Finite Group Databasebeta

← #37543343

Subgroups — #37543343

Order 23168.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19153841C211yesmaximal, characteristic, complemented
2C2×M16×D181C211yesmaximal, characteristic, complemented
3#26041786C211yesmaximal, characteristic, complemented
4C2×#19153791C211yesmaximal, characteristic, complemented
5#26041531C212yesmaximal, complemented
6#26041528C212yesmaximal, complemented
7#26041514C212yesmaximal, complemented
8#26041511C212yesmaximal, complemented
9#26041319C211yesmaximal, characteristic, complemented
10#26041285C211yesmaximal, characteristic, complemented
11#417×C181C211yesmaximal, characteristic, complemented
12C22×D724—21no—
13C22×D724C2211yescharacteristic
14C22×C4×D181C2211yescharacteristic
15#19153841—24no—
16C2×#15668436C2212yescomplemented
17C2×#15668434C2212yescomplemented
18C2×#15668433C2211yescharacteristic
19C2×#15668432C2212yescomplemented
20M16×D181—24no—
21C2×C8×D181C2212yescomplemented
22#19153797—21no—
23#19153792C2211yescharacteristic, complemented
24#19153792C2211yescharacteristic, complemented
25#19153791—21no—
26#19153791—21no—
27#19153791C2212yes—
28#19153791C2212yes—
29#19153787C2211yescharacteristic, complemented
30#19153787C2211yescharacteristic, complemented
31#19153784—21no—
32C4×C22⋊D181—21no—
33C4×C22⋊D181—21no—
34#19153775C2211yescharacteristic
35C2×#15668426—21no—
36#19153749C2212yescomplemented
37#19153747C2212yescomplemented
38#19153738C2212yescomplemented
39#19153738C2212yescomplemented
40C2×#41×C181C2211yescharacteristic
41C2×M16×C181C2211yescharacteristic
42C2×C4⋊C4×C181C2211yescharacteristic
43#103×C181C2212yescomplemented
44#102×C181C2212yescomplemented
45C23×D181—21no—
46C23×D181D411yescharacteristic
47C2×D724—22no—
48C2×D724—22no—
49C2×D724—22no—
50C2×D724—21no—
51C2×D724—22no—
52C2×D724—44no—
53C2×D724C2312yes—
54C2×C22⋊D181—21no—
55C2×C22⋊D181—21no—
56C22×Dic181D411yescharacteristic
57C2×C4×D181—22no—
58C2×C4×D181—22no—
59C2×C4×D181D411yescharacteristic
60C2×C4×D181D411yescharacteristic
61C2×C4×D181C2311yescharacteristic
62C2×C4×D181C2311yescharacteristic
63#15668436—24no—
64#15668434—24no—
65#15668433D414yescomplemented
66#15668432—24no—
67C2×#13089678C2312yes—
68C8×D181—24no—
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71#15668428—21no—
72#15668428—21no—
73C22⋊Dic181—21no—
74C22⋊Dic181—21no—
75C22⋊Dic181—21no—
76#15668426—21no—
77#15668426—21no—
78#15668426—22no—
79#15668426—22no—
80C4×Dic181—21no—
81#41×C181—22no—
82#41×C181—22no—
83C2×Q8×C181C2311yescharacteristic
84C2×D4×C181—21no—
85C2×D4×C181C2311yescharacteristic
86C22×C4×C181—21no—
87C22×C4×C181—21no—
88C22×C4×C181C2311yescharacteristic
89M16×C181—22no—
90C2×C8×C181C2312yes—
91C4⋊C4×C181—21no—
92C4⋊C4×C181C2312yes—
93C22×D181—22no—
94C22×D181—22no—
95C22×D181—22no—
96C22×D181—41no—
97C22×D181—42no—
98C22×D181—42no—
99C22×D181—42no—
100C22×D181C2×D411yescharacteristic
101C22×D181C2×D411yescharacteristic
102D724—22no—
103D724—22no—
104D724—44no—
105D724—44no—
106D724—42no—
107C22⋊D181—42no—
108C22⋊D181—42no—
109C2×Dic181—22no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C2×Dic181C2×D411yescharacteristic
113C2×Dic181C2×D411yescharacteristic
114C4×D181—22no—
115C4×D181—22no—
116C4×D181—21no—
117C4×D181—21no—
118C4×D181—21no—
119C4×D181—21no—
120#13089678C2×D414yescomplemented
121C23×C181—21no—
122C23×C181C2×D411yescharacteristic
123Q8×C181—21no—
124Q8×C181—21no—
125D4×C181—21no—
126D4×C181—21no—
127D4×C181—42no—
128C2×C4×C181—22no—
129C2×C4×C181—21no—
130C2×C4×C181—42no—
131C2×C4×C181—42no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D412yes—
134C2×C4×C181#4112yes—
135C2×C4×C181C2411yescharacteristic
136C8×C181—22no—
137C22×C181—21no—
138C22×C181—21no—
139C22×C181—41no—
140C22×C181—41no—
141C22×C181—41no—
142C22×C181#12112yes—
143C22×C181C22×D411yescharacteristic
144C2×D181—21no—
145C2×D181—21no—
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—21no—
149C2×D181—44no—
150C2×D181—42no—
151C2×D181—42no—
152C2×D181—42no—
153C2×D181—84no—
154C2×D181AGL(1,ℤ/8)12yescomplemented
155C4×C181—41no—
156C4×C181—42no—
157C4×C181#12112yes—
