Finite Group Databasebeta

← #37549563

Subgroups — #37549563

Order 23168.

618 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#19153881C212yesmaximal, complemented
2C2×#19153880C212yesmaximal, complemented
3#26042250C211yesmaximal, characteristic, complemented
4C4×C22⋊#10624382C211yesmaximal, characteristic, complemented
5C2×#41×D181C211yesmaximal, characteristic
6C2×#15668459C2212yescomplemented
7C2×C22⋊#10624382—21no—
8C2×C4×#10624382C2212yescomplemented
9#19153881—24no—
10#19153880—24no—
11#19153873—21no—
12C22⋊#13089685—21no—
13(C2×#10624382)⋊C4—21no—
14C2×#15668450C2211yescharacteristic
15C2×#15668449—21no—
16C2×#15668449C411yescharacteristic, complemented
17C2×#15668448—21no—
18C2×#15668448C2211yescharacteristic
19#41×D181—24no—
20#41×D181—24no—
21C2×Q8×D181C411yescharacteristic, complemented
22C2×D4×D181—21no—
23C2×D4×D181C411yescharacteristic, complemented
24C22×C4×D181—21no—
25C22×C4×D181C2211yescharacteristic
26C2×#41×C181C411yescharacteristic, complemented
27C22×#10624382—22no—
28#15668459—22no—
29#15668459—22no—
30C2×#13089685—22no—
31C22⋊#10624382—24no—
32C4×#10624382—22no—
33C4×#10624382—22no—
34C23×D181—21no—
35C23×D181D411yescharacteristic
36#15668450—22no—
37#15668450D414yes—
38#15668449—28no—
39#15668449—22no—
40#15668449—22no—
41#15668448—28no—
42#15668448—22no—
43#15668448D414yes—
44C2×Dic362—21no—
45C2×Dic362C2×C411yescharacteristic
46Q8×D181—22no—
47Q8×D181—22no—
48C2×D724—21no—
49C2×D724C2×C411yescharacteristic
50C2×C22⋊D181—21no—
51C2×C22⋊D181—21no—
52C2×C22⋊D181—21no—
53D4×D181—28no—
54D4×D181—22no—
55D4×D181—22no—
56C22×Dic181—21no—
57C22×Dic181D411yescharacteristic
58C2×C4×D181—28no—
59C2×C4×D181—21no—
60C2×C4×D181—21no—
61C2×C4×D181—22no—
62C2×C4×D181—22no—
63C2×C4×D181—21no—
64C2×C4×D181C2×C411yescharacteristic
65C2×C4×D181D411yescharacteristic
66C2×C4×D181D411yescharacteristic
67C2×C4×D181C2311yescharacteristic
68#41×C181—22no—
69#41×C181—22no—
70C2×Q8×C181C2×C411yescharacteristic
71C2×D4×C181—21no—
72C2×D4×C181C2×C411yescharacteristic
73C22×C4×C181—21no—
74C22×C4×C181C2×C411yescharacteristic
75C2×#10624382—24no—
76C2×#10624382—42no—
77C2×#10624382—42no—
78#13089685—42no—
79C22×D181—24no—
80C22×D181—21no—
81C22×D181—22no—
82C22×D181—22no—
83C22×D181—41no—
84C22×D181—42no—
85C22×D181—42no—
86C22×D181C22⋊C411yescharacteristic
87C22×D181C2×D411yescharacteristic
88Dic362—24no—
89Dic362—21no—
90Dic362C22⋊C412yescomplemented
91D724—24no—
92D724—21no—
93D724C22⋊C412yescomplemented
94C22⋊D181—24no—
95C22⋊D181—24no—
96C22⋊D181—24no—
97C2×Dic181—21no—
98C2×Dic181—24no—
99C2×Dic181—22no—
100C2×Dic181C22⋊C411yescharacteristic
101C2×Dic181C2×D411yescharacteristic
102C4×D181—24no—
103C4×D181—24no—
104C4×D181—22no—
105C4×D181—21no—
106C4×D181—21no—
107C4×D181—22no—
108C4×D181—24no—
109C4×D181C2×D412yes—
110C4×D181C2×D412yes—
111C23×C181—21no—
112C23×C181C22⋊C411yescharacteristic
113Q8×C181—21no—
114Q8×C181C22⋊C412yescomplemented
115D4×C181—24no—
116D4×C181—21no—
117D4×C181C22⋊C412yescomplemented
118C2×C4×C181—24no—
119C2×C4×C181—21no—
120C2×C4×C181—21no—
121C2×C4×C181—21no—
122C2×C4×C181C22⋊C411yescharacteristic
123C2×C4×C181C22×C411yescharacteristic
124C22×C181—21no—
125C22×C181—22no—
126C22×C181—21no—
127C22×C181—21no—
128C22×C181—41no—
129C22×C181—41no—
130C22×C181C2×C22⋊C411yescharacteristic
131C2×D181—21no—
132C2×D181—21no—
133C2×D181—21no—
134C2×D181—21no—
135C2×D181—22no—
136C2×D181—44no—
137C2×D181—48no—
138C2×D181—41no—
139C2×D181—41no—
140C2×D181#10412yescomplemented
141C2×D181#12012yes—
142#10624382—44no—
143C4×C181—21no—
144C4×C181—22no—
145C4×C181C2×C22⋊C411yescharacteristic
146C4×C181C2×C22⋊C411yescharacteristic
147Dic181—22no—
148Dic181—21no—
149Dic181#12012yes—
150C2×C181—41no—
151C2×C181—42no—
152C2×C181#37911yescharacteristic
153C2×C181C2×#10411yescharacteristic
154C2×C181#42111yescharacteristic
