Finite Group Databasebeta

← #34056481

Subgroups — #34056481

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#23750434C212yesmaximal, complemented
2#23750433C212yesmaximal, complemented
3#23749999C211yesmaximal, characteristic, complemented
4#23749891C212yesmaximal, complemented
5#23749890C212yesmaximal, complemented
6#23749807C211yesmaximal, characteristic, complemented
7#23749794C212yesmaximal, complemented
8#23749739C212yesmaximal, complemented
9#456×C149C211yesmaximal, characteristic, complemented
10#18011959C2211yescharacteristic
11#18011957C2212yescomplemented
12#18011943—22no—
13#18011942—22no—
14#18011942C2212yescomplemented
15#18011939C2212yes—
16#18011938—22no—
17#18011936—22no—
18#18011936C2212yes—
19#18011934—22no—
20#18011931—22no—
21#18011931C2212yes—
22C2×#14558723—22no—
23C2×#14558723C2212yescomplemented
24C2×#14558723C2212yescomplemented
25SD16×D149—22no—
26#18011919C2212yescomplemented
27#18011918C2212yescomplemented
28#18011908C2211yescharacteristic
29#18011846C2212yescomplemented
30#18011839C2212yescomplemented
31#18011830C2211yescharacteristic, complemented
32#18011827C2212yescomplemented
33C4×#11924746C2211yescharacteristic, complemented
342−1+4×C149C2211yescharacteristic
35#137×C149C2212yescomplemented
36C4⋊Q8×C149C2211yescharacteristic, complemented
37#104×C149C2212yescomplemented
38#101×C149C2211yescharacteristic, complemented
39#14558737—22no—
40#14558737—22no—
41#14558736—22no—
42#14558736—41no—
43#14558736C2312yes—
44Q8×D149—22no—
45C2×D596—22no—
46C2×D596—41no—
47C2×D596C2312yes—
48D4×D149—22no—
49D4×D149—42no—
50#14558726—22no—
51#14558726D412yescomplemented
52#14558725—22no—
53#14558724D412yescomplemented
54#14558723—22no—
55#14558723—22no—
56#14558723—22no—
57#14558723—22no—
58#14558723—22no—
59#14558723—22no—
60#14558723D412yescomplemented
61D1192—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×C149—41no—
72#41×C149C2312yes—
73C2×Q8×C149—22no—
74C2×Q8×C149C2311yescharacteristic
75C2×Q8×C149C2311yescharacteristic
76Q16×C149—22no—
77SD16×C149—22no—
78M16×C149C2312yes—
79C4⋊C4×C149—41no—
80C42×C149C2311yescharacteristic
81C22×D149—22no—
82C22×D149—42no—
83Dic298—41no—
84Dic298C2×D412yes—
85D596—22no—
86D596—22no—
87D596—42no—
88D596—41no—
89D596—82no—
90D596C2×D412yes—
91C22⋊D149—22no—
92C22⋊D149—42no—
93C4×D149—22no—
94C4×D149—42no—
95C4×D149—42no—
96#11924746—22no—
97#11924746C2×D412yes—
98#11924746C2×D412yes—
99Q8×C149—22no—
100Q8×C149—22no—
101Q8×C149—41no—
102Q8×C149C2×D412yes—
103D4×C149—22no—
104D4×C149—41no—
105D4×C149C2×D412yes—
106C2×C4×C149—21no—
107C2×C4×C149—22no—
108C2×C4×C149—21no—
109C2×C4×C149—21no—
110C2×C4×C149—42no—
111C2×C4×C149C2411yescharacteristic
112C8×C149—22no—
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×C149—41no—
122C4×C149—41no—
123C4×C149C22×D412yes—
124Dic149—22no—
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—1491no—
138#456D14911yescharacteristic, complemented
139#453—1492no—
140#443—1492no—
1412−1+4—1491no—
1422−1+4C2×D14911yescharacteristic
143H(5,𝔽2)—1492no—
144#137—2982no—
145#137—2982no—
146#137—1492no—
147#137C2×D14912yescomplemented
148AGL(1,ℤ/8)—2982no—
