Finite Group Databasebeta

← #34078585

Subgroups — #34078585

Order 19328.

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

normal subgroups only

This group has 584 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
1C2×#18012763C212yesmaximal, complemented
2#127×D151C211yesmaximal, characteristic, complemented
3C4×#14559003C211yesmaximal, characteristic, complemented
4#23752413C211yesmaximal, characteristic, complemented
5#23752411C212yesmaximal, complemented
6#23752409C211yesmaximal, characteristic, complemented
7#23752407C211yesmaximal, characteristic, complemented
8#23752405C212yesmaximal, complemented
9#23752234C211yesmaximal, characteristic, complemented
10#23752215C212yesmaximal, complemented
11#459×C151C211yesmaximal, characteristic, complemented
12C2×#14559003C2211yescharacteristic
13C2×D4×D151—21no—
14C2×D4×D151—21no—
15C2×D4×D151C2212yes—
16#18012763—28no—
17C2×#14558993C2212yescomplemented
18C2×#14558992C2212yescomplemented
19C2×#14558989C2212yescomplemented
20C2×#14558987C2212yes—
21#18012736C2211yescharacteristic
22#18012735—21no—
23#18012735—21no—
24#18012735C2212yes—
25C4×Dic302C2211yescharacteristic, complemented
26#18012704C2211yescharacteristic, complemented
27D4×Dic151C2211yescharacteristic, complemented
28C4×C22⋊D151—21no—
29(C22⋊D151)⋊C4—21no—
30#18012697C2211yescharacteristic
31#18012695—21no—
32C2×C4×Dic151—21no—
33C42×D151C2211yescharacteristic
34C4⋊(C151⋊C8)C2211yescharacteristic, complemented
35Dic151⋊C8C2212yescomplemented
36#18012661C2212yescomplemented
37#18012660C2212yescomplemented
38#18012659C2212yescomplemented
39C2×D8×C151C2212yescomplemented
40#127×C151C2211yescharacteristic
41C4×D4×C151C2211yescharacteristic, complemented
42C4⋊C8×C151C2211yescharacteristic, complemented
43#102×C151C2212yescomplemented
44C23×D151—22no—
45#14559003—22no—
46#14559003D414yescomplemented
47C2×Dic302C2311yescharacteristic
48C2×D604—21no—
49C2×D604—21no—
50C2×D604C2312yes—
51C2×C22⋊D151—21no—
52C2×C22⋊D151—22no—
53C2×C22⋊D151—22no—
54C2×C22⋊D151—22no—
55D4×D151—24no—
56D4×D151—24no—
57D4×D151—44no—
58D4×D151—44no—
59C22×Dic151—21no—
60C22×Dic151—21no—
61C2×C4×D151D411yescharacteristic
62C2×C4×D151D411yescharacteristic
63C2×C4×D151C2311yescharacteristic
64#14558993—24no—
65#14558992—24no—
66#14558989—24no—
67#14558987—24no—
68C2×C151⋊C8C2312yes—
69C4⋊Dic151C2311yescharacteristic
70Dic151⋊C4—21no—
71Dic151⋊C4—21no—
72C22⋊Dic151—21no—
73(C2×D151)⋊C4—21no—
74C4×Dic151—21no—
75C4×Dic151—21no—
76C4×Dic151—21no—
77C4×Dic151D411yescharacteristic
78C4×Dic151D411yescharacteristic
79C4×Dic151C2311yescharacteristic
80C2×D4×C151—21no—
81C2×D4×C151—21no—
82C2×D4×C151C2312yes—
83C2×D4×C151C2311yescharacteristic
84C22×C4×C151—21no—
85D8×C151—24no—
86C2×C8×C151C2312yes—
87C4⋊C4×C151C2311yescharacteristic
88C22⋊C4×C151—21no—
89C42×C151C2311yescharacteristic
90C22×D151—22no—
91C22×D151—44no—
92C22×D151—44no—
93C22×D151—44no—
94C22×D151C2×D411yescharacteristic
95Dic302—21no—
