Finite Group Databasebeta

← #37532649

Subgroups — #37532649

Order 23168.

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

normal subgroups only

This group has 272 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#26041878C211yesmaximal, characteristic, complemented
2#26041413C211yesmaximal, characteristic
3#26041193C211yesmaximal, characteristic, complemented
4#26041016C211yesmaximal, characteristic, complemented
5#26041014C211yesmaximal, characteristic, complemented
6#26041004C211yesmaximal, characteristic, complemented
7#450×C181C211yesmaximal, characteristic, complemented
8C2×#15668434—21no—
9#19153811—21no—
10#19153807—21no—
11#19153801—21no—
12#19153800—21no—
13#19153792—21no—
14#19153790C2211yescharacteristic
15#19153776—21no—
16C2×#15668428C2211yescharacteristic
17C2×C22⋊Dic181—21no—
18C2×C22⋊Dic181—21no—
19#19153737—21no—
20C22⋊#13089678C2211yescharacteristic
21#19153723C2211yescharacteristic
22C22×D4×C181C2211yescharacteristic
23C2×SD16×C181—21no—
24C22⋊Q8×C181C2211yescharacteristic
25#102×C181—21no—
26C22⋊C8×C181C2211yescharacteristic
27C2×D724D411yescharacteristic
28C2×C22⋊D181—21no—
29C22×Dic181—21no—
30C22×Dic181D411yescharacteristic
31C22×Dic181D411yescharacteristic
32C2×C4×D181—21no—
33#15668434—42no—
34C2×#13089678—21no—
35C4⋊Dic181—21no—
36C4⋊Dic181—21no—
37C4⋊Dic181D411yescharacteristic
38#15668428—21no—
39#15668428—21no—
40#15668428—21no—
41C22⋊Dic181—42no—
42C22⋊Dic181—42no—
43#15668426—21no—
44#15668426—21no—
45#15668426—21no—
46#15668426—21no—
47C4×Dic181—21no—
48C24×C181—21no—
49C2×Q8×C181D411yescharacteristic
50C2×D4×C181—21no—
51C2×D4×C181—21no—
52C2×D4×C181—42no—
53C22×C4×C181C2311yescharacteristic
54SD16×C181—42no—
55C2×C8×C181—21no—
56C4⋊C4×C181—21no—
57C4⋊C4×C181D411yescharacteristic
58C22⋊C4×C181—21no—
59C22×D181—21no—
60D724—41no—
61C22⋊D181—41no—
62C22⋊D181—41no—
63C2×Dic181—21no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—21no—
67C2×Dic181—21no—
68C2×Dic181—41no—
69C2×Dic181—41no—
70C2×Dic181—41no—
71C2×Dic181—41no—
72C4×D181—81no—
73#13089678—41no—
74C23×C181—22no—
75C23×C181—41no—
76C23×C181—41no—
77C23×C181—41no—
78C23×C181—41no—
79C23×C181—41no—
80C23×C181—41no—
81C23×C181C2×D411yescharacteristic
82Q8×C181—41no—
83D4×C181—22no—
84D4×C181—22no—
85D4×C181—41no—
86D4×C181—41no—
87C2×C4×C181—21no—
88C2×C4×C181—21no—
89C2×C4×C181—41no—
90C2×C4×C181C2×D411yescharacteristic
91C2×C4×C181C2×D411yescharacteristic
92C8×C181—41no—
93C22×C181—21no—
94C22×C181—21no—
95C22×C181—42no—
96C22×C181—42no—
97C22×C181—42no—
98C22×C181—81no—
99C22×C181—81no—
100C22×C181—81no—
101C22×C181—81no—
102C22×C181#12011yescharacteristic
103C22×C181#13511yescharacteristic
104C22×C181AGL(1,ℤ/8)11yescharacteristic
105C2×D181—41no—
106C2×D181—41no—
107C2×D181—41no—
108C4×C181—21no—
109C4×C181—21no—
110C4×C181—41no—
111C4×C181—41no—
112Dic181—41no—
113Dic181—41no—
114Dic181—41no—
115Dic181—81no—
116C2×C181—21no—
117C2×C181—21no—
118C2×C181—82no—
119C2×C181#44911yescharacteristic
120C2×C181#45411yescharacteristic
121C2×C181B(4,𝔽2)11yescharacteristic
122D181—81no—
123C181#159711yescharacteristic, complemented, Sylow
124#1597—1811nomaximal, Sylow
125#539—1811no—
126#450D18111yescharacteristic, complemented
127#447—1811no—
128#394—1811no—
129#328—1811no—
130#327—1811no—
131#324—1811no—
132C22×D4C2×D18111yescharacteristic
133C2×SD16—21no—
134C2×D8—3621no—
135#126—3621no—
136#125—3621no—
137#123—3621no—
138#123—3621no—
139C22⋊Q8—3621no—
140C22⋊Q8C2×D18111yescharacteristic
141#121—1811no—
142C4×D4—3621no—
143C2×C4⋊C4—1811no—
144C2×C22⋊C4—3621no—
145C2×C22⋊C4—3621no—
146#102—21no—
147#102—3621no—
148C22⋊C8—1811no—
149C22⋊C8C2×D18111yescharacteristic
150#95—1811no—
151C24—21no—
152C2×Q8C22⋊D18111yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—3621no—
157C2×D4—1811no—
158C22×C4—3621no—
159C22×C4—3621no—
160C22×C4—1811no—
161C22×C4—1811no—
162C22×C4C22×D18111yescharacteristic
163SD16—42no—
164D8—7242no—
165C2×C8—21no—
166C2×C8—3621no—
167C4⋊C4—21no—
168C4⋊C4—3621no—
169C4⋊C4—3621no—
170C4⋊C4—3621no—
171C4⋊C4—3621no—
172C4⋊C4—3621no—
173C4⋊C4—1811no—
174C4⋊C4C22⋊D18111yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—3621no—
180C22⋊C4—7242no—
181C22⋊C4—7242no—
182C42—3621no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—3621no—
191C23D4×D18111yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—7241no—
198D4—7241no—
199D4—7241no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—3621no—
204C2×C4—3621no—
205C2×C4—3621no—
206C2×C4—3621no—
207C2×C4—3621no—
208C2×C4—14481no—
209C2×C4—7241no—
210C2×C4—7241no—
211C2×C4—7241no—
212C2×C4—7241no—
213C2×C4D4×D18111yescharacteristic
214C2×C4C2×C22⋊D18111yescharacteristic
215C8—41no—
216C8—7241no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—7241no—
227C22—7241no—
228C22—7241no—
229C22#1915378411yescharacteristic
230C22#1915383711yescharacteristic
231C22#1915384311yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—14481no—
237C4—7241no—
238C4—7241no—
239C4—7241no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—14481no—
244C2#2604142311yescharacteristic
245C2#2604145011yescharacteristic
246C2#2604146511yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#450×C181C2
socleC22×C181#120
prime radical#26041413C2
hypercenter#450D181
hypocenterC181#1597
LCSG ▷ C22×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26041878C2semidirect
3. #26041193C2semidirect
4. #26041016C2semidirect
5. #26041014C2semidirect
6. #26041004C2semidirect
7. #450×C181C2semidirect
123. C181#1597semidirect
126. #450D181semidirect

Maximal subgroups

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

subgroupindex
#260418782
#260414132
#260411932
#260410162
#260410142
#260410042
#450×C1812
#1597181