Finite Group Databasebeta

← #34056487

Subgroups — #34056487

Order 19072.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#23750429C212yesmaximal, complemented
2#23750428C212yesmaximal, complemented
3#23750001C211yesmaximal, characteristic, complemented
4#23749894C212yesmaximal, complemented
5#23749893C212yesmaximal, complemented
6#23749794C212yesmaximal, complemented
7#23749792C211yesmaximal, characteristic, complemented
8#23749739C212yesmaximal, complemented
9#453×C149C211yesmaximal, characteristic, complemented
10#18011957C2212yescomplemented
11#18011956C2211yescharacteristic, complemented
12#18011939C2212yescomplemented
13#18011938—22no—
14#18011937—22no—
15#18011937C2212yescomplemented
16#18011936—22no—
17#18011936C2212yescomplemented
18#18011935—22no—
19#18011932—22no—
20#18011931—22no—
21#18011931C2212yes—
22C2×#14558721—22no—
23C2×#14558721C2212yescomplemented
24C2×#14558721C2212yescomplemented
25D8×D149—22no—
26#18011919C2212yescomplemented
27#18011918C2212yescomplemented
28#18011908C2211yescharacteristic, complemented
29#18011846C2212yescomplemented
30#18011839C2212yescomplemented
31#18011828C2211yescharacteristic, complemented
32#18011827C2212yescomplemented
33C4×#11924746C2211yescharacteristic, complemented
34H(5,𝔽2)×C149C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C149C2212yescomplemented
36#127×C149C2211yescharacteristic, complemented
37#104×C149C2212yescomplemented
38#100×C149C2211yescharacteristic, complemented
39#14558737—22no—
40#14558736—22no—
41#14558736C2312yes—
42#14558735—22no—
43C2×D596—22no—
44C2×D596—41no—
45C2×D596C2312yes—
46C2×C22⋊D149—41no—
47D4×D149—22no—
48D4×D149—22no—
49D4×D149—42no—
50#14558726D412yescomplemented
51#14558725—22no—
52#14558724—22no—
53#14558724D412yescomplemented
54#14558723—22no—
55#14558723D412yescomplemented
56D1192—22no—
57#14558721—22no—
58#14558721—22no—
59#14558721—22no—
60#14558721—22no—
61#14558721—22no—
62#14558721—22no—
63#14558721D412yescomplemented
64#14558720—22no—
65#14558720C2312yes—
66#14558719—22no—
67C2×#11924746—22no—
68C2×#11924746C2312yes—
69C8×D149—22no—
70#41×C149—22no—
71#41×C149C2312yes—
72C2×D4×C149—22no—
73C2×D4×C149—41no—
74C2×D4×C149—41no—
75C2×D4×C149C2311yescharacteristic
76C2×D4×C149C2311yescharacteristic
77SD16×C149—22no—
78D8×C149—22no—
79M16×C149C2312yes—
80C42×C149C2311yescharacteristic
81C22×D149—22no—
82C22×D149—41no—
83C22×D149—42no—
84Dic298C2×D412yes—
85D596—22no—
86D596—22no—
87D596—42no—
88D596—82no—
89D596C2×D412yes—
90C22⋊D149—22no—
91C22⋊D149—42no—
92C22⋊D149—42no—
93C2×Dic149—41no—
94C4×D149—22no—
95C4×D149—42no—
96#11924746—22no—
97#11924746C2×D412yes—
98#11924746C2×D412yes—
99C23×C149—21no—
100C23×C149—21no—
101C23×C149—41no—
102Q8×C149C2×D412yes—
103D4×C149—22no—
104D4×C149—22no—
105D4×C149—22no—
106D4×C149—42no—
107D4×C149—82no—
108D4×C149C2×D412yes—
109C2×C4×C149—21no—
110C2×C4×C149—22no—
111C2×C4×C149—41no—
112C2×C4×C149C2411yescharacteristic
113C8×C149—22no—
114C22×C149—22no—
115C22×C149—41no—
116C22×C149—41no—
117C22×C149—81no—
118C22×C149—81no—
119C22×C149—161no—
120C22×C149H(5,𝔽2)11yescharacteristic
121C2×D149—22no—
122C2×D149—42no—
123C2×D149—82no—
124C2×D149—161no—
125C2×D149—162no—
126C4×C149—22no—
127C4×C149—22no—
128C4×C149C22×D412yes—
129Dic149—22no—
130C2×C149—21no—
131C2×C149—42no—
132C2×C149—81no—
133C2×C149—81no—
134C2×C149D4211yescharacteristic
135D149—42no—
136D149—82no—
137C149#329211yescharacteristic, complemented, Sylow
138#3292—1491nomaximal, Sylow
139#576—1492no—
140#576—1492no—
141#493—1491no—
142#472—1492no—
143#472—1492no—
144#453—1491no—
145#453—1492no—
146#453D14911yescharacteristic, complemented
