Finite Group Databasebeta

← #34056486

Subgroups — #34056486

Order 19072.

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

normal subgroups only

This group has 460 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#23750436C212yesmaximal, complemented
2#23750435C212yesmaximal, complemented
3#23749997C211yesmaximal, characteristic, complemented
4#23749887C212yesmaximal, complemented
5#23749886C212yesmaximal, complemented
6#23749809C212yesmaximal, complemented
7#23749792C211yesmaximal, characteristic, complemented
8#23749739C212yesmaximal, complemented
9#453×C149C211yesmaximal, characteristic, complemented
10#18011958C2212yes—
11#18011956C2211yescharacteristic, complemented
12#18011944C2212yescomplemented
13#18011941—22no—
14#18011941C2212yescomplemented
15#18011940—22no—
16#18011937—22no—
17#18011937C2212yes—
18#18011935—22no—
19#18011931—22no—
20#18011931C2212yes—
21#18011930—22no—
22C2×#14558724—22no—
23C2×#14558724C2212yescomplemented
24C2×#14558724C2212yescomplemented
25SD16×D149—22no—
26#18011919C2212yescomplemented
27#18011918C2212yescomplemented
28#18011911C2211yescharacteristic, complemented
29#18011846C2212yescomplemented
30#18011839C2212yescomplemented
31#18011829C2212yescomplemented
32#18011828C2211yescharacteristic, complemented
33C4×#11924746C2211yescharacteristic, complemented
34H(5,𝔽2)×C149C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C149C2212yes—
36#127×C149C2211yescharacteristic
37#104×C149C2212yescomplemented
38#100×C149C2211yescharacteristic, complemented
39#14558736—22no—
40#14558736C2312yes—
41#14558735—22no—
42#14558735—22no—
43#14558735—42no—
44C2×Dic298—22no—
45C2×Dic298C2312yes—
46Q8×D149—22no—
47C2×C22⋊D149—41no—
48D4×D149—22no—
49Dic596—22no—
50#14558726—22no—
51#14558726D412yescomplemented
52#14558725—22no—
53#14558724—22no—
54#14558724—22no—
55#14558724—22no—
56#14558724—22no—
57#14558724—22no—
58#14558724—22no—
59#14558724D412yescomplemented
60#14558723D412yescomplemented
61#14558721—22no—
62#14558721D412yescomplemented
63#14558720—22no—
64#14558720C2312yes—
65#14558719—22no—
66C2×#11924746—22no—
67C2×#11924746C2312yes—
68C8×D149—22no—
69C4⋊Dic149—41no—
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—41no—
82Dic298—22no—
83Dic298—22no—
84Dic298—42no—
85Dic298C2×D412yes—
86D596C2×D412yes—
87C22⋊D149—22no—
88C22⋊D149—42no—
89C22⋊D149—42no—
90C2×Dic149—22no—
91C2×Dic149—41no—
92C2×Dic149—42no—
93C4×D149—22no—
94C4×D149—42no—
95#11924746—22no—
96#11924746C2×D412yes—
97#11924746C2×D412yes—
98C23×C149—21no—
99C23×C149—21no—
100C23×C149—41no—
101Q8×C149C2×D412yes—
102D4×C149—22no—
103D4×C149—22no—
104D4×C149—22no—
105D4×C149—42no—
106D4×C149—82no—
107D4×C149C2×D412yes—
108C2×C4×C149—21no—
109C2×C4×C149—22no—
110C2×C4×C149—41no—
111C2×C4×C149C2411yescharacteristic
112C8×C149—22no—
113C22×C149—22no—
114C22×C149—41no—
115C22×C149—41no—
116C22×C149—81no—
117C22×C149—81no—
118C22×C149—161no—
119C22×C149H(5,𝔽2)11yescharacteristic
120C2×D149—22no—
121C2×D149—161no—
122C4×C149—22no—
123C4×C149—22no—
124C4×C149C22×D412yes—
125Dic149—22no—
126Dic149—42no—
127C2×C149—21no—
128C2×C149—42no—
129C2×C149—81no—
130C2×C149—81no—
131C2×C149D4211yescharacteristic
132D149—42no—
133C149#329111yescharacteristic, complemented, Sylow
134#3291—1491nomaximal, Sylow
135#577—1492no—
136#577—1492no—
137#492—1491no—
138#471—1492no—
139#471—1492no—
140#456—1492no—
141#453—1491no—
