Finite Group Databasebeta

← #37543097

Subgroups — #37543097

Order 23168.

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

normal subgroups only

This group has 704 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
1#26042115C211yesmaximal, characteristic
2C2×#19153847C211yesmaximal, characteristic, complemented
3#26041853C211yesmaximal, characteristic, complemented
4#26041460C214yesmaximal, complemented
5#26041455C214yesmaximal, complemented
6#26041296C211yesmaximal, characteristic, complemented
7#26041295C211yesmaximal, characteristic, complemented
8#26041261C211yesmaximal, characteristic, complemented
9C2×#137×C181C211yesmaximal, characteristic, complemented
10#19153864—21no—
11#19153863—21no—
12#19153863C2212yes—
13#19153862—21no—
14#19153862C2212yes—
15#19153861—22no—
16#19153861—21no—
17C2×#15668449—22no—
18C2×#15668449C2211yescharacteristic
19C2×#15668449C2211yescharacteristic
20#41×D181—22no—
21#41×D181—22no—
22#19153847—22no—
23#19153847C2214yescomplemented
24C2×#15668439—21no—
25C2×#15668436—21no—
26C2×#15668433C2211yescharacteristic, complemented
27#19153814C2212yescomplemented
28#19153805C2212yescomplemented
29#19153799—22no—
30#19153792—22no—
31#19153775C2211yescharacteristic, complemented
32#19153743C2214yescomplemented
33#19153742C2214yescomplemented
34#19153734C2212yescomplemented
35#19153732C2212yescomplemented
36C2×#41×C181C2211yescharacteristic
37C22×Q8×C181C2211yescharacteristic, complemented
38#137×C181—22no—
39#137×C181C2214yescomplemented
40C2×Q16×C181—21no—
41C2×SD16×C181—21no—
42C2×M16×C181C2211yescharacteristic, complemented
43#15668450—22no—
44#15668450—22no—
45#15668449—24no—
46#15668449—24no—
47#15668449—22no—
48#15668449—22no—
49#15668449—21no—
50#15668449—21no—
51#15668449—21no—
52#15668449—21no—
53#15668449—21no—
54#15668449—21no—
55#15668449D412yes—
56#15668449C2312yes—
57#15668448—24no—
58#15668448—22no—
59#15668448—24no—
60#15668448—22no—
61C2×Dic362—22no—
62C2×Dic362D411yescharacteristic
63C2×Dic362C2311yescharacteristic
64Q8×D181—22no—
65Q8×D181—22no—
66C2×D724—22no—
67C2×D724D411yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—24no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181D412yes—
72D4×D181—24no—
73D4×D181—22no—
74D4×D181—24no—
75D4×D181—22no—
76C2×C4×D181—24no—
77C2×C4×D181—21no—
78C2×C4×D181D412yes—
79#15668439—24no—
80#15668436—24no—
81#15668433—21no—
82#15668433C2312yes—
83C2×#13089678—21no—
84C4⋊Dic181—21no—
85#15668428—24no—
86C22⋊Dic181—21no—
87#15668426—24no—
88C4×Dic181C2312yes—
89#41×C181—21no—
90#41×C181—21no—
91#41×C181D412yes—
92#41×C181C2312yes—
93C2×Q8×C181—21no—
94C2×Q8×C181—21no—
95C2×Q8×C181—21no—
96C2×Q8×C181—21no—
97C2×Q8×C181—21no—
98C2×Q8×C181D411yescharacteristic
99C2×Q8×C181C2312yes—
100C2×D4×C181—21no—
101C2×D4×C181D411yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181—21no—
104C22×C4×C181C2311yescharacteristic
105Q16×C181—24no—
106SD16×C181—24no—
107M16×C181—21no—
108M16×C181C2312yes—
109C2×C8×C181—21no—
110C22×D181—24no—
111C22×D181—21no—
112C22×D181C2×D412yes—
113Dic362—24no—
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117Dic362C2×D412yes—
