Finite Group Databasebeta

← #34055140

Subgroups — #34055140

Order 19072.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#18011938C212yesmaximal, complemented
2C2×#18011935C212yesmaximal, complemented
3#23750174C211yesmaximal, characteristic, complemented
4#127×D149C211yesmaximal, characteristic, complemented
5#23750015C211yesmaximal, characteristic, complemented
6#23750014C211yesmaximal, characteristic, complemented
7#23750012C211yesmaximal, characteristic, complemented
8#23750000C212yesmaximal, complemented
9#23749996C212yesmaximal, complemented
10#23749577C211yesmaximal, characteristic, complemented
11#496×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2211yescharacteristic
13C2×#14558735C2211yescharacteristic
14C2×D4×D149—22no—
15C2×D4×D149C2212yes—
16#18011938—28no—
17#18011935—28no—
18C2×#14558725C2212yescomplemented
19C2×#14558724C2212yescomplemented
20C2×#14558723C2212yescomplemented
21C2×D1192C2212yescomplemented
22C2×#14558721C2212yescomplemented
23C2×#14558721C2212yescomplemented
24C2×#14558719C2212yes—
25#18011909C2211yescharacteristic, complemented
26#18011908C2211yescharacteristic
27#18011907—22no—
28#18011907C2212yes—
29#18011907C2211yescharacteristic, complemented
30#18011899C2211yescharacteristic, complemented
31#18011884—22no—
32#18011871—22no—
33C42×D149C2211yescharacteristic
34#18011822C2211yescharacteristic, complemented
35C8×Dic149C2212yescomplemented
36C2×SD16×C149C2212yescomplemented
37C2×D8×C149C2212yescomplemented
38#127×C149C2211yescharacteristic
39#124×C149C2211yescharacteristic, complemented
40#97×C149C2211yescharacteristic, complemented
41C23×D149—22no—
42#14558737—24no—
43#14558735—24no—
44C2×Dic298C2311yescharacteristic
45C2×D596—21no—
46C2×D596—22no—
47C2×D596C2312yes—
48C2×D596C2311yescharacteristic
49C2×C22⋊D149—21no—
50C2×C22⋊D149—22no—
51C2×C22⋊D149—24no—
52C2×C22⋊D149—22no—
53D4×D149—24no—
54D4×D149—24no—
55D4×D149—48no—
56C22×Dic149—21no—
57C2×C4×D149—21no—
58C2×C4×D149D412yes—
59C2×C4×D149C2311yescharacteristic
60#14558725—24no—
61#14558724—24no—
62#14558723—24no—
63D1192—24no—
64#14558721—24no—
65#14558721—24no—
66#14558719D418yescomplemented
67C2×#11924746C2312yes—
68#14558715—22no—
69#14558713—22no—
70C4×Dic149D412yes—
71C4×Dic149C2311yescharacteristic
72C2×Q8×C149C2311yescharacteristic
73C2×D4×C149—22no—
74C2×D4×C149C2312yes—
75C2×D4×C149C2311yescharacteristic
76SD16×C149—24no—
77D8×C149—24no—
78C2×C8×C149C2312yes—
79C22⋊C4×C149—22no—
80C42×C149C2311yescharacteristic
81C22×D149—21no—
82C22×D149—22no—
83C22×D149—48no—
84C22×D149—44no—
85C22×D149C2×D411yescharacteristic
86Dic298—22no—
87D596—22no—
88D596—22no—
89D596—22no—
90D596—44no—
91D596—42no—
92C22⋊D149—42no—
93C22⋊D149—44no—
94C22⋊D149—48no—
95C22⋊D149—44no—
96C2×Dic149—21no—
97C2×Dic149—42no—
98C2×Dic149C2×D411yescharacteristic
99C2×Dic149C2×D412yes—
100C4×D149—24no—
101C4×D149—42no—
102C4×D149C2×D414yes—
103#11924746C2×D414yescomplemented
104C23×C149—21no—
105C23×C149—22no—
106Q8×C149—22no—
107D4×C149—22no—
108D4×C149—22no—
109D4×C149—22no—
110D4×C149—44no—
111C2×C4×C149—21no—
112C2×C4×C149C2×D412yes—
113C2×C4×C149C2411yescharacteristic
114C8×C149C2×D414yescomplemented
115C22×C149—41no—
116C22×C149—42no—
117C22×C149—44no—
118C22×C149—42no—
119C22×C149C22×D411yescharacteristic
120C2×D149—22no—
121C2×D149—41no—
122C2×D149—42no—
123C2×D149—44no—
124C2×D149—42no—
125C2×D149—88no—
126C2×D149#12712yes—
127C4×C149—22no—
128C4×C149—41no—
129C4×C149C22×D412yes—
130Dic149—41no—
131Dic149#12712yes—
132Dic149AGL(1,ℤ/8)14yescomplemented
133C2×C149—82no—
134C2×C149—81no—
135C2×C149C2×#12711yescharacteristic
136C2×C149C2×AGL(1,ℤ/8)12yes—
137D149—22no—
138D149—81no—
139D149—82no—
140C149#315111yescharacteristic, complemented, Sylow
141#3151—1491nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1492no—
143C2×AGL(1,ℤ/8)—1492no—
