Finite Group Databasebeta

← #34054000

Subgroups — #34054000

Order 19072.

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×#18011956C211yesmaximal, characteristic, complemented
2C2×#18011937C211yesmaximal, characteristic, complemented
3C2×#18011933C211yesmaximal, characteristic, complemented
4#23750206C211yesmaximal, characteristic, complemented
5#23749788C212yesmaximal, complemented
6#23749777C212yesmaximal, complemented
7#23749771C212yesmaximal, complemented
8#23749760C212yesmaximal, complemented
9#23749624C211yesmaximal, characteristic, complemented
10#23749591C211yesmaximal, characteristic, complemented
11C2×#102×C149C211yesmaximal, characteristic, complemented
12#18011956—24no—
13#18011956—24no—
14C2×#14558736C2211yescharacteristic
15C2×#14558736C2211yescharacteristic
16C2×#14558736C2211yescharacteristic, complemented
17C2×#14558735—21no—
18C2×#14558735C2212yes—
19C22×C22⋊D149—22no—
20C2×D4×D149—21no—
21C2×D4×D149C2212yes—
22#18011937—24no—
23#18011933—24no—
24C2×Dic596C2211yescharacteristic, complemented
25C2×#14558725C2211yescharacteristic, complemented
26C2×#14558725C2211yescharacteristic, complemented
27C2×#14558724C2212yescomplemented
28C2×D1192C2211yescharacteristic, complemented
29C2×#14558721C2212yescomplemented
30C2×#14558720C2211yescharacteristic, complemented
31#18011891—21no—
32#18011891C2212yescomplemented
33#18011886—21no—
34#18011886C2212yescomplemented
35#18011885—22no—
36#18011834C2212yescomplemented
37#18011831C2212yescomplemented
38#18011823C2212yescomplemented
39#18011823C2212yescomplemented
40C22×D4×C149C2211yescharacteristic
41C22×C8×C149C2211yescharacteristic, complemented
42C2×C4⋊C4×C149C2211yescharacteristic, complemented
43#102×C149C2212yescomplemented
44#102×C149C2212yescomplemented
45C23×D149—22no—
46#14558736—22no—
47#14558736—22no—
48#14558736—22no—
49#14558736—22no—
50#14558736—22no—
51#14558736D414yescomplemented
52#14558735—28no—
53#14558735—24no—
54#14558735—24no—
55C2×Dic298D411yescharacteristic
56C2×Dic298C2311yescharacteristic
57C2×Dic298C2311yescharacteristic
58C2×D596—22no—
59C2×D596D411yescharacteristic
60C2×D596C2311yescharacteristic
61C2×D596C2311yescharacteristic
62C2×C22⋊D149—216no—
63C2×C22⋊D149—22no—
64C2×C22⋊D149—22no—
65C2×C22⋊D149—21no—
66C2×C22⋊D149—21no—
67C2×C22⋊D149D412yes—
68D4×D149—28no—
69D4×D149—24no—
70D4×D149—24no—
71C22×Dic149—22no—
72C2×C4×D149—21no—
73C2×C4×D149—21no—
74C2×C4×D149C2312yes—
75Dic596—22no—
76#14558725—22no—
77#14558725—22no—
78#14558724—24no—
79D1192—22no—
80#14558721—24no—
81#14558720—22no—
82C2×#11924746C2312yes—
83C4⋊Dic149—22no—
84#14558713—22no—
85#14558713—22no—
86C24×C149—21no—
87C2×D4×C149—21no—
88C2×D4×C149—22no—
89C2×D4×C149D414yescomplemented
90C2×D4×C149C2312yes—
91C22×C4×C149—21no—
92C22×C4×C149C2311yescharacteristic
93C2×C8×C149—22no—
94C2×C8×C149C2311yescharacteristic
95C2×C8×C149C2311yescharacteristic
96C4⋊C4×C149—21no—
97C4⋊C4×C149C2312yes—
98C22×D149—21no—
99C22×D149—28no—
100C22×D149—21no—
101C22×D149—82no—
102C22×D149—82no—
103C22×D149C2×D412yes—
104Dic298—22no—
105Dic298—21no—
106Dic298—21no—