158C4×C181C22×D411yescharacteristic
159C4×C181C2×#4111yescharacteristic
160Dic181—21no—
161Dic181—21no—
162Dic181—41no—
163Dic181—41no—
164C2×C181—22no—
165C2×C181—81no—
166C2×C181C2×#12111yescharacteristic
167C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C181#57511yescharacteristic
169D181—22no—
170D181—41no—
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#428—1811no—
183#418—1811no—
184#417D18111yescharacteristic, complemented
185C2×#41C2×D18111yescharacteristic
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—1811no—
203#121—1811no—
204#121—1812no—
205#121—1812no—
206#120—3621no—
207C4×D4—3621no—
208C4×D4—3621no—
209#117—1811no—
210C2×C4⋊C4C2×D18111yescharacteristic
211C2×C22⋊C4—3621no—
212#107—1812no—
213#106—1812no—
214#103C2×D18112yescomplemented
215#102—1812no—
216#102—1812no—
217#102C2×D18112yescomplemented
218C24—3621no—
219C24—1811no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D18111yescharacteristic
223C2×D4—21no—
224C2×D4—3622no—
225C2×D4—3622no—
226C2×D4—3621no—
227C2×D4—3621no—
228C2×D4—3622no—
229C2×D4—3621no—
230C2×D4—3622no—
231C2×D4—1812no—
232C2×D4—7244no—
233C2×D4C22×D18111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—3622no—
237C22×C4—3622no—
238C22×C4—1811no—
239C22×C4—1811no—
240C22×C4—1811no—
241C22×C4—1811no—
242C22×C4—1811no—
243C22×C4C22×D18111yescharacteristic
244SD16—3624no—
245D8—3624no—
246M16—22no—
247M16—3624no—
248M16—1814no—
249C2×C8—3624no—
250C2×C8—1812no—
251C2×C8C22×D18112yes—
252C4⋊C4—21no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4C22×D18112yes—
258C22⋊C4—3621no—
259C22⋊C4—3621no—
260C22⋊C4—3622no—
261C22⋊C4—3622no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C42—3621no—
266C23—21no—
267C23—3622no—
268C23—3622no—
269C23—3622no—
270C23—1811no—
271C23—1811no—
272C23—7241no—
273C23—7242no—
274C23—7242no—
275C23—7242no—
276C23D4×D18111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—3622no—
283D4—3622no—
284D4—7244no—
285D4—7244no—
286D4—7242no—
287D4—7242no—
288D4—7242no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
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—3621no—
302C2×C4—1811no—
303C2×C4—1811no—
304C2×C4D4×D18111yescharacteristic
305C2×C4D4×D18112yes—
306C2×C4#1566844912yes—
307C2×C4C23×D18111yescharacteristic
308C8—22no—
309C8—1814no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—3621no—
316C22—3621no—
317C22—3621no—
318C22—3621no—
319C22—3621no—
320C22—14484no—
321C22—1812no—
322C22—7244no—
323C22—7242no—
324C22—7242no—
325C22—7242no—
326C22#1915378912yes—
327C22C2×D4×D18111yescharacteristic
328C4—41no—
329C4—42no—
330C4—3621no—
331C4—3621no—
332C4—7241no—
333C4—7241no—
334C4#1915378912yes—
335C4C2×D4×D18111yescharacteristic
336C4C2×#1566844911yescharacteristic
337C2—22no—
338C2—81no—
339C2—3622no—
340C2—14482no—
341C2—7241no—
342C2C2×#1915378911yescharacteristic
343C2AGL(1,ℤ/8)×D18111yescharacteristic
344C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#417×C181C2
socleC22×C181C22×D4
hypercenter#417D181
hypocenterC181#3055
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19153841C2semidirect
2. C2×M16×D181C2semidirect
3. #26041786C2semidirect
4. C2×#19153791C2semidirect
5. #26041531C2semidirect
6. #26041528C2semidirect
7. #26041514C2semidirect
8. #26041511C2semidirect
9. #26041319C2semidirect
10. #26041285C2semidirect
11. #417×C181C2semidirect
16. C2×#15668436C22semidirect
17. C2×#15668434C22semidirect
19. C2×#15668432C22semidirect
21. C2×C8×D181C22semidirect
23. #19153792C22semidirect
24. #19153792C22semidirect
29. #19153787C22semidirect
30. #19153787C22semidirect
36. #19153749C22semidirect
37. #19153747C22semidirect
38. #19153738C22semidirect
39. #19153738C22semidirect
43. #103×C181C22semidirect
44. #102×C181C22semidirect
65. #15668433D4semidirect
120. #13089678C2×D4semidirect
154. C2×D181AGL(1,ℤ/8)semidirect
172. C181#3055semidirect
184. #417D181semidirect
214. #103C2×D181semidirect
217. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538412
C2×M16×D1812
#260417862
C2×#191537912
#260415312
#260415282
#260415142
#260415112
#260413192
#260412852
#417×C1812
#3055181