Finite Group Databasebeta

← #34079559

Subgroups — #34079559

Order 19328.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#18012766C212yesmaximal, complemented
2C2×#18012763C212yesmaximal, complemented
3#23752761C211yesmaximal, characteristic, complemented
4#127×D151C211yesmaximal, characteristic, complemented
5#23752602C211yesmaximal, characteristic, complemented
6#23752601C211yesmaximal, characteristic, complemented
7#23752599C211yesmaximal, characteristic, complemented
8#23752587C212yesmaximal, complemented
9#23752583C212yesmaximal, complemented
10#23752164C211yesmaximal, characteristic, complemented
11#496×C151C211yesmaximal, characteristic, complemented
12C2×#14559005C2211yescharacteristic
13C2×#14559003C2211yescharacteristic
14C2×D4×D151—22no—
15C2×D4×D151C2212yes—
16#18012766—28no—
17#18012763—28no—
18C2×#14558993C2212yescomplemented
19C2×#14558992C2212yescomplemented
20C2×#14558991C2212yescomplemented
21C2×D1208C2212yescomplemented
22C2×#14558989C2212yescomplemented
23C2×#14558989C2212yescomplemented
24C2×#14558987C2212yes—
25#18012737C2211yescharacteristic, complemented
26#18012736C2211yescharacteristic
27#18012735—22no—
28#18012735C2212yes—
29#18012735C2211yescharacteristic, complemented
30#18012727C2211yescharacteristic, complemented
31#18012712—22no—
32(C22⋊D151)⋊C4—22no—
33C42×D151C2211yescharacteristic
34#18012650C2211yescharacteristic, complemented
35C8×Dic151C2212yescomplemented
36C2×SD16×C151C2212yescomplemented
37C2×D8×C151C2212yescomplemented
38#127×C151C2211yescharacteristic
39#124×C151C2211yescharacteristic, complemented
40#97×C151C2211yescharacteristic, complemented
41C23×D151—22no—
42#14559005—24no—
43#14559003—24no—
44C2×Dic302C2311yescharacteristic
45C2×D604—21no—
46C2×D604—22no—
47C2×D604C2311yescharacteristic
48C2×D604C2312yes—
49C2×C22⋊D151—21no—
50C2×C22⋊D151—22no—
51C2×C22⋊D151—22no—
52C2×C22⋊D151—24no—
53D4×D151—24no—
54D4×D151—24no—
55D4×D151—48no—
56C22×Dic151—21no—
57C2×C4×D151—21no—
58C2×C4×D151D412yes—
59C2×C4×D151C2311yescharacteristic
60#14558993—24no—
61#14558992—24no—
62#14558991—24no—
63D1208—24no—
64#14558989—24no—
65#14558989—24no—
66#14558987D418yescomplemented
67C2×C151⋊C8C2312yes—
68Dic151⋊C4—22no—
69(C2×D151)⋊C4—22no—
70C4×Dic151D412yes—
71C4×Dic151C2311yescharacteristic
72C2×Q8×C151C2311yescharacteristic
73C2×D4×C151—22no—
74C2×D4×C151C2312yes—
75C2×D4×C151C2311yescharacteristic
76SD16×C151—24no—
77D8×C151—24no—
78C2×C8×C151C2312yes—
79C22⋊C4×C151—22no—
80C42×C151C2311yescharacteristic
81C22×D151—21no—
82C22×D151—22no—
83C22×D151—48no—
84C22×D151—44no—
85C22×D151C2×D411yescharacteristic
86Dic302—22no—
87D604—22no—
88D604—22no—
89D604—22no—
90D604—44no—
91D604—42no—
92C22⋊D151—42no—
93C22⋊D151—44no—
94C22⋊D151—48no—
95C22⋊D151—44no—
96C2×Dic151—21no—
97C2×Dic151—42no—
98C2×Dic151C2×D411yescharacteristic
99C2×Dic151C2×D412yes—
100C4×D151—24no—
101C4×D151—42no—
102C4×D151C2×D414yes—
103C151⋊C8C2×D414yescomplemented
104C23×C151—21no—
105C23×C151—22no—
106Q8×C151—22no—
107D4×C151—22no—
108D4×C151—22no—
109D4×C151—22no—
110D4×C151—44no—
111C2×C4×C151—21no—
112C2×C4×C151C2×D412yes—
113C2×C4×C151C2411yescharacteristic
114C8×C151C2×D414yescomplemented
115C22×C151—41no—
116C22×C151—42no—
117C22×C151—44no—
118C22×C151—42no—
119C22×C151C22×D411yescharacteristic
120C2×D151—22no—
121C2×D151—41no—
122C2×D151—42no—
123C2×D151—44no—
124C2×D151—42no—
125C2×D151—88no—
126C2×D151#12712yes—
127C4×C151—22no—
128C4×C151—41no—
129C4×C151C22×D412yes—
130Dic151—41no—
131Dic151#12712yes—
132Dic151AGL(1,ℤ/8)14yescomplemented
133C2×C151—82no—
134C2×C151—81no—
135C2×C151C2×#12711yescharacteristic
136C2×C151C2×AGL(1,ℤ/8)12yes—
137D151—22no—
138D151—81no—
139D151—82no—
140C151#315111yescharacteristic, complemented, Sylow
141#3151—1511nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1512no—
