Finite Group Databasebeta

← #34056921

Subgroups — #34056921

Order 19072.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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#23750349C211yesmaximal, characteristic, complemented
2#23750260C211yesmaximal, characteristic
3#23749948C212yesmaximal, complemented
4#23749902C211yesmaximal, characteristic, complemented
5#23749870C212yesmaximal, complemented
6#23749822C211yesmaximal, characteristic, complemented
7#23749777C212yesmaximal, complemented
8C4×#14558725C211yesmaximal, characteristic, complemented
9#23749704C211yesmaximal, characteristic, complemented
10#23749628C212yesmaximal, complemented
11#486×C149C211yesmaximal, characteristic, complemented
12C2×D4×D149—21no—
13C2×D4×D149C2212yes—
14C2×#14558725C2211yescharacteristic
15C2×#14558724C2212yescomplemented
16C2×#14558719C2212yescomplemented
17#18011906—22no—
18#18011900—21no—
19#18011891C2212yescomplemented
20#18011882—21no—
21#18011882C2212yes—
22#18011879—22no—
23Q8×Dic149—21no—
24C4×Dic298C2211yescharacteristic, complemented
25C4×D596C2211yescharacteristic
26C4×D596C2211yescharacteristic
27#18011869C2212yescomplemented
28#18011844C2212yescomplemented
29#18011843C2211yescharacteristic, complemented
30C4⋊#11924746C2211yescharacteristic, complemented
31#18011836C2211yescharacteristic, complemented
32#18011833C2211yescharacteristic
33#18011832C2212yescomplemented
34#18011831C2212yes—
35#18011823C2212yescomplemented
36C4⋊Q8×C149C2211yescharacteristic, complemented
37#127×C149C2211yescharacteristic
38#102×C149C2212yescomplemented
39#102×C149C2212yescomplemented
40C4×C8×C149C2211yescharacteristic, complemented
41C23×D149—22no—
42C2×Dic298C2311yescharacteristic
43C2×D596D411yescharacteristic
44C2×D596C2311yescharacteristic
45C2×C22⋊D149—22no—
46C2×C22⋊D149—22no—
47D4×D149—24no—
48D4×D149—42no—
49D4×D149—44no—
50C2×C4×D149—21no—
51C2×C4×D149C2312yes—
52#14558725—22no—
53#14558724—42no—
54#14558719—42no—
55C2×#11924746C2312yes—
56C4⋊Dic149—22no—
57C4⋊Dic149D411yescharacteristic
58C4⋊Dic149C2311yescharacteristic
59C4⋊Dic149C2311yescharacteristic
60#14558715—21no—
61C22⋊Dic149—22no—
62#14558713—22no—
63#14558713—22no—
64#14558713—21no—
65#14558713D412yes—
66C4×Dic149—22no—
67C4×Dic149—21no—
68C2×Q8×C149—21no—
69C2×D4×C149—21no—
70C2×D4×C149—21no—
71C2×D4×C149C2312yes—
72C2×C8×C149C2311yescharacteristic
73C2×C8×C149C2311yescharacteristic
74C4⋊C4×C149—21no—
75C4⋊C4×C149C2312yes—
76C42×C149C2311yescharacteristic
77C22×D149—21no—
78C22×D149—44no—
79C22×D149—44no—
80C22×D149—82no—
81C22×D149C2×D412yes—
82Dic298—21no—
83Dic298—21no—
84D596—22no—
85D596—21no—
86D596#4112yes—
87C22⋊D149—44no—
88C22⋊D149—44no—
89C2×Dic149—21no—
90C2×Dic149—21no—
91C2×Dic149—21no—
92C2×Dic149C2×D412yes—
93C4×D149—22no—
94C4×D149—22no—
95C4×D149—42no—
96#11924746—22no—
97C23×C149—22no—
98Q8×C149—42no—
99D4×C149—22no—
100D4×C149—22no—
101D4×C149—42no—
102D4×C149—81no—
103C2×C4×C149—22no—
104C2×C4×C149C2×D411yescharacteristic
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2411yescharacteristic
107C8×C149—21no—
108C8×C149—21no—
109C22×C149—42no—
110C22×C149—42no—
111C22×C149—42no—
112C22×C149C22×D411yescharacteristic
113C2×D149—22no—
114C2×D149—22no—
115C2×D149—22no—
116C2×D149—22no—
117C2×D149—41no—
118C2×D149—41no—
119C2×D149—88no—
120C4×C149—21no—
121C4×C149—42no—
122C4×C149AGL(1,ℤ/8)12yes—
123C4×C149C2×#4111yescharacteristic
124C4×C149H(5,𝔽2)11yescharacteristic
125Dic149—22no—
126Dic149—41no—
127Dic149—41no—
128Dic149—41no—
129C2×C149—82no—
130C2×C149#54611yescharacteristic
131C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C149#57711yescharacteristic
133D149—42no—
134D149—42no—
135C149#332411yescharacteristic, complemented, Sylow
136#3324—1491nomaximal, Sylow
