Finite Group Databasebeta

← #34078480

Subgroups — #34078480

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#23753049C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D151C212yesmaximal, complemented
3#23752780C211yesmaximal, characteristic, complemented
4#23752387C211yesmaximal, characteristic, complemented
5#23752385C211yesmaximal, characteristic, complemented
6#23752384C211yesmaximal, characteristic, complemented
7#23752381C212yesmaximal, complemented
8#23752378C212yesmaximal, complemented
9#23752227C212yesmaximal, complemented
10#23752195C211yesmaximal, characteristic, complemented
11#454×C151C211yesmaximal, characteristic, complemented
12#18012787—22no—
13#18012787C2211yescharacteristic, complemented
14#18012785—22no—
15#18012785—22no—
16#18012785C2212yescomplemented
17C2×#14559005—22no—
18C2×#14559005C2211yescharacteristic
19#41×D151—22no—
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#18012736C2211yescharacteristic, complemented
32#18012735C2211yescharacteristic, complemented
33#18012728C2211yescharacteristic, complemented
34#18012712—22no—
35#18012709—22no—
36#18012696C2211yescharacteristic, complemented
37#18012667C2212yescomplemented
38#18012666C2212yescomplemented
39#18012665C2212yescomplemented
40#18012658C2211yescharacteristic, complemented
41#18012655C2212yescomplemented
422−1+4×C151C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C151C2212yescomplemented
44#124×C151C2211yescharacteristic, complemented
45#104×C151C2212yescomplemented
46#101×C151C2211yescharacteristic, complemented
47C23×D151—21no—
48#14559005—22no—
49#14559005—24no—
50#14559005—22no—
51#14559005—22no—
52#14559005—22no—
53#14559005—22no—
54#14559005D412yescomplemented
55#14559004—24no—
56#14559004—24no—
57#14559004—22no—
58#14559004—22no—
59#14559004—22no—
60#14559004C2312yes—
61#14559003—22no—
62#14559003—22no—
63#14559003D412yescomplemented
64Q8×D151—22no—
65Q8×D151—22no—
66Q8×D151D412yescomplemented
67C2×D604—24no—
68C2×D604—22no—
69C2×D604—22no—
70C2×D604D412yes—
71C2×D604C2311yescharacteristic
72C2×D604C2311yescharacteristic
73C2×C22⋊D151—22no—
74C2×C22⋊D151—21no—
75D4×D151—24no—
76D4×D151—22no—
77D4×D151—24no—
78D4×D151—22no—
79D4×D151—22no—
80D4×D151—22no—
81D4×D151—22no—
82D4×D151D412yescomplemented
83C2×C4×D151—24no—
84C2×C4×D151—22no—
85C2×C4×D151D412yes—
86C2×C4×D151C2311yescharacteristic
87#14558993—22no—
88#14558992—22no—
89#14558991—22no—
90D1208—22no—
91#14558989—22no—
92#14558989—22no—
93#14558988C2312yes—
94#14558987—22no—
95C8×D151—22no—
96Dic151⋊C4—21no—
97(C2×D151)⋊C4—21no—
98(C2×D151)⋊C4—22no—
99C4×Dic151C2311yescharacteristic
100#41×C151—22no—
101#41×C151—22no—
102#41×C151C2312yes—
103C2×Q8×C151—22no—
104C2×Q8×C151C2311yescharacteristic
105C2×D4×C151C2311yescharacteristic
106SD16×C151—22no—
107D8×C151—22no—
108M16×C151C2312yes—
109C22⋊C4×C151—22no—
110C42×C151C2311yescharacteristic
111C22×D151—24no—
112C22×D151—22no—
113C22×D151—21no—
114C22×D151—42no—
115C22×D151—82no—
116C22×D151C2×D412yes—
117C22×D151C2×D411yescharacteristic
118Dic302—22no—
119Dic302—22no—
120Dic302C2×D412yes—
121D604—24no—
