Finite Group Databasebeta

← #34078413

Subgroups — #34078413

Order 19328.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18012784C211yesmaximal, characteristic, complemented
2C2×#18012759C211yesmaximal, characteristic, complemented
3#23752790C211yesmaximal, characteristic, complemented
4#23752371C212yesmaximal, complemented
5#23752361C212yesmaximal, complemented
6#23752354C212yesmaximal, complemented
7#23752344C212yesmaximal, complemented
8#23752214C211yesmaximal, characteristic, complemented
9C2×#18012660C211yesmaximal, characteristic, complemented
10#23752180C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
12#18012784—24no—
13#18012784—24no—
14C2×#14559004C2211yescharacteristic
15C2×#14559004C2211yescharacteristic
16C2×#14559003—21no—
17C2×#14559003C2212yes—
18C22×C22⋊D151—22no—
19C2×D4×D151—21no—
20C2×D4×D151C2212yes—
21#18012759—24no—
22C2×#14558994C2211yescharacteristic, complemented
23C2×#14558992C2211yescharacteristic, complemented
24C2×#14558991C2211yescharacteristic, complemented
25C2×#14558989C2211yescharacteristic, complemented
26C22×C151⋊C8C2211yescharacteristic, complemented
27#18012718—21no—
28#18012715C2212yescomplemented
29#18012713—21no—
30#18012711—22no—
31#18012710C2212yescomplemented
32C2×C4⋊Dic151C2211yescharacteristic, complemented
33#18012664C2212yescomplemented
34#18012661C2212yescomplemented
35#18012660C2212yescomplemented
36#18012660C2212yescomplemented
37(C2×D151)⋊C8C2212yescomplemented
38(C2×D151)⋊C8C2212yescomplemented
39C2×#41×C151C2211yescharacteristic, complemented
40C22×D4×C151C2211yescharacteristic
41AGL(1,ℤ/8)×C151—24no—
42C2×SD16×C151C2212yescomplemented
43C2×D8×C151C2212yescomplemented
44C2×M16×C151C2211yescharacteristic, complemented
45C23×D151—22no—
46#14559004—22no—
47#14559004—22no—
48#14559004—22no—
49#14559004D414yescomplemented
50#14559003—28no—
51#14559003—24no—
52#14559003—24no—
53C2×Dic302D411yescharacteristic
54C2×Dic302C2311yescharacteristic
55C2×D604D411yescharacteristic
56C2×D604C2311yescharacteristic
57C2×C22⋊D151—216no—
58C2×C22⋊D151—22no—
59C2×C22⋊D151—22no—
60C2×C22⋊D151—22no—
61C2×C22⋊D151—21no—
62C2×C22⋊D151D412yes—
63D4×D151—28no—
64D4×D151—24no—
65D4×D151—24no—
66C22×Dic151—22no—
67C22×Dic151—21no—
68C2×C4×D151—21no—
69C2×C4×D151C2312yes—
70#14558994—22no—
71#14558992—22no—
72#14558991—22no—
73#14558989—22no—
74C2×C151⋊C8—22no—
75C2×C151⋊C8C2311yescharacteristic
76C2×C151⋊C8C2311yescharacteristic
77C4⋊Dic151—21no—
78C4⋊Dic151C2312yes—
79Dic151⋊C4—22no—
80C22⋊Dic151—22no—
81(C2×D151)⋊C4—22no—
82C24×C151—21no—
83#41×C151—22no—
84#41×C151—22no—
85C2×Q8×C151C2311yescharacteristic
86C2×D4×C151—21no—
87C2×D4×C151—21no—
88C2×D4×C151—22no—
89C2×D4×C151D414yescomplemented
90C2×D4×C151C2311yescharacteristic
91C2×D4×C151C2312yes—
92C22×C4×C151—21no—
93C22×C4×C151C2311yescharacteristic
94SD16×C151—24no—
95D8×C151—24no—
96M16×C151—22no—
97C2×C8×C151C2312yes—
98C22×D151—21no—
99C22×D151—28no—
100C22×D151—82no—
101C22×D151—82no—
102C22×D151C2×D412yes—
103Dic302—22no—
104Dic302—21no—
105Dic302C2×D412yes—
