Finite Group Databasebeta

← #34078419

Subgroups — #34078419

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×#18012765C211yesmaximal, characteristic, complemented
3C2×#18012761C211yesmaximal, characteristic, complemented
4#23752793C211yesmaximal, characteristic, complemented
5#23752375C212yesmaximal, complemented
6#23752364C212yesmaximal, complemented
7#23752358C212yesmaximal, complemented
8#23752347C212yesmaximal, complemented
9#23752211C211yesmaximal, characteristic, complemented
10#23752178C211yesmaximal, characteristic, complemented
11C2×#102×C151C211yesmaximal, characteristic, complemented
12#18012784—24no—
13#18012784—24no—
14C2×#14559004C2211yescharacteristic
15C2×#14559004C2211yescharacteristic
16C2×#14559004C2211yescharacteristic, complemented
17C2×#14559003—21no—
18C2×#14559003C2212yes—
19C22×C22⋊D151—22no—
20C2×D4×D151—21no—
21C2×D4×D151C2212yes—
22#18012765—24no—
23#18012761—24no—
24C2×Dic604C2211yescharacteristic, complemented
25C2×#14558993C2211yescharacteristic, complemented
26C2×#14558993C2211yescharacteristic, complemented
27C2×#14558992C2212yescomplemented
28C2×D1208C2211yescharacteristic, complemented
29C2×#14558989C2212yescomplemented
30C2×#14558988C2211yescharacteristic, complemented
31#18012719—21no—
32#18012719C2212yescomplemented
33#18012714—21no—
34#18012714C2212yescomplemented
35#18012713—22no—
36#18012662C2212yescomplemented
37#18012659C2212yescomplemented
38(C2×D151)⋊C8C2212yescomplemented
39(C2×D151)⋊C8C2212yescomplemented
40C22×D4×C151C2211yescharacteristic
41C22×C8×C151C2211yescharacteristic, complemented
42C2×C4⋊C4×C151C2211yescharacteristic, complemented
43#102×C151C2212yescomplemented
44#102×C151C2212yescomplemented
45C23×D151—22no—
46#14559004—22no—
47#14559004—22no—
48#14559004—22no—
49#14559004—22no—
50#14559004—22no—
51#14559004D414yescomplemented
52#14559003—28no—
53#14559003—24no—
54#14559003—24no—
55C2×Dic302D411yescharacteristic
56C2×Dic302C2311yescharacteristic
57C2×Dic302C2311yescharacteristic
58C2×D604—22no—
59C2×D604D411yescharacteristic
60C2×D604C2311yescharacteristic
61C2×D604C2311yescharacteristic
62C2×C22⋊D151—216no—
63C2×C22⋊D151—22no—
64C2×C22⋊D151—22no—
65C2×C22⋊D151—21no—
66C2×C22⋊D151—21no—
67C2×C22⋊D151D412yes—
68D4×D151—28no—
69D4×D151—24no—
70D4×D151—24no—
71C22×Dic151—22no—
72C2×C4×D151—21no—
73C2×C4×D151—21no—
74C2×C4×D151C2312yes—
75Dic604—22no—
76#14558993—22no—
77#14558993—22no—
78#14558992—24no—
79D1208—22no—
80#14558989—24no—
81#14558988—22no—
82C2×C151⋊C8C2312yes—
83C4⋊Dic151—22no—
84(C2×D151)⋊C4—22no—
85(C2×D151)⋊C4—22no—
86C24×C151—21no—
87C2×D4×C151—21no—
88C2×D4×C151—22no—
89C2×D4×C151D414yescomplemented
90C2×D4×C151C2312yes—
91C22×C4×C151—21no—
92C22×C4×C151C2311yescharacteristic
93C2×C8×C151—22no—
94C2×C8×C151C2311yescharacteristic
95C2×C8×C151C2311yescharacteristic
96C4⋊C4×C151—21no—
97C4⋊C4×C151C2312yes—
98C22×D151—21no—
99C22×D151—28no—
100C22×D151—21no—
101C22×D151—82no—
102C22×D151—82no—
103C22×D151C2×D412yes—
104Dic302—22no—
105Dic302—21no—
106Dic302—21no—
107Dic302—21no—
108Dic302C2×D412yes—
109D604—22no—
110D604—21no—