118D724—24no—
119D724—21no—
120D724—21no—
121D724—21no—
122D724C2×D412yes—
123C22⋊D181—24no—
124C22⋊D181—24no—
125C22⋊D181—22no—
126C22⋊D181—22no—
127C22⋊D181—22no—
128C22⋊D181—22no—
129C2×Dic181—24no—
130C2×Dic181—21no—
131C2×Dic181—22no—
132C2×Dic181C2×D412yes—
133C4×D181—24no—
134C4×D181—24no—
135C4×D181—22no—
136C4×D181—22no—
137C4×D181—22no—
138C4×D181—22no—
139#13089678—22no—
140C23×C181—21no—
141C23×C181C2×D411yescharacteristic
142Q8×C181—21no—
143Q8×C181—22no—
144Q8×C181—22no—
145Q8×C181—41no—
146Q8×C181—41no—
147Q8×C181C2×D412yes—
148D4×C181—22no—
149D4×C181—22no—
150D4×C181—21no—
151D4×C181C2×D412yes—
152C2×C4×C181—21no—
153C2×C4×C181—22no—
154C2×C4×C181—22no—
155C2×C4×C181—22no—
156C2×C4×C181—41no—
157C2×C4×C181—41no—
158C2×C4×C181C2×D411yescharacteristic
159C2×C4×C181C2×D411yescharacteristic
160C2×C4×C181C2×D411yescharacteristic
161C2×C4×C181C2×D411yescharacteristic
162C2×C4×C181C2×D411yescharacteristic
163C2×C4×C181C2411yescharacteristic
164C8×C181—22no—
165C22×C181—22no—
166C22×C181—41no—
167C22×C181—81no—
168C22×C181#12011yescharacteristic
169C22×C181#12011yescharacteristic
170C22×C181C22×D411yescharacteristic
171C2×D181—22no—
172C2×D181—22no—
173C2×D181—42no—
174C2×D181—81no—
175C2×D181—84no—
176C4×C181—22no—
177C4×C181—42no—
178C4×C181#12011yescharacteristic
179C4×C181#12011yescharacteristic
180C4×C181C22×D411yescharacteristic
181C4×C181C22×D411yescharacteristic
182Dic181—22no—
183Dic181—22no—
184Dic181—42no—
185C2×C181—21no—
186C2×C181—21no—
187C2×C181—21no—
188C2×C181—42no—
189C2×C181C2×#12011yescharacteristic
190D181—42no—
191D181—42no—
192C181#302211yescharacteristic, complemented, Sylow
193#3022—1811nomaximal, Sylow
194#585—1811no—
195C2×#137—1811no—
196C2×#137D18111yescharacteristic, complemented
197#536—1811no—
198#456—1814no—
199#455—1814no—
200#421—1811no—
201#421—1811no—
202#413—1811no—
2032−1+4—3621no—
2042−1+4—3621no—
2052−1+4—1812no—
206H(5,𝔽2)—3622no—
207H(5,𝔽2)—3621no—
208H(5,𝔽2)—3621no—
209H(5,𝔽2)—1812no—
210C2×#41—3622no—
211C2×#41—3622no—
212C2×#41—3622no—
213C2×#41—1811no—
214C2×#41—1811no—
215C2×#41C2×D18111yescharacteristic
216C22×Q8C2×D18111yescharacteristic, complemented
217#137—22no—
218#137—3622no—
219#137—1814no—
220#137C2×D18114yescomplemented
221C2×Q16—21no—
222C2×Q16—3621no—
223C2×SD16—21no—
224C2×SD16—3621no—
225C2×M16—1811no—
226C2×M16C2×D18111yescharacteristic, complemented
227C4⋊Q8—1812no—
228#124—1812no—
229#123—3622no—
230C22⋊Q8—3622no—
231#117—1811no—
232#104—1814no—
233#104—1814no—
234#101—1812no—
235#100—1812no—
236#41—21no—
237#41—21no—
238#41—3624no—
239#41—3624no—
240#41—3622no—
241#41—3622no—
242#41—3624no—
243#41—3622no—
244#41—3622no—
245#41—3622no—
246#41—3622no—
247#41—3624no—
248#41—3621no—
249#41—3621no—
250#41—3621no—
251#41—3621no—
252#41—3621no—
253#41—3621no—
254#41—1812no—
255#41—1812no—
256#41C22⋊D18112yes—
257#41C22×D18112yes—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—3622no—
264C2×Q8—3622no—
265C2×Q8—3622no—
266C2×Q8—1811no—
267C2×Q8—1811no—
268C2×Q8C22⋊D18111yescharacteristic
269C2×Q8C22×D18112yes—
270C2×D4—21no—
