Finite Group Databasebeta

← #37543352

Subgroups — #37543352

Order 23168.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19153849C211yesmaximal, characteristic, complemented
2C2×M16×D181C211yesmaximal, characteristic, complemented
3#26041791C211yesmaximal, characteristic
4C2×#19153796C211yesmaximal, characteristic, complemented
5#26041538C212yesmaximal, complemented
6#26041526C212yesmaximal, complemented
7#26041521C212yesmaximal, complemented
8#26041509C212yesmaximal, complemented
9#26041287C211yesmaximal, characteristic, complemented
10#26041278C211yesmaximal, characteristic, complemented
11#428×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×Dic362C2211yescharacteristic
14C22×C4×D181C2211yescharacteristic
15#19153849—24no—
16C2×Dic724C2212yescomplemented
17C2×#15668438C2212yescomplemented
18C2×#15668433C2211yescharacteristic
19C2×#15668432C2212yes—
20M16×D181—24no—
21C2×C8×D181C2212yes—
22#19153800—21no—
23#19153796—21no—
24#19153796—21no—
25#19153796C2212yes—
26#19153796C2211yescharacteristic
27#19153796C2212yes—
28#19153796C2211yescharacteristic
29#19153791C2211yescharacteristic
30#19153791C2211yescharacteristic
31#19153786—21no—
32C4×D724—21no—
33C4×D724—21no—
34C2×C4⋊Dic181—21no—
35C2×#15668426—21no—
36#19153739C2212yescomplemented
37#19153739C2212yescomplemented
38#19153739C2212yescomplemented
39#19153736C2212yescomplemented
40C2×M16×C181C2211yescharacteristic
41#117×C181C2211yescharacteristic
42C2×C4⋊C4×C181C2211yescharacteristic
43#107×C181C2212yescomplemented
44#106×C181C2212yescomplemented
45C23×D181D411yescharacteristic
46#15668449—22no—
47#15668449—22no—
48C2×Dic362—21no—
49C2×Dic362—22no—
50C2×Dic362—22no—
51C2×Dic362C2312yes—
52C2×Dic362C2311yescharacteristic
53C2×D724—21no—
54C2×D724—21no—
55C2×D724C2311yescharacteristic
56C2×C22⋊D181—21no—
57C22×Dic181—21no—
58C22×Dic181D411yescharacteristic
59C2×C4×D181—22no—
60C2×C4×D181—21no—
61C2×C4×D181—22no—
62C2×C4×D181D411yescharacteristic
63C2×C4×D181D411yescharacteristic
64C2×C4×D181C2311yescharacteristic
65C2×C4×D181C2311yescharacteristic
66Dic724—24no—
67#15668438—24no—
68#15668433—22no—
69#15668432—24no—
70C2×#13089678C2312yes—
71C8×D181—24no—
72C4⋊Dic181—22no—
73C4⋊Dic181—22no—
74C4⋊Dic181—21no—
75C4⋊Dic181—21no—
76#15668426—22no—
77#15668426—22no—
78#15668426—21no—
79#15668426—21no—
80#15668426—21no—
81#15668426—21no—
82C22×C4×C181—21no—
83C22×C4×C181C2311yescharacteristic
84M16×C181D414yescomplemented
85C2×C8×C181C2312yes—
86C4⋊C4×C181—21no—
87C4⋊C4×C181C2311yescharacteristic
88C4⋊C4×C181C2312yes—
89C4⋊C4×C181C2311yescharacteristic
90C22⋊C4×C181—21no—
91C42×C181—21no—
92C22×D181—22no—
93C22×D181—22no—
94C22×D181—21no—
95C22×D181—41no—
96C22×D181C2×D411yescharacteristic
97C22×D181C2×D411yescharacteristic
98Dic362—22no—
99Dic362—21no—
100Dic362—22no—
101Dic362—21no—
102Dic362—42no—
103D724—21no—
104D724—21no—
105D724—42no—
106D724—42no—
107C22⋊D181—42no—
108C2×Dic181—22no—
109C2×Dic181—21no—
110C2×Dic181—22no—
111C2×Dic181—42no—
112C2×Dic181C2×D411yescharacteristic
113C2×Dic181C2×D411yescharacteristic
114C4×D181—22no—
115C4×D181—22no—
116C4×D181—21no—
117C4×D181—21no—
118C4×D181—21no—
119C4×D181—21no—
120C4×D181—42no—
121#13089678—22no—
122C23×C181C2×D411yescharacteristic
123C2×C4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181—22no—
126C2×C4×C181—42no—
127C2×C4×C181C2×D411yescharacteristic
128C2×C4×C181C2×D412yes—
129C2×C4×C181#4112yes—
130C2×C4×C181C2411yescharacteristic
131C8×C181C2×D414yes—
132C22×C181—21no—
133C22×C181—21no—
134C22×C181#12112yes—
135C22×C181C22×D411yescharacteristic
136C2×D181—21no—
137C2×D181—21no—
138C2×D181—21no—
139C2×D181—21no—
140C2×D181—21no—
141C2×D181—44no—
142C2×D181—41no—
143C2×D181—41no—
144C2×D181—41no—
145C2×D181#13712yescomplemented
146C4×C181—41no—
147C4×C181—42no—
148C4×C181—41no—
149C4×C181#12112yes—
150C4×C181C22×D411yescharacteristic
151C4×C181C2×#4111yescharacteristic
152Dic181—21no—
153Dic181—21no—
154Dic181—41no—
155Dic181—42no—
156C2×C181—22no—
157C2×C181C2×#12111yescharacteristic
158C2×C181C2×#13711yescharacteristic
159C2×C181#57511yescharacteristic
160D181—22no—
161D181—41no—