122D604—22no—
123D604—22no—
124D604—24no—
125D604—22no—
126D604—22no—
127D604—82no—
128D604C2×D412yes—
129C22⋊D151—24no—
130C22⋊D151—22no—
131C22⋊D151—22no—
132C22⋊D151—22no—
133C22⋊D151—41no—
134C22⋊D151—41no—
135C22⋊D151—82no—
136C2×Dic151—22no—
137C2×Dic151—21no—
138C2×Dic151C2×D411yescharacteristic
139C4×D151—24no—
140C4×D151—24no—
141C4×D151—22no—
142C4×D151—22no—
143C4×D151—22no—
144C4×D151—22no—
145C4×D151—22no—
146C4×D151C2×D412yes—
147C4×D151C2×D412yes—
148C151⋊C8—22no—
149C23×C151—21no—
150Q8×C151—22no—
151Q8×C151—22no—
152Q8×C151C2×D412yes—
153D4×C151—22no—
154D4×C151—22no—
155D4×C151—22no—
156D4×C151C2×D412yes—
157C2×C4×C151—24no—
158C2×C4×C151—22no—
159C2×C4×C151—21no—
160C2×C4×C151C2×D412yes—
161C2×C4×C151C2411yescharacteristic
162C8×C151—22no—
163C22×C151—22no—
164C22×C151—42no—
165C22×C151—41no—
166C22×C151C22×D411yescharacteristic
167C2×D151—22no—
168C2×D151—22no—
169C2×D151—41no—
170C2×D151—42no—
171C2×D151—42no—
172C2×D151—41no—
173C2×D151—82no—
174C2×D151—84no—
175C2×D151—84no—
176C2×D151#12012yes—
177C4×C151—22no—
178C4×C151—22no—
179C4×C151—41no—
180C4×C151C22×D412yes—
181Dic151—22no—
182Dic151—41no—
183Dic151#12011yescharacteristic
184Dic151#12011yescharacteristic
185C2×C151—21no—
186C2×C151—42no—
187C2×C151—81no—
188C2×C151C2×#12011yescharacteristic
189D151—22no—
190D151—42no—
191D151—42no—
192D151—81no—
193C151#302111yescharacteristic, complemented, Sylow
194#3021—1511nomaximal, Sylow
195#585—1511no—
196C2×AGL(1,ℤ/8)—1512no—
197#535—1511no—
198#454—1511no—
199#454—1511no—
200#454—1511no—
201#454D15111yescharacteristic, complemented
202#453—1512no—
203#453—1512no—
204#421—1512no—
205#413—1511no—
2062−1+4—3022no—
2072−1+4—1511no—
2082−1+4C2×D15111yescharacteristic, complemented
209H(5,𝔽2)—3022no—
210H(5,𝔽2)—3022no—
211H(5,𝔽2)—1512no—
212C2×#41—3022no—
213C2×#41—3022no—
214C2×#41—3022no—
215C2×#41—1512no—
216C2×#41—1511no—
217C22×D4—1511no—
218AGL(1,ℤ/8)—3022no—
219AGL(1,ℤ/8)—3022no—
220AGL(1,ℤ/8)—1512no—
221AGL(1,ℤ/8)—1512no—
222AGL(1,ℤ/8)—1512no—
223AGL(1,ℤ/8)C2×D15112yescomplemented
224C2×SD16—3022no—
225C2×D8—3022no—
226C2×M16—1512no—
227#127—1511no—
228#127—1511no—
229#124—1511no—
230#124C2×D15111yescharacteristic, complemented
231#121—3022no—
232#120—3022no—
233#117—1511no—
234#104—1512no—
235#104—1512no—
236#104—1512no—
237#104C2×D15112yescomplemented
238#101—1511no—
239#101C2×D15111yescharacteristic, complemented
240#100—1512no—
241C24—3021no—
242#41—22no—
243#41—22no—
244#41—3024no—
245#41—3024no—
246#41—3022no—
247#41—3022no—
248#41—3022no—
249#41—3022no—
250#41—3024no—
251#41—3022no—
252#41—3022no—
253#41—3022no—
254#41—3022no—
255#41—3022no—
256#41—3022no—
257#41—1512no—
258#41—1512no—
259#41—1512no—
260#41C22×D15112yes—
261C2×Q8—22no—
262C2×Q8—3022no—
263C2×Q8—3022no—
264C2×Q8—1512no—
265C2×Q8C22×D15111yescharacteristic
266C2×D4—3024no—
267C2×D4—3022no—
268C2×D4—3022no—
269C2×D4—3024no—
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×D4—1511no—
282C2×D4C22×D15111yescharacteristic
283C22×C4—3024no—
284C22×C4—3022no—
285C22×C4—1512no—
286C22×C4—1511no—
287SD16—22no—
288SD16—3022no—
