Finite Group Databasebeta

← #34078338

Subgroups — #34078338

Order 19328.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D4×D151C211yesmaximal, characteristic, complemented
2C2×#18012766C212yesmaximal, complemented
3#23752725C211yesmaximal, characteristic, complemented
4#23752362C211yesmaximal, characteristic, complemented
5#23752361C212yesmaximal, complemented
6#23752348C211yesmaximal, characteristic, complemented
7#23752347C212yesmaximal, complemented
8#23752346C211yesmaximal, characteristic, complemented
9#23752215C212yesmaximal, complemented
10#23752174C211yesmaximal, characteristic, complemented
11#450×C151C211yesmaximal, characteristic, complemented
12C24×D151—21no—
13C2×#14559005C2211yescharacteristic, complemented
14C22×D604C2211yescharacteristic
15C22×C22⋊D151—21no—
16C2×D4×D151—28no—
17C2×D4×D151—21no—
18C2×D4×D151—21no—
19C2×D4×D151C2212yes—
20C2×D4×D151C2212yes—
21C22×C4×D151C2211yescharacteristic
22#18012766—28no—
23C2×#14558991C2212yescomplemented
24C2×D1208C2212yescomplemented
25C2×#14558989C2212yescomplemented
26C2×#14558987C2212yescomplemented
27#18012724—21no—
28#18012715C2211yescharacteristic, complemented
29#18012714C2211yescharacteristic, complemented
30#18012713C2211yescharacteristic, complemented
31#18012709—21no—
32#18012701—21no—
33(C22⋊D151)⋊C4—21no—
34#18012697C2211yescharacteristic, complemented
35#18012661C2212yescomplemented
36#18012660C2212yescomplemented
37#18012659C2212yescomplemented
38C22⋊(C151⋊C8)C2211yescharacteristic, complemented
39(C2×D151)⋊C8C2212yescomplemented
40C22×D4×C151C2211yescharacteristic
41C2×SD16×C151C2212yescomplemented
42C22⋊Q8×C151C2211yescharacteristic, complemented
43#102×C151C2212yescomplemented
44C22⋊C8×C151C2211yescharacteristic, complemented
45C23×D151—28no—
46C23×D151—22no—
47C23×D151—22no—
48C23×D151—21no—
49C23×D151—44no—
50C23×D151—44no—
51C23×D151D411yescharacteristic
52#14559005—24no—
53C2×D604—24no—
54C2×D604—21no—
55C2×D604—21no—
56C2×D604—21no—
57C2×D604C2312yes—
58C2×D604C2311yescharacteristic, complemented
59C2×C22⋊D151—28no—
60C2×C22⋊D151—22no—
61C2×C22⋊D151—22no—
62C2×C22⋊D151—21no—
63D4×D151—28no—
64D4×D151—28no—
65D4×D151—24no—
66D4×D151—24no—
67D4×D151—24no—
68D4×D151D418yescomplemented
69C22×Dic151D411yescharacteristic
70C2×C4×D151—24no—
71C2×C4×D151—21no—
72C2×C4×D151D411yescharacteristic
73C2×C4×D151D411yescharacteristic
74C2×C4×D151C2311yescharacteristic
75C2×C4×D151C2311yescharacteristic
76#14558991—24no—
77D1208—24no—
78#14558989—24no—
79#14558987—24no—
80C2×C151⋊C8C2312yescomplemented
81C4⋊Dic151C2311yescharacteristic, complemented
82Dic151⋊C4—21no—
83(C2×D151)⋊C4—21no—
84(C2×D151)⋊C4—21no—
85(C2×D151)⋊C4—21no—
86(C2×D151)⋊C4—21no—
87C4×Dic151—21no—
88C24×C151—21no—
89C2×Q8×C151C2311yescharacteristic, complemented
90C2×D4×C151—24no—
91C2×D4×C151—21no—
92C2×D4×C151C2312yes—
93C22×C4×C151C2311yescharacteristic
94SD16×C151—24no—
95C2×C8×C151C2312yescomplemented
96C4⋊C4×C151—21no—
97C4⋊C4×C151C2311yescharacteristic, complemented
98C22⋊C4×C151—21no—
99C22×D151—22no—
100C22×D151—22no—
101C22×D151—22no—
102C22×D151—28no—
103C22×D151—24no—
104C22×D151—22no—
105C22×D151—28no—
106C22×D151—21no—
107C22×D151—432no—
108C22×D151—44no—
109C22×D151—44no—
110C22×D151—44no—
111C22×D151—42no—
112C22×D151—42no—
113C22×D151—44no—
114C22×D151C2×D411yescharacteristic