147#443—1492no—
148H(5,𝔽2)—1492no—
149H(5,𝔽2)—1491no—
150H(5,𝔽2)C2×D14911yescharacteristic, complemented
151AGL(1,ℤ/8)—2982no—
152AGL(1,ℤ/8)—2982no—
153AGL(1,ℤ/8)—2982no—
154AGL(1,ℤ/8)—2982no—
155AGL(1,ℤ/8)—1492no—
156AGL(1,ℤ/8)—1492no—
157AGL(1,ℤ/8)—1492no—
158AGL(1,ℤ/8)C2×D14912yescomplemented
159#135—2982no—
160#135—2982no—
161#135—1492no—
162C2×D8—2982no—
163C2×D8—2982no—
164C2×D8—1492no—
165C2×D8—1492no—
166#131—1492no—
167#131—1492no—
168#127—1491no—
169#127C2×D14911yescharacteristic, complemented
170#108—1492no—
171#104—1492no—
172#104C2×D14912yescomplemented
173#100—1491no—
174#100—1492no—
175#100C2×D14911yescharacteristic, complemented
176C4×C8—1491no—
177#41—22no—
178#41—2982no—
179#41—2982no—
180#41—2982no—
181#41—1492no—
182#41C22×D14912yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2982no—
187C2×D4—2982no—
188C2×D4—2982no—
189C2×D4—5961no—
190C2×D4—5962no—
191C2×D4—5961no—
192C2×D4—1492no—
193C2×D4C22×D14911yescharacteristic
194C2×D4C22×D14911yescharacteristic
195Q16—1492no—
196SD16—22no—
197SD16—2982no—
198SD16—2982no—
199SD16—2982no—
200SD16—1492no—
201SD16—1492no—
202D8—22no—
203D8—2982no—
204D8—2982no—
205D8—2982no—
206D8—2982no—
207D8—2982no—
208D8—2982no—
209D8—2982no—
210D8—1492no—
211M16—2982no—
212M16—2982no—
213M16—1492no—
214M16C22×D14912yes—
215C2×C8—2982no—
216C2×C8—2982no—
217C2×C8—1492no—
218C42C22×D14911yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—2982no—
223C23—5962no—
224C23—5961no—
225Q8—1492no—
226Q8D4×D14912yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2982no—
233D4—2982no—
234D4—2982no—
235D4—5962no—
236D4—5962no—
237D4—5962no—
238D4—1492no—
239D4—11922no—
240D4D4×D14912yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—2982no—
245C2×C4—5962no—
246C2×C4—5961no—
247C2×C4C23×D14911yescharacteristic
248C8—22no—
249C8—2982no—
250C8—1492no—
251C8—1492no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—2982no—
259C22—23842no—
260C22—23841no—
261C22—5962no—
262C22—11922no—
263C22#1801195611yescharacteristic
264C4—22no—
265C4—22no—
266C4—2982no—
267C4C2×D4×D14912yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—5962no—
273C2—11922no—
274C2#2375025711yescharacteristic
subgroupG/H
centerC2#23750257
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#453×C149C2
socleC2×C149D42
hypercenter#453D149
hypocenterC149#3292
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23750429C2semidirect
2. #23750428C2semidirect
3. #23750001C2semidirect
4. #23749894C2semidirect
5. #23749893C2semidirect
6. #23749794C2semidirect
7. #23749792C2semidirect
8. #23749739C2semidirect
9. #453×C149C2semidirect
10. #18011957C22semidirect
11. #18011956C22semidirect
12. #18011939C22semidirect
15. #18011937C22semidirect
17. #18011936C22semidirect
23. C2×#14558721C22semidirect
24. C2×#14558721C22semidirect
26. #18011919C22semidirect
27. #18011918C22semidirect
28. #18011908C22semidirect
29. #18011846C22semidirect
30. #18011839C22semidirect
31. #18011828C22semidirect
32. #18011827C22semidirect
33. C4×#11924746C22semidirect
34. H(5,𝔽2)×C149C22semidirect
35. AGL(1,ℤ/8)×C149C22semidirect
36. #127×C149C22semidirect
37. #104×C149C22semidirect
38. #100×C149C22semidirect
50. #14558726D4semidirect
53. #14558724D4semidirect
55. #14558723D4semidirect
63. #14558721D4semidirect
137. C149#3292semidirect
146. #453D149semidirect
150. H(5,𝔽2)C2×D149semidirect
158. AGL(1,ℤ/8)C2×D149semidirect
169. #127C2×D149semidirect
172. #104C2×D149semidirect
175. #100C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504292
#237504282
#237500012
#237498942
#237498932
#237497942
#237497922
#237497392
#453×C1492
#3292149