144#532—1491no—
145C2×#127—1491no—
146#496—1491no—
147#496—1491no—
148#496—1491no—
149#496D14911yescharacteristic, complemented
150#493—1492no—
151#492—1492no—
152C4×M16—1491no—
153C2×#41—1491no—
154C2×#41—1491no—
155C22×D4—2982no—
156C22×D4—1492no—
157AGL(1,ℤ/8)—2988no—
158AGL(1,ℤ/8)—2988no—
159C2×SD16—1492no—
160C2×SD16—1492no—
161C2×SD16—1492no—
162C2×SD16C2×D14912yescomplemented
163C2×D8—1492no—
164C2×D8—1492no—
165C2×D8—1492no—
166C2×D8C2×D14912yescomplemented
167C2×M16—1492no—
168C4⋊Q8—1491no—
169#127—2982no—
170#127—1492no—
171#127—1491no—
172#127—1491no—
173#127C2×D14911yescharacteristic
174#124—1491no—
175#124C2×D14911yescharacteristic, complemented
176#121—2982no—
177C4×D4—2982no—
178C2×C42—1491no—
179#97—1491no—
180#97C2×D14911yescharacteristic, complemented
181C4×C8—1492no—
182C24—2982no—
183#41—2984no—
184#41—2984no—
185C2×Q8—1491no—
186C2×Q8C22×D14911yescharacteristic
187C2×D4—22no—
188C2×D4—2981no—
189C2×D4—2981no—
190C2×D4—2984no—
191C2×D4—2984no—
192C2×D4—2982no—
193C2×D4—2982no—
194C2×D4—2984no—
195C2×D4—2982no—
196C2×D4—5968no—
197C2×D4—1491no—
198C2×D4—1492no—
199C2×D4C22×D14911yescharacteristic
200C2×D4C22×D14912yes—
201C22×C4—2981no—
202C22×C4—2981no—
203C22×C4—1492no—
204C22×C4—1491no—
205SD16—24no—
206SD16—2984no—
207SD16—2984no—
208SD16—2984no—
209D8—24no—
210D8—2984no—
211D8—2984no—
212D8—2984no—
213M16—1498no—
214C2×C8—1492no—
215C2×C8C22×D14912yes—
216C4⋊C4—2982no—
217C22⋊C4—22no—
218C22⋊C4—2982no—
219C42—1492no—
220C42—1491no—
221C42C22×D14911yescharacteristic
222C23—21no—
223C23—22no—
224C23—2981no—
225C23—2982no—
226C23—5968no—
227C23—5964no—
228C23—1491no—
229Q8—22no—
230Q8—2982no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—2982no—
236D4—2982no—
237D4—2982no—
238D4—5964no—
239D4—5962no—
240D4—5962no—
241D4—5964no—
242D4—5964no—
243D4—5968no—
244C2×C4—21no—
245C2×C4—2984no—
246C2×C4—2981no—
247C2×C4—5962no—
248C2×C4—5962no—
249C2×C4—1492no—
250C2×C4—1491no—
251C2×C4—1494no—
252C2×C4D4×D14912yes—
253C2×C4C23×D14911yescharacteristic
254C8—1494no—
255C8D4×D14914yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2982no—
261C22—5964no—
262C22—5962no—
263C22—5961no—
264C22—5962no—
265C22—1492no—
266C22—11928no—
267C22C2×D4×D14911yescharacteristic
268C4—22no—
269C4—41no—
270C4—5961no—
271C4—1494no—
272C4—1492no—
273C4C2×D4×D14912yes—
274C2—82no—
275C2—81no—
276C2—2982no—
277C2—11922no—
278C2—11921no—
279C2#127×D14911yescharacteristic
280C2AGL(1,ℤ/8)×D14912yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#496×C149C2
socleC22×C149C22×D4
hypercenter#496D149
hypocenterC149#3151
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#18011938C2semidirect
2. C2×#18011935C2semidirect
3. #23750174C2semidirect
4. #127×D149C2semidirect
5. #23750015C2semidirect
6. #23750014C2semidirect
7. #23750012C2semidirect
8. #23750000C2semidirect
9. #23749996C2semidirect
10. #23749577C2semidirect
11. #496×C149C2semidirect
18. C2×#14558725C22semidirect
19. C2×#14558724C22semidirect
20. C2×#14558723C22semidirect
21. C2×D1192C22semidirect
22. C2×#14558721C22semidirect
23. C2×#14558721C22semidirect
25. #18011909C22semidirect
29. #18011907C22semidirect
30. #18011899C22semidirect
34. #18011822C22semidirect
35. C8×Dic149C22semidirect
36. C2×SD16×C149C22semidirect
37. C2×D8×C149C22semidirect
39. #124×C149C22semidirect
40. #97×C149C22semidirect
66. #14558719D4semidirect
103. #11924746C2×D4semidirect
114. C8×C149C2×D4semidirect
132. Dic149AGL(1,ℤ/8)semidirect
140. C149#3151semidirect
149. #496D149semidirect
162. C2×SD16C2×D149semidirect
166. C2×D8C2×D149semidirect
175. #124C2×D149semidirect
180. #97C2×D149semidirect
255. C8D4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119382
C2×#180119352
#237501742
#127×D1492
#237500152
#237500142
#237500122
#237500002
#237499962
#237495772
#496×C1492
#3151149