Finite Group Databasebeta

← #37543068

Subgroups — #37543068

Order 23168.

724 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 133 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#26042116C211yesmaximal, characteristic, complemented
2#26042075C212yesmaximal, complemented
3#26041828C211yesmaximal, characteristic, complemented
4#26041464C211yesmaximal, characteristic, complemented
5#26041462C211yesmaximal, characteristic, complemented
6#26041458C212yesmaximal, complemented
7#26041451C211yesmaximal, characteristic, complemented
8#26041445C212yesmaximal, complemented
9#104×D181C212yesmaximal, complemented
10#26041262C211yesmaximal, characteristic, complemented
11#454×C181C211yesmaximal, characteristic, complemented
12#19153864—22no—
13#19153864C2211yescharacteristic
14#19153862—22no—
15#19153862—22no—
16#19153862C2212yes—
17C2×#15668450—22no—
18C2×#15668450C2211yescharacteristic
19C2×#15668448C2211yescharacteristic, complemented
20#41×D181—22no—
21#41×D181—22no—
22#41×D181C2212yes—
23#19153849C2212yescomplemented
24#19153846C2212yescomplemented
25#19153845—22no—
26#19153842C2212yescomplemented
27#19153840—22no—
28#19153837—22no—
29#19153835—22no—
30M16×D181C2212yescomplemented
31#19153814C2211yescharacteristic, complemented
32#19153812C2211yescharacteristic, complemented
33#19153804C2211yescharacteristic, complemented
34#19153789—22no—
35#19153776—22no—
36C42×D181C2211yescharacteristic, complemented
37#19153744C2212yescomplemented
38#19153743C2212yescomplemented
39#19153742C2212yescomplemented
40#19153735C2211yescharacteristic, complemented
41#19153732C2212yescomplemented
422−1+4×C181C2211yescharacteristic
43AGL(1,ℤ/8)×C181C2212yescomplemented
44#124×C181C2211yescharacteristic, complemented
45#104×C181C2212yescomplemented
46#101×C181C2211yescharacteristic, complemented
47#15668450—22no—
48#15668450—24no—
49#15668450—22no—
50#15668450—22no—
51#15668450—22no—
52#15668450—22no—
53#15668450D412yescomplemented
54#15668449—24no—
55#15668449—24no—
56#15668449—22no—
57#15668449—22no—
58#15668449—22no—
59#15668449C2312yes—
60#15668448—22no—
61#15668448—22no—
62#15668448—22no—
63#15668448—22no—
64#15668448D412yescomplemented
65C2×Dic362C2311yescharacteristic
66Q8×D181—22no—
67Q8×D181—22no—
68Q8×D181D412yescomplemented
69C2×D724—24no—
70C2×D724—22no—
71C2×D724D412yes—
72C2×D724C2311yescharacteristic
73C2×C22⋊D181—22no—
74C2×C22⋊D181—21no—
75D4×D181—24no—
76D4×D181—22no—
77D4×D181—24no—
78D4×D181—22no—
79D4×D181—22no—
80D4×D181D412yescomplemented
81C22×Dic181—21no—
82C2×C4×D181—24no—
83C2×C4×D181—22no—
84C2×C4×D181—21no—
85C2×C4×D181D412yes—
86C2×C4×D181C2311yescharacteristic
87Dic724—22no—
88#15668439—22no—
89#15668438—22no—
90#15668437—22no—
91#15668437—22no—
92#15668434—22no—
93#15668433C2312yes—
94#15668432—22no—
95C8×D181—22no—
96#15668428—22no—
97#15668426—22no—
98C4×Dic181—21no—
99C4×Dic181C2311yescharacteristic
100#41×C181—22no—
101#41×C181—22no—
102#41×C181C2312yes—
103C2×Q8×C181—22no—
104C2×Q8×C181C2311yescharacteristic
105C2×D4×C181C2311yescharacteristic
106SD16×C181—22no—
107D8×C181—22no—
108M16×C181C2312yes—
109C22⋊C4×C181—22no—
110C42×C181C2311yescharacteristic
