Finite Group Databasebeta

← #37543341

Subgroups — #37543341

Order 23168.

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

normal subgroups only

This group has 548 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×#19153844C211yesmaximal, characteristic, complemented
2C2×M16×D181C211yesmaximal, characteristic, complemented
3#26041791C211yesmaximal, characteristic, complemented
4C2×#19153791C211yesmaximal, characteristic, complemented
5#26041533C212yesmaximal, complemented
6#26041526C212yesmaximal, complemented
7#26041516C212yesmaximal, complemented
8#26041509C212yesmaximal, complemented
9#26041283C211yesmaximal, characteristic, complemented
10#26041278C211yesmaximal, characteristic, complemented
11#428×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13C22×D724—21no—
14C22×D724C2211yescharacteristic
15C22×C4×D181C2211yescharacteristic
16#19153844—24no—
17C2×#15668438C2212yescomplemented
18C2×D1448C2212yescomplemented
19C2×#15668433C2211yescharacteristic
20C2×#15668432C2212yescomplemented
21M16×D181—24no—
22C2×C8×D181C2212yescomplemented
23#19153800—21no—
24#19153796C2211yescharacteristic, complemented
25#19153796C2211yescharacteristic, complemented
26#19153791—21no—
27#19153791—21no—
28#19153791C2212yes—
29#19153791C2212yes—
30#19153791C2211yescharacteristic, complemented
31#19153791C2211yescharacteristic, complemented
32#19153786—21no—
33C4×D724—21no—
34C4×D724—21no—
35C2×#15668426—21no—
36#19153739C2212yescomplemented
37#19153736C2212yescomplemented
38#19153736C2212yescomplemented
39#19153736C2212yescomplemented
40C2×M16×C181C2211yescharacteristic
41#117×C181C2211yescharacteristic
42C2×C4⋊C4×C181C2211yescharacteristic
43#107×C181C2212yescomplemented
44#106×C181C2212yescomplemented
45C23×D181—21no—
46C23×D181D411yescharacteristic
47#15668449—22no—
48#15668449—22no—
49C2×Dic362C2311yescharacteristic
50C2×D724—21no—
51C2×D724—21no—
52C2×D724—22no—
53C2×D724—22no—
54C2×D724—22no—
55C2×D724—21no—
56C2×D724—22no—
57C2×D724—44no—
58C2×D724C2312yes—
59C2×D724C2311yescharacteristic
60C2×C22⋊D181—21no—
61C22×Dic181D411yescharacteristic
62C2×C4×D181—22no—
63C2×C4×D181—21no—
64C2×C4×D181—22no—
65C2×C4×D181D411yescharacteristic
66C2×C4×D181D411yescharacteristic
67C2×C4×D181C2311yescharacteristic
68C2×C4×D181C2311yescharacteristic
69#15668438—24no—
70D1448—24no—
71#15668433—22no—
72#15668432—24no—
73C2×#13089678C2312yes—
74C8×D181—24no—
75C4⋊Dic181—21no—
76C4⋊Dic181—21no—
77#15668426—21no—
78#15668426—21no—
79#15668426—22no—
80#15668426—22no—
81#15668426—21no—
82#15668426—21no—
83C22×C4×C181—21no—
84C22×C4×C181C2311yescharacteristic
85M16×C181D414yescomplemented
86C2×C8×C181C2312yes—
87C4⋊C4×C181—21no—
88C4⋊C4×C181C2311yescharacteristic
89C4⋊C4×C181C2312yes—
90C4⋊C4×C181C2311yescharacteristic
91C22⋊C4×C181—21no—
92C42×C181—21no—
93C22×D181—22no—
94C22×D181—22no—
95C22×D181—21no—
96C22×D181—22no—
97C22×D181—41no—
98C22×D181—42no—
99C22×D181—42no—
100C22×D181—42no—
101C22×D181C2×D411yescharacteristic
