Finite Group Databasebeta

← #37543101

Subgroups — #37543101

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×#19153849C211yesmaximal, characteristic, complemented
3C2×#19153846C211yesmaximal, characteristic, complemented
4#26041855C211yesmaximal, characteristic, complemented
5#26041463C214yesmaximal, complemented
6#26041458C214yesmaximal, complemented
7#26041297C211yesmaximal, characteristic, complemented
8#26041261C211yesmaximal, characteristic, complemented
9#421×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—
22C22×Dic362C2211yescharacteristic, complemented
23#19153849—22no—
24#19153849C2214yescomplemented
25#19153846—22no—
26#19153846C2214yescomplemented
27C2×Dic724—21no—
28C2×#15668439—21no—
29C2×#15668438—21no—
30C2×#15668437—21no—
31C2×#15668433C2211yescharacteristic, complemented
32#19153816C2212yescomplemented
33#19153804C2212yescomplemented
34#19153800—22no—
35#19153796—22no—
36#19153744C2214yescomplemented
37#19153734C2212yescomplemented
38#19153732C2212yescomplemented
39C2×#41×C181C2211yescharacteristic
40C2×M16×C181C2211yescharacteristic, complemented
41#117×C181C2211yescharacteristic, complemented
42#104×C181C2214yescomplemented
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×Dic362—21no—
63C2×Dic362—21no—
64C2×Dic362—21no—
65C2×Dic362—21no—
66C2×Dic362—21no—
67C2×Dic362D411yescharacteristic
68C2×Dic362C2312yes—
69C2×Dic362C2311yescharacteristic
70Q8×D181—22no—
71Q8×D181—22no—
72C2×D724—22no—
73C2×D724D411yescharacteristic
74C2×D724C2311yescharacteristic
75C2×C22⋊D181—24no—
76C2×C22⋊D181—21no—
77C2×C22⋊D181D412yes—
78D4×D181—24no—
79D4×D181—22no—
80D4×D181—24no—
81D4×D181—22no—
82C22×Dic181—21no—
83C2×C4×D181—24no—
84C2×C4×D181—21no—
85C2×C4×D181D412yes—
86Dic724—24no—
87#15668439—24no—
88#15668438—24no—
89#15668437—24no—
90#15668433—21no—
91#15668433C2312yes—
92C2×#13089678—21no—
93C4⋊Dic181—24no—
94#15668426—24no—
95#41×C181—21no—
96#41×C181—21no—
97#41×C181D412yes—
98#41×C181C2312yes—
99C2×Q8×C181D411yescharacteristic
100C2×D4×C181—21no—
101C2×D4×C181D411yescharacteristic
102C22×C4×C181—21no—
103C22×C4×C181C2311yescharacteristic
104M16×C181—21no—
105M16×C181C2312yes—
106C2×C8×C181—21no—
107C4⋊C4×C181—21no—
108C22⋊C4×C181—21no—
109C42×C181C2312yes—
110C22×D181—24no—
111C22×D181—21no—
112C22×D181C2×D412yes—
113Dic362—24no—
114Dic362—21no—
115Dic362—21no—
116Dic362—21no—
117Dic362—22no—
118Dic362—22no—
119Dic362—41no—
120Dic362—41no—
121Dic362C2×D412yes—
122D724—24no—
123D724—21no—
124D724—21no—
125D724—21no—
126D724C2×D412yes—
127C22⋊D181—24no—
128C22⋊D181—24no—
129C22⋊D181—22no—
130C22⋊D181—22no—
131C22⋊D181—22no—
132C22⋊D181—22no—
133C2×Dic181—24no—
134C2×Dic181—21no—
135C2×Dic181—22no—
136C2×Dic181—41no—
137C2×Dic181—41no—
138C2×Dic181C2×D412yes—
139C4×D181—24no—
140C4×D181—24no—
141C4×D181—22no—
142C4×D181—22no—
143C4×D181—22no—
144C4×D181—22no—
145#13089678—22no—
146C23×C181—21no—
147C23×C181C2×D411yescharacteristic
148Q8×C181—21no—
149Q8×C181C2×D412yes—
150D4×C181—22no—
151D4×C181—22no—
152D4×C181—21no—
153D4×C181C2×D412yes—
154C2×C4×C181—21no—
155C2×C4×C181—22no—
156C2×C4×C181—22no—
157C2×C4×C181—22no—
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×#137—1811no—
197#536—1811no—
198#456—1814no—
199#455—1814no—
200#421—1811no—
201#421D18111yescharacteristic, complemented
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×Q8—1811no—
217#137—3622no—
218#137—3622no—
219#137—1814no—
220#137—1814no—
221C2×Q16—3621no—
