Finite Group Databasebeta

← #454×D151 #34078443

Subgroups — #454×D151 #34078443

Order 19328.

728 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 135 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
12−1+4×D151C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D151C212yesmaximal, complemented
3#124×D151C211yesmaximal, characteristic, complemented
4#23752387C211yesmaximal, characteristic, complemented
5#23752386C212yesmaximal, complemented
6#23752385C211yesmaximal, characteristic, complemented
7#23752384C211yesmaximal, characteristic, complemented
8#23752383C212yesmaximal, complemented
9#104×D151C212yesmaximal, complemented
10#101×D151C211yesmaximal, characteristic, complemented
11#454×C151C211yesmaximal, characteristic, complemented
12#18012787—22no—
13#18012787C2211yescharacteristic, complemented
14#18012786—22no—
15#18012786—22no—
16#18012786C2212yescomplemented
17#41×D151—22no—
18#41×D151—22no—
19#41×D151C2212yes—
20C2×Q8×D151—22no—
21C2×Q8×D151C2211yescharacteristic
22C2×D4×D151C2211yescharacteristic
23#18012767C2212yescomplemented
24#18012766—22no—
25#18012765C2212yescomplemented
26#18012764C2212yescomplemented
27#18012763—22no—
28SD16×D151—22no—
29D8×D151—22no—
30M16×D151C2212yes—
31#18012728C2211yescharacteristic, complemented
32#18012727C2211yescharacteristic, complemented
33#18012726C2211yescharacteristic, complemented
34#18012725—22no—
35C22⋊C4×D151—22no—
36C42×D151C2211yescharacteristic
37#18012667C2212yescomplemented
38#18012666C2212yescomplemented
39#18012665C2212yescomplemented
40#18012658C2211yescharacteristic, complemented
41#18012657C2212yescomplemented
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—22no—
50#14559005D412yes—
51#14559004—24no—
52#14559004—24no—
53#14559004—22no—
54#14559004—22no—
55#14559004—22no—
56#14559004C2312yes—
57#14559003—24no—
58#14559003—22no—
59#14559003—24no—
60#14559003—22no—
61#14559003—22no—
62#14559003D412yes—
63C2×Dic302—24no—
64C2×Dic302—22no—
65C2×Dic302D412yes—
66C2×Dic302C2311yescharacteristic
67Q8×D151—22no—
68Q8×D151—24no—
69Q8×D151—22no—
70Q8×D151—22no—
71Q8×D151—22no—
72Q8×D151—22no—
73Q8×D151D412yes—
74C2×D604C2311yescharacteristic
75C2×C22⋊D151—21no—
76D4×D151—22no—
77D4×D151—22no—
78D4×D151—22no—
79D4×D151—22no—
80D4×D151D412yes—
81C2×C4×D151—24no—
82C2×C4×D151—22no—
83C2×C4×D151—21no—
84C2×C4×D151D412yes—
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—
95C22⋊Dic151—22no—
96(C2×D151)⋊C4—22no—
97(C2×D151)⋊C4—22no—
98C4×Dic151—21no—
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—22no—
112C22×D151—21no—
113C22×D151—44no—
114C22×D151—41no—
115C22×D151—41no—
116C22×D151C2×D411yescharacteristic
117Dic302—22no—
118Dic302—24no—
119Dic302—22no—
120Dic302—24no—
121Dic302—22no—
122Dic302C2×D412yes—
123D604—22no—
124D604—22no—
125D604—22no—
126D604C2×D412yes—
127C22⋊D151—24no—
128C22⋊D151—22no—
129C22⋊D151—22no—
130C22⋊D151—22no—
131C22⋊D151—42no—
132C2×Dic151—24no—
133C2×Dic151—22no—
134C2×Dic151—21no—
135C2×Dic151C2×D412yes—