96Dic302C2×D412yescomplemented
97D604—22no—
98D604—22no—
99D604—42no—
100D604—42no—
101C22⋊D151—24no—
102C22⋊D151—44no—
103C22⋊D151—44no—
104C22⋊D151—44no—
105C2×Dic151—21no—
106C2×Dic151—24no—
107C2×Dic151—21no—
108C2×Dic151—42no—
109C2×Dic151C2×D411yescharacteristic
110C2×Dic151C2×D411yescharacteristic
111C2×Dic151C2×D411yescharacteristic
112C4×D151—22no—
113C4×D151—22no—
114C4×D151C2×D412yes—
115C4×D151#4112yes—
116C151⋊C8—22no—
117C23×C151—21no—
118C23×C151—22no—
119D4×C151—22no—
120D4×C151—21no—
121D4×C151—22no—
122D4×C151—42no—
123D4×C151—42no—
124D4×C151C2×D412yescomplemented
125C2×C4×C151—21no—
126C2×C4×C151—42no—
127C2×C4×C151C2×D411yescharacteristic
128C2×C4×C151C2×D411yescharacteristic
129C2×C4×C151C2411yescharacteristic
130C8×C151—22no—
131C22×C151—22no—
132C22×C151—41no—
133C22×C151—41no—
134C22×C151—42no—
135C22×C151—42no—
136C22×C151—42no—
137C22×C151C22×D411yescharacteristic
138C2×D151—21no—
139C2×D151—21no—
140C2×D151—42no—
141C2×D151—42no—
142C2×D151—42no—
143C2×D151—88no—
144C2×D151#12112yes—
145C4×C151—21no—
146C4×C151—41no—
147C4×C151AGL(1,ℤ/8)12yescomplemented
148C4×C151C22×D411yescharacteristic
149C4×C151C2×#4111yescharacteristic
150Dic151—22no—
151Dic151—21no—
152Dic151—41no—
153Dic151#12112yes—
154Dic151AGL(1,ℤ/8)12yescomplemented
155C2×C151—42no—
156C2×C151—82no—
157C2×C151C2×#12111yescharacteristic
158C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
160D151—22no—
161D151—82no—
162C151#303811yescharacteristic, complemented, Sylow
163#3038—1511nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1512no—
165C2×#127—1511no—
166C4×#41—1511no—
167#460—1511no—
168#460—1511no—
169#460—1512no—
170#459—1512no—
171#459—1511no—
172#459D15111yescharacteristic, complemented
173#423—1511no—
174#418—1512no—
175C2×#41—1511no—
176C22×D4—3021no—
177C22×D4—3021no—
178C22×D4—1512no—
179AGL(1,ℤ/8)—3028no—
180C2×SD16—1512no—
181C2×SD16—1512no—
182C2×D8—1512no—
183C2×D8C2×D15112yescomplemented
184C2×M16—1512no—
185#127—3021no—
186#127—3021no—
187#127—1512no—
188#127—1511no—
189#127C2×D15111yescharacteristic
190C4×Q8—1511no—
191C4×Q8—1511no—
192C4×D4—3021no—
193C4×D4—3021no—
194C4×D4—1511no—
195C4×D4C2×D15111yescharacteristic, complemented
196#117—3021no—
197#117—1511no—
198C2×C42—3021no—
199C2×C42—1511no—
200C4⋊C8—1512no—
201C4⋊C8—1511no—
202C4⋊C8C2×D15111yescharacteristic, complemented
203#102—1512no—
204#102—1512no—
205#102—1512no—
206#102C2×D15112yescomplemented
207C24—3022no—
208#41—3022no—
209#41—1514no—
210C2×Q8—1511no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—3021no—
214C2×D4—3024no—
215C2×D4—3024no—
216C2×D4—3022no—
217C2×D4—3022no—
218C2×D4—3022no—
219C2×D4—3021no—
220C2×D4—3021no—
221C2×D4—6044no—
222C2×D4—6044no—
223C2×D4—1512no—
224C2×D4C22×D15112yes—
225C2×D4C22×D15111yescharacteristic
226C22×C4—21no—
227C22×C4—3021no—
228C22×C4—3021no—
229C22×C4—1511no—
230C22×C4—1511no—
231C22×C4—1511no—
232SD16—3024no—
233SD16—3024no—
234D8—24no—
235D8—3024no—
236M16—3024no—
237C2×C8—1512no—
