Finite Group Databasebeta

← #37543076

Subgroups — #37543076

Order 23168.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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×#19153862C211yesmaximal, characteristic, complemented
2#26042077C211yesmaximal, characteristic, complemented
3#26041843C211yesmaximal, characteristic, complemented
4#26041455C214yesmaximal, complemented
5#26041445C214yesmaximal, complemented
6#26041296C211yesmaximal, characteristic, complemented
7#26041295C211yesmaximal, characteristic, complemented
8C2×#19153732C211yesmaximal, characteristic, complemented
9#575×C181C211yesmaximal, characteristic, complemented
10#19153862—22no—
11#19153862—22no—
12#19153862—22no—
13#19153862C2214yes—
14C2×#15668450—21no—
15C2×#15668449—21no—
16C2×#15668449C2211yescharacteristic
17C22×D724—21no—
18C22×D724C2211yescharacteristic
19C2×D4×D181—22no—
20C2×D4×D181—21no—
21#19153847C2212yescomplemented
22#19153846—21no—
23#19153842C2212yescomplemented
24#19153841—21no—
25#19153836—22no—
26C2×#15668433C2211yescharacteristic, complemented
27#19153812C2212yescomplemented
28#19153805C2212yescomplemented
29#19153790—22no—
30#19153784—22no—
31#19153775C2211yescharacteristic, complemented
32#19153743C2214yescomplemented
33#19153742C2214yescomplemented
34#19153732C2214yescomplemented
35C2×#41×C181C2211yescharacteristic
36C2×#41×C181C2211yescharacteristic, complemented
37#137×C181—21no—
38#137×C181C2212yescomplemented
39AGL(1,ℤ/8)×C181—21no—
40AGL(1,ℤ/8)×C181C2212yescomplemented
41#135×C181—22no—
42C2×M16×C181C2211yescharacteristic, complemented
43C23×D181—22no—
44C23×D181D412yes—
45#15668450—24no—
46#15668450—24no—
47#15668449—24no—
48#15668449—24no—
49#15668449—21no—
50#15668449—21no—
51#15668449D412yescomplemented
52#15668449C2312yes—
53C2×Dic362D411yescharacteristic, complemented
54C2×D724—22no—
55C2×D724—24no—
56C2×D724—24no—
57C2×D724—21no—
58C2×D724—21no—
59C2×D724—21no—
60C2×D724—21no—
61C2×D724D412yes—
62C2×D724D411yescharacteristic, complemented
63C2×D724C2312yes—
64C2×C22⋊D181—22no—
65C2×C22⋊D181—24no—
66C2×C22⋊D181—21no—
67D4×D181—28no—
68D4×D181—24no—
69D4×D181—28no—
70D4×D181—24no—
71C2×C4×D181—22no—
72C2×C4×D181—21no—
73#15668439—22no—
74#15668437—22no—
75#15668436—22no—
76#15668434—22no—
77#15668433—21no—
78#15668433C2312yescomplemented
79C2×#13089678—21no—
80C4⋊Dic181—21no—
81C22⋊Dic181—21no—
82#15668426—24no—
83C4×Dic181C2312yescomplemented
84#41×C181—21no—
85#41×C181—21no—
86#41×C181—21no—
87#41×C181—21no—
88#41×C181—21no—
89#41×C181—21no—
90#41×C181D412yescomplemented
91#41×C181C2312yes—
92C2×Q8×C181D411yescharacteristic, complemented
93C2×Q8×C181C2311yescharacteristic, complemented
94C2×D4×C181—21no—
95C2×D4×C181—21no—
96C2×D4×C181D411yescharacteristic, complemented
97C2×D4×C181C2311yescharacteristic, complemented
98C22×C4×C181—21no—
99C22×C4×C181—21no—
100C22×C4×C181C2311yescharacteristic
101Q16×C181—22no—
102SD16×C181—22no—
103SD16×C181—22no—
104D8×C181—22no—
105M16×C181—21no—
106M16×C181C2312yescomplemented
107C2×C8×C181—21no—
108C22×D181—22no—
109C22×D181—21no—
110C22×D181—28no—
111C22×D181—22no—
112C22×D181—82no—
113C22×D181—84no—
114C22×D181C2×D414yes—
115Dic362—21no—
116Dic362C2×D412yes—
117D724—28no—
118D724—22no—
119D724—22no—
120D724—21no—
121D724—22no—
122D724—22no—
