Finite Group Databasebeta

← #34079558

Subgroups — #34079558

Order 19328.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#18012767C211yesmaximal, characteristic, complemented
2C2×#18012765C211yesmaximal, characteristic, complemented
3C2×#18012764C211yesmaximal, characteristic, complemented
4#23752758C211yesmaximal, characteristic, complemented
5C2×#18012735C211yesmaximal, characteristic, complemented
6#23752600C212yesmaximal, complemented
7#23752598C212yesmaximal, complemented
8#23752587C212yesmaximal, complemented
9#23752583C212yesmaximal, complemented
10M16×Dic151C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
12C2×#14559004C2211yescharacteristic
13C22×D604C2211yescharacteristic
14C22×C22⋊D151—22no—
15#18012767—24no—
16#18012765—24no—
17#18012764—24no—
18C2×#14558993C2212yescomplemented
19C2×#14558992C2212yescomplemented
20C2×#14558991C2212yescomplemented
21C2×D1208C2212yescomplemented
22C2×#14558989C2212yescomplemented
23C2×#14558989C2212yescomplemented
24C2×#14558988C2211yescharacteristic
25#18012737C2211yescharacteristic, complemented
26#18012735—22no—
27#18012735C2212yes—
28#18012735C2212yes—
29#18012735C2211yescharacteristic, complemented
30#18012728C2211yescharacteristic, complemented
31#18012726C2211yescharacteristic, complemented
32#18012711—22no—
33C4×C22⋊D151—22no—
34C2×C4×Dic151C2211yescharacteristic
35#18012649C2212yescomplemented
36C8×Dic151C2212yescomplemented
37C2×#41×C151C2211yescharacteristic
38C22×D4×C151C2211yescharacteristic
39AGL(1,ℤ/8)×C151—24no—
40C2×SD16×C151C2212yescomplemented
41C2×D8×C151C2212yescomplemented
42C2×M16×C151C2211yescharacteristic
43C23×D151—21no—
44#14559004—22no—
45#14559004—22no—
46C2×Dic302C2311yescharacteristic
47C2×D604—22no—
48C2×D604—22no—
49C2×D604—21no—
50C2×D604C2311yescharacteristic
51C2×D604C2312yes—
52C2×C22⋊D151—22no—
53C2×C22⋊D151—24no—
54C2×C22⋊D151—24no—
55C2×C22⋊D151—21no—
56C2×C22⋊D151—48no—
57C22×Dic151D412yes—
58C2×C4×D151—21no—
59#14558993—24no—
60#14558992—24no—
61#14558991—24no—
62D1208—24no—
63#14558989—24no—
64#14558989—24no—
65#14558988D414yescomplemented
66C2×C151⋊C8C2312yes—
67Dic151⋊C4—22no—
68C22⋊Dic151—22no—
69(C2×D151)⋊C4—22no—
70C4×Dic151D412yes—
71C4×Dic151C2311yescharacteristic
72C4×Dic151C2311yescharacteristic
73C24×C151—21no—
74#41×C151—22no—
75#41×C151—22no—
76C2×Q8×C151C2311yescharacteristic
77C2×D4×C151—21no—
78C2×D4×C151—22no—
79C2×D4×C151—22no—
80C2×D4×C151—21no—
81C2×D4×C151C2311yescharacteristic
82C2×D4×C151C2312yes—
83C22×C4×C151—21no—
84C22×C4×C151C2311yescharacteristic
85SD16×C151—24no—
86D8×C151—24no—
87M16×C151D414yescomplemented
88C2×C8×C151C2312yes—
89C22×D151—21no—
90C22×D151—22no—
91C22×D151—44no—
92C22×D151—42no—
93Dic302—21no—
94Dic302—21no—
95D604—21no—
96D604—22no—
97D604—21no—
98D604—22no—
99D604—42no—
100C22⋊D151—48no—
101C22⋊D151—42no—
102C22⋊D151—44no—
103C22⋊D151—48no—
104C2×Dic151—24no—
105C2×Dic151—21no—
106C2×Dic151C2×D412yes—
107C2×Dic151C2×D412yes—
108C4×D151—42no—
109C151⋊C8C2×D414yescomplemented
110C23×C151—22no—
111C23×C151—21no—
112C23×C151—44no—
113C23×C151—42no—
114C23×C151C2×D411yescharacteristic
115Q8×C151—21no—
116Q8×C151—21no—
117D4×C151—21no—
118D4×C151—21no—
119D4×C151—22no—
120D4×C151—22no—
121D4×C151—42no—
122D4×C151—42no—
123C2×C4×C151—21no—
124C2×C4×C151—42no—
125C2×C4×C151C2×D411yescharacteristic
126C2×C4×C151C2×D412yes—
127C2×C4×C151C2×D412yes—
128C2×C4×C151C2411yescharacteristic
129C8×C151C2×D414yescomplemented
130C22×C151—22no—
131C22×C151—41no—
132C22×C151—42no—
133C22×C151—44no—
134C22×C151—42no—
135C22×C151—84no—
136C22×C151#12712yes—
137C22×C151C22×D411yescharacteristic
138C2×D151—41no—
139C2×D151—42no—
140C2×D151—44no—
141C2×D151—42no—
142C2×D151—84no—
143C4×C151—41no—
144C4×C151#12712yes—
145C4×C151C22×D411yescharacteristic
146C4×C151C22×D411yescharacteristic
147Dic151—22no—
148Dic151—41no—
149Dic151AGL(1,ℤ/8)14yescomplemented
150C2×C151—22no—
151C2×C151—81no—
152C2×C151—82no—
153C2×C151C2×#12711yescharacteristic
154C2×C151C2×AGL(1,ℤ/8)12yes—
155D151—81no—
156D151—82no—
157C151#315111yescharacteristic, complemented, Sylow
158#3151—1511nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1511no—
