Finite Group Databasebeta

← #34056485

Subgroups — #34056485

Order 19072.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#23750437C212yesmaximal, complemented
2#23750432C212yesmaximal, complemented
3#23749998C211yesmaximal, characteristic, complemented
4#23749889C212yesmaximal, complemented
5#23749888C212yesmaximal, complemented
6#23749809C212yesmaximal, complemented
7#23749794C212yesmaximal, complemented
8#23749743C211yesmaximal, characteristic, complemented
9#443×C149C211yesmaximal, characteristic, complemented
10#18011958C2212yes—
11#18011957C2212yescomplemented
12#18011944—22no—
13#18011944C2212yescomplemented
14#18011943—22no—
15#18011941C2212yescomplemented
16#18011939—22no—
17#18011939C2212yes—
18#18011936C2212yes—
19#18011935—22no—
20#18011933—22no—
21#18011933C2211yescharacteristic
22#18011932—22no—
23C2×#14558725—22no—
24C2×#14558725C2212yescomplemented
25C2×#14558725C2212yescomplemented
26SD16×D149—22no—
27#18011917C2212yescomplemented
28#18011911C2211yescharacteristic, complemented
29#18011908C2211yescharacteristic
30#18011847C2211yescharacteristic, complemented
31#18011839C2212yescomplemented
32#18011829C2212yescomplemented
33#18011827C2212yescomplemented
34#135×C149C2211yescharacteristic
35#131×C149C2212yescomplemented
36#108×C149C2211yescharacteristic, complemented
37#104×C149C2212yescomplemented
38C4×C8×C149C2211yescharacteristic, complemented
39#14558737—22no—
40#14558736—22no—
41#14558736—22no—
42#14558736C2312yes—
43#14558735—22no—
44#14558735—42no—
45C2×Dic298—22no—
46C2×Dic298C2312yes—
47Q8×D149—22no—
48C2×D596—22no—
49C2×D596—41no—
50C2×D596C2312yes—
51D4×D149—22no—
52D4×D149—42no—
53Dic596—22no—
54Dic596D411yescharacteristic, complemented
55#14558726—22no—
56#14558725—22no—
57#14558725—22no—
58#14558725—22no—
59#14558725—22no—
60#14558725—22no—
61#14558725—22no—
62#14558725D412yescomplemented
63#14558724—22no—
64#14558723—22no—
65D1192—22no—
66D1192D411yescharacteristic, complemented
67#14558721—22no—
68#14558720C2312yes—
69#14558719—22no—
70C8×D149—22no—
71C4⋊Dic149—41no—
72#41×C149C2312yes—
73Q16×C149D411yescharacteristic, complemented
74SD16×C149D412yescomplemented
75D8×C149D411yescharacteristic, complemented
76M16×C149—22no—
77M16×C149C2312yes—
78C2×C8×C149—22no—
79C2×C8×C149C2311yescharacteristic
80C2×C8×C149C2311yescharacteristic
81C42×C149C2311yescharacteristic
82C22×D149—22no—
83C22×D149—42no—
84Dic298—22no—
85Dic298—22no—
86Dic298—42no—
87Dic298C2×D412yes—
88D596—22no—
89D596—22no—
90D596—42no—
91D596—82no—
92D596C2×D412yes—
93C22⋊D149—22no—
94C22⋊D149—42no—
95C22⋊D149—42no—
96C2×Dic149—22no—
97C2×Dic149—42no—
98C4×D149—22no—
99C4×D149—42no—
100C4×D149—42no—
101#11924746—22no—
102Q8×C149C2×D412yes—
103D4×C149—22no—
104D4×C149C2×D412yes—
105C2×C4×C149—21no—
106C2×C4×C149—22no—
107C2×C4×C149C2411yescharacteristic
108C8×C149—22no—
109C8×C149C2×D411yescharacteristic
110C8×C149C2×D411yescharacteristic
111C8×C149C2×D411yescharacteristic
112C8×C149C2×D411yescharacteristic
113C22×C149—22no—
114C22×C149H(5,𝔽2)11yescharacteristic
115C2×D149—22no—
116C2×D149—42no—
117C2×D149—82no—
118C2×D149—162no—
119C4×C149—22no—
120C4×C149—22no—
121C4×C149C22×D411yescharacteristic
122C4×C149C22×D411yescharacteristic
123Dic149—22no—
124Dic149—42no—
125C2×C149—21no—
126C2×C149—42no—
127C2×C149D4211yescharacteristic
128D149—42no—
129D149—82no—
130C149#329111yescharacteristic, complemented, Sylow
131#3291—1491nomaximal, Sylow
132#577—1492no—
133#577—1492no—
134#492—1491no—
135#471—1492no—
136#471—1492no—
137#456—1492no—
138#453—1492no—
139#443—1491no—
140#443D14911yescharacteristic, complemented
1412−1+4—1492no—
142H(5,𝔽2)—1492no—
143#137—2982no—