123D724C2×D412yes—
124C22⋊D181—28no—
125C22⋊D181—22no—
126C22⋊D181—22no—
127C22⋊D181—84no—
128C2×Dic181—21no—
129C2×Dic181—22no—
130C4×D181—28no—
131C4×D181—22no—
132C4×D181—22no—
133#13089678—22no—
134C23×C181—21no—
135C23×C181—21no—
136C23×C181C2×D411yescharacteristic
137Q8×C181—21no—
138Q8×C181—21no—
139Q8×C181—21no—
140Q8×C181C2×D412yes—
141D4×C181—22no—
142D4×C181—22no—
143D4×C181—21no—
144D4×C181—21no—
145D4×C181—21no—
146D4×C181—41no—
147D4×C181—41no—
148D4×C181C2×D412yes—
149C2×C4×C181—21no—
150C2×C4×C181—22no—
151C2×C4×C181—22no—
152C2×C4×C181—21no—
153C2×C4×C181—41no—
154C2×C4×C181—41no—
155C2×C4×C181C2×D411yescharacteristic
156C2×C4×C181C2×D411yescharacteristic
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2×D411yescharacteristic
160C2×C4×C181C2411yescharacteristic
161C8×C181—22no—
162C22×C181—22no—
163C22×C181—41no—
164C22×C181—42no—
165C22×C181—42no—
166C22×C181—41no—
167C22×C181#12011yescharacteristic
168C22×C181#12011yescharacteristic
169C22×C181C22×D411yescharacteristic
170C2×D181—24no—
171C2×D181—22no—
172C2×D181—44no—
173C2×D181—42no—
174C2×D181—44no—
175C2×D181—82no—
176C2×D181—88no—
177C4×C181—22no—
178C4×C181—41no—
179C4×C181#12011yescharacteristic
180C4×C181#12011yescharacteristic
181C4×C181C22×D411yescharacteristic
182C4×C181C22×D411yescharacteristic
183Dic181—22no—
184Dic181—42no—
185C2×C181—21no—
186C2×C181—21no—
187C2×C181—21no—
188C2×C181—42no—
189C2×C181—81no—
190C2×C181C2×#12011yescharacteristic
191D181—44no—
192D181—42no—
193C181#301911yescharacteristic, complemented, Sylow
194#3019—1811nomaximal, Sylow
195C2×H(5,𝔽2)—1811no—
196#575—1811no—
197#575D18111yescharacteristic, complemented
198#535—1811no—
199#455—1814no—
200#453—1814no—
201#421—1811no—
202#421—1811no—
203C2×#100—1811no—
204H(5,𝔽2)—3622no—
205H(5,𝔽2)—3622no—
206H(5,𝔽2)—3622no—
207H(5,𝔽2)—1814no—
208C2×#41—3621no—
209C2×#41—3621no—
210C2×#41—1811no—
211C2×#41C2×D18111yescharacteristic
212C2×#41C2×D18111yescharacteristic, complemented
213C22×D4—3621no—
214C22×D4—3622no—
215C22×D4—3621no—
216C22×D4—1811no—
217#137—21no—
218#137—3621no—
219#137—1812no—
220#137C2×D18112yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—3621no—
223AGL(1,ℤ/8)—1812no—
224AGL(1,ℤ/8)C2×D18112yescomplemented
225#135—22no—
226#135—3622no—
227C2×M16—1811no—
228C2×M16C2×D18111yescharacteristic, complemented
229#127—1812no—
230#124—1812no—
231#121—3622no—
232#120—3622no—
233#117—1811no—
234#104—1814no—
235#104—1814no—
236#100—1814no—
237C24—3622no—
238C24—1812no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—3624no—
246#41—3624no—
247#41—3624no—
248#41—3624no—
249#41—3621no—
250#41—3621no—
251#41—1812no—
252#41—1812no—
253#41C22⋊D18112yescomplemented
254#41C22×D18112yes—
255C2×Q8—1811no—
256C2×Q8C22⋊D18111yescharacteristic, complemented
257C2×Q8C22×D18111yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—3622no—
261C2×D4—3624no—
262C2×D4—3624no—
263C2×D4—3628no—
264C2×D4—3624no—
265C2×D4—3628no—
266C2×D4—3622no—
267C2×D4—3621no—
268C2×D4—3624no—
269C2×D4—3624no—
270C2×D4—3621no—
271C2×D4—3621no—
272C2×D4—3621no—
273C2×D4—3621no—
274C2×D4—1812no—
275C2×D4—1811no—
276C2×D4—1812no—
277C2×D4C22⋊D18111yescharacteristic, complemented
278C2×D4C22×D18111yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3622no—
282C22×C4—3621no—
