Finite Group Databasebeta

← #34078478

Subgroups — #34078478

Order 19328.

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

normal subgroups only

This group has 736 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#23753047C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D151C212yesmaximal, complemented
3#23752775C211yesmaximal, characteristic, complemented
4#23752386C212yesmaximal, complemented
5#23752383C212yesmaximal, complemented
6#23752382C211yesmaximal, characteristic, complemented
7#23752380C211yesmaximal, characteristic, complemented
8#23752379C211yesmaximal, characteristic, complemented
9#23752227C212yesmaximal, complemented
10#23752193C211yesmaximal, characteristic, complemented
11#453×C151C211yesmaximal, characteristic, complemented
12#18012786—22no—
13#18012786C2212yescomplemented
14#18012784—22no—
15#18012784—22no—
16#18012784C2211yescharacteristic, complemented
17C2×#14559003—22no—
18C2×#14559003—22no—
19C2×#14559003C2211yescharacteristic
20#41×D151—22no—
21#41×D151C2212yes—
22C2×D4×D151C2211yescharacteristic, complemented
23#18012767C2212yescomplemented
24#18012766—22no—
25#18012765C2212yescomplemented
26#18012764C2212yescomplemented
27#18012763—22no—
28SD16×D151—22no—
29D8×D151—22no—
30M16×D151C2212yescomplemented
31#18012735C2211yescharacteristic, complemented
32#18012727C2211yescharacteristic, complemented
33#18012726C2211yescharacteristic, complemented
34#18012711—22no—
35#18012707—22no—
36#18012696C2211yescharacteristic, complemented
37#18012667C2212yescomplemented
38#18012666C2212yescomplemented
39#18012665C2212yescomplemented
40#18012657C2212yescomplemented
41#18012656C2211yescharacteristic, complemented
42H(5,𝔽2)×C151C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C151C2212yescomplemented
44#127×C151C2211yescharacteristic, complemented
45#104×C151C2212yescomplemented
46#100×C151C2211yescharacteristic, complemented
47C23×D151—21no—
48#14559005D412yescomplemented
49#14559004—24no—
50#14559004—22no—
51#14559004—22no—
52#14559004C2312yes—
53#14559003—24no—
54#14559003—24no—
55#14559003—24no—
56#14559003—22no—
57#14559003—22no—
58#14559003—24no—
59#14559003—22no—
60#14559003—22no—
61#14559003—22no—
62#14559003D412yescomplemented
63C2×Dic302—22no—
64C2×Dic302—22no—
65C2×Dic302C2311yescharacteristic
66Q8×D151—22no—
67Q8×D151D412yescomplemented
68C2×D604C2311yescharacteristic
69C2×C22⋊D151—24no—
70C2×C22⋊D151—22no—
71C2×C22⋊D151—21no—
72C2×C22⋊D151—22no—
73C2×C22⋊D151—22no—
74C2×C22⋊D151D412yes—
75D4×D151—24no—
76D4×D151—22no—
77D4×D151—22no—
78D4×D151—22no—
79D4×D151—22no—
80D4×D151D412yescomplemented
81C22×Dic151—22no—
82C22×Dic151D412yes—
83C2×C4×D151—22no—
84C2×C4×D151—22no—
85C2×C4×D151C2311yescharacteristic
86#14558993—22no—
87#14558992—22no—
88#14558991—22no—
89D1208—22no—
90#14558989—22no—
91#14558989—22no—
92#14558988C2312yes—
93#14558987—22no—
94C8×D151—22no—
95Dic151⋊C4—21no—
96C22⋊Dic151—22no—
97(C2×D151)⋊C4—21no—
98(C2×D151)⋊C4—22no—
99C4×Dic151C2311yescharacteristic
100#41×C151—22no—
101#41×C151C2312yes—
102C2×D4×C151—22no—
103C2×D4×C151—22no—
104C2×D4×C151—22no—
105C2×D4×C151C2311yescharacteristic
106C2×D4×C151C2311yescharacteristic
107SD16×C151—22no—
108D8×C151—22no—
109M16×C151C2312yes—
110C42×C151C2311yescharacteristic
111C22×D151—22no—