102C22×D181C2×D411yescharacteristic
103Dic362—21no—
104Dic362—21no—
105D724—21no—
106D724—22no—
107D724—22no—
108D724—21no—
109D724—42no—
110D724—44no—
111D724—42no—
112D724—44no—
113D724—42no—
114C22⋊D181—42no—
115C2×Dic181—22no—
116C2×Dic181—21no—
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C4×D181—22no—
120C4×D181—22no—
121C4×D181—21no—
122C4×D181—21no—
123C4×D181—21no—
124C4×D181—21no—
125C4×D181—42no—
126#13089678—22no—
127C23×C181C2×D411yescharacteristic
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—22no—
131C2×C4×C181—42no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D412yes—
134C2×C4×C181#4112yes—
135C2×C4×C181C2411yescharacteristic
136C8×C181C2×D414yescomplemented
137C22×C181—21no—
138C22×C181—21no—
139C22×C181#12112yes—
140C22×C181C22×D411yescharacteristic
141C2×D181—21no—
142C2×D181—21no—
143C2×D181—21no—
144C2×D181—21no—
145C2×D181—21no—
146C2×D181—44no—
147C2×D181—41no—
148C2×D181—42no—
149C2×D181—42no—
150C2×D181—41no—
151C2×D181—41no—
152C2×D181—42no—
153C2×D181—84no—
154C2×D181AGL(1,ℤ/8)12yescomplemented
155C4×C181—42no—
156C4×C181—41no—
157C4×C181—41no—
158C4×C181#12112yes—
159C4×C181C22×D411yescharacteristic
160C4×C181C2×#4111yescharacteristic
161Dic181—21no—
162Dic181—21no—
163Dic181—41no—
164C2×C181—22no—
165C2×C181C2×#12111yescharacteristic
166C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C181#57511yescharacteristic
168D181—22no—
169D181—41no—
170D181—81no—
171D181—82no—
172C181#305511yescharacteristic, complemented, Sylow
173#3055—1811nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1811no—
175C22×M16—1811no—
176#525—1811no—
177C2×#121—1811no—
178#469—1812no—
179#468—1812no—
180#466—1812no—
181#465—1812no—
182#428D18111yescharacteristic, complemented
183#418—1811no—
184#417—1811no—
185C2×#41—1811no—
186C22×D4—3621no—
187C22×D4—1811no—
188C23×C4—1811no—
189AGL(1,ℤ/8)—3624no—
190C2×SD16—1812no—
191C2×D8—1812no—
192C2×M16—3624no—
193C2×M16—1811no—
194C2×M16—1812no—
195C2×M16C2×D18111yescharacteristic
196C22×C8—1812no—
197#123—3621no—
198C22⋊Q8—1811no—
199C22⋊Q8—1811no—
200#121—3621no—
201#121—3621no—
202#121—1812no—
203#121—1812no—
204#121—1811no—
205#121—1811no—
206#120—3621no—
207C4×D4—3621no—
208C4×D4—3621no—
209#117C2×D18111yescharacteristic
210C2×C4⋊C4C2×D18111yescharacteristic
211C2×C22⋊C4—3621no—
212#107C2×D18112yescomplemented
213#106C2×D18112yescomplemented
214#103—1812no—
215#102—1812no—
216#102—1812no—
217#102—1812no—
218C24—3621no—
219C24—1811no—
220#41—3622no—
221#41—3622no—
222C2×Q8—1811no—
223C2×D4—3621no—
224C2×D4—3621no—
225C2×D4—3622no—
226C2×D4—3622no—
227C2×D4—3621no—
228C2×D4—3621no—
229C2×D4—3622no—
230C2×D4—3622no—
231C2×D4—1811no—
232C2×D4—1812no—
233C2×D4—7244no—
234C22×C4—21no—
235C22×C4—3622no—
236C22×C4—3622no—
237C22×C4—3621no—
238C22×C4—1811no—
239C22×C4—1811no—
240C22×C4—1811no—
241C22×C4—1811no—
242C22×C4—1811no—
243C22×C4C22×D18111yescharacteristic
244SD16—3624no—
245D8—3624no—
246M16—3622no—
247M16—3624no—