111D604—21no—
112D604—21no—
113D604—82no—
114D604C2×D412yes—
115C22⋊D151—28no—
116C22⋊D151—28no—
117C22⋊D151—24no—
118C22⋊D151—24no—
119C22⋊D151—42no—
120C2×Dic151—28no—
121C2×Dic151—21no—
122C2×Dic151—21no—
123C2×Dic151C2×D412yes—
124C4×D151—24no—
125C4×D151—22no—
126C4×D151—22no—
127C4×D151—42no—
128C151⋊C8—22no—
129C23×C151—22no—
130C23×C151—24no—
131C23×C151—81no—
132C23×C151—81no—
133C23×C151C2×D411yescharacteristic
134D4×C151—24no—
135D4×C151—22no—
136D4×C151C2×D414yes—
137C2×C4×C151—22no—
138C2×C4×C151—81no—
139C2×C4×C151C2×D411yescharacteristic
140C2×C4×C151C2×D412yes—
141C2×C4×C151C2×D412yes—
142C2×C4×C151C2411yescharacteristic
143C8×C151—21no—
144C8×C151—21no—
145C22×C151—21no—
146C22×C151—21no—
147C22×C151—24no—
148C22×C151—42no—
149C22×C151—42no—
150C22×C151—84no—
151C22×C151#12012yes—
152C22×C151C22×D411yescharacteristic
153C2×D151—22no—
154C2×D151—22no—
155C2×D151—22no—
156C2×D151—22no—
157C2×D151—41no—
158C2×D151—41no—
159C2×D151—41no—
160C2×D151—41no—
161C2×D151—41no—
162C2×D151—88no—
163C4×C151—42no—
164C4×C151#12012yes—
165C4×C151C22×D411yescharacteristic
166C4×C151C22×D411yescharacteristic
167Dic151—22no—
168Dic151—22no—
169Dic151—41no—
170C2×C151—22no—
171C2×C151—44no—
172C2×C151C2×#12011yescharacteristic
173C2×C151#57611yescharacteristic
174C2×C151#57711yescharacteristic
175D151—42no—
176D151—42no—
177D151—81no—
178C151#301211yescharacteristic, complemented, Sylow
179#3012—1511nomaximal, Sylow
180C2×H(5,𝔽2)—1511no—
181C2×AGL(1,ℤ/8)—1511no—
182C2×#135—1511no—
183#537—1511no—
184#452—1512no—
185#450—1512no—
186#449—1512no—
187#447—1512no—
188#417—1511no—
189C2×#102D15111yescharacteristic, complemented
190#408—1511no—
191H(5,𝔽2)—3024no—
192H(5,𝔽2)—3024no—
193C2×#41—3021no—
194C2×#41—1512no—
195C2×#41—1511no—
196C2×#41—1511no—
197C2×#41—1511no—
198C22×D4—3021no—
199C22×D4—3022no—
200C22×D4—1512no—
201C22×D4C2×D15111yescharacteristic
202AGL(1,ℤ/8)—3024no—
203#135—3024no—
204C2×Q16—1511no—
205C2×SD16—1512no—
206C2×SD16—1511no—
207C2×SD16—1511no—
208C2×D8—1512no—
209C2×D8—1511no—
210C2×M16—1511no—
211C22×C8C2×D15111yescharacteristic, complemented
212C22⋊Q8—3021no—
213C22⋊Q8—1512no—
214#121—3021no—
215#121—3022no—
216#121—1512no—
217C2×C4⋊C4C2×D15111yescharacteristic, complemented
218#103—1512no—
219#102—1512no—
220#102C2×D15112yescomplemented
221#102C2×D15112yescomplemented
222C22⋊C8—1512no—
223C22⋊C8—1512no—
224C24—21no—
225C24—3022no—
226#41—3028no—
227#41—3024no—
228#41—3024no—
229#41—3022no—
230#41—3022no—
231#41—3022no—
232#41—3022no—
233#41—3022no—
234#41—1514no—
235C2×Q8—1511no—
236C2×Q8—1511no—
237C2×Q8—1511no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—3022no—
241C2×D4—3028no—
242C2×D4—3024no—
243C2×D4—30216no—
244C2×D4—3022no—
245C2×D4—3021no—
246C2×D4—3024no—
247C2×D4—3021no—
248C2×D4—3022no—
249C2×D4—1512no—
250C2×D4—1511no—
251C2×D4—1511no—
252C2×D4—1511no—
253C2×D4D60414yescomplemented
254C2×D4C22×D15112yes—
255C22×C4—21no—
256C22×C4—3022no—
257C22×C4—3021no—
258C22×C4—3021no—
259C22×C4—1512no—
260C22×C4C22×D15111yescharacteristic
261Q16—3022no—
262SD16—3022no—