112C22×D151—22no—
113C22×D151—21no—
114C22×D151—42no—
115C22×D151—82no—
116C22×D151C2×D411yescharacteristic
117Dic302—24no—
118Dic302—22no—
119Dic302—22no—
120Dic302C2×D412yes—
121D604—22no—
122D604—22no—
123D604C2×D412yes—
124C22⋊D151—24no—
125C22⋊D151—24no—
126C22⋊D151—22no—
127C22⋊D151—22no—
128C22⋊D151—24no—
129C22⋊D151—22no—
130C22⋊D151—41no—
131C22⋊D151—41no—
132C22⋊D151—82no—
133C2×Dic151—22no—
134C2×Dic151—24no—
135C2×Dic151—22no—
136C2×Dic151—22no—
137C2×Dic151—22no—
138C2×Dic151—21no—
139C2×Dic151C2×D412yes—
140C2×Dic151C2×D411yescharacteristic
141C4×D151—24no—
142C4×D151—22no—
143C4×D151—22no—
144C4×D151—22no—
145C4×D151C2×D412yes—
146C4×D151C2×D412yes—
147C151⋊C8—22no—
148C23×C151—22no—
149C23×C151—21no—
150C23×C151C2×D412yes—
151Q8×C151C2×D412yes—
152D4×C151—24no—
153D4×C151—22no—
154D4×C151—22no—
155D4×C151—22no—
156D4×C151—82no—
157D4×C151C2×D412yes—
158C2×C4×C151—22no—
159C2×C4×C151—22no—
160C2×C4×C151—21no—
161C2×C4×C151C2411yescharacteristic
162C8×C151—22no—
163C22×C151—22no—
164C22×C151—22no—
165C22×C151—42no—
166C22×C151—42no—
167C22×C151—41no—
168C22×C151—82no—
169C22×C151C22×D411yescharacteristic
170C2×D151—22no—
171C2×D151—41no—
172C2×D151—42no—
173C2×D151—41no—
174C2×D151—82no—
175C2×D151—84no—
176C2×D151—82no—
177C2×D151#12012yes—
178C4×C151—22no—
179C4×C151—41no—
180C4×C151C22×D412yes—
181Dic151—22no—
182Dic151—22no—
183Dic151—41no—
184Dic151#12011yescharacteristic
185Dic151#12011yescharacteristic
186C2×C151—21no—
187C2×C151—42no—
188C2×C151—42no—
189C2×C151—81no—
190C2×C151C2×#12011yescharacteristic
191D151—22no—
192D151—42no—
193D151—81no—
194C151#302111yescharacteristic, complemented, Sylow
195#3021—1511nomaximal, Sylow
196#585—1511no—
197C2×AGL(1,ℤ/8)—1512no—
198#535—1511no—
199#454—1512no—
200#454—1512no—
201#453—1511no—
202#453—1511no—
203#453—1511no—
204#453D15111yescharacteristic, complemented
205#421—1512no—
206#413—1511no—
2072−1+4—3022no—
2082−1+4—1512no—
209H(5,𝔽2)—3022no—
210H(5,𝔽2)—3022no—
211H(5,𝔽2)—1511no—
212H(5,𝔽2)C2×D15111yescharacteristic, complemented
213C2×#41—3022no—
214C2×#41—3022no—
215C2×#41—3022no—
216C2×#41—1512no—
217C2×#41—1511no—
218C22×D4—1511no—
219AGL(1,ℤ/8)—3022no—
220AGL(1,ℤ/8)—3022no—
221AGL(1,ℤ/8)—1512no—
222AGL(1,ℤ/8)—1512no—
223AGL(1,ℤ/8)—1512no—
224AGL(1,ℤ/8)C2×D15112yescomplemented
225C2×SD16—3022no—
226C2×D8—3022no—
227C2×M16—1512no—
228#127—1511no—
229#127C2×D15111yescharacteristic, complemented
230#124—1511no—
231#124—1511no—
232#121—3022no—
233#120—3022no—
234#117—1511no—
235#104—1512no—
236#104—1512no—
237#104—1512no—
238#104C2×D15112yescomplemented
239#101—1512no—
240#100—1511no—
241#100C2×D15111yescharacteristic, complemented
242C24—3021no—
243#41—22no—
244#41—3024no—
245#41—3022no—
246#41—3022no—
247#41—3024no—
248#41—3024no—
249#41—3024no—
250#41—3022no—
251#41—3022no—
252#41—3022no—
253#41—3022no—
254#41—3024no—
255#41—3022no—
256#41—1512no—
257#41—1512no—
258#41—1512no—
259#41C22×D15112yes—
260C2×Q8—3022no—
261C2×Q8—3022no—
262C2×Q8—3022no—
263C2×Q8—1512no—
264C2×Q8—1511no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—3024no—
269C2×D4—3022no—
270C2×D4—3022no—
271C2×D4—3024no—
272C2×D4—3022no—
273C2×D4—3022no—
274C2×D4—3022no—
275C2×D4—3022no—
276C2×D4—3022no—
277C2×D4—3021no—
278C2×D4—1512no—
279C2×D4—1512no—
280C2×D4—1511no—