149AGL(1,ℤ/8)—2982no—
150AGL(1,ℤ/8)—1492no—
151AGL(1,ℤ/8)—1492no—
152#135—2982no—
153#135—2982no—
154#135—1492no—
155C2×SD16—2982no—
156C2×SD16—2982no—
157C2×SD16—1492no—
158C2×SD16—1492no—
159#131—1492no—
160#131—1492no—
161C4⋊Q8C2×D14911yescharacteristic, complemented
162#127—1491no—
163#108—1492no—
164#104—1492no—
165#104C2×D14912yescomplemented
166#101—1491no—
167#101C2×D14911yescharacteristic, complemented
168#100—1492no—
169C4×C8—1491no—
170#41—22no—
171#41—41no—
172#41—2982no—
173#41—2982no—
174#41—2982no—
175#41—5961no—
176#41—1492no—
177#41C22×D14912yes—
178C2×Q8—22no—
179C2×Q8—2982no—
180C2×Q8C22×D14911yescharacteristic
181C2×Q8C22×D14911yescharacteristic
182C2×D4—2982no—
183C2×D4—2982no—
184C2×D4—5961no—
185C2×D4—5962no—
186C2×D4—1492no—
187Q16—22no—
188Q16—2982no—
189Q16—1492no—
190SD16—22no—
191SD16—2982no—
192SD16—2982no—
193SD16—2982no—
194SD16—2982no—
195SD16—2982no—
196SD16—2982no—
197SD16—2982no—
198SD16—1492no—
199SD16—1492no—
200D8—2982no—
201D8—2982no—
202D8—1492no—
203M16—2982no—
204M16—2982no—
205M16—1492no—
206M16C22×D14912yes—
207C2×C8—2982no—
208C2×C8—2982no—
209C2×C8—1492no—
210C4⋊C4—41no—
211C42C22×D14911yescharacteristic
212C23—2982no—
213C23—5962no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—5961no—
218Q8—1492no—
219Q8D4×D14912yes—
220D4—22no—
221D4—41no—
222D4—2982no—
223D4—2982no—
224D4—2982no—
225D4—5962no—
226D4—5961no—
227D4—5962no—
228D4—1492no—
229D4—11922no—
230D4D4×D14912yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—2982no—
237C2×C4—5962no—
238C2×C4—5962no—
239C2×C4C23×D14911yescharacteristic
240C8—22no—
241C8—2982no—
242C8—1492no—
243C8—1492no—
244C22—22no—
245C22—2982no—
246C22—23842no—
247C22—5962no—
248C22—11922no—
249C22#1801195611yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2982no—
255C4C2×D4×D14912yes—
256C2—21no—
257C2—42no—
258C2—5962no—
259C2—11922no—
260C2#2375025711yescharacteristic
subgroupG/H
centerC2#23750257
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#456×C149C2
socleC2×C149D42
hypercenter#456D149
hypocenterC149#3291
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23750434C2semidirect
2. #23750433C2semidirect
3. #23749999C2semidirect
4. #23749891C2semidirect
5. #23749890C2semidirect
6. #23749807C2semidirect
7. #23749794C2semidirect
8. #23749739C2semidirect
9. #456×C149C2semidirect
11. #18011957C22semidirect
14. #18011942C22semidirect
23. C2×#14558723C22semidirect
24. C2×#14558723C22semidirect
26. #18011919C22semidirect
27. #18011918C22semidirect
29. #18011846C22semidirect
30. #18011839C22semidirect
31. #18011830C22semidirect
32. #18011827C22semidirect
33. C4×#11924746C22semidirect
35. #137×C149C22semidirect
36. C4⋊Q8×C149C22semidirect
37. #104×C149C22semidirect
38. #101×C149C22semidirect
51. #14558726D4semidirect
53. #14558724D4semidirect
60. #14558723D4semidirect
63. #14558721D4semidirect
130. C149#3291semidirect
138. #456D149semidirect
147. #137C2×D149semidirect
161. C4⋊Q8C2×D149semidirect
165. #104C2×D149semidirect
167. #101C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504342
#237504332
#237499992
#237498912
#237498902
#237498072
#237497942
#237497392
#456×C1492
#3291149