263SD16—3022no—
264SD16—3024no—
265D8—3022no—
266D8—3024no—
267M16—3022no—
268C2×C8—22no—
269C2×C8—1512no—
270C2×C8C22×D15111yescharacteristic
271C2×C8C22×D15111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3022no—
274C4⋊C4C22×D15112yes—
275C22⋊C4—3022no—
276C22⋊C4—3022no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—3028no—
282C23—3021no—
283C23—3021no—
284C23—1512no—
285C23—12082no—
286C23—12082no—
287C23D4×D15111yescharacteristic
288Q8—3022no—
289Q8—3021no—
290Q8—3021no—
291Q8—3021no—
292Q8—1512no—
293D4—24no—
294D4—22no—
295D4—3028no—
296D4—3028no—
297D4—3024no—
298D4—3024no—
299D4—3022no—
300D4—3021no—
301D4—3021no—
302D4—3021no—
303D4—6042no—
304D4—1512no—
305D4—12082no—
306D4C2×D60414yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—3028no—
310C2×C4—3021no—
311C2×C4—3024no—
312C2×C4—3022no—
313C2×C4—3022no—
314C2×C4—3021no—
315C2×C4—6042no—
316C2×C4—1512no—
317C2×C4D4×D15111yescharacteristic
318C2×C4D4×D15112yes—
319C2×C4C2×D60412yes—
320C2×C4C23×D15111yescharacteristic
321C8—21no—
322C8—21no—
323C8—3022no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—3022no—
331C22—3022no—
332C22—3022no—
333C22—3022no—
334C22—6041no—
335C22—6041no—
336C22—6041no—
337C22—6041no—
338C22—6041no—
339C22—12088no—
340C22#1801270912yes—
341C22C2×D4×D15111yescharacteristic
342C4—42no—
343C4—3022no—
344C4—3022no—
345C4—6041no—
346C4#1801270912yes—
347C4C2×D4×D15111yescharacteristic
348C4C22×D60411yescharacteristic
349C2—22no—
350C2—44no—
351C2—6042no—
352C2—6042no—
353C2—12081no—
354C2C2×#1801270911yescharacteristic
355C2#2375301311yescharacteristic
356C2#2375301811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×#102×C151C2
socleC22×C151C22×D4
hypercenterC2×#102D151
hypocenterC151#3012
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#18012784C2semidirect
2. C2×#18012765C2semidirect
3. C2×#18012761C2semidirect
4. #23752793C2semidirect
5. #23752375C2semidirect
6. #23752364C2semidirect
7. #23752358C2semidirect
8. #23752347C2semidirect
9. #23752211C2semidirect
10. #23752178C2semidirect
11. C2×#102×C151C2semidirect
16. C2×#14559004C22semidirect
24. C2×Dic604C22semidirect
25. C2×#14558993C22semidirect
26. C2×#14558993C22semidirect
27. C2×#14558992C22semidirect
28. C2×D1208C22semidirect
29. C2×#14558989C22semidirect
30. C2×#14558988C22semidirect
32. #18012719C22semidirect
34. #18012714C22semidirect
36. #18012662C22semidirect
37. #18012659C22semidirect
38. (C2×D151)⋊C8C22semidirect
39. (C2×D151)⋊C8C22semidirect
41. C22×C8×C151C22semidirect
42. C2×C4⋊C4×C151C22semidirect
43. #102×C151C22semidirect
44. #102×C151C22semidirect
51. #14559004D4semidirect
89. C2×D4×C151D4semidirect
178. C151#3012semidirect
189. C2×#102D151semidirect
211. C22×C8C2×D151semidirect
217. C2×C4⋊C4C2×D151semidirect
220. #102C2×D151semidirect
221. #102C2×D151semidirect
253. C2×D4D604semidirect

Maximal subgroups

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

subgroupindex
C2×#180127842
C2×#180127652
C2×#180127612
#237527932
#237523752
#237523642
#237523582
#237523472
#237522112
#237521782
C2×#102×C1512
#3012151