281C2×D4C22×D15111yescharacteristic
282C2×D4C22×D15111yescharacteristic
283C22×C4—3022no—
284C22×C4—3022no—
285C22×C4—3022no—
286C22×C4—1512no—
287C22×C4—1511no—
288SD16—22no—
289SD16—3022no—
290SD16—3022no—
291SD16—3022no—
292D8—22no—
293D8—3022no—
294D8—3022no—
295D8—3022no—
296M16—3022no—
297M16—1512no—
298M16C22×D15112yes—
299C2×C8—3022no—
300C4⋊C4—3021no—
301C22⋊C4—3021no—
302C22⋊C4—3022no—
303C22⋊C4—3022no—
304C42—1511no—
305C42C22×D15111yescharacteristic
306C23—22no—
307C23—21no—
308C23—3022no—
309C23—3022no—
310C23—3021no—
311C23—6042no—
312C23—1511no—
313C23—12082no—
314C23D4×D15112yes—
315Q8—3024no—
316Q8—3022no—
317Q8—3022no—
318Q8—1512no—
319Q8D4×D15112yes—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—82no—
325D4—3022no—
326D4—3024no—
327D4—3024no—
328D4—3022no—
329D4—3024no—
330D4—3022no—
331D4—3022no—
332D4—3022no—
333D4—6041no—
334D4—6041no—
335D4—1512no—
336D4—12082no—
337D4D4×D15112yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—3024no—
342C2×C4—3022no—
343C2×C4—3022no—
344C2×C4—3022no—
345C2×C4—3024no—
346C2×C4—3022no—
347C2×C4—3022no—
348C2×C4—3022no—
349C2×C4—3022no—
350C2×C4—3021no—
351C2×C4—1512no—
352C2×C4—1511no—
353C2×C4—1512no—
354C2×C4—1512no—
355C2×C4C23×D15111yescharacteristic
356C8—22no—
357C8—3022no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—82no—
364C22—3022no—
365C22—6041no—
366C22—6042no—
367C22—6041no—
368C22—1512no—
369C22—12082no—
370C22—12082no—
371C22—12084no—
372C22C2×D4×D15111yescharacteristic
373C4—22no—
374C4—41no—
375C4—3022no—
376C4—3022no—
377C4—6041no—
378C4—1511no—
379C4—1511no—
380C4C2×D4×D15112yes—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—81no—
385C2—3022no—
386C2—6042no—
387C2—12081no—
388C2#120×D15111yescharacteristic
subgroupG/H
centerC2#120×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#453×C151C2
socleC2×C151C2×#120
hypercenter#453D151
hypocenterC151#3021
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23753047C2semidirect
2. AGL(1,ℤ/8)×D151C2semidirect
3. #23752775C2semidirect
4. #23752386C2semidirect
5. #23752383C2semidirect
6. #23752382C2semidirect
7. #23752380C2semidirect
8. #23752379C2semidirect
9. #23752227C2semidirect
10. #23752193C2semidirect
11. #453×C151C2semidirect
13. #18012786C22semidirect
16. #18012784C22semidirect
22. C2×D4×D151C22semidirect
23. #18012767C22semidirect
25. #18012765C22semidirect
26. #18012764C22semidirect
30. M16×D151C22semidirect
31. #18012735C22semidirect
32. #18012727C22semidirect
33. #18012726C22semidirect
36. #18012696C22semidirect
37. #18012667C22semidirect
38. #18012666C22semidirect
39. #18012665C22semidirect
40. #18012657C22semidirect
41. #18012656C22semidirect
42. H(5,𝔽2)×C151C22semidirect
43. AGL(1,ℤ/8)×C151C22semidirect
44. #127×C151C22semidirect
45. #104×C151C22semidirect
46. #100×C151C22semidirect
48. #14559005D4semidirect
62. #14559003D4semidirect
67. Q8×D151D4semidirect
80. D4×D151D4semidirect
194. C151#3021semidirect
204. #453D151semidirect
212. H(5,𝔽2)C2×D151semidirect
224. AGL(1,ℤ/8)C2×D151semidirect
229. #127C2×D151semidirect
238. #104C2×D151semidirect
241. #100C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530472
AGL(1,ℤ/8)×D1512
#237527752
#237523862
#237523832
#237523822
#237523802
#237523792
#237522272
#237521932
#453×C1512
#3021151