143C2×AGL(1,ℤ/8)—1512no—
144#532—1511no—
145C2×#127—1511no—
146#496—1511no—
147#496—1511no—
148#496—1511no—
149#496D15111yescharacteristic, complemented
150#493—1512no—
151#492—1512no—
152C4×M16—1511no—
153C2×#41—1511no—
154C2×#41—1511no—
155C22×D4—3022no—
156C22×D4—1512no—
157AGL(1,ℤ/8)—3028no—
158AGL(1,ℤ/8)—3028no—
159C2×SD16—1512no—
160C2×SD16—1512no—
161C2×SD16—1512no—
162C2×SD16C2×D15112yescomplemented
163C2×D8—1512no—
164C2×D8—1512no—
165C2×D8—1512no—
166C2×D8C2×D15112yescomplemented
167C2×M16—1512no—
168C4⋊Q8—1511no—
169#127—3022no—
170#127—1512no—
171#127—1511no—
172#127—1511no—
173#127C2×D15111yescharacteristic
174#124—1511no—
175#124C2×D15111yescharacteristic, complemented
176#121—3022no—
177C4×D4—3022no—
178C2×C42—1511no—
179#97—1511no—
180#97C2×D15111yescharacteristic, complemented
181C4×C8—1512no—
182C24—3022no—
183#41—3024no—
184#41—3024no—
185C2×Q8—1511no—
186C2×Q8C22×D15111yescharacteristic
187C2×D4—22no—
188C2×D4—3021no—
189C2×D4—3021no—
190C2×D4—3024no—
191C2×D4—3024no—
192C2×D4—3022no—
193C2×D4—3022no—
194C2×D4—3024no—
195C2×D4—3022no—
196C2×D4—6048no—
197C2×D4—1511no—
198C2×D4—1512no—
199C2×D4C22×D15111yescharacteristic
200C2×D4C22×D15112yes—
201C22×C4—3021no—
202C22×C4—3021no—
203C22×C4—1512no—
204C22×C4—1511no—
205SD16—24no—
206SD16—3024no—
207SD16—3024no—
208SD16—3024no—
209D8—24no—
210D8—3024no—
211D8—3024no—
212D8—3024no—
213M16—1518no—
214C2×C8—1512no—
215C2×C8C22×D15112yes—
216C4⋊C4—3022no—
217C22⋊C4—22no—
218C22⋊C4—3022no—
219C42—1512no—
220C42—1511no—
221C42C22×D15111yescharacteristic
222C23—21no—
223C23—22no—
224C23—3021no—
225C23—3022no—
226C23—6048no—
227C23—6044no—
228C23—1511no—
229Q8—22no—
230Q8—3022no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3022no—
236D4—3022no—
237D4—3022no—
238D4—6044no—
239D4—6042no—
240D4—6042no—
241D4—6044no—
242D4—6048no—
243D4—6044no—
244C2×C4—21no—
245C2×C4—3024no—
246C2×C4—3021no—
247C2×C4—6042no—
248C2×C4—6042no—
249C2×C4—1512no—
250C2×C4—1511no—
251C2×C4—1514no—
252C2×C4D4×D15112yes—
253C2×C4C23×D15111yescharacteristic
254C8—1514no—
255C8D4×D15114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—3022no—
261C22—6044no—
262C22—6042no—
263C22—6041no—
264C22—6042no—
265C22—1512no—
266C22—12088no—
267C22C2×D4×D15111yescharacteristic
268C4—22no—
269C4—41no—
270C4—6041no—
271C4—1514no—
272C4—1512no—
273C4C2×D4×D15112yes—
274C2—82no—
275C2—81no—
276C2—3022no—
277C2—12082no—
278C2—12081no—
279C2#127×D15111yescharacteristic
280C2AGL(1,ℤ/8)×D15112yes—
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#496×C151C2
socleC22×C151C22×D4
hypercenter#496D151
hypocenterC151#3151
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C2×#18012763C2semidirect
3. #23752761C2semidirect
4. #127×D151C2semidirect
5. #23752602C2semidirect
6. #23752601C2semidirect
7. #23752599C2semidirect
8. #23752587C2semidirect
9. #23752583C2semidirect
10. #23752164C2semidirect
11. #496×C151C2semidirect
18. C2×#14558993C22semidirect
19. C2×#14558992C22semidirect
20. C2×#14558991C22semidirect
21. C2×D1208C22semidirect
22. C2×#14558989C22semidirect
23. C2×#14558989C22semidirect
25. #18012737C22semidirect
29. #18012735C22semidirect
30. #18012727C22semidirect
34. #18012650C22semidirect
35. C8×Dic151C22semidirect
36. C2×SD16×C151C22semidirect
37. C2×D8×C151C22semidirect
39. #124×C151C22semidirect
40. #97×C151C22semidirect
66. #14558987D4semidirect
103. C151⋊C8C2×D4semidirect
114. C8×C151C2×D4semidirect
132. Dic151AGL(1,ℤ/8)semidirect
140. C151#3151semidirect
149. #496D151semidirect
162. C2×SD16C2×D151semidirect
166. C2×D8C2×D151semidirect
175. #124C2×D151semidirect
180. #97C2×D151semidirect
255. C8D4×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C2×#180127632
#237527612
#127×D1512
#237526022
#237526012
#237525992
#237525872
#237525832
#237521642
#496×C1512
#3151151