160C2×AGL(1,ℤ/8)—1511no—
161C2×AGL(1,ℤ/8)—1511no—
162C2×AGL(1,ℤ/8)D15111yescharacteristic, complemented
163#532—1511no—
164C2×#127—1511no—
165#496—1512no—
166#496—1512no—
167#493—1512no—
168#492—1512no—
169C4×M16—1511no—
170C2×#41—1511no—
171C2×#41C2×D15111yescharacteristic
172C22×D4—3022no—
173C22×D4—1511no—
174C22×D4C2×D15111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3024no—
177AGL(1,ℤ/8)—3024no—
178AGL(1,ℤ/8)—3024no—
179C2×SD16—1512no—
180C2×SD16—1512no—
181C2×SD16—1512no—
182C2×SD16C2×D15112yescomplemented
183C2×D8—1512no—
184C2×D8—1512no—
185C2×D8—1512no—
186C2×D8C2×D15112yescomplemented
187C2×M16—1511no—
188C2×M16C2×D15111yescharacteristic
189C4⋊Q8—1511no—
190#127—3022no—
191#127—1512no—
192#127—1512no—
193#127—1511no—
194#124—1511no—
195#124—1511no—
196#121—3022no—
197C4×D4—3022no—
198C2×C42—1511no—
199#97—1512no—
200C4×C8—1512no—
201C24—21no—
202C24—3021no—
203#41—22no—
204#41—22no—
205#41—3022no—
206#41—3022no—
207C2×Q8—1511no—
208C2×Q8C22×D15111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3021no—
214C2×D4—3022no—
215C2×D4—3024no—
216C2×D4—3024no—
217C2×D4—3022no—
218C2×D4—3021no—
219C2×D4—3022no—
220C2×D4—6048no—
221C2×D4—1511no—
222C2×D4—1512no—
223C2×D4C22×D15111yescharacteristic
224C2×D4C22×D15112yes—
225C22×C4—21no—
226C22×C4—3021no—
227C22×C4—1512no—
228C22×C4C22×D15111yescharacteristic
229SD16—24no—
230SD16—3024no—
231SD16—3024no—
232SD16—3024no—
233D8—24no—
234D8—3024no—
235D8—3024no—
236D8—3024no—
237M16—1514no—
238M16C22⋊D15114yescomplemented
239C2×C8—1512no—
240C2×C8C22×D15112yes—
241C4⋊C4—3022no—
242C22⋊C4—3022no—
243C22⋊C4—3022no—
244C42—1512no—
245C42—1511no—
246C42—1511no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3021no—
252C23—3022no—
253C23—6044no—
254C23—6042no—
255C23D4×D15111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3021no—
259Q8—3021no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3021no—
267D4—3022no—
268D4—3021no—
269D4—3022no—
270D4—6048no—
271D4—6042no—
272D4—6048no—
273D4—6044no—
274D4—6042no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3024no—
278C2×C4—3021no—
279C2×C4—6042no—
280C2×C4—1512no—
281C2×C4—1512no—
282C2×C4D4×D15111yescharacteristic
283C2×C4D4×D15112yes—
284C2×C4C2×C22⋊D15112yes—
285C2×C4C23×D15111yescharacteristic
286C8—1514no—
287C8C2×C22⋊D15114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—6044no—
295C22—6042no—
296C22—6041no—
297C22—6042no—
298C22—12084no—
299C22#1801273512yes—
300C22C2×D4×D15111yescharacteristic
301C4—41no—
302C4—3022no—
303C4—6041no—
304C4—1514no—
305C4#1801273512yes—
306C4C2×D4×D15111yescharacteristic
307C4C22×C22⋊D15111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—12082no—
312C2—12081no—
313C2C2×#1801273511yescharacteristic
314C2AGL(1,ℤ/8)×D15112yes—
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×AGL(1,ℤ/8)×C151C2
socleC22×C151C22×D4
hypercenterC2×AGL(1,ℤ/8)D151
hypocenterC151#3151
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18012767C2semidirect
2. C2×#18012765C2semidirect
3. C2×#18012764C2semidirect
4. #23752758C2semidirect
5. C2×#18012735C2semidirect
6. #23752600C2semidirect
7. #23752598C2semidirect
8. #23752587C2semidirect
9. #23752583C2semidirect
10. M16×Dic151C2semidirect
11. C2×AGL(1,ℤ/8)×C151C2semidirect
18. C2×#14558993C22semidirect
19. C2×#14558992C22semidirect
20. C2×#14558991C22semidirect
21. C2×D1208C22semidirect
22. C2×#14558989C22semidirect
23. C2×#14558989C22semidirect
25. #18012737C22semidirect
29. #18012735C22semidirect
30. #18012728C22semidirect
31. #18012726C22semidirect
35. #18012649C22semidirect
36. C8×Dic151C22semidirect
40. C2×SD16×C151C22semidirect
41. C2×D8×C151C22semidirect
65. #14558988D4semidirect
87. M16×C151D4semidirect
109. C151⋊C8C2×D4semidirect
129. C8×C151C2×D4semidirect
149. Dic151AGL(1,ℤ/8)semidirect
157. C151#3151semidirect
162. C2×AGL(1,ℤ/8)D151semidirect
182. C2×SD16C2×D151semidirect
186. C2×D8C2×D151semidirect
238. M16C22⋊D151semidirect
287. C8C2×C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127672
C2×#180127652
C2×#180127642
#237527582
C2×#180127352
#237526002
#237525982
#237525872
#237525832
M16×Dic1512
C2×AGL(1,ℤ/8)×C1512
#3151151