Finite Group Databasebeta

← #34055176

Subgroups — #34055176

Order 19072.

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

normal subgroups only

This group has 596 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×#18011939C211yesmaximal, characteristic, complemented
2C2×#18011931C211yesmaximal, characteristic, complemented
3C22×#14558721C211yesmaximal, characteristic, complemented
4#23750192C211yesmaximal, characteristic, complemented
5#23750189C211yesmaximal, characteristic, complemented
6#23750013C212yesmaximal, complemented
7#23750011C212yesmaximal, complemented
8#23750006C212yesmaximal, complemented
9#23750000C212yesmaximal, complemented
10#23749583C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C22×D596C2211yescharacteristic
14#18011939—24no—
15#18011931—24no—
16C2×#14558726C2211yescharacteristic, complemented
17C2×#14558725C2212yescomplemented
18C2×#14558724C2211yescharacteristic, complemented
19C2×#14558723C2211yescharacteristic, complemented
20C2×D1192C2212yescomplemented
21C2×#14558721—24no—
22C2×#14558721C2212yes—
23C2×#14558721C2212yes—
24C2×#14558721C2211yescharacteristic, complemented
25C22×#11924746C2211yescharacteristic
26#18011907C2212yescomplemented
27#18011907C2212yescomplemented
28#18011900C2212yescomplemented
29#18011898C2212yescomplemented
30#18011885—22no—
31#18011882—22no—
32#18011880—22no—
33#18011879—22no—
34#18011870C2211yescharacteristic
35#18011821C2212yescomplemented
36C8×Dic149C2212yescomplemented
37C2×#41×C149C2211yescharacteristic
38C22×D4×C149C2211yescharacteristic
39AGL(1,ℤ/8)×C149—24no—
40C2×SD16×C149C2212yescomplemented
41C2×D8×C149C2212yescomplemented
42C2×M16×C149C2211yescharacteristic
43C23×D149—21no—
44#14558736—22no—
45#14558736—22no—
46C2×Dic298C2311yescharacteristic
47C2×D596—22no—
48C2×D596—22no—
49C2×D596—21no—
50C2×D596C2311yescharacteristic
51C2×D596C2312yes—
52C2×C22⋊D149—24no—
53C2×C22⋊D149—24no—
54C2×C22⋊D149—21no—
55C2×C4×D149—21no—
56#14558726—22no—
57#14558725—24no—
58#14558724—22no—
59#14558723—22no—
60D1192—24no—
61#14558721—24no—
62#14558721—24no—
63#14558721—22no—
64C2×#11924746D414yescomplemented
65C2×#11924746C2311yescharacteristic
66C2×#11924746C2311yescharacteristic
67C4⋊Dic149D412yes—
68C22⋊Dic149—24no—
69C22⋊Dic149D412yes—
70#14558713—24no—
71C4×Dic149C2312yes—
72C24×C149—21no—
73#41×C149—22no—
74#41×C149—22no—
75C2×Q8×C149C2311yescharacteristic
76C2×D4×C149—21no—
77C2×D4×C149—22no—
78C2×D4×C149—22no—
79C2×D4×C149—21no—
80C2×D4×C149C2311yescharacteristic
81C2×D4×C149C2312yes—
82C22×C4×C149—21no—
83C22×C4×C149C2311yescharacteristic
84SD16×C149—24no—
85D8×C149—24no—
86M16×C149D414yescomplemented
87C2×C8×C149C2312yes—
88C22×D149—21no—
89C22×D149—22no—
90C22×D149—42no—
91C22×D149—82no—
92Dic298—21no—
93Dic298—21no—
94D596—22no—
95D596—21no—
96D596—21no—
97D596—22no—
98D596—42no—
99C22⋊D149—42no—
100C22⋊D149—84no—
101C22⋊D149—84no—
102C2×Dic149—21no—
103C2×Dic149C2×D414yes—
104C4×D149—42no—
105#11924746C2×D412yes—
106#11924746C2×D412yes—
107C23×C149—21no—
108C23×C149—22no—
109C23×C149—42no—
110C23×C149—82no—
111C23×C149C2×D411yescharacteristic
112Q8×C149—21no—
113Q8×C149—21no—
114D4×C149—22no—
115D4×C149—21no—
116D4×C149—21no—
117D4×C149—22no—
118D4×C149—42no—
119D4×C149—42no—
120C2×C4×C149—21no—
121C2×C4×C149—42no—
122C2×C4×C149C2×D411yescharacteristic
123C2×C4×C149C2×D412yes—
124C2×C4×C149C2×D412yes—
125C2×C4×C149C2411yescharacteristic
126C8×C149C2×D414yes—
127C22×C149—22no—
128C22×C149—41no—
129C22×C149—42no—
130C22×C149—44no—
131C22×C149—42no—
132C22×C149—84no—
133C22×C149#12712yes—
134C22×C149C22×D411yescharacteristic
135C2×D149—41no—
136C2×D149—42no—
137C2×D149—42no—
138C2×D149—44no—
139C2×D149—84no—
140C4×C149—41no—
141C4×C149#12712yes—
142C4×C149C22×D411yescharacteristic
143C4×C149C22×D411yescharacteristic
144Dic149—24no—
145Dic149—41no—
146C2×C149—22no—
147C2×C149—81no—
148C2×C149—82no—
149C2×C149C2×#12711yescharacteristic
150C2×C149#57612yes—
151D149—81no—
152D149—82no—
153C149#315511yescharacteristic, complemented, Sylow