283C22×C4C22×D18111yescharacteristic
284Q16—22no—
285Q16—3622no—
286SD16—22no—
287SD16—22no—
288SD16—3622no—
289SD16—3622no—
290D8—22no—
291D8—3622no—
292M16—21no—
293M16—3621no—
294M16—1812no—
295M16C22×D18112yescomplemented
296C2×C8—21no—
297C2×C8—3621no—
298C4⋊C4—3621no—
299C22⋊C4—3624no—
300C22⋊C4—3621no—
301C42—1812no—
302C23—21no—
303C23—21no—
304C23—3628no—
305C23—3622no—
306C23—3622no—
307C23—3621no—
308C23—14484no—
309C23—14482no—
310C23—1814no—
311C23D4×D18111yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—3621no—
316Q8—1812no—
317Q8C2×C22⋊D18112yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—3628no—
326D4—3628no—
327D4—3622no—
328D4—3622no—
329D4—3622no—
330D4—3621no—
331D4—3622no—
332D4—3622no—
333D4—3622no—
334D4—14484no—
335D4—1812no—
336D4C2×C22⋊D18112yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—3628no—
344C2×C4—3621no—
345C2×C4—3622no—
346C2×C4—3622no—
347C2×C4—3622no—
348C2×C4D4×D18111yescharacteristic
349C2×C4D4×D18111yescharacteristic
350C2×C4D4×D18111yescharacteristic
351C2×C4C2×C22⋊D18111yescharacteristic
352C2×C4C2×C22⋊D18111yescharacteristic
353C2×C4C23×D18111yescharacteristic
354C8—22no—
355C8—3622no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—3624no—
362C22—3622no—
363C22—14482no—
364C22—14488no—
365C22—7244no—
366C22—7242no—
367C22—7244no—
368C22#1915378411yescharacteristic
369C22#1915378411yescharacteristic
370C22C2×D4×D18111yescharacteristic
371C4—22no—
372C4—41no—
373C4—3622no—
374C4—7242no—
375C4#1915378411yescharacteristic
376C4#1915378411yescharacteristic
377C4C2×D4×D18111yescharacteristic
378C4C22×C22⋊D18111yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—7242no—
385C2—7244no—
386C2C2×#1915378411yescharacteristic
subgroupG/H
centerC2C2×#19153784
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#575×C181C2
socleC2×C181C2×#120
hypercenter#575D181
hypocenterC181#3019
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. #26042077C2semidirect
3. #26041843C2semidirect
4. #26041455C2semidirect
5. #26041445C2semidirect
6. #26041296C2semidirect
7. #26041295C2semidirect
8. C2×#19153732C2semidirect
9. #575×C181C2semidirect
21. #19153847C22semidirect
23. #19153842C22semidirect
26. C2×#15668433C22semidirect
27. #19153812C22semidirect
28. #19153805C22semidirect
31. #19153775C22semidirect
32. #19153743C22semidirect
33. #19153742C22semidirect
34. #19153732C22semidirect
36. C2×#41×C181C22semidirect
38. #137×C181C22semidirect
40. AGL(1,ℤ/8)×C181C22semidirect
42. C2×M16×C181C22semidirect
51. #15668449D4semidirect
53. C2×Dic362D4semidirect
62. C2×D724D4semidirect
78. #15668433C23semidirect
83. C4×Dic181C23semidirect
90. #41×C181D4semidirect
92. C2×Q8×C181D4semidirect
93. C2×Q8×C181C23semidirect
96. C2×D4×C181D4semidirect
97. C2×D4×C181C23semidirect
106. M16×C181C23semidirect
193. C181#3019semidirect
197. #575D181semidirect
212. C2×#41C2×D181semidirect
220. #137C2×D181semidirect
224. AGL(1,ℤ/8)C2×D181semidirect
228. C2×M16C2×D181semidirect
253. #41C22⋊D181semidirect
256. C2×Q8C22⋊D181semidirect
257. C2×Q8C22×D181semidirect
277. C2×D4C22⋊D181semidirect
278. C2×D4C22×D181semidirect
295. M16C22×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538622
#260420772
#260418432
#260414552
#260414452
#260412962
#260412952
C2×#191537322
#575×C1812
#3019181