111C22×D181—24no—
112C22×D181—22no—
113C22×D181C2×D412yes—
114C22×D181C2×D411yescharacteristic
115Dic362—22no—
116Dic362—22no—
117Dic362—22no—
118Dic362C2×D412yes—
119D724—24no—
120D724—22no—
121D724—22no—
122D724—24no—
123D724—22no—
124D724C2×D412yes—
125C22⋊D181—24no—
126C22⋊D181—22no—
127C22⋊D181—22no—
128C22⋊D181—22no—
129C22⋊D181—44no—
130C22⋊D181—42no—
131C2×Dic181—21no—
132C2×Dic181—22no—
133C2×Dic181—21no—
134C2×Dic181—41no—
135C2×Dic181—41no—
136C2×Dic181C2×D411yescharacteristic
137C4×D181—24no—
138C4×D181—24no—
139C4×D181—22no—
140C4×D181—22no—
141C4×D181—22no—
142C4×D181—22no—
143C4×D181—22no—
144C4×D181—42no—
145C4×D181C2×D412yes—
146C4×D181C2×D412yes—
147#13089678—22no—
148C23×C181—21no—
149Q8×C181—22no—
150Q8×C181—22no—
151Q8×C181C2×D412yes—
152D4×C181—22no—
153D4×C181—22no—
154D4×C181—22no—
155D4×C181C2×D412yes—
156C2×C4×C181—24no—
157C2×C4×C181—22no—
158C2×C4×C181—21no—
159C2×C4×C181C2×D412yes—
160C2×C4×C181C2411yescharacteristic
161C8×C181—22no—
162C22×C181—22no—
163C22×C181—42no—
164C22×C181—41no—
165C22×C181C22×D411yescharacteristic
166C2×D181—22no—
167C2×D181—22no—
168C2×D181—41no—
169C2×D181—42no—
170C2×D181—82no—
171C2×D181—84no—
172C2×D181#12012yes—
173C4×C181—22no—
174C4×C181—22no—
175C4×C181—41no—
176C4×C181C22×D412yes—
177Dic181—22no—
178Dic181—22no—
179Dic181—41no—
180Dic181#12011yescharacteristic
181Dic181#12011yescharacteristic
182C2×C181—21no—
183C2×C181—42no—
184C2×C181—81no—
185C2×C181C2×#12011yescharacteristic
186D181—22no—
187D181—42no—
188D181—42no—
189C181#301811yescharacteristic, complemented, Sylow
190#3018—1811nomaximal, Sylow
191#585—1811no—
192#575—1812no—
193#532—1811no—
194#456—1811no—
195#456—1811no—
196#455—1812no—
197#454—1811no—
198#454D18111yescharacteristic, complemented
199#453—1812no—
200C2×#104—1812no—
201#413—1811no—
2022−1+4—3622no—
2032−1+4—1811no—
2042−1+4C2×D18111yescharacteristic
205H(5,𝔽2)—3622no—
206H(5,𝔽2)—3622no—
207H(5,𝔽2)—1812no—
208C2×#41—3622no—
209C2×#41—3622no—
210C2×#41—3622no—
211C2×#41—1812no—
212C2×#41—1811no—
213C2×#41—1811no—
214#137—3622no—
215#137—1812no—
216#137—1812no—
217AGL(1,ℤ/8)—3622no—
218AGL(1,ℤ/8)—1812no—
219AGL(1,ℤ/8)C2×D18112yescomplemented
220#135—3622no—
221#135—3622no—
222C2×M16—1812no—
223C4⋊Q8—1811no—
224#127—1811no—
225#124—1811no—
226#124C2×D18111yescharacteristic, complemented
227#121—3622no—
228C4×D4—3622no—
229C2×C42—1811no—
230#104—1812no—
231#104—1812no—
232#104—1812no—
233#104C2×D18112yescomplemented
234#101—1811no—
235#101C2×D18111yescharacteristic, complemented
236#100—1812no—
237#41—22no—
238#41—22no—
239#41—3624no—
240#41—3624no—
241#41—3622no—
242#41—3622no—
243#41—3622no—
244#41—3622no—
245#41—3624no—
246#41—3622no—
247#41—3622no—
248#41—3622no—
249#41—3622no—
250#41—3622no—
251#41—3622no—
252#41—3622no—
253#41—3622no—
254#41—1812no—
255#41—1812no—
256#41—1812no—
257#41C22×D18112yes—
258C2×Q8—22no—
259C2×Q8—3622no—
260C2×Q8—3622no—
261C2×Q8—1812no—
262C2×Q8—1811no—
263C2×Q8C22×D18111yescharacteristic
264C2×D4—3624no—
265C2×D4—3622no—
266C2×D4—3624no—
267C2×D4—3622no—
268C2×D4—3624no—
269C2×D4—3622no—
270C2×D4—3622no—
271C2×D4—3622no—
272C2×D4—3621no—