155D181—22no—
156D181—22no—
157D181—42no—
158C181#178911yescharacteristic, complemented, Sylow
159#1789—1811nomaximal, Sylow
160C22×#41—1811no—
161C2×#104—1812no—
162C2×#104—1812no—
163#407—1811no—
164C4×C22⋊C4—1811no—
165C2×#41—3621no—
166C2×#41—3624no—
167C2×#41—3624no—
168C2×#41—3621no—
169C2×#41—1811no—
170C2×#41—1811no—
171C2×#41—1811no—
172C2×#41#1062438211yescharacteristic, complemented
173C22×Q8—1811no—
174C22×D4—3621no—
175C22×D4—1811no—
176C23×C4—3621no—
177C23×C4—1811no—
178C2×M16—1812no—
179C2×C22⋊C4—3621no—
180C2×C42—1812no—
181#104—3624no—
182#104—3624no—
183C22⋊C8—3621no—
184C22⋊C8—3621no—
185#95—3621no—
186C24—3621no—
187C24—1811no—
188#41—22no—
189#41—22no—
190#41—3628no—
191#41—3622no—
192#41—3622no—
193#41—3628no—
194#41—3622no—
195#41—3622no—
196#41—1814no—
197#41—1814no—
198C2×Q8—3621no—
199C2×Q8—3622no—
200C2×Q8—3622no—
201C2×Q8—1811no—
202C2×Q8C2×#1062438211yescharacteristic
203C2×D4—21no—
204C2×D4—3621no—
205C2×D4—3628no—
206C2×D4—3622no—
207C2×D4—3621no—
208C2×D4—3621no—
209C2×D4—3622no—
210C2×D4—3621no—
211C2×D4—1811no—
212C2×D4C2×#1062438211yescharacteristic
213C22×C4—21no—
214C22×C4—3628no—
215C22×C4—3621no—
216C22×C4—3621no—
217C22×C4—3622no—
218C22×C4—3622no—
219C22×C4—3621no—
220C22×C4—3621no—
221C22×C4—3622no—
222C22×C4—1811no—
223C22×C4—1811no—
224C22×C4—1811no—
225C22×C4—1811no—
226C22×C4—1811no—
227C22×C4C2×#1062438211yescharacteristic
228M16—3622no—
229M16—3622no—
230C2×C8—3622no—
231C22⋊C4—3624no—
232C42—3622no—
233C42—3622no—
234C23—21no—
235C23—3624no—
236C23—3622no—
237C23—3622no—
238C23—3621no—
239C23—1811no—
240C23—1811no—
241C23—7241no—
242C23—7242no—
243C23—7242no—
244C23C22⋊#1062438211yescharacteristic
245Q8—21no—
246Q8—3624no—
247Q8—3621no—
248Q8—1812no—
249Q8C22⋊#1062438212yescomplemented
250D4—24no—
251D4—21no—
252D4—3624no—
253D4—3624no—
254D4—3624no—
255D4—3624no—
256D4—3621no—
257D4—1812no—
258D4C22⋊#1062438212yescomplemented
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—3624no—
264C2×C4—3624no—
265C2×C4—3622no—
266C2×C4—3622no—
267C2×C4—3624no—
268C2×C4—3621no—
269C2×C4—3624no—
270C2×C4—3622no—
271C2×C4—3621no—
272C2×C4—3621no—
273C2×C4—3624no—
274C2×C4—1811no—
275C2×C4—1811no—
276C2×C4—1812no—
277C2×C4—1812no—
278C2×C4—7242no—
279C2×C4—7242no—
280C2×C4C22⋊#1062438211yescharacteristic
281C2×C4C22×#1062438211yescharacteristic
282C8—7242no—
283C22—21no—
284C22—22no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—3621no—
290C22—3621no—
291C22—3621no—
292C22—3621no—
293C22—3622no—
294C22—1812no—
295C22—1812no—
296C22—7244no—
297C22—7248no—
298C22—7241no—
299C22—7241no—
300C22C2×C22⋊#1062438211yescharacteristic
301C4—21no—
302C4—22no—
303C4—3622no—
304C4—3621no—
305C4—1812no—
306C4—7244no—
307C4C2×C22⋊#1062438211yescharacteristic
308C4C2×C22⋊#1062438211yescharacteristic
309C2—41no—
310C2—42no—
311C2—3622no—
312C2—3622no—
313C2—7242no—
314C2#2604222011yescharacteristic
315C2D181⋊#10411yescharacteristic
316C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC2×#41×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC2×#41×D181C2
cosocleC2×C4×D181C23
hypercenterC2×#41#10624382
hypocenterC181#1789
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19153881C2semidirect
2. C2×#19153880C2semidirect
3. #26042250C2semidirect
4. C4×C22⋊#10624382C2semidirect
6. C2×#15668459C22semidirect
8. C2×C4×#10624382C22semidirect
16. C2×#15668449C4semidirect
21. C2×Q8×D181C4semidirect
23. C2×D4×D181C4semidirect
26. C2×#41×C181C4semidirect
90. Dic362C22⋊C4semidirect
93. D724C22⋊C4semidirect
114. Q8×C181C22⋊C4semidirect
117. D4×C181C22⋊C4semidirect
140. C2×D181#104semidirect
158. C181#1789semidirect
172. C2×#41#10624382semidirect
249. Q8C22⋊#10624382semidirect
258. D4C22⋊#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538812
C2×#191538802
#260422502
C4×C22⋊#106243822
C2×#41×D1812
#1789181