154#3155—1491nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1491no—
156C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
157C2×#135—1491no—
158C22×D8—1491no—
159#535—1491no—
160#535—1491no—
161#496—1492no—
162#496—1492no—
163#495—1492no—
164#493—1492no—
165#405—1491no—
166C2×#41—1491no—
167C2×#41C2×D14911yescharacteristic
168C22×D4—1491no—
169C22×D4C2×D14911yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2984no—
172#135—2984no—
173C2×Q16—1491no—
174C2×SD16—1491no—
175C2×SD16—1491no—
176C2×SD16—1492no—
177C2×SD16C2×D14912yescomplemented
178C2×D8—2984no—
179C2×D8—1492no—
180C2×D8—1492no—
181C2×D8—1491no—
182C2×D8—1492no—
183C2×D8C2×D14912yescomplemented
184C2×M16C2×D14911yescharacteristic
185C22×C8—1491no—
186#127—1492no—
187#127—1492no—
188#124—1492no—
189#124—1492no—
190#121—2982no—
191#121—2982no—
192#120—2982no—
193#120—2982no—
194#117—1491no—
195#97—1492no—
196C4×C8—1492no—
197C24—21no—
198C24—2981no—
199#41—22no—
200#41—22no—
201#41—2982no—
202#41—2982no—
203C2×Q8—1491no—
204C2×Q8C22×D14911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2981no—
210C2×D4—2984no—
211C2×D4—2984no—
212C2×D4—2982no—
213C2×D4—2981no—
214C2×D4—2982no—
215C2×D4—1492no—
216C2×D4—1491no—
217C2×D4C22×D14912yes—
218C2×D4C22×D14911yescharacteristic
219C22×C4—21no—
220C22×C4—2981no—
221C22×C4C22×D14911yescharacteristic
222Q16—2982no—
223SD16—24no—
224SD16—2982no—
225SD16—2982no—
226SD16—2984no—
227D8—24no—
228D8—2984no—
229D8—2984no—
230D8—2982no—
231D8—2984no—
232M16C22⋊D14914yescomplemented
233C2×C8—1494no—
234C2×C8—1491no—
235C2×C8—1491no—
236C2×C8C22×D14912yes—
237C4⋊C4—1492no—
238C22⋊C4—2984no—
239C22⋊C4—2984no—
240C22⋊C4—1492no—
241C42—1492no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—2982no—
247C23—2981no—
248C23—5962no—
249C23—11922no—
250C23D4×D14911yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2981no—
254Q8—2981no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2982no—
262D4—2981no—
263D4—2981no—
264D4—2982no—
265D4—5962no—
266D4—5962no—
267D4—11924no—
268D4—11924no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2981no—
272C2×C4—5962no—
273C2×C4—1494no—
274C2×C4D4×D14911yescharacteristic
275C2×C4D4×D14912yes—
276C2×C4C2×C22⋊D14912yes—
277C2×C4C23×D14911yescharacteristic
278C8—1492no—
279C8—1492no—
280C8C2×C22⋊D14914yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—5961no—
288C22—5962no—
289C22—5962no—
290C22—5964no—
291C22—11924no—
292C22#1801190712yes—
293C22C2×D4×D14911yescharacteristic
294C4—41no—
295C4—2984no—
296C4—5961no—
297C4#1801190712yes—
298C4C2×D4×D14911yescharacteristic
299C4C22×C22⋊D14911yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—11922no—
304C2—11921no—
305C2C2×#1801190711yescharacteristic
306C2#2375042912yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×AGL(1,ℤ/8)×C149C2
socleC22×C149C22×D4
hypercenterC2×AGL(1,ℤ/8)D149
hypocenterC149#3155
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
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×#18011939C2semidirect
2. C2×#18011931C2semidirect
3. C22×#14558721C2semidirect
4. #23750192C2semidirect
5. #23750189C2semidirect
6. #23750013C2semidirect
7. #23750011C2semidirect
8. #23750006C2semidirect
9. #23750000C2semidirect
10. #23749583C2semidirect
11. C2×AGL(1,ℤ/8)×C149C2semidirect
16. C2×#14558726C22semidirect
17. C2×#14558725C22semidirect
18. C2×#14558724C22semidirect
19. C2×#14558723C22semidirect
20. C2×D1192C22semidirect
24. C2×#14558721C22semidirect
26. #18011907C22semidirect
27. #18011907C22semidirect
28. #18011900C22semidirect
29. #18011898C22semidirect
35. #18011821C22semidirect
36. C8×Dic149C22semidirect
40. C2×SD16×C149C22semidirect
41. C2×D8×C149C22semidirect
64. C2×#11924746D4semidirect
86. M16×C149D4semidirect
153. C149#3155semidirect
156. C2×AGL(1,ℤ/8)D149semidirect
177. C2×SD16C2×D149semidirect
183. C2×D8C2×D149semidirect
232. M16C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119392
C2×#180119312
C22×#145587212
#237501922
#237501892
#237500132
#237500112
#237500062
#237500002
#237495832
C2×AGL(1,ℤ/8)×C1492
#3155149