273C2×D4—1812no—
274C2×D4—1812no—
275C2×D4—1811no—
276C2×D4C22×D18111yescharacteristic
277C22×C4—3624no—
278C22×C4—3622no—
279C22×C4—3621no—
280C22×C4—3621no—
281C22×C4—1812no—
282C22×C4—1811no—
283Q16—3622no—
284Q16—3622no—
285SD16—22no—
286SD16—3622no—
287SD16—3622no—
288SD16—3622no—
289D8—22no—
290D8—3622no—
291M16—3622no—
292M16—1812no—
293M16C22×D18112yes—
294C2×C8—3622no—
295C4⋊C4—3622no—
296C22⋊C4—22no—
297C22⋊C4—3622no—
298C42—3621no—
299C42—1811no—
300C42C22×D18111yescharacteristic
301C23—21no—
302C23—3624no—
303C23—3622no—
304C23—1812no—
305C23—1811no—
306Q8—22no—
307Q8—22no—
308Q8—3622no—
309Q8—3622no—
310Q8—3622no—
311Q8—1812no—
312Q8D4×D18112yes—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—3624no—
317D4—3622no—
318D4—3622no—
319D4—3624no—
320D4—3624no—
321D4—3622no—
322D4—3622no—
323D4—3622no—
324D4—3622no—
325D4—1812no—
326D4—7244no—
327D4—7242no—
328D4D4×D18112yes—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—3624no—
333C2×C4—3624no—
334C2×C4—3622no—
335C2×C4—3622no—
336C2×C4—3622no—
337C2×C4—3622no—
338C2×C4—3621no—
339C2×C4—3622no—
340C2×C4—3622no—
341C2×C4—3621no—
342C2×C4—1811no—
343C2×C4—1812no—
344C2×C4—1812no—
345C2×C4—7242no—
346C2×C4—7241no—
347C2×C4—7241no—
348C2×C4D4×D18112yes—
349C2×C4C23×D18111yescharacteristic
350C8—22no—
351C8—3622no—
352C22—22no—
353C22—42no—
354C22—41no—
355C22—3622no—
356C22—3622no—
357C22—14482no—
358C22—14484no—
359C22—1812no—
360C22—7241no—
361C22—7242no—
362C22C2×D4×D18111yescharacteristic
363C4—22no—
364C4—22no—
365C4—41no—
366C4—3622no—
367C4—3622no—
368C4—1811no—
369C4—1811no—
370C4—7241no—
371C4C2×D4×D18112yes—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—3622no—
376C2—7242no—
377C2—7242no—
378C2#120×D18111yescharacteristic
subgroupG/H
centerC2#120×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#454×C181C2
socleC2×C181C2×#120
hypercenter#454D181
hypocenterC181#3018
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26042116C2semidirect
2. #26042075C2semidirect
3. #26041828C2semidirect
4. #26041464C2semidirect
5. #26041462C2semidirect
6. #26041458C2semidirect
7. #26041451C2semidirect
8. #26041445C2semidirect
9. #104×D181C2semidirect
10. #26041262C2semidirect
11. #454×C181C2semidirect
19. C2×#15668448C22semidirect
23. #19153849C22semidirect
24. #19153846C22semidirect
26. #19153842C22semidirect
30. M16×D181C22semidirect
31. #19153814C22semidirect
32. #19153812C22semidirect
33. #19153804C22semidirect
36. C42×D181C22semidirect
37. #19153744C22semidirect
38. #19153743C22semidirect
39. #19153742C22semidirect
40. #19153735C22semidirect
41. #19153732C22semidirect
43. AGL(1,ℤ/8)×C181C22semidirect
44. #124×C181C22semidirect
45. #104×C181C22semidirect
46. #101×C181C22semidirect
53. #15668450D4semidirect
64. #15668448D4semidirect
68. Q8×D181D4semidirect
80. D4×D181D4semidirect
189. C181#3018semidirect
198. #454D181semidirect
219. AGL(1,ℤ/8)C2×D181semidirect
226. #124C2×D181semidirect
233. #104C2×D181semidirect
235. #101C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260421162
#260420752
#260418282
#260414642
#260414622
#260414582
#260414512
#260414452
#104×D1812
#260412622
#454×C1812
#3018181