106D604—22no—
107D604—21no—
108D604C2×D412yes—
109C22⋊D151—28no—
110C22⋊D151—28no—
111C22⋊D151—24no—
112C22⋊D151—24no—
113C22⋊D151—82no—
114C2×Dic151—28no—
115C2×Dic151—21no—
116C2×Dic151—22no—
117C2×Dic151—81no—
118C2×Dic151C2×D412yes—
119C4×D151—24no—
120C4×D151—22no—
121C4×D151—22no—
122C151⋊C8—21no—
123C151⋊C8—21no—
124C23×C151—22no—
125C23×C151—24no—
126C23×C151—21no—
127C23×C151—81no—
128C23×C151—81no—
129C23×C151C2×D411yescharacteristic
130Q8×C151—21no—
131Q8×C151—21no—
132D4×C151—24no—
133D4×C151—22no—
134D4×C151—21no—
135D4×C151—21no—
136D4×C151—42no—
137D4×C151C2×D414yes—
138C2×C4×C151—21no—
139C2×C4×C151—42no—
140C2×C4×C151C2×D411yescharacteristic
141C2×C4×C151C2×D412yes—
142C2×C4×C151C2×D412yes—
143C2×C4×C151C2411yescharacteristic
144C8×C151—22no—
145C22×C151—21no—
146C22×C151—21no—
147C22×C151—24no—
148C22×C151—42no—
149C22×C151—42no—
150C22×C151—41no—
151C22×C151—41no—
152C22×C151—41no—
153C22×C151—84no—
154C22×C151#12012yes—
155C22×C151C22×D411yescharacteristic
156C2×D151—22no—
157C2×D151—22no—
158C2×D151—22no—
159C2×D151—22no—
160C2×D151—41no—
161C2×D151—41no—
162C2×D151—88no—
163C4×C151—41no—
164C4×C151#12012yes—
165C4×C151C22×D411yescharacteristic
166C4×C151C22×D411yescharacteristic
167Dic151—22no—
168Dic151—22no—
169Dic151—42no—
170C2×C151—22no—
171C2×C151—44no—
172C2×C151—81no—
173C2×C151C2×#12011yescharacteristic
174C2×C151#57611yescharacteristic
175C2×C151#57711yescharacteristic
176D151—42no—
177D151—42no—
178C151#301211yescharacteristic, complemented, Sylow
179#3012—1511nomaximal, Sylow
180C2×H(5,𝔽2)—1511no—
181C2×AGL(1,ℤ/8)D15111yescharacteristic, complemented
182C2×#135—1511no—
183#537—1511no—
184#452—1512no—
185#450—1512no—
186#449—1512no—
187#447—1512no—
188#417—1511no—
189C2×#102—1511no—
190#408—1511no—
191H(5,𝔽2)—3024no—
192H(5,𝔽2)—3024no—
193C2×#41—3021no—
194C2×#41—1512no—
195C2×#41—1511no—
196C2×#41—1511no—
197C2×#41C2×D15111yescharacteristic, complemented
198C22×D4—3021no—
199C22×D4—3022no—
200C22×D4—1512no—
201C22×D4C2×D15111yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—3024no—
204C2×Q16—1511no—
205C2×SD16—1511no—
206C2×SD16—1511no—
207C2×SD16C2×D15112yescomplemented
208C2×D8—1511no—
209C2×D8C2×D15112yescomplemented
210C2×M16C2×D15111yescharacteristic, complemented
211C22×C8—1511no—
212C22⋊Q8—3021no—
213C22⋊Q8—1512no—
214#121—3021no—
215#121—3022no—
216#121—1512no—
217C2×C4⋊C4—1511no—
218#103—1512no—
219#102—1512no—
220#102—1512no—
221#102—1512no—
222C22⋊C8—1512no—
223C22⋊C8—1512no—
224C24—21no—
225C24—3022no—
226#41—22no—
227#41—22no—
228#41—3028no—
229#41—3024no—
230#41—3024no—
231#41—3022no—
232#41—3022no—
233#41—3022no—
234#41—1514no—
235C2×Q8—1511no—
236C2×Q8—1511no—
237C2×Q8C22×D15111yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—3028no—
242C2×D4—3024no—
243C2×D4—30216no—
244C2×D4—3022no—
245C2×D4—3021no—
246C2×D4—3024no—
247C2×D4—3022no—
248C2×D4—3022no—
249C2×D4—1512no—
250C2×D4—1511no—
251C2×D4—1511no—
252C2×D4C22⋊D15114yescomplemented
253C2×D4C22×D15111yescharacteristic
254C2×D4C22×D15112yes—