142#453D14911yescharacteristic, complemented
143#443—1492no—
1442−1+4—1492no—
145H(5,𝔽2)—1491no—
146H(5,𝔽2)C2×D14911yescharacteristic, complemented
147#137—2982no—
148#137—2982no—
149#137—1492no—
150#137—1492no—
151AGL(1,ℤ/8)—2982no—
152AGL(1,ℤ/8)—2982no—
153AGL(1,ℤ/8)—1492no—
154AGL(1,ℤ/8)C2×D14912yes—
155#135—2982no—
156#135—2982no—
157#135—1492no—
158C2×SD16—2982no—
159C2×SD16—2982no—
160C2×SD16—1492no—
161C2×SD16—1492no—
162#131—1492no—
163#131—1492no—
164C4⋊Q8—1491no—
165#127C2×D14911yescharacteristic
166#108—1492no—
167#104—1492no—
168#104C2×D14912yescomplemented
169#101—1492no—
170#100—1491no—
171#100C2×D14911yescharacteristic, complemented
172C4×C8—1491no—
173#41—22no—
174#41—2982no—
175#41—2982no—
176#41—2982no—
177#41—5962no—
178#41—1492no—
179#41C22×D14912yes—
180C2×Q8—2982no—
181C2×Q8—2982no—
182C2×Q8—1492no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2982no—
187C2×D4—5961no—
188C2×D4C22×D14911yescharacteristic
189C2×D4C22×D14911yescharacteristic
190Q16—2982no—
191Q16—2982no—
192Q16—1492no—
193SD16—22no—
194SD16—2982no—
195SD16—2982no—
196SD16—2982no—
197SD16—2982no—
198SD16—2982no—
199SD16—2982no—
200SD16—2982no—
201SD16—1492no—
202SD16—1492no—
203D8—22no—
204D8—2982no—
205D8—1492no—
206M16—2982no—
207M16—2982no—
208M16—1492no—
209M16C22×D14912yes—
210C2×C8—2982no—
211C2×C8—2982no—
212C2×C8—1492no—
213C4⋊C4—5961no—
214C42C22×D14911yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—5961no—
219Q8—2982no—
220Q8—2982no—
221Q8—5962no—
222Q8—1492no—
223Q8D4×D14912yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—2982no—
230D4—5962no—
231D4—5962no—
232D4—1492no—
233D4D4×D14912yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—2982no—
238C2×C4—2982no—
239C2×C4—5962no—
240C2×C4—5961no—
241C2×C4—5962no—
242C2×C4C23×D14911yescharacteristic
243C8—22no—
244C8—2982no—
245C8—1492no—
246C8—1492no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—2982no—
254C22—23841no—
255C22#1801195611yescharacteristic
256C4—22no—
257C4—22no—
258C4—2982no—
259C4—5962no—
260C4C2×D4×D14912yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—5962no—
266C2#2375025711yescharacteristic
subgroupG/H
centerC2#23750257
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#453×C149C2
socleC2×C149D42
hypercenter#453D149
hypocenterC149#3291
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. #23750436C2semidirect
2. #23750435C2semidirect
3. #23749997C2semidirect
4. #23749887C2semidirect
5. #23749886C2semidirect
6. #23749809C2semidirect
7. #23749792C2semidirect
8. #23749739C2semidirect
9. #453×C149C2semidirect
11. #18011956C22semidirect
12. #18011944C22semidirect
14. #18011941C22semidirect
23. C2×#14558724C22semidirect
24. C2×#14558724C22semidirect
26. #18011919C22semidirect
27. #18011918C22semidirect
28. #18011911C22semidirect
29. #18011846C22semidirect
30. #18011839C22semidirect
31. #18011829C22semidirect
32. #18011828C22semidirect
33. C4×#11924746C22semidirect
34. H(5,𝔽2)×C149C22semidirect
37. #104×C149C22semidirect
38. #100×C149C22semidirect
51. #14558726D4semidirect
59. #14558724D4semidirect
60. #14558723D4semidirect
62. #14558721D4semidirect
133. C149#3291semidirect
142. #453D149semidirect
146. H(5,𝔽2)C2×D149semidirect
168. #104C2×D149semidirect
171. #100C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504362
#237504352
#237499972
#237498872
#237498862
#237498092
#237497922
#237497392
#453×C1492
#3291149