115C22×D151C2×D411yescharacteristic
116D604—24no—
117D604—22no—
118D604—22no—
119D604—42no—
120D604—42no—
121D604C2×D414yes—
122C22⋊D151—28no—
123C22⋊D151—28no—
124C22⋊D151—42no—
125C2×Dic151—22no—
126C2×Dic151—21no—
127C2×Dic151C2×D411yescharacteristic
128C2×Dic151C2×D411yescharacteristic
129C4×D151—22no—
130C4×D151—22no—
131C4×D151—22no—
132C4×D151—42no—
133C4×D151C2×D414yes—
134C151⋊C8—22no—
135C23×C151—24no—
136C23×C151—22no—
137C23×C151—42no—
138C23×C151—42no—
139C23×C151C2×D411yescharacteristic
140Q8×C151—22no—
141D4×C151—24no—
142D4×C151—22no—
143D4×C151C2×D414yes—
144C2×C4×C151—22no—
145C2×C4×C151—21no—
146C2×C4×C151—21no—
147C2×C4×C151C2×D411yescharacteristic
148C2×C4×C151C2411yescharacteristic
149C8×C151—22no—
150C22×C151—21no—
151C22×C151—21no—
152C22×C151—24no—
153C22×C151—48no—
154C22×C151—42no—
155C22×C151—42no—
156C22×C151AGL(1,ℤ/8)12yescomplemented
157C22×C151C22×D411yescharacteristic
158C2×D151—28no—
159C2×D151—21no—
160C2×D151—21no—
161C2×D151—21no—
162C2×D151—21no—
163C2×D151—24no—
164C2×D151—416no—
165C2×D151—416no—
166C2×D151—48no—
167C2×D151—42no—
168C2×D151—42no—
169C2×D151—41no—
170C2×D151—41no—
171C2×D151—41no—
172C2×D151#12012yes—
173C2×D151AGL(1,ℤ/8)12yescomplemented
174C4×C151—21no—
175C4×C151—41no—
176C4×C151—41no—
177C4×C151C22×D412yes—
178Dic151—21no—
179Dic151—41no—
180Dic151#12012yes—
181C2×C151—22no—
182C2×C151—44no—
183C2×C151C2×#12011yescharacteristic
184C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
186D151—22no—
187D151—22no—
188D151—44no—
189D151—81no—
190C151#300311yescharacteristic, complemented, Sylow
191#3003—1511nomaximal, Sylow
192C23×D4—1511no—
193C2×AGL(1,ℤ/8)—1512no—
194#525—1511no—
195#450—1512no—
196#450—1511no—
197#450D15111yescharacteristic, complemented
198#447—1511no—
199#447—1512no—
200#447—1511no—
201#418—1512no—
202#407—1511no—
203C25—3021no—
204C2×#41—1511no—
205C22×D4—3028no—
206C22×D4—3021no—
207C22×D4—3021no—
208C22×D4—3021no—
209C22×D4—1512no—
210C22×D4—1512no—
211C22×D4—1511no—
212C22×D4C2×D15111yescharacteristic
213C23×C4—1511no—
214AGL(1,ℤ/8)—3028no—
215C2×SD16—1512no—
216C2×SD16C2×D15112yescomplemented
217C2×D8—1512no—
218C2×D8—1512no—
219C2×M16—1512no—
220#123—3021no—
221C22⋊Q8—1511no—
222C22⋊Q8C2×D15111yescharacteristic, complemented
223#121—1511no—
224#121—1511no—
225#120—3021no—
226C4×D4—3021no—
227C4×D4—3021no—
228#117—1511no—
229#102—1512no—
230#102—1512no—
231#102—1512no—
232#102C2×D15112yescomplemented
233C22⋊C8—1511no—
234C22⋊C8—1512no—
235C22⋊C8C2×D15111yescharacteristic, complemented
236C24—21no—
237C24—3028no—
238C24—3022no—
239C24—3022no—
240C24—3021no—
241C24—6044no—
242C24—6044no—
243C24—1511no—
244#41—3024no—
245C2×Q8C22×D15111yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—3024no—
249C2×D4—3021no—
250C2×D4—3021no—
251C2×D4—3028no—
252C2×D4—3028no—
253C2×D4—3024no—
254C2×D4—3028no—
255C2×D4—3024no—
256C2×D4—3021no—
257C2×D4—3024no—
258C2×D4—3022no—
259C2×D4—3022no—
260C2×D4—3021no—
261C2×D4—1518no—
262C2×D4—1511no—
263C2×D4—1512no—
264C2×D4C22×D15112yes—
265C22×C4—3024no—
266C22×C4—3021no—
267C22×C4—1511no—
268C22×C4—1511no—
269C22×C4—1511no—
270C22×C4—1511no—
271C22×C4—1511no—
272C22×C4C22×D15111yescharacteristic
273SD16—24no—
274SD16—3024no—
275D8—3024no—
276D8—3024no—
277M16—3024no—
278C2×C8—1512no—
