Finite Group Databasebeta

← #34080908

Subgroups — #34080908

Order 19328.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#23753017C211yesmaximal, characteristic, complemented
2#23753016C212yesmaximal, complemented
3#23753013C211yesmaximal, characteristic, complemented
4#23752587C211yesmaximal, characteristic, complemented
5#23752483C211yesmaximal, characteristic, complemented
6#23752481C212yesmaximal, complemented
7#23752382C212yesmaximal, complemented
8#23752378C212yesmaximal, complemented
9#23752328C211yesmaximal, characteristic, complemented
10#23752327C211yesmaximal, characteristic, complemented
11#472×C151C211yesmaximal, characteristic, complemented
12#18012785C2212yescomplemented
13#18012784C2212yescomplemented
14#18012767—22no—
15#18012767C2212yescomplemented
16#18012767C2212yescomplemented
17#18012766—22no—
18#18012765C2212yescomplemented
19#18012763—22no—
20#18012763—22no—
21#18012761—21no—
22#18012761C2211yescharacteristic
23#18012760—22no—
24#18012759C2211yescharacteristic
25#18012758—21no—
26C2×D1208—21no—
27C2×D1208C2211yescharacteristic, complemented
28C2×#14558989C2212yescomplemented
29D8×D151—21no—
30D8×D151—22no—
31#18012747C2212yescomplemented
32#18012745C2211yescharacteristic, complemented
33#18012735C2212yescomplemented
34#18012674C2211yescharacteristic, complemented
35#18012666C2211yescharacteristic, complemented
36#18012665C2211yescharacteristic, complemented
37#18012665C2212yescomplemented
38#18012655C2212yescomplemented
39C8×Dic151C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C151C2212yescomplemented
41C2×D8×C151C2211yescharacteristic, complemented
42#131×C151C2211yescharacteristic, complemented
43#108×C151C2211yescharacteristic, complemented
44#100×C151C2212yescomplemented
45#14559005—22no—
46#14559004—22no—
47#14559004C2311yescharacteristic
48#14559004C2312yes—
49#14559003—22no—
50#14559003—22no—
51C2×D604—22no—
52C2×D604C2312yes—
53C2×C22⋊D151—42no—
54C2×C22⋊D151—42no—
55D4×D151—22no—
56D4×D151—22no—
57D4×D151—22no—
58D4×D151—42no—
59Dic604D411yescharacteristic, complemented
60#14558994D411yescharacteristic, complemented
61#14558993—22no—
62#14558993—22no—
63#14558993D412yescomplemented
64#14558992—22no—
65#14558992D411yescharacteristic, complemented
66#14558991D411yescharacteristic, complemented
67D1208—22no—
68D1208—22no—
69D1208—21no—
70D1208—21no—
71D1208D411yescharacteristic, complemented
72#14558989—22no—
73#14558989—22no—
74#14558989—22no—
75#14558989D411yescharacteristic, complemented
76#14558988C2311yescharacteristic
77#14558987—22no—
78#14558987—21no—
79C2×C151⋊C8C2311yescharacteristic
80C8×D151—22no—
81C8×D151—21no—
82C4×Dic151C2311yescharacteristic
83#41×C151C2311yescharacteristic
84C2×D4×C151C2312yes—
85SD16×C151—22no—
86D8×C151—22no—
87D8×C151—21no—
88D8×C151—21no—
89M16×C151—21no—
90M16×C151C2311yescharacteristic
91M16×C151C2312yes—
92C2×C8×C151—21no—
93C2×C8×C151C2311yescharacteristic
94C22×D151—22no—
95C22×D151—42no—
96C22×D151—42no—
97Dic302C2×D412yes—
98Dic302C2×D411yescharacteristic
99D604—22no—
100D604—22no—
101D604—42no—
102D604C2×D412yes—
103D604C2×D411yescharacteristic
104C22⋊D151—22no—
105C22⋊D151—21no—
106C22⋊D151—42no—
107C22⋊D151—42no—
108C22⋊D151—42no—
109C22⋊D151—84no—
110C2×Dic151—21no—
111C2×Dic151—42no—
112C4×D151—21no—
113C4×D151—22no—
114C4×D151—42no—
115C151⋊C8—21no—
116C151⋊C8C2×D411yescharacteristic
117C151⋊C8C2×D411yescharacteristic
118C23×C151—22no—
119Q8×C151C2×D411yescharacteristic
120D4×C151—21no—
121D4×C151—22no—
122D4×C151—22no—
123D4×C151C2×D411yescharacteristic
124C2×C4×C151—21no—
125C2×C4×C151C2411yescharacteristic
126C8×C151—22no—
127C8×C151—21no—
128C8×C151C2×D411yescharacteristic
129C8×C151C2×D411yescharacteristic
130C22×C151—21no—
131C22×C151—42no—
132C22×C151—82no—
133C22×C151H(5,𝔽2)11yescharacteristic
134C2×D151—21no—
135C2×D151—22no—
136C2×D151—42no—
137C2×D151—82no—
138C2×D151—162no—
139C2×D151—162no—
140C4×C151—21no—
141C4×C151C22×D411yescharacteristic
142C4×C151C22×D411yescharacteristic
143Dic151—22no—
144Dic151—22no—
145Dic151—21no—
146C2×C151—21no—
147C2×C151—41no—
148C2×C151—82no—
149C2×C151D4211yescharacteristic
150D151—41no—
151D151—42no—
152D151—82no—
153C151#329211yescharacteristic, complemented, Sylow
154#3292—1511nomaximal, Sylow
155#576—1512no—
156#576—1511no—
157#576—1511no—