289SD16—3022no—
290SD16—3022no—
291D8—22no—
292D8—3022no—
293D8—3022no—
294D8—3022no—
295M16—3022no—
296M16—1512no—
297M16C22×D15112yes—
298C2×C8—3022no—
299C4⋊C4—3021no—
300C22⋊C4—22no—
301C22⋊C4—3021no—
302C22⋊C4—3022no—
303C42—1511no—
304C42C22×D15111yescharacteristic
305C23—21no—
306C23—3024no—
307C23—3022no—
308C23—3021no—
309C23—6042no—
310C23—1512no—
311C23—1511no—
312C23—12082no—
313Q8—22no—
314Q8—22no—
315Q8—3022no—
316Q8—3022no—
317Q8—1512no—
318Q8D4×D15112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—3024no—
323D4—3022no—
324D4—3022no—
325D4—3024no—
326D4—3024no—
327D4—3022no—
328D4—3022no—
329D4—3022no—
330D4—3022no—
331D4—3022no—
332D4—6041no—
333D4—6041no—
334D4—1512no—
335D4—12082no—
336D4—12082no—
337D4D4×D15112yes—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—3024no—
342C2×C4—3024no—
343C2×C4—3022no—
344C2×C4—3022no—
345C2×C4—3022no—
346C2×C4—3022no—
347C2×C4—3022no—
348C2×C4—3022no—
349C2×C4—3021no—
350C2×C4—1511no—
351C2×C4—1512no—
352C2×C4—1512no—
353C2×C4D4×D15112yes—
354C2×C4C23×D15111yescharacteristic
355C8—22no—
356C8—3022no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—3022no—
361C22—3022no—
362C22—6041no—
363C22—6042no—
364C22—6042no—
365C22—6041no—
366C22—1512no—
367C22—12082no—
368C22—12084no—
369C22—12084no—
370C22C2×D4×D15111yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—3022no—
375C4—6041no—
376C4—1511no—
377C4—1511no—
378C4C2×D4×D15112yes—
379C2—21no—
380C2—42no—
381C2—81no—
382C2—3022no—
383C2—6042no—
384C2—6042no—
385C2—12081no—
386C2#120×D15111yescharacteristic
subgroupG/H
centerC2#120×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#454×C151C2
socleC2×C151C2×#120
hypercenter#454D151
hypocenterC151#3021
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23753049C2semidirect
2. AGL(1,ℤ/8)×D151C2semidirect
3. #23752780C2semidirect
4. #23752387C2semidirect
5. #23752385C2semidirect
6. #23752384C2semidirect
7. #23752381C2semidirect
8. #23752378C2semidirect
9. #23752227C2semidirect
10. #23752195C2semidirect
11. #454×C151C2semidirect
13. #18012787C22semidirect
16. #18012785C22semidirect
22. C2×D4×D151C22semidirect
23. #18012767C22semidirect
25. #18012765C22semidirect
26. #18012764C22semidirect
30. M16×D151C22semidirect
31. #18012736C22semidirect
32. #18012735C22semidirect
33. #18012728C22semidirect
36. #18012696C22semidirect
37. #18012667C22semidirect
38. #18012666C22semidirect
39. #18012665C22semidirect
40. #18012658C22semidirect
41. #18012655C22semidirect
42. 2−1+4×C151C22semidirect
43. AGL(1,ℤ/8)×C151C22semidirect
44. #124×C151C22semidirect
45. #104×C151C22semidirect
46. #101×C151C22semidirect
54. #14559005D4semidirect
63. #14559003D4semidirect
66. Q8×D151D4semidirect
82. D4×D151D4semidirect
193. C151#3021semidirect
201. #454D151semidirect
208. 2−1+4C2×D151semidirect
223. AGL(1,ℤ/8)C2×D151semidirect
230. #124C2×D151semidirect
237. #104C2×D151semidirect
239. #101C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530492
AGL(1,ℤ/8)×D1512
#237527802
#237523872
#237523852
#237523842
#237523812
#237523782
#237522272
#237521952
#454×C1512
#3021151