238C2×C8C22×D15112yes—
239C4⋊C4—3021no—
240C4⋊C4—3021no—
241C4⋊C4—1511no—
242C4⋊C4C22×D15111yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—3021no—
245C22⋊C4—3021no—
246C42—3021no—
247C42—3021no—
248C42—3021no—
249C42—1511no—
250C42—1511no—
251C42—1511no—
252C42C22×D15111yescharacteristic
253C23—21no—
254C23—22no—
255C23—3022no—
256C23—6044no—
257C23—6044no—
258C23—6044no—
259C23—1511no—
260Q8—3021no—
261Q8—1512no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—3024no—
268D4—3022no—
269D4—3022no—
270D4—6042no—
271D4—6044no—
272D4—6044no—
273D4—6044no—
274D4—6042no—
275D4D4×D15112yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—3022no—
279C2×C4—3021no—
280C2×C4—3024no—
281C2×C4—3022no—
282C2×C4—3021no—
283C2×C4—6042no—
284C2×C4—1511no—
285C2×C4—1511no—
286C2×C4—1511no—
287C2×C4—1512no—
288C2×C4—1512no—
289C2×C4D4×D15111yescharacteristic
290C2×C4D4×D15111yescharacteristic
291C2×C4C23×D15111yescharacteristic
292C8—22no—
293C8—3022no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—3021no—
301C22—3021no—
302C22—6042no—
303C22—6042no—
304C22—6042no—
305C22—1512no—
306C22—12088no—
307C22C2×D4×D15111yescharacteristic
308C4—21no—
309C4—41no—
310C4—3021no—
311C4—3022no—
312C4—6041no—
313C4—1512no—
314C4—1512no—
315C4#1801276312yescomplemented
316C4C2×D4×D15111yescharacteristic
317C4#41×D15111yescharacteristic
318C2—42no—
319C2—82no—
320C2—3022no—
321C2—12082no—
322C2#121×D15111yescharacteristic
323C2AGL(1,ℤ/8)×D15111yescharacteristic
324C2C2×#1801276311yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#459×C151C2
socleC22×C151C22×D4
hypercenter#459D151
hypocenterC151#3038
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#18012763C2semidirect
2. #127×D151C2semidirect
3. C4×#14559003C2semidirect
4. #23752413C2semidirect
5. #23752411C2semidirect
6. #23752409C2semidirect
7. #23752407C2semidirect
8. #23752405C2semidirect
9. #23752234C2semidirect
10. #23752215C2semidirect
11. #459×C151C2semidirect
17. C2×#14558993C22semidirect
18. C2×#14558992C22semidirect
19. C2×#14558989C22semidirect
25. C4×Dic302C22semidirect
26. #18012704C22semidirect
27. D4×Dic151C22semidirect
34. C4⋊(C151⋊C8)C22semidirect
35. Dic151⋊C8C22semidirect
36. #18012661C22semidirect
37. #18012660C22semidirect
38. #18012659C22semidirect
39. C2×D8×C151C22semidirect
41. C4×D4×C151C22semidirect
42. C4⋊C8×C151C22semidirect
43. #102×C151C22semidirect
46. #14559003D4semidirect
96. Dic302C2×D4semidirect
124. D4×C151C2×D4semidirect
147. C4×C151AGL(1,ℤ/8)semidirect
154. Dic151AGL(1,ℤ/8)semidirect
162. C151#3038semidirect
172. #459D151semidirect
183. C2×D8C2×D151semidirect
195. C4×D4C2×D151semidirect
202. C4⋊C8C2×D151semidirect
206. #102C2×D151semidirect
275. D4D4×D151semidirect
315. C4#18012763semidirect

Maximal subgroups

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

subgroupindex
C2×#180127632
#127×D1512
C4×#145590032
#237524132
#237524112
#237524092
#237524072
#237524052
#237522342
#237522152
#459×C1512
#3038151