222C2×Q16—3621no—
223C2×SD16—3621no—
224C2×SD16—3621no—
225C2×M16—1811no—
226C2×M16C2×D18111yescharacteristic, complemented
227C4⋊Q8—1812no—
228#124—1812no—
229#123—3622no—
230C22⋊Q8—3622no—
231#117C2×D18111yescharacteristic, complemented
232#104—1814no—
233#104C2×D18114yescomplemented
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#41D72412yes—
257#41C22×D18112yes—
258C2×Q8—3622no—
259C2×Q8—3622no—
260C2×Q8—3622no—
261C2×Q8—3621no—
262C2×Q8—3621no—
263C2×Q8—3621no—
264C2×Q8—3621no—
265C2×Q8—3621no—
266C2×Q8—1811no—
267C2×Q8—1812no—
268C2×Q8—1811no—
269C2×Q8D72411yescharacteristic
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×D4D72411yescharacteristic
282C22×C4—21no—
283C22×C4—3624no—
284C22×C4—3621no—
285C22×C4—3621no—
286C22×C4—1812no—
287C22×C4C22×D18111yescharacteristic
288Q16—3624no—
289Q16—3624no—
290SD16—3624no—
291SD16—3624no—
292M16—21no—
293M16—3621no—
294M16—1812no—
295M16C22×D18112yes—
296C2×C8—21no—
297C2×C8—3621no—
298C4⋊C4—21no—
299C4⋊C4—3624no—
300C22⋊C4—21no—
301C22⋊C4—3624no—
302C42C22×D18112yes—
303C23—21no—
304C23—3624no—
305C23—3621no—
306C23—1812no—
307C23D4×D18111yescharacteristic
308Q8—21no—
309Q8—3624no—
310Q8—3621no—
311Q8—3621no—
312Q8—3621no—
313Q8—3622no—
314Q8—3622no—
315Q8—1812no—
316Q8—7241no—
317Q8—7241no—
318Q8C2×D72412yes—
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×D72412yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—3624no—
339C2×C4—3624no—
340C2×C4—3624no—
341C2×C4—3621no—
342C2×C4—3622no—
343C2×C4—3622no—
344C2×C4—3622no—
345C2×C4—3622no—
346C2×C4—3622no—
347C2×C4—1812no—
348C2×C4—7241no—
349C2×C4—7241no—
350C2×C4D4×D18111yescharacteristic
351C2×C4D4×D18111yescharacteristic
352C2×C4D4×D18111yescharacteristic
353C2×C4C2×D72411yescharacteristic
354C2×C4C2×D72411yescharacteristic
355C2×C4C23×D18111yescharacteristic
356C8—22no—
357C8—3622no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—3622no—
362C22—3622no—
363C22—14481no—
364C22—14484no—
365C22—7242no—
366C22#1915378611yescharacteristic
367C22#1915378611yescharacteristic
368C22C2×D4×D18111yescharacteristic
369C4—22no—
370C4—42no—
371C4—3622no—
372C4—3622no—
373C4—7242no—
374C4#1915378611yescharacteristic
375C4#1915378611yescharacteristic
376C4C2×D4×D18111yescharacteristic
377C4C22×D72411yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—7242no—
383C2—7242no—
384C2C2×#1915378611yescharacteristic
subgroupG/H
centerC2C2×#19153786
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#421×C181C2
socleC2×C181C2×#120
prime radical#26042115C2
hypercenter#421D181
hypocenterC181#3022
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#19153849C2semidirect
3. C2×#19153846C2semidirect
4. #26041855C2semidirect
5. #26041463C2semidirect
6. #26041458C2semidirect
7. #26041297C2semidirect
8. #26041261C2semidirect
9. #421×C181C2semidirect
22. C22×Dic362C22semidirect
24. #19153849C22semidirect
26. #19153846C22semidirect
31. C2×#15668433C22semidirect
32. #19153816C22semidirect
33. #19153804C22semidirect
36. #19153744C22semidirect
37. #19153734C22semidirect
38. #19153732C22semidirect
40. C2×M16×C181C22semidirect
41. #117×C181C22semidirect
42. #104×C181C22semidirect
192. C181#3022semidirect
201. #421D181semidirect
226. C2×M16C2×D181semidirect
231. #117C2×D181semidirect
233. #104C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421152
C2×#191538492
C2×#191538462
#260418552
#260414632
#260414582
#260412972
#260412612
#421×C1812
#3022181