136C2×Dic151C2×D411yescharacteristic
137C4×D151—24no—
138C4×D151—24no—
139C4×D151—22no—
140C4×D151—22no—
141C4×D151—22no—
142C4×D151—22no—
143C4×D151—22no—
144C4×D151—42no—
145C4×D151C2×D412yes—
146C4×D151C2×D412yes—
147C151⋊C8—22no—
148C23×C151—21no—
149Q8×C151—22no—
150Q8×C151—22no—
151Q8×C151C2×D412yes—
152D4×C151—22no—
153D4×C151—22no—
154D4×C151—22no—
155D4×C151C2×D412yes—
156C2×C4×C151—24no—
157C2×C4×C151—22no—
158C2×C4×C151—21no—
159C2×C4×C151C2×D412yes—
160C2×C4×C151C2411yescharacteristic
161C8×C151—22no—
162C22×C151—22no—
163C22×C151—42no—
164C22×C151—41no—
165C22×C151C22×D411yescharacteristic
166C2×D151—22no—
167C2×D151—22no—
168C2×D151—44no—
169C2×D151—42no—
170C2×D151—41no—
171C2×D151—82no—
172C2×D151—82no—
173C2×D151#12011yescharacteristic
174C2×D151#12011yescharacteristic
175C4×C151—22no—
176C4×C151—22no—
177C4×C151—41no—
178C4×C151C22×D412yes—
179Dic151—22no—
180Dic151—22no—
181Dic151—41no—
182Dic151#12012yes—
183C2×C151—21no—
184C2×C151—42no—
185C2×C151—81no—
186C2×C151C2×#12011yescharacteristic
187D151—21no—
188D151—42no—
189D151—81no—
190D151#45412yesdirect product factor
191C151C2×#45411yescharacteristic, complemented, Sylow
192C2×#454—1511nomaximal, Sylow
193C2×2−1+4—1511no—
194C2×AGL(1,ℤ/8)—1512no—
195C2×#124—1511no—
196#454—1511no—
197#454—1512no—
198#454—1511no—
199#454—1512no—
200#454—1511no—
201#454D15111yescharacteristic, direct product factor
202C2×#104—1512no—
203C2×#101—1511no—
2042−1+4—3022no—
2052−1+4—3022no—
2062−1+4—3022no—
2072−1+4—1512no—
2082−1+4—1511no—
2092−1+4C2×D15111yescharacteristic, complemented
210C2×#41—3022no—
211C2×#41—3022no—
212C2×#41—1512no—
213C22×Q8—3022no—
214C22×Q8—1511no—
215C22×D4—1511no—
216AGL(1,ℤ/8)—3022no—
217AGL(1,ℤ/8)—3022no—
218AGL(1,ℤ/8)—1512no—
219AGL(1,ℤ/8)—1512no—
220AGL(1,ℤ/8)—1512no—
221AGL(1,ℤ/8)C2×D15112yescomplemented
222C2×SD16—3022no—
223C2×D8—3022no—
224C2×M16—1512no—
225#124—3022no—
226#124—1511no—
227#124—1511no—
228#124—1511no—
229#124C2×D15111yescharacteristic, complemented
230C2×C22⋊C4—3022no—
231C2×C42—1511no—
232#104—1512no—
233#104—1512no—
234#104—1512no—
235#104C2×D15112yescomplemented
236#101—1511no—
237#101—1512no—
238#101C2×D15111yescharacteristic, complemented
239C24—3021no—
240#41—22no—
241#41—22no—
242#41—3024no—
243#41—3024no—
244#41—3022no—
245#41—3022no—
246#41—3024no—
247#41—3022no—
248#41—3022no—
249#41—3022no—
250#41—3022no—
251#41—3024no—
252#41—3022no—
253#41—3022no—
254#41—1512no—
255#41—1512no—
256#41—1512no—
257#41C22×D15112yes—
258C2×Q8—22no—
259C2×Q8—3024no—
260C2×Q8—3022no—
261C2×Q8—3022no—
262C2×Q8—3024no—
263C2×Q8—3022no—
264C2×Q8—3022no—
265C2×Q8—3022no—
266C2×Q8—3022no—
267C2×Q8—1512no—
268C2×Q8—1512no—
269C2×Q8—1511no—
270C2×Q8C22×D15111yescharacteristic
271C2×D4—3022no—
272C2×D4—3022no—
273C2×D4—3022no—
274C2×D4—3022no—
275C2×D4—3021no—
276C2×D4—1512no—
277C2×D4—1511no—
278C2×D4C22×D15111yescharacteristic
279C22×C4—3024no—
280C22×C4—3022no—
281C22×C4—3021no—
282C22×C4—1512no—
283C22×C4—1511no—