107Dic298—21no—
108Dic298C2×D412yes—
109D596—22no—
110D596—21no—
111D596—21no—
112D596—21no—
113D596—82no—
114D596C2×D412yes—
115C22⋊D149—28no—
116C22⋊D149—28no—
117C22⋊D149—24no—
118C22⋊D149—24no—
119C22⋊D149—42no—
120C2×Dic149—28no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149C2×D412yes—
124C4×D149—24no—
125C4×D149—22no—
126C4×D149—22no—
127C4×D149—42no—
128#11924746—22no—
129C23×C149—22no—
130C23×C149—24no—
131C23×C149—81no—
132C23×C149—81no—
133C23×C149C2×D411yescharacteristic
134D4×C149—24no—
135D4×C149—22no—
136D4×C149C2×D414yes—
137C2×C4×C149—22no—
138C2×C4×C149—81no—
139C2×C4×C149C2×D411yescharacteristic
140C2×C4×C149C2×D412yes—
141C2×C4×C149C2×D412yes—
142C2×C4×C149C2411yescharacteristic
143C8×C149—21no—
144C8×C149—21no—
145C22×C149—21no—
146C22×C149—21no—
147C22×C149—24no—
148C22×C149—42no—
149C22×C149—42no—
150C22×C149—84no—
151C22×C149#12012yes—
152C22×C149C22×D411yescharacteristic
153C2×D149—22no—
154C2×D149—22no—
155C2×D149—22no—
156C2×D149—22no—
157C2×D149—41no—
158C2×D149—41no—
159C2×D149—41no—
160C2×D149—41no—
161C2×D149—41no—
162C2×D149—88no—
163C4×C149—42no—
164C4×C149#12012yes—
165C4×C149C22×D411yescharacteristic
166C4×C149C22×D411yescharacteristic
167Dic149—22no—
168Dic149—22no—
169Dic149—41no—
170C2×C149—22no—
171C2×C149—44no—
172C2×C149C2×#12011yescharacteristic
173C2×C149#57611yescharacteristic
174C2×C149#57711yescharacteristic
175D149—42no—
176D149—42no—
177D149—81no—
178C149#301211yescharacteristic, complemented, Sylow
179#3012—1491nomaximal, Sylow
180C2×H(5,𝔽2)—1491no—
181C2×AGL(1,ℤ/8)—1491no—
182C2×#135—1491no—
183#537—1491no—
184#452—1492no—
185#450—1492no—
186#449—1492no—
187#447—1492no—
188#417—1491no—
189C2×#102D14911yescharacteristic, complemented
190#408—1491no—
191H(5,𝔽2)—2984no—
192H(5,𝔽2)—2984no—
193C2×#41—2981no—
194C2×#41—1492no—
195C2×#41—1491no—
196C2×#41—1491no—
197C2×#41—1491no—
198C22×D4—2981no—
199C22×D4—2982no—
200C22×D4—1492no—
201C22×D4C2×D14911yescharacteristic
202AGL(1,ℤ/8)—2984no—
203#135—2984no—
204C2×Q16—1491no—
205C2×SD16—1492no—
206C2×SD16—1491no—
207C2×SD16—1491no—
208C2×D8—1492no—
209C2×D8—1491no—
210C2×M16—1491no—
211C22×C8C2×D14911yescharacteristic, complemented
212C22⋊Q8—2981no—
213C22⋊Q8—1492no—
214#121—2981no—
215#121—2982no—
216#121—1492no—
217C2×C4⋊C4C2×D14911yescharacteristic, complemented
218#103—1492no—
219#102—1492no—
220#102C2×D14912yescomplemented
221#102C2×D14912yescomplemented
222C22⋊C8—1492no—
223C22⋊C8—1492no—
224C24—21no—
225C24—2982no—
226#41—2988no—
227#41—2984no—
228#41—2984no—
229#41—2982no—
230#41—2982no—
231#41—2982no—
232#41—2982no—
233#41—2982no—
234#41—1494no—
235C2×Q8—1491no—
236C2×Q8—1491no—
237C2×Q8—1491no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—2982no—
241C2×D4—2988no—
242C2×D4—2984no—
243C2×D4—29816no—
244C2×D4—2982no—
245C2×D4—2981no—
246C2×D4—2984no—
247C2×D4—2981no—
248C2×D4—2982no—
249C2×D4—1492no—
250C2×D4—1491no—
251C2×D4—1491no—
252C2×D4—1491no—
253C2×D4D59614yescomplemented
254C2×D4C22×D14912yes—
255C22×C4—21no—
256C22×C4—2982no—
257C22×C4—2981no—
258C22×C4—2981no—
259C22×C4—1492no—