137#556—1491no—
138D42—1491no—
139#486D14911yescharacteristic, complemented
140#482—1492no—
141#474—1491no—
142#468—1492no—
143#460—1491no—
144#450—1492no—
145C4×SD16—1491no—
146#436—1491no—
147#418—1492no—
148C22×D4—2981no—
149C22×D4—1492no—
150C2×SD16—1492no—
151C2×SD16—1491no—
152C2×M16—1492no—
153C4⋊Q8C2×D14911yescharacteristic, complemented
154#127C2×D14911yescharacteristic
155#126—2982no—
156#124—2981no—
157C22⋊Q8—1492no—
158#121—2981no—
159#121—1492no—
160#120—2982no—
161C4×Q8—2981no—
162C4×Q8—1491no—
163C4×D4—1491no—
164C4×D4—1491no—
165#117—1492no—
166#107—1492no—
167#106—1491no—
168C4⋊C8—1491no—
169#103—1491no—
170#102—1492no—
171#102—1491no—
172#102—1492no—
173#102C2×D14912yescomplemented
174#102C2×D14912yescomplemented
175C22⋊C8—1492no—
176C4×C8C2×D14911yescharacteristic, complemented
177C24—2982no—
178C2×Q8—21no—
179C2×Q8—1491no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—2984no—
183C2×D4—2982no—
184C2×D4—2982no—
185C2×D4—5962no—
186C2×D4—5964no—
187C2×D4—1491no—
188C2×D4—1491no—
189C2×D4C22×D14912yes—
190C22×C4—2981no—
191C22×C4—1492no—
192SD16—2982no—
193SD16—5962no—
194M16—5962no—
195C2×C8—1492no—
196C2×C8C22×D14911yescharacteristic
197C2×C8C22×D14911yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—2982no—
200C4⋊C4—2981no—
201C4⋊C4—1491no—
202C4⋊C4—1491no—
203C4⋊C4—1491no—
204C4⋊C4C22×D14912yes—
205C22⋊C4—2982no—
206C22⋊C4—2982no—
207C22⋊C4—2982no—
208C22⋊C4—2981no—
209C22⋊C4—1492no—
210C42—2982no—
211C42—2981no—
212C42C22×D14911yescharacteristic
213C23—22no—
214C23—2981no—
215C23—5964no—
216C23—5964no—
217C23—1492no—
218C23—11922no—
219Q8—42no—
220Q8—2981no—
221Q8—2981no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—81no—
226D4—2982no—
227D4—2981no—
228D4—5964no—
229D4—5964no—
230D4—1492no—
231C2×C4—22no—
232C2×C4—2982no—
233C2×C4—2982no—
234C2×C4—2981no—
235C2×C4—2981no—
236C2×C4—2981no—
237C2×C4—5962no—
238C2×C4—1492no—
239C2×C4D4×D14911yescharacteristic
240C2×C4D4×D14911yescharacteristic
241C2×C4C23×D14911yescharacteristic
242C8—21no—
243C8—21no—
244C8—2982no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—2982no—
249C22—2982no—
250C22—2982no—
251C22—2982no—
252C22—5961no—
253C22—5961no—
254C22—11928no—
255C22C2×D4×D14911yescharacteristic
256C4—21no—
257C4—42no—
258C4—2982no—
259C4—5961no—
260C4—5961no—
261C4—5961no—
262C4#1801193512yes—
263C4#41×D14911yescharacteristic
264C4#1801195711yescharacteristic
265C2—82no—
266C2—5962no—
267C2—5962no—
268C2#2375026811yescharacteristic
269C2C2×#1801193511yescharacteristic
270C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#486×C149C2
socleC22×C149C22×D4
prime radical#23750260C2
hypercenter#486D149
hypocenterC149#3324
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #23750349C2semidirect
3. #23749948C2semidirect
4. #23749902C2semidirect
5. #23749870C2semidirect
6. #23749822C2semidirect
7. #23749777C2semidirect
8. C4×#14558725C2semidirect
9. #23749704C2semidirect
10. #23749628C2semidirect
11. #486×C149C2semidirect
15. C2×#14558724C22semidirect
16. C2×#14558719C22semidirect
19. #18011891C22semidirect
24. C4×Dic298C22semidirect
27. #18011869C22semidirect
28. #18011844C22semidirect
29. #18011843C22semidirect
30. C4⋊#11924746C22semidirect
31. #18011836C22semidirect
33. #18011832C22semidirect
35. #18011823C22semidirect
36. C4⋊Q8×C149C22semidirect
38. #102×C149C22semidirect
39. #102×C149C22semidirect
40. C4×C8×C149C22semidirect
135. C149#3324semidirect
139. #486D149semidirect
153. C4⋊Q8C2×D149semidirect
173. #102C2×D149semidirect
174. #102C2×D149semidirect
176. C4×C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503492
#237502602
#237499482
#237499022
#237498702
#237498222
#237497772
C4×#145587252
#237497042
#237496282
#486×C1492
#3324149