144#137—2982no—
145#137—1492no—
146#137—1492no—
147AGL(1,ℤ/8)—2982no—
148AGL(1,ℤ/8)—2982no—
149AGL(1,ℤ/8)—1492no—
150AGL(1,ℤ/8)—1492no—
151#135—2982no—
152#135—2982no—
153#135—1491no—
154#135C2×D14911yescharacteristic
155C2×SD16—2982no—
156C2×SD16—2982no—
157C2×SD16—1492no—
158C2×SD16—1492no—
159#131—1492no—
160#131C2×D14912yescomplemented
161C4⋊Q8—1491no—
162#127—1491no—
163#108—1491no—
164#108C2×D14911yescharacteristic, complemented
165#104—1492no—
166#104C2×D14912yescomplemented
167#101—1492no—
168#100—1492no—
169C4×C8C2×D14911yescharacteristic, complemented
170#41—2982no—
171#41—2982no—
172#41—2982no—
173#41—2982no—
174#41—5962no—
175#41—1492no—
176#41C22×D14912yes—
177C2×Q8—2982no—
178C2×Q8—2982no—
179C2×Q8—1492no—
180C2×D4—2982no—
181C2×D4—2982no—
182C2×D4—5961no—
183C2×D4—5962no—
184C2×D4—1492no—
185Q16—2982no—
186Q16—2982no—
187Q16—1491no—
188Q16D59611yescharacteristic, complemented
189SD16—2982no—
190SD16—2982no—
191SD16—2982no—
192SD16—2982no—
193SD16—2982no—
194SD16—2982no—
195SD16—2982no—
196SD16—2982no—
197SD16—1492no—
198SD16D59612yescomplemented
199D8—2982no—
200D8—2982no—
201D8—1491no—
202D8D59611yescharacteristic, complemented
203M16—22no—
204M16—2982no—
205M16—1492no—
206M16C22×D14912yes—
207C2×C8—22no—
208C2×C8—2982no—
209C2×C8C22×D14911yescharacteristic
210C2×C8C22×D14911yescharacteristic
211C4⋊C4—5961no—
212C42C22×D14911yescharacteristic
213C23—2982no—
214C23—5962no—
215Q8—2982no—
216Q8—2982no—
217Q8—5962no—
218Q8—1492no—
219Q8C2×D59612yes—
220D4—22no—
221D4—2982no—
222D4—2982no—
223D4—2982no—
224D4—5962no—
225D4—5962no—
226D4—5962no—
227D4—1492no—
228D4—11922no—
229D4C2×D59612yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—5962no—
235C2×C4—5962no—
236C2×C4—5962no—
237C2×C4C23×D14911yescharacteristic
238C8—22no—
239C8—2982no—
240C8D4×D14911yescharacteristic
241C8D4×D14911yescharacteristic
242C8C2×D59611yescharacteristic
243C8C2×D59611yescharacteristic
244C22—22no—
245C22—2982no—
246C22—23842no—
247C22—5962no—
248C22—11922no—
249C22#1801195711yescharacteristic
250C4—22no—
251C4—22no—
252C4—2982no—
253C4—5962no—
254C4C2×D4×D14911yescharacteristic
255C4C22×D59611yescharacteristic
256C2—21no—
257C2—42no—
258C2—5962no—
259C2—11922no—
260C2D4×D59611yescharacteristic
subgroupG/H
centerC2D4×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#443×C149C2
socleC2×C149D42
hypercenter#443D149
hypocenterC149#3291
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #23750437C2semidirect
2. #23750432C2semidirect
3. #23749998C2semidirect
4. #23749889C2semidirect
5. #23749888C2semidirect
6. #23749809C2semidirect
7. #23749794C2semidirect
8. #23749743C2semidirect
9. #443×C149C2semidirect
11. #18011957C22semidirect
13. #18011944C22semidirect
15. #18011941C22semidirect
24. C2×#14558725C22semidirect
25. C2×#14558725C22semidirect
27. #18011917C22semidirect
28. #18011911C22semidirect
30. #18011847C22semidirect
31. #18011839C22semidirect
32. #18011829C22semidirect
33. #18011827C22semidirect
35. #131×C149C22semidirect
36. #108×C149C22semidirect
37. #104×C149C22semidirect
38. C4×C8×C149C22semidirect
54. Dic596D4semidirect
62. #14558725D4semidirect
66. D1192D4semidirect
73. Q16×C149D4semidirect
74. SD16×C149D4semidirect
75. D8×C149D4semidirect
130. C149#3291semidirect
140. #443D149semidirect
160. #131C2×D149semidirect
164. #108C2×D149semidirect
166. #104C2×D149semidirect
169. C4×C8C2×D149semidirect
188. Q16D596semidirect
198. SD16D596semidirect
202. D8D596semidirect

Maximal subgroups

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

subgroupindex
#237504372
#237504322
#237499982
#237498892
#237498882
#237498092
#237497942
#237497432
#443×C1492
#3291149