255C22×C4—21no—
256C22×C4—3022no—
257C22×C4—3021no—
258C22×C4—3021no—
259C22×C4—1512no—
260C22×C4C22×D15111yescharacteristic
261Q16—3022no—
262SD16—24no—
263SD16—3022no—
264SD16—3022no—
265D8—24no—
266D8—3022no—
267M16—22no—
268C2×C8—3022no—
269C2×C8—1511no—
270C2×C8—1511no—
271C2×C8C22×D15112yes—
272C4⋊C4—3022no—
273C4⋊C4—3021no—
274C4⋊C4—1512no—
275C22⋊C4—3022no—
276C22⋊C4—3022no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—3028no—
283C23—3021no—
284C23—1512no—
285C23—12082no—
286C23—12082no—
287C23D4×D15111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3022no—
291Q8—3021no—
292Q8—1512no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—3028no—
299D4—3028no—
300D4—3024no—
301D4—3024no—
302D4—3022no—
303D4—3021no—
304D4—1512no—
305D4—12082no—
306D4C2×C22⋊D15114yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3028no—
310C2×C4—3021no—
311C2×C4—3024no—
312C2×C4—3022no—
313C2×C4—3022no—
314C2×C4—3022no—
315C2×C4—1512no—
316C2×C4—12081no—
317C2×C4D4×D15111yescharacteristic
318C2×C4D4×D15112yes—
319C2×C4C2×C22⋊D15112yes—
320C2×C4C23×D15111yescharacteristic
321C8—22no—
322C8—3021no—
323C8—3021no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—3022no—
334C22—3022no—
335C22—3022no—
336C22—3022no—
337C22—6041no—
338C22—6041no—
339C22—12088no—
340C22#1801270712yes—
341C22C2×D4×D15111yescharacteristic
342C4—41no—
343C4—3022no—
344C4—3022no—
345C4—6042no—
346C4#1801270712yes—
347C4C2×D4×D15111yescharacteristic
348C4C22×C22⋊D15111yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—6042no—
353C2—6042no—
354C2C2×#1801270711yescharacteristic
355C2#2375301611yescharacteristic
356C2#2375302211yescharacteristic
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#3012
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×#18012784C2semidirect
2. C2×#18012759C2semidirect
3. #23752790C2semidirect
4. #23752371C2semidirect
5. #23752361C2semidirect
6. #23752354C2semidirect
7. #23752344C2semidirect
8. #23752214C2semidirect
9. C2×#18012660C2semidirect
10. #23752180C2semidirect
11. C2×AGL(1,ℤ/8)×C151C2semidirect
22. C2×#14558994C22semidirect
23. C2×#14558992C22semidirect
24. C2×#14558991C22semidirect
25. C2×#14558989C22semidirect
26. C22×C151⋊C8C22semidirect
28. #18012715C22semidirect
31. #18012710C22semidirect
32. C2×C4⋊Dic151C22semidirect
33. #18012664C22semidirect
34. #18012661C22semidirect
35. #18012660C22semidirect
36. #18012660C22semidirect
37. (C2×D151)⋊C8C22semidirect
38. (C2×D151)⋊C8C22semidirect
39. C2×#41×C151C22semidirect
42. C2×SD16×C151C22semidirect
43. C2×D8×C151C22semidirect
44. C2×M16×C151C22semidirect
49. #14559004D4semidirect
89. C2×D4×C151D4semidirect
178. C151#3012semidirect
181. C2×AGL(1,ℤ/8)D151semidirect
197. C2×#41C2×D151semidirect
207. C2×SD16C2×D151semidirect
209. C2×D8C2×D151semidirect
210. C2×M16C2×D151semidirect
252. C2×D4C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127842
C2×#180127592
#237527902
#237523712
#237523612
#237523542
#237523442
#237522142
C2×#180126602
#237521802
C2×AGL(1,ℤ/8)×C1512
#3012151