279C2×C8C22×D15112yescomplemented
280C4⋊C4—21no—
281C4⋊C4—3021no—
282C4⋊C4—1511no—
283C4⋊C4C22×D15111yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—3021no—
286C22⋊C4—3021no—
287C22⋊C4—3021no—
288C22⋊C4—3021no—
289C42—3021no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—3022no—
295C23—3022no—
296C23—3024no—
297C23—3028no—
298C23—3028no—
299C23—3022no—
300C23—3022no—
301C23—3021no—
302C23—6042no—
303C23—60432no—
304C23—6044no—
305C23—6044no—
306C23—6044no—
307C23—6044no—
308C23—6042no—
309C23—1511no—
310C23—1511no—
311C23D4×D15111yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—3024no—
316D4—3028no—
317D4—3028no—
318D4—3022no—
319D4—3022no—
320D4—6042no—
321D4—6042no—
322D4—6042no—
323D4—1514no—
324D4D4×D15114yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3022no—
329C2×C4—3022no—
330C2×C4—3022no—
331C2×C4—3022no—
332C2×C4—3021no—
333C2×C4—6042no—
334C2×C4—1511no—
335C2×C4—1511no—
336C2×C4—1514no—
337C2×C4D4×D15111yescharacteristic
338C2×C4C23×D15111yescharacteristic
339C8—22no—
340C8—3022no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—3028no—
348C22—3021no—
349C22—3021no—
350C22—3021no—
351C22—3021no—
352C22—3024no—
353C22—6048no—
354C22—60416no—
355C22—60416no—
356C22—6042no—
357C22—6042no—
358C22—6041no—
359C22—6041no—
360C22—6041no—
361C22—1512no—
362C22—1512no—
363C22#1801276612yescomplemented
364C22C2×D4×D15111yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3021no—
369C4—6041no—
370C4—1512no—
371C4C2×D4×D15112yes—
372C2—22no—
373C2—44no—
374C2—3022no—
375C2—3022no—
376C2—6044no—
377C2—12081no—
378C2#120×D15111yescharacteristic
379C2AGL(1,ℤ/8)×D15111yescharacteristic
380C2C2×#1801276611yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
hypercenter#450D151
hypocenterC151#3003
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D151C2semidirect
2. C2×#18012766C2semidirect
3. #23752725C2semidirect
4. #23752362C2semidirect
5. #23752361C2semidirect
6. #23752348C2semidirect
7. #23752347C2semidirect
8. #23752346C2semidirect
9. #23752215C2semidirect
10. #23752174C2semidirect
11. #450×C151C2semidirect
13. C2×#14559005C22semidirect
23. C2×#14558991C22semidirect
24. C2×D1208C22semidirect
25. C2×#14558989C22semidirect
26. C2×#14558987C22semidirect
28. #18012715C22semidirect
29. #18012714C22semidirect
30. #18012713C22semidirect
34. #18012697C22semidirect
35. #18012661C22semidirect
36. #18012660C22semidirect
37. #18012659C22semidirect
38. C22⋊(C151⋊C8)C22semidirect
39. (C2×D151)⋊C8C22semidirect
41. C2×SD16×C151C22semidirect
42. C22⋊Q8×C151C22semidirect
43. #102×C151C22semidirect
44. C22⋊C8×C151C22semidirect
58. C2×D604C23semidirect
68. D4×D151D4semidirect
80. C2×C151⋊C8C23semidirect
81. C4⋊Dic151C23semidirect
89. C2×Q8×C151C23semidirect
95. C2×C8×C151C23semidirect
97. C4⋊C4×C151C23semidirect
156. C22×C151AGL(1,ℤ/8)semidirect
173. C2×D151AGL(1,ℤ/8)semidirect
190. C151#3003semidirect
197. #450D151semidirect
216. C2×SD16C2×D151semidirect
222. C22⋊Q8C2×D151semidirect
232. #102C2×D151semidirect
235. C22⋊C8C2×D151semidirect
245. C2×Q8C22×D151semidirect
279. C2×C8C22×D151semidirect
283. C4⋊C4C22×D151semidirect
363. C22#18012766semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1512
C2×#180127662
#237527252
#237523622
#237523612
#237523482
#237523472
#237523462
#237522152
#237521742
#450×C1512
#3003151