260C22×C4C22×D14911yescharacteristic
261Q16—2982no—
262SD16—2982no—
263SD16—2982no—
264SD16—2984no—
265D8—2982no—
266D8—2984no—
267M16—2982no—
268C2×C8—22no—
269C2×C8—1492no—
270C2×C8C22×D14911yescharacteristic
271C2×C8C22×D14911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2982no—
274C4⋊C4C22×D14912yes—
275C22⋊C4—2982no—
276C22⋊C4—2982no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—2988no—
282C23—2981no—
283C23—2981no—
284C23—1492no—
285C23—11922no—
286C23—11922no—
287C23D4×D14911yescharacteristic
288Q8—2981no—
289Q8—2982no—
290Q8—2981no—
291Q8—2981no—
292Q8—1492no—
293D4—22no—
294D4—24no—
295D4—2988no—
296D4—2988no—
297D4—2984no—
298D4—2984no—
299D4—2981no—
300D4—2981no—
301D4—2981no—
302D4—2982no—
303D4—5962no—
304D4—1492no—
305D4—11922no—
306D4C2×D59614yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—2988no—
310C2×C4—2981no—
311C2×C4—2982no—
312C2×C4—2982no—
313C2×C4—2984no—
314C2×C4—2981no—
315C2×C4—5962no—
316C2×C4—1492no—
317C2×C4D4×D14912yes—
318C2×C4D4×D14911yescharacteristic
319C2×C4C2×D59612yes—
320C2×C4C23×D14911yescharacteristic
321C8—21no—
322C8—21no—
323C8—2982no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—2982no—
331C22—2982no—
332C22—2982no—
333C22—2982no—
334C22—5961no—
335C22—5961no—
336C22—5961no—
337C22—5961no—
338C22—5961no—
339C22—11928no—
340C22#1801188112yes—
341C22C2×D4×D14911yescharacteristic
342C4—42no—
343C4—2982no—
344C4—2982no—
345C4—5961no—
346C4#1801188112yes—
347C4C2×D4×D14911yescharacteristic
348C4C22×D59611yescharacteristic
349C2—22no—
350C2—44no—
351C2—5962no—
352C2—5962no—
353C2—11921no—
354C2C2×#1801188111yescharacteristic
355C2#2375042611yescharacteristic
356C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×#102×C149C2
socleC22×C149C22×D4
hypercenterC2×#102D149
hypocenterC149#3012
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#18011956C2semidirect
2. C2×#18011937C2semidirect
3. C2×#18011933C2semidirect
4. #23750206C2semidirect
5. #23749788C2semidirect
6. #23749777C2semidirect
7. #23749771C2semidirect
8. #23749760C2semidirect
9. #23749624C2semidirect
10. #23749591C2semidirect
11. C2×#102×C149C2semidirect
16. C2×#14558736C22semidirect
24. C2×Dic596C22semidirect
25. C2×#14558725C22semidirect
26. C2×#14558725C22semidirect
27. C2×#14558724C22semidirect
28. C2×D1192C22semidirect
29. C2×#14558721C22semidirect
30. C2×#14558720C22semidirect
32. #18011891C22semidirect
34. #18011886C22semidirect
36. #18011834C22semidirect
37. #18011831C22semidirect
38. #18011823C22semidirect
39. #18011823C22semidirect
41. C22×C8×C149C22semidirect
42. C2×C4⋊C4×C149C22semidirect
43. #102×C149C22semidirect
44. #102×C149C22semidirect
51. #14558736D4semidirect
89. C2×D4×C149D4semidirect
178. C149#3012semidirect
189. C2×#102D149semidirect
211. C22×C8C2×D149semidirect
217. C2×C4⋊C4C2×D149semidirect
220. #102C2×D149semidirect
221. #102C2×D149semidirect
253. C2×D4D596semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
C2×#180119372
C2×#180119332
#237502062
#237497882
#237497772
#237497712
#237497602
#237496242
#237495912
C2×#102×C1492
#3012149