Finite Group Databasebeta

← #34055139

Subgroups — #34055139

Order 19072.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#18011937C211yesmaximal, characteristic, complemented
3C2×#18011936C211yesmaximal, characteristic, complemented
4#23750171C211yesmaximal, characteristic, complemented
5C2×#18011907C211yesmaximal, characteristic, complemented
6#23750013C212yesmaximal, complemented
7#23750011C212yesmaximal, complemented
8#23750000C212yesmaximal, complemented
9#23749996C212yesmaximal, complemented
10M16×Dic149C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C22×D596C2211yescharacteristic
14C22×C22⋊D149—22no—
15#18011939—24no—
16#18011937—24no—
17#18011936—24no—
18C2×#14558725C2212yescomplemented
19C2×#14558724C2212yescomplemented
20C2×#14558723C2212yescomplemented
21C2×D1192C2212yescomplemented
22C2×#14558721C2212yescomplemented
23C2×#14558721C2212yescomplemented
24C2×#14558720C2211yescharacteristic
25#18011909C2211yescharacteristic, complemented
26#18011907—22no—
27#18011907C2212yes—
28#18011907C2212yes—
29#18011907C2211yescharacteristic, complemented
30#18011900C2211yescharacteristic, complemented
31#18011898C2211yescharacteristic, complemented
32#18011883—22no—
33C4×C22⋊D149—22no—
34C2×C4×Dic149C2211yescharacteristic
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×D596C2312yes—
51C2×D596C2311yescharacteristic
52C2×C22⋊D149—24no—
53C2×C22⋊D149—24no—
54C2×C22⋊D149—22no—
55C2×C22⋊D149—21no—
56C2×C22⋊D149—48no—
57C22×Dic149D412yes—
58C2×C4×D149—21no—
59#14558725—24no—
60#14558724—24no—
61#14558723—24no—
62D1192—24no—
63#14558721—24no—
64#14558721—24no—
65#14558720D414yescomplemented
66C2×#11924746C2312yes—
67#14558715—22no—
68C22⋊Dic149—22no—
69#14558713—22no—
70C4×Dic149D412yes—
71C4×Dic149C2311yescharacteristic
72C4×Dic149C2311yescharacteristic
73C24×C149—21no—
74#41×C149—22no—
75#41×C149—22no—
76C2×Q8×C149C2311yescharacteristic
77C2×D4×C149—21no—
78C2×D4×C149—22no—
79C2×D4×C149—22no—
80C2×D4×C149—21no—
81C2×D4×C149C2312yes—
82C2×D4×C149C2311yescharacteristic
83C22×C4×C149—21no—
84C22×C4×C149C2311yescharacteristic
85SD16×C149—24no—
86D8×C149—24no—
87M16×C149D414yescomplemented
88C2×C8×C149C2312yes—
89C22×D149—21no—
90C22×D149—22no—
91C22×D149—44no—
92C22×D149—42no—
93Dic298—21no—
94Dic298—21no—
95D596—21no—
96D596—22no—
97D596—21no—
98D596—22no—
99D596—42no—
100C22⋊D149—48no—
101C22⋊D149—42no—
102C22⋊D149—44no—
103C22⋊D149—48no—
104C2×Dic149—24no—
105C2×Dic149—21no—
106C2×Dic149C2×D412yes—
107C2×Dic149C2×D412yes—
108C4×D149—42no—
109#11924746C2×D414yescomplemented
110C23×C149—22no—
111C23×C149—21no—
112C23×C149—44no—
113C23×C149—42no—
114C23×C149C2×D411yescharacteristic
115Q8×C149—21no—
116Q8×C149—21no—
117D4×C149—21no—
118D4×C149—22no—
119D4×C149—21no—
120D4×C149—22no—
121D4×C149—42no—
122D4×C149—42no—
123C2×C4×C149—21no—
124C2×C4×C149—42no—
125C2×C4×C149C2×D411yescharacteristic
126C2×C4×C149C2×D412yes—
127C2×C4×C149C2×D412yes—
128C2×C4×C149C2411yescharacteristic
129C8×C149C2×D414yescomplemented
130C22×C149—22no—
131C22×C149—41no—
132C22×C149—42no—
133C22×C149—44no—
134C22×C149—42no—
135C22×C149—84no—
136C22×C149#12712yes—
137C22×C149C22×D411yescharacteristic
138C2×D149—41no—
139C2×D149—42no—
140C2×D149—44no—
141C2×D149—42no—
142C2×D149—84no—
143C4×C149—41no—
144C4×C149#12712yes—
145C4×C149C22×D411yescharacteristic
146C4×C149C22×D411yescharacteristic
147Dic149—22no—
148Dic149—41no—
149Dic149AGL(1,ℤ/8)14yescomplemented
150C2×C149—22no—
151C2×C149—81no—
152C2×C149—82no—
153C2×C149C2×#12711yescharacteristic
154C2×C149C2×AGL(1,ℤ/8)12yes—
155D149—81no—
156D149—82no—
157C149#315111yescharacteristic, complemented, Sylow
158#3151—1491nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1491no—
160C2×AGL(1,ℤ/8)—1491no—
161C2×AGL(1,ℤ/8)—1491no—
162C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
163#532—1491no—
164C2×#127—1491no—
165#496—1492no—
166#496—1492no—