158#493—1511no—
159#472—1511no—
160#472—1512no—
161#472D15111yescharacteristic, complemented
162#453—1512no—
163#453—1512no—
164#443—1511no—
165#443—1511no—
166H(5,𝔽2)—1512no—
167H(5,𝔽2)—1512no—
168AGL(1,ℤ/8)—3022no—
169AGL(1,ℤ/8)—3022no—
170AGL(1,ℤ/8)—3022no—
171AGL(1,ℤ/8)—3022no—
172AGL(1,ℤ/8)—1512no—
173AGL(1,ℤ/8)—1512no—
174AGL(1,ℤ/8)—1512no—
175AGL(1,ℤ/8)C2×D15112yescomplemented
176#135—3021no—
177#135—3021no—
178#135—3022no—
179#135—1511no—
180#135—1511no—
181C2×D8—3021no—
182C2×D8—3022no—
183C2×D8—3021no—
184C2×D8—1512no—
185C2×D8—1511no—
186C2×D8C2×D15111yescharacteristic, complemented
187#131—1512no—
188#131—1511no—
189#131C2×D15111yescharacteristic, complemented
190#127—1512no—
191#108—1511no—
192#108C2×D15111yescharacteristic, complemented
193#104—1511no—
194#104—1512no—
195#104—1511no—
196#100—1512no—
197#100C2×D15112yescomplemented
198C4×C8—1511no—
199#41—3022no—
200#41—3022no—
201#41—3022no—
202#41—3022no—
203#41—1511no—
204#41—1512no—
205#41C22×D15111yescharacteristic
206C2×D4—3022no—
207C2×D4—3022no—
208C2×D4—3022no—
209C2×D4—3022no—
210C2×D4—6042no—
211C2×D4—6042no—
212C2×D4—6042no—
213C2×D4—1512no—
214C2×D4C22×D15112yes—
215Q16—1511no—
216Q16—1511no—
217SD16—22no—
218SD16—3022no—
219SD16—3022no—
220SD16—3022no—
221SD16—1511no—
222SD16—1511no—
223SD16—1512no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3022no—
228D8—3022no—
229D8—3022no—
230D8—3022no—
231D8—3022no—
232D8—3021no—
233D8—3021no—
234D8—1511no—
235D8—1511no—
236M16—21no—
237M16—3022no—
238M16—3021no—
239M16—1511no—
240M16C22×D15112yes—
241M16C22×D15111yescharacteristic
242C2×C8—21no—
243C2×C8—3022no—
244C2×C8—3021no—
245C2×C8—1511no—
246C2×C8C22×D15111yescharacteristic
247C42—1511no—
248C23—22no—
249C23—3022no—
250C23—6042no—
251C23—6042no—
252Q8—1511no—
253Q8—1512no—
254Q8D4×D15111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—3022no—
259D4—3021no—
260D4—3022no—
261D4—3022no—
262D4—6042no—
263D4—6042no—
264D4—6042no—
265D4—6042no—
266D4—1511no—
267D4—1512no—
268D4—12084no—
269D4D4×D15111yescharacteristic
270C2×C4—21no—
271C2×C4—3021no—
272C2×C4—3022no—
273C2×C4—3021no—
274C2×C4—6042no—
275C2×C4—6042no—
276C2×C4C23×D15111yescharacteristic
277C8—21no—
278C8—22no—
279C8—3021no—
280C8—1511no—
281C8—1511no—
282C8D4×D15111yescharacteristic
283C8D4×D15111yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—3022no—
288C22—3021no—
289C22—6042no—
290C22—24162no—
291C22—24162no—
292C22—12082no—
293C22#1801278511yescharacteristic
294C4—21no—
295C4—3022no—
296C4—3022no—
297C4—3021no—
298C4C2×D4×D15111yescharacteristic
299C4C2×D4×D15111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—6042no—
304C2—6041no—
305C2—12082no—
306C2#2375284611yescharacteristic
subgroupG/H
centerC2#23752846
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#472×C151C2
socleC2×C151D42
hypercenter#472D151
hypocenterC151#3292
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #23753017C2semidirect
2. #23753016C2semidirect
3. #23753013C2semidirect
4. #23752587C2semidirect
5. #23752483C2semidirect
6. #23752481C2semidirect
7. #23752382C2semidirect
8. #23752378C2semidirect
9. #23752328C2semidirect
10. #23752327C2semidirect
11. #472×C151C2semidirect
12. #18012785C22semidirect
13. #18012784C22semidirect
15. #18012767C22semidirect
16. #18012767C22semidirect
18. #18012765C22semidirect
27. C2×D1208C22semidirect
28. C2×#14558989C22semidirect
31. #18012747C22semidirect
32. #18012745C22semidirect
33. #18012735C22semidirect
34. #18012674C22semidirect
35. #18012666C22semidirect
36. #18012665C22semidirect
37. #18012665C22semidirect
38. #18012655C22semidirect
39. C8×Dic151C22semidirect
40. AGL(1,ℤ/8)×C151C22semidirect
41. C2×D8×C151C22semidirect
42. #131×C151C22semidirect
43. #108×C151C22semidirect
44. #100×C151C22semidirect
59. Dic604D4semidirect
60. #14558994D4semidirect
63. #14558993D4semidirect
65. #14558992D4semidirect
66. #14558991D4semidirect
71. D1208D4semidirect
75. #14558989D4semidirect
153. C151#3292semidirect
161. #472D151semidirect
175. AGL(1,ℤ/8)C2×D151semidirect
186. C2×D8C2×D151semidirect
189. #131C2×D151semidirect
192. #108C2×D151semidirect
197. #100C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530172
#237530162
#237530132
#237525872
#237524832
#237524812
#237523822
#237523782
#237523282
#237523272
#472×C1512
#3292151