271C2×D4—3622no—
272C2×D4—3624no—
273C2×D4—3622no—
274C2×D4—3624no—
275C2×D4—3624no—
276C2×D4—3621no—
277C2×D4—3622no—
278C2×D4—1812no—
279C2×D4—1811no—
280C2×D4—1811no—
281C2×D4C22⋊D18111yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—3624no—
285C22×C4—3621no—
286C22×C4—1812no—
287C22×C4C22×D18111yescharacteristic
288Q16—24no—
289Q16—3624no—
290SD16—24no—
291SD16—3624no—
292M16—21no—
293M16—3621no—
294M16—1812no—
295M16C22×D18112yes—
296C2×C8—21no—
297C2×C8—3621no—
298C4⋊C4—3624no—
299C4⋊C4—3621no—
300C22⋊C4—3624no—
301C22⋊C4—3621no—
302C42—1812no—
303C23—21no—
304C23—3624no—
305C23—3621no—
306C23—1812no—
307C23D4×D18111yescharacteristic
308Q8—21no—
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313Q8—3624no—
314Q8—3621no—
315Q8—3621no—
316Q8—3621no—
317Q8—1812no—
318Q8C2×C22⋊D18112yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—3624no—
323D4—3624no—
324D4—3624no—
325D4—3622no—
326D4—3622no—
327D4—3622no—
328D4—3622no—
329D4—3621no—
330D4—3621no—
331D4—3621no—
332D4—1812no—
333D4C2×C22⋊D18112yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—3624no—
341C2×C4—3624no—
342C2×C4—3624no—
343C2×C4—3621no—
344C2×C4—3622no—
345C2×C4—3622no—
346C2×C4—3622no—
347C2×C4—3622no—
348C2×C4—3622no—
349C2×C4—1812no—
350C2×C4D4×D18111yescharacteristic
351C2×C4D4×D18111yescharacteristic
352C2×C4D4×D18111yescharacteristic
353C2×C4C2×C22⋊D18111yescharacteristic
354C2×C4C2×C22⋊D18111yescharacteristic
355C2×C4C23×D18111yescharacteristic
356C8—22no—
357C8—3622no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—3622no—
362C22—3622no—
363C22—14481no—
364C22—14484no—
365C22—7242no—
366C22#1915378411yescharacteristic
367C22#1915378411yescharacteristic
368C22C2×D4×D18111yescharacteristic
369C4—22no—
370C4—42no—
371C4—3622no—
372C4—3622no—
373C4—7242no—
374C4#1915378411yescharacteristic
375C4#1915378411yescharacteristic
376C4C2×D4×D18111yescharacteristic
377C4C22×C22⋊D18111yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—7242no—
383C2—7242no—
384C2C2×#1915378411yescharacteristic
subgroupG/H
centerC2C2×#19153784
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#137×C181C2
socleC2×C181C2×#120
prime radical#26042115C2
hypercenterC2×#137D181
hypocenterC181#3022
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19153847C2semidirect
3. #26041853C2semidirect
4. #26041460C2semidirect
5. #26041455C2semidirect
6. #26041296C2semidirect
7. #26041295C2semidirect
8. #26041261C2semidirect
9. C2×#137×C181C2semidirect
23. #19153847C22semidirect
26. C2×#15668433C22semidirect
27. #19153814C22semidirect
28. #19153805C22semidirect
31. #19153775C22semidirect
32. #19153743C22semidirect
33. #19153742C22semidirect
34. #19153734C22semidirect
35. #19153732C22semidirect
37. C22×Q8×C181C22semidirect
39. #137×C181C22semidirect
42. C2×M16×C181C22semidirect
192. C181#3022semidirect
196. C2×#137D181semidirect
216. C22×Q8C2×D181semidirect
220. #137C2×D181semidirect
226. C2×M16C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421152
C2×#191538472
#260418532
#260414602
#260414552
#260412962
#260412952
#260412612
C2×#137×C1812
#3022181