167#493—1492no—
168#492—1492no—
169C4×M16—1491no—
170C2×#41—1491no—
171C2×#41C2×D14911yescharacteristic
172C22×D4—2982no—
173C22×D4—1491no—
174C22×D4C2×D14911yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2984no—
177AGL(1,ℤ/8)—2984no—
178AGL(1,ℤ/8)—2984no—
179C2×SD16—1492no—
180C2×SD16—1492no—
181C2×SD16—1492no—
182C2×SD16C2×D14912yescomplemented
183C2×D8—1492no—
184C2×D8—1492no—
185C2×D8—1492no—
186C2×D8C2×D14912yescomplemented
187C2×M16—1491no—
188C2×M16C2×D14911yescharacteristic
189C4⋊Q8—1491no—
190#127—2982no—
191#127—1492no—
192#127—1492no—
193#127—1491no—
194#124—1491no—
195#124—1491no—
196#121—2982no—
197C4×D4—2982no—
198C2×C42—1491no—
199#97—1492no—
200C4×C8—1492no—
201C24—21no—
202C24—2981no—
203#41—22no—
204#41—22no—
205#41—2982no—
206#41—2982no—
207C2×Q8—1491no—
208C2×Q8C22×D14911yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2981no—
214C2×D4—2982no—
215C2×D4—2984no—
216C2×D4—2984no—
217C2×D4—2982no—
218C2×D4—2981no—
219C2×D4—2982no—
220C2×D4—5968no—
221C2×D4—1491no—
222C2×D4—1492no—
223C2×D4C22×D14911yescharacteristic
224C2×D4C22×D14912yes—
225C22×C4—21no—
226C22×C4—2981no—
227C22×C4—1492no—
228C22×C4C22×D14911yescharacteristic
229SD16—24no—
230SD16—2984no—
231SD16—2984no—
232SD16—2984no—
233D8—24no—
234D8—2984no—
235D8—2984no—
236D8—2984no—
237M16—1494no—
238M16C22⋊D14914yescomplemented
239C2×C8—1492no—
240C2×C8C22×D14912yes—
241C4⋊C4—2982no—
242C22⋊C4—2982no—
243C22⋊C4—2982no—
244C42—1492no—
245C42—1491no—
246C42—1491no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—2981no—
252C23—2982no—
253C23—5964no—
254C23—5962no—
255C23D4×D14911yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2981no—
259Q8—2981no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2981no—
267D4—2982no—
268D4—2981no—
269D4—2982no—
270D4—5968no—
271D4—5962no—
272D4—5964no—
273D4—5968no—
274D4—5962no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2984no—
278C2×C4—2981no—
279C2×C4—5962no—
280C2×C4—1492no—
281C2×C4—1492no—
282C2×C4D4×D14911yescharacteristic
283C2×C4D4×D14912yes—
284C2×C4C2×C22⋊D14912yes—
285C2×C4C23×D14911yescharacteristic
286C8—1494no—
287C8C2×C22⋊D14914yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—5964no—
295C22—5962no—
296C22—5961no—
297C22—5962no—
298C22—11924no—
299C22#1801190712yes—
300C22C2×D4×D14911yescharacteristic
301C4—41no—
302C4—2982no—
303C4—5961no—
304C4—1494no—
305C4#1801190712yes—
306C4C2×D4×D14911yescharacteristic
307C4C22×C22⋊D14911yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—11922no—
312C2—11921no—
313C2C2×#1801190711yescharacteristic
314C2AGL(1,ℤ/8)×D14912yes—
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#3151
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×#18011937C2semidirect
3. C2×#18011936C2semidirect
4. #23750171C2semidirect
5. C2×#18011907C2semidirect
6. #23750013C2semidirect
7. #23750011C2semidirect
8. #23750000C2semidirect
9. #23749996C2semidirect
10. M16×Dic149C2semidirect
11. C2×AGL(1,ℤ/8)×C149C2semidirect
18. C2×#14558725C22semidirect
19. C2×#14558724C22semidirect
20. C2×#14558723C22semidirect
21. C2×D1192C22semidirect
22. C2×#14558721C22semidirect
23. C2×#14558721C22semidirect
25. #18011909C22semidirect
29. #18011907C22semidirect
30. #18011900C22semidirect
31. #18011898C22semidirect
35. #18011821C22semidirect
36. C8×Dic149C22semidirect
40. C2×SD16×C149C22semidirect
41. C2×D8×C149C22semidirect
65. #14558720D4semidirect
87. M16×C149D4semidirect
109. #11924746C2×D4semidirect
129. C8×C149C2×D4semidirect
149. Dic149AGL(1,ℤ/8)semidirect
157. C149#3151semidirect
162. C2×AGL(1,ℤ/8)D149semidirect
182. C2×SD16C2×D149semidirect
186. C2×D8C2×D149semidirect
238. M16C22⋊D149semidirect
287. C8C2×C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119392
C2×#180119372
C2×#180119362
#237501712
C2×#180119072
#237500132
#237500112
#237500002
#237499962
M16×Dic1492
C2×AGL(1,ℤ/8)×C1492
#3151149