4×C181—21no—
83C2×C4×C181—21no—
84C2×C4×C181—21no—
85C2×C4×C181—21no—
86C2×C4×C181—21no—
87C2×C4×C181—21no—
88C2×C4×C181—41no—
89C2×C4×C181—41no—
90C2×C4×C181—81no—
91C2×C4×C181C2×D411yescharacteristic
92C2×C4×C181C2×D411yescharacteristic
93C22×C181—21no—
94C22×C181—21no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181#12011yescharacteristic
99C22×C181#13511yescharacteristic
100C22×C181AGL(1,ℤ/8)11yescharacteristic
101C2×D181—42no—
102C2×D181—42no—
103C2×D181—42no—
104C2×D181—81no—
105C2×D181—81no—
106C2×D181—81no—
107C2×D181—81no—
108C4×C181—21no—
109C4×C181—21no—
110C4×C181—41no—
111C4×C181—41no—
112C4×C181—41no—
113C4×C181—41no—
114C4×C181—41no—
115Dic181—81no—
116C2×C181—21no—
117C2×C181—21no—
118C2×C181—81no—
119C2×C181#44911yescharacteristic
120C2×C181#45411yescharacteristic
121C2×C181B(4,𝔽2)11yescharacteristic
122D181—82no—
123C181#159711yescharacteristic, complemented, Sylow
124#1597—1811nomaximal, Sylow
125#539D18111yescharacteristic, complemented
126#450—1811no—
127#447—1811no—
128#394—1811no—
129#328—1811no—
130#327—1811no—
131#324—1811no—
132C22×D4—1811no—
133C2×SD16—3621no—
134C2×D8—3621no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D18111yescharacteristic
141#121C2×D18111yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D18111yescharacteristic
144C2×C22⋊C4—3621no—
145C2×C22⋊C4—3621no—
146#102—3621no—
147#102—3621no—
148C22⋊C8—1811no—
149C22⋊C8—1811no—
150#95—1811no—
151C24—3621no—
152C2×Q8C22⋊D18111yescharacteristic
153C2×D4—21no—
154C2×D4—3621no—
155C2×D4—3621no—
156C2×D4—7242no—
157C2×D4C22⋊D18111yescharacteristic
158C22×C4—21no—
159C22×C4—3621no—
160C22×C4C22⋊D18111yescharacteristic
161C22×C4C22⋊D18111yescharacteristic
162C22×C4C22×D18111yescharacteristic
163SD16—7242no—
164D8—7242no—
165C2×C8—3621no—
166C2×C8—3621no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D18111yescharacteristic
174C4⋊C4C22⋊D18111yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—7242no—
181C22⋊C4—7242no—
182C42—21no—
183C23—21no—
184C23—3622no—
185C23—7241no—
186C23—7241no—
187C23—7241no—
188C23—7241no—
189C23—7241no—
190C23—7241no—
191C23C2×C22⋊D18111yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—3622no—
197D4—3622no—
198D4—7241no—
199D4—7241no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—7241no—
211C2×C4—7241no—
212C2×C4—7241no—
213C2×C4C2×C22⋊D18111yescharacteristic
214C2×C4C2×C22⋊D18111yescharacteristic
215C8—7241no—
216C8—7241no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—14481no—
223C22—14481no—
224C22—14481no—
225C22—14481no—
226C22—7242no—
227C22—7242no—
228C22—7242no—
229C22#1915378511yescharacteristic
230C22#1915383611yescharacteristic
231C22#1915384111yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—14481no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—14482no—
244C2#2604142411yescharacteristic
245C2#2604145211yescharacteristic
246C2#2604146611yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#539×C181C2
socleC22×C181#120
prime radical#26041412C2
hypercenter#539D181
hypocenterC181#1597
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #26041429C2semidirect
3. #26041196C2semidirect
4. #26041016C2semidirect
5. #26041013C2semidirect
6. #26041005C2semidirect
7. #539×C181C2semidirect
123. C181#1597semidirect
125. #539D181semidirect

Maximal subgroups

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

subgroupindex
#260414292
#260414122
#260411962
#260410162
#260410132
#260410052
#539×C1812
#1597181