248M16D72414yescomplemented
249C2×C8—3624no—
250C2×C8—1812no—
251C2×C8C22×D18112yes—
252C4⋊C4—21no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4C22×D18111yescharacteristic
256C4⋊C4C22×D18111yescharacteristic
257C4⋊C4C22×D18112yes—
258C22⋊C4—21no—
259C22⋊C4—3621no—
260C22⋊C4—3621no—
261C22⋊C4—3622no—
262C22⋊C4—3622no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C42—21no—
266C23—3622no—
267C23—3622no—
268C23—3621no—
269C23—3622no—
270C23—1811no—
271C23—1811no—
272C23—7241no—
273C23—7242no—
274C23—7242no—
275C23—7242no—
276C23D4×D18111yescharacteristic
277Q8—3621no—
278Q8—3621no—
279D4—3621no—
280D4—3622no—
281D4—3622no—
282D4—3621no—
283D4—7244no—
284D4—7242no—
285D4—7244no—
286D4—7242no—
287D4—7242no—
288D4—7242no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—3622no—
294C2×C4—3622no—
295C2×C4—3622no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3621no—
300C2×C4—3621no—
301C2×C4—1811no—
302C2×C4—1811no—
303C2×C4—7242no—
304C2×C4D4×D18111yescharacteristic
305C2×C4C2×D72412yes—
306C2×C4#1566845012yes—
307C2×C4C23×D18111yescharacteristic
308C8—3622no—
309C8C2×D72414yescomplemented
310C22—21no—
311C22—21no—
312C22—3621no—
313C22—3621no—
314C22—3621no—
315C22—3621no—
316C22—3621no—
317C22—14484no—
318C22—1812no—
319C22—7244no—
320C22—7241no—
321C22—7242no—
322C22—7242no—
323C22—7241no—
324C22—7241no—
325C22—7242no—
326C22#1915379112yes—
327C22C2×D4×D18111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—3621no—
332C4—3621no—
333C4—7241no—
334C4#1915379112yes—
335C4C22×D72411yescharacteristic
336C4C2×#1566845011yescharacteristic
337C2—22no—
338C2—3622no—
339C2—14482no—
340C2—14481no—
341C2—7241no—
342C2C2×#1915379111yescharacteristic
343C2AGL(1,ℤ/8)×D18111yescharacteristic
344C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
hypercenter#428D181
hypocenterC181#3055
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×#19153844C2semidirect
2. C2×M16×D181C2semidirect
3. #26041791C2semidirect
4. C2×#19153791C2semidirect
5. #26041533C2semidirect
6. #26041526C2semidirect
7. #26041516C2semidirect
8. #26041509C2semidirect
9. #26041283C2semidirect
10. #26041278C2semidirect
11. #428×C181C2semidirect
17. C2×#15668438C22semidirect
18. C2×D1448C22semidirect
20. C2×#15668432C22semidirect
22. C2×C8×D181C22semidirect
24. #19153796C22semidirect
25. #19153796C22semidirect
30. #19153791C22semidirect
31. #19153791C22semidirect
36. #19153739C22semidirect
37. #19153736C22semidirect
38. #19153736C22semidirect
39. #19153736C22semidirect
43. #107×C181C22semidirect
44. #106×C181C22semidirect
85. M16×C181D4semidirect
136. C8×C181C2×D4semidirect
154. C2×D181AGL(1,ℤ/8)semidirect
172. C181#3055semidirect
182. #428D181semidirect
212. #107C2×D181semidirect
213. #106C2×D181semidirect
248. M16D724semidirect
309. C8C2×D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538442
C2×M16×D1812
#260417912
C2×#191537912
#260415332
#260415262
#260415162
#260415092
#260412832
#260412782
#428×C1812
#3055181