284SD16—22no—
285SD16—3022no—
286SD16—3022no—
287SD16—3022no—
288D8—22no—
289D8—3022no—
290D8—3022no—
291D8—3022no—
292M16—3022no—
293M16—1512no—
294M16C22×D15112yes—
295C2×C8—3022no—
296C22⋊C4—22no—
297C22⋊C4—3022no—
298C22⋊C4—3022no—
299C22⋊C4—3022no—
300C42—3021no—
301C42—1511no—
302C42C22×D15111yescharacteristic
303C23—21no—
304C23—3022no—
305C23—3021no—
306C23—6044no—
307C23—6041no—
308C23—6041no—
309C23—1511no—
310Q8—22no—
311Q8—22no—
312Q8—3024no—
313Q8—3022no—
314Q8—3022no—
315Q8—3024no—
316Q8—3022no—
317Q8—1512no—
318Q8D4×D15112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—3022no—
323D4—3022no—
324D4—3024no—
325D4—3022no—
326D4—3022no—
327D4—3022no—
328D4—3022no—
329D4—6042no—
330D4—1512no—
331D4D4×D15112yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—3024no—
336C2×C4—3024no—
337C2×C4—3022no—
338C2×C4—3022no—
339C2×C4—3022no—
340C2×C4—3022no—
341C2×C4—3024no—
342C2×C4—3022no—
343C2×C4—3022no—
344C2×C4—3021no—
345C2×C4—6042no—
346C2×C4—1512no—
347C2×C4—1511no—
348C2×C4—1512no—
349C2×C4—1512no—
350C2×C4D4×D15112yes—
351C2×C4C23×D15111yescharacteristic
352C8—22no—
353C8—3022no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—3022no—
358C22—3022no—
359C22—6042no—
360C22—6044no—
361C22—6041no—
362C22—1511no—
363C22—1511no—
364C22—12082no—
365C22—12082no—
366C22C2×D4×D15111yescharacteristic
367C4—22no—
368C4—22no—
369C4—41no—
370C4—3022no—
371C4—3022no—
372C4—6041no—
373C4—1512no—
374C4C2×D4×D15112yes—
375C2—21no—
376C2—42no—
377C2—81no—
378C2—3021no—
379C2—6042no—
380C2—1512no—
381C2—12081no—
382C2#120×D15111yescharacteristic
subgroupG/H
centerC2#120×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#454×C151C2
socleC2×C151C2×#120
hypercenter#454D151
hypocenterC151C2×#454
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. 2−1+4×D151C2semidirect
2. AGL(1,ℤ/8)×D151C2semidirect
3. #124×D151C2semidirect
4. #23752387C2semidirect
5. #23752386C2semidirect
6. #23752385C2semidirect
7. #23752384C2semidirect
8. #23752383C2semidirect
9. #104×D151C2semidirect
10. #101×D151C2semidirect
11. #454×C151C2semidirect
13. #18012787C22semidirect
16. #18012786C22semidirect
23. #18012767C22semidirect
25. #18012765C22semidirect
26. #18012764C22semidirect
31. #18012728C22semidirect
32. #18012727C22semidirect
33. #18012726C22semidirect
37. #18012667C22semidirect
38. #18012666C22semidirect
39. #18012665C22semidirect
40. #18012658C22semidirect
41. #18012657C22semidirect
42. 2−1+4×C151C22semidirect
43. AGL(1,ℤ/8)×C151C22semidirect
44. #124×C151C22semidirect
45. #104×C151C22semidirect
46. #101×C151C22semidirect
190. D151#454direct (N × H)
191. C151C2×#454semidirect
201. #454D151direct (N × H)
209. 2−1+4C2×D151semidirect
221. AGL(1,ℤ/8)C2×D151semidirect
229. #124C2×D151semidirect
235. #104C2×D151semidirect
238. #101C2×D151semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D1512
AGL(1,ℤ/8)×D1512
#124×D1512
#237523872
#237523862
#237523852
#237523842
#237523832
#104×D1512
#101×D1512
#454×C1512
C2×#454151