162D181—81no—
163C181#305611yescharacteristic, complemented, Sylow
164#3056—1811nomaximal, Sylow
165C2×#137—1811no—
166C22×M16—1811no—
167#525—1811no—
168C2×C22⋊Q8—1811no—
169#470—1812no—
170#468—1812no—
171#467—1812no—
172#465—1812no—
173#428D18111yescharacteristic, complemented
174#419—1811no—
175#417—1811no—
176C2×#41—1811no—
177C22×Q8—1811no—
178C23×C4—1811no—
179#137—3624no—
180C2×Q16—1812no—
181C2×SD16—1812no—
182C2×M16—3624no—
183C2×M16—1811no—
184C2×M16—1812no—
185C2×M16C2×D18111yescharacteristic
186C22×C8—1812no—
187#123—3621no—
188C22⋊Q8—3621no—
189C22⋊Q8—3621no—
190C22⋊Q8—1812no—
191C22⋊Q8—1812no—
192C22⋊Q8—1811no—
193C22⋊Q8—1811no—
194#121—1811no—
195#121—1811no—
196#120—3621no—
197C4×D4—3621no—
198C4×D4—3621no—
199#117C2×D18111yescharacteristic
200C2×C4⋊C4—3621no—
201C2×C4⋊C4C2×D18111yescharacteristic
202C2×C22⋊C4—3621no—
203#107C2×D18112yescomplemented
204#106C2×D18112yescomplemented
205#103—1812no—
206#103—1812no—
207#103—1812no—
208#102—1812no—
209C24—1811no—
210#41—3622no—
211#41—3622no—
212C2×Q8—3621no—
213C2×Q8—3622no—
214C2×Q8—3622no—
215C2×Q8—1812no—
216C2×Q8—1811no—
217C2×D4—3621no—
218C2×D4—3621no—
219C2×D4—3621no—
220C2×D4—1811no—
221C22×C4—21no—
222C22×C4—3622no—
223C22×C4—3622no—
224C22×C4—3621no—
225C22×C4—3621no—
226C22×C4—1811no—
227C22×C4—1811no—
228C22×C4—1811no—
229C22×C4—1811no—
230C22×C4—1811no—
231C22×C4C22×D18111yescharacteristic
232Q16—3624no—
233SD16—3624no—
234M16—3622no—
235M16—3624no—
236M16D72414yescomplemented
237C2×C8—3624no—
238C2×C8—1812no—
239C2×C8C22×D18112yes—
240C4⋊C4—21no—
241C4⋊C4—3622no—
242C4⋊C4—3622no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4C22×D18112yes—
246C4⋊C4C22×D18111yescharacteristic
247C4⋊C4C22×D18111yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—3622no—
250C22⋊C4—3622no—
251C22⋊C4—3621no—
252C22⋊C4—3621no—
253C22⋊C4—3621no—
254C22⋊C4—3621no—
255C42—21no—
256C23—3622no—
257C23—3622no—
258C23—3621no—
259C23—1811no—
260C23—1811no—
261C23—7241no—
262C23D4×D18111yescharacteristic
263Q8—3622no—
264Q8—3621no—
265Q8—3622no—
266Q8—3621no—
267Q8—7242no—
268D4—3621no—
269D4—3621no—
270D4—7242no—
271D4—7242no—
272D4—7242no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3622no—
278C2×C4—3622no—
279C2×C4—3622no—
280C2×C4—3621no—
281C2×C4—3621no—
282C2×C4—3621no—
283C2×C4—3621no—
284C2×C4—3621no—
285C2×C4—3622no—
286C2×C4—1811no—
287C2×C4—1811no—
288C2×C4—7242no—
289C2×C4—7242no—
290C2×C4D4×D18111yescharacteristic
291C2×C4C2×D72412yes—
292C2×C4#1566845012yes—
293C2×C4C23×D18111yescharacteristic
294C8—3622no—
295C8C2×D72414yes—
296C22—21no—
297C22—21no—
298C22—3621no—
299C22—3621no—
300C22—3621no—
301C22—3621no—
302C22—3621no—
303C22—1812no—
304C22—7244no—
305C22—7241no—
306C22—7241no—
307C22—7241no—
308C22#1915379112yes—
309C22C2×D4×D18111yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3621no—
314C4—3621no—
315C4—7241no—
316C4—7242no—
317C4#1915379112yes—
318C4C22×D72411yescharacteristic
319C4C2×#1566845011yescharacteristic
320C2—22no—
321C2—3622no—
322C2—14481no—
323C2—7241no—
324C2C2×#1915379111yescharacteristic
325C2#137×D18111yescharacteristic
326C2#2604207511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
prime radical#26041791C2
hypercenter#428D181
hypocenterC181#3056
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19153849C2semidirect
2. C2×M16×D181C2semidirect
4. C2×#19153796C2semidirect
5. #26041538C2semidirect
6. #26041526C2semidirect
7. #26041521C2semidirect
8. #26041509C2semidirect
9. #26041287C2semidirect
10. #26041278C2semidirect
11. #428×C181C2semidirect
16. C2×Dic724C22semidirect
17. C2×#15668438C22semidirect
36. #19153739C22semidirect
37. #19153739C22semidirect
38. #19153739C22semidirect
39. #19153736C22semidirect
43. #107×C181C22semidirect
44. #106×C181C22semidirect
84. M16×C181D4semidirect
145. C2×D181#137semidirect
163. C181#3056semidirect
173. #428D181semidirect
203. #107C2×D181semidirect
204. #106C2×D181semidirect
236. M16D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538492
C2×M16×D1812
#260417912
C2×#191537962
#260415382
#260415262
#260415212
#260415092
#260412872
#260412782
#428×C1812
#3056181