Finite Group Databasebeta

← #37549751

Subgroups — #37549751

Order 23168.

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

normal subgroups only

This group has 304 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#26042359C211yesmaximal, characteristic
2#26042278C211yesmaximal, characteristic
3#26042272C211yesmaximal, characteristic, complemented
4D181⋊#101C211yesmaximal, characteristic, complemented
5#26042161C211yesmaximal, characteristic, complemented
6#26042148C211yesmaximal, characteristic, complemented
7C2×#19153848C211yesmaximal, characteristic
8#19153904—22no—
9#19153904—22no—
10D181⋊M16—22no—
11D181⋊M16C2211yescharacteristic
12D181⋊M16C2211yescharacteristic
13D181⋊M16C2211yescharacteristic
14#19153895C2211yescharacteristic
15#19153881—22no—
16#19153879—22no—
17#19153877—22no—
18C2×#15668450C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153848—22no—
21#19153848—22no—
22C2×#15668439C411yescharacteristic, complemented
23C2×#15668438C411yescharacteristic, complemented
24C2×#15668436C411yescharacteristic, complemented
25C2×#15668432C2211yescharacteristic
26C2×Q16×C181C411yescharacteristic, complemented
27#15668459—24no—
28#15668459—21no—
29#15668459—21no—
30#15668459—21no—
31#15668458—21no—
32#15668458—21no—
33#15668458D412yes—
34D181⋊C8—24no—
35D181⋊C8—21no—
36D181⋊C8—21no—
37D181⋊C8D412yes—
38C4⋊#10624382—21no—
39C22⋊#10624382—21no—
40C4×#10624382—21no—
41#15668450—21no—
42#15668450—21no—
43#15668450D412yes—
44#15668450D412yes—
45C2×Dic362—21no—
46C2×Dic362C2×C411yescharacteristic
47Q8×D181—21no—
48Q8×D181—21no—
49Q8×D181—21no—
50Q8×D181—21no—
51C2×D724—21no—
52C2×D724C2×C411yescharacteristic
53C2×C4×D181—21no—
54C2×C4×D181—21no—
55C2×C4×D181C2311yescharacteristic
56#15668439—22no—
57#15668438—22no—
58#15668436—22no—
59#15668432—21no—
60#15668432—21no—
61C2×#13089678C2×C411yescharacteristic
62C2×Q8×C181C2×C411yescharacteristic
63C2×Q8×C181C2×C411yescharacteristic
64Q16×C181—22no—
65C2×C8×C181C2×C411yescharacteristic
66C2×#10624382—41no—
67#13089685—22no—
68#13089685—22no—
69#13089685—41no—
70C22×D181—21no—
71C22×D181#4111yescharacteristic
72Dic362—21no—
73Dic362—21no—
74Dic362—41no—
75Dic362—41no—
76D724—22no—
77D724—22no—
78D724—21no—
79D724C22⋊C412yes—
80C2×Dic181—21no—
81C2×Dic181C2×D411yescharacteristic
82C4×D181—22no—
83C4×D181—22no—
84C4×D181—41no—
85C4×D181—41no—
86C4×D181C2×D411yescharacteristic
87C4×D181C2×D411yescharacteristic
88C4×D181C2×D411yescharacteristic
89C4×D181#4111yescharacteristic
90#13089678—21no—
91Q8×C181—21no—
92Q8×C181—21no—
93Q8×C181—21no—
94Q8×C181C22⋊C412yes—
95C2×C4×C181—21no—
96C2×C4×C181—21no—
97C2×C4×C181C22×C411yescharacteristic
98C8×C181—21no—
99C22×C181C4×D411yescharacteristic
100C2×D181—22no—
101C2×D181—41no—
102C2×D181—81no—
103C2×D181#12111yescharacteristic
104C2×D181#12311yescharacteristic
105#10624382—81no—
106C4×C181—22no—
107C4×C181—41no—
108C4×C181C2×C22⋊C411yescharacteristic
109C4×C181C4×D411yescharacteristic
110Dic181—41no—
111Dic181#12011yescharacteristic
112Dic181#12111yescharacteristic
113C2×C181—21no—
114C2×C181#38611yescharacteristic
115D181—21no—
116D181—21no—
117D181—42no—
118C181#190211yescharacteristic, complemented, Sylow
119#1902—1811nomaximal, Sylow
120C2×#137—1811no—
121#568—1811no—
122#421—1811no—
123#419—1811no—
124C2×#101—1811no—
125#344—1811no—
126#338—1811no—
127C2×#41—1811no—
128C22×Q8—1811no—
129#137—3622no—
130#137—3622no—
131C2×Q16—1811no—
132C2×Q16#1062438211yescharacteristic, complemented
133C2×SD16—1811no—
134C2×SD16—1811no—
135#131—3622no—
136#131—3622no—
137C2×M16—3622no—
138C2×M16—1811no—
139C2×M16—1811no—
140C2×M16—1811no—
141C2×M16—1811no—
142#117—1811no—
143#104—3622no—
144#103—3622no—
145#101—3622no—
146#41—3621no—
147#41—3621no—
148#41—1812no—
149#41—1812no—
150C2×Q8—3621no—
151C2×Q8—3621no—
152C2×Q8—3621no—
153C2×Q8—3621no—
154C2×Q8—3621no—
155C2×Q8—1811no—
156C2×Q8C2×#1062438211yescharacteristic
157C2×Q8C2×#1062438211yescharacteristic
158C2×D4—3621no—
159C2×D4—1811no—
160C22×C4—3621no—
161C22×C4—3621no—
162C22×C4—1811no—
163Q16—22no—
164Q16—3622no—
165SD16—3622no—
166SD16—3622no—
167M16—3621no—
168M16—3621no—
169M16—3624no—
170M16—3621no—
171M16—3621no—
172M16—3621no—
173M16—3621no—
174M16—3621no—
175M16—1812no—
176C2×C8—3624no—
177C2×C8—3621no—
178C2×C8—3621no—
179C2×C8—1812no—
180C2×C8—1811no—
181C2×C8C2×#1062438211yescharacteristic
182C4⋊C4—3621no—
183C22⋊C4—3621no—
184C42—3621no—
185C23—3621no—
186C23—1811no—
187Q8—21no—
188Q8—21no—
189Q8—21no—
190Q8—3621no—
191Q8—3621no—
192Q8—7241no—
193Q8—7241no—
194Q8C22⋊#1062438212yes—
195D4—3622no—
196D4—3622no—
197D4—3621no—
198D4—1812no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—3622no—
202C2×C4—3622no—
203C2×C4—3621no—
204C2×C4—1811no—
205C2×C4—1811no—
206C2×C4—1811no—
207C2×C4—1811no—
208C2×C4—1811no—
209C2×C4—7241no—
210C2×C4—7241no—
211C2×C4—7241no—
212C2×C4C22×#1062438211yescharacteristic
213C8—21no—
214C8—3622no—
215C8—3622no—
216C8—3621no—
217C8—7241no—
218C22—3622no—
219C22—14481no—
220C22—1811no—
221C22—1811no—
222C22—7241no—
223C22D4×#1062438211yescharacteristic
224C4—22no—
225C4—41no—
226C4—14481no—
227C4—1811no—
228C4—1811no—
229C4—7241no—
230C4C2×C22⋊#1062438211yescharacteristic
231C4D4×#1062438211yescharacteristic
232C2—21no—
233C2—3621no—
234C2—3621no—
235C2—7242no—
236C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC2×Q16×C181C4
socleC2×C181#386
prime radicalC2×#15668450C22
cosocleC2×C4×D181C23
hypercenterC2×Q16#10624382
hypocenterC181#1902
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
3. #26042272C2semidirect
4. D181⋊#101C2semidirect
5. #26042161C2semidirect
6. #26042148C2semidirect
22. C2×#15668439C4semidirect
23. C2×#15668438C4semidirect
24. C2×#15668436C4semidirect
26. C2×Q16×C181C4semidirect
118. C181#1902semidirect
132. C2×Q16#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423592
#260422782
#260422722
D181⋊#1012
#260421612
#260421482
C2×#191538482
#1902181