Finite Group Databasebeta

← #34056312

Subgroups — #34056312

Order 19072.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#18011938C212yesmaximal, complemented
2#23750297C211yesmaximal, characteristic, complemented
3D4×D596C211yesmaximal, characteristic, complemented
4#23750002C211yesmaximal, characteristic, complemented
5#23749879C212yesmaximal, complemented
6#23749825C211yesmaximal, characteristic, complemented
7#23749820C211yesmaximal, characteristic, complemented
8#23749773C212yesmaximal, complemented
9#23749760C212yesmaximal, complemented
10#23749704C211yesmaximal, characteristic, complemented
11C4×SD16×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2212yescomplemented
13C22×D596—21no—
14C2×D4×D149C2212yes—
15#18011938—28no—
16C2×#14558723C2212yescomplemented
17C2×D1192C2212yescomplemented
18C2×D1192C2212yescomplemented
19C2×#14558721C2212yes—
20C2×#14558719C2212yescomplemented
21#18011908—21no—
22#18011908C2211yescharacteristic
23#18011908C2211yescharacteristic, complemented
24#18011886—22no—
25#18011886C2212yes—
26#18011886C2212yescomplemented
27#18011885—21no—
28#18011881—22no—
29C4×D596—21no—
30C4×D596C2211yescharacteristic
31C4⋊#11924746C2211yescharacteristic, complemented
32#18011831C2212yescomplemented
33#18011831C2212yescomplemented
34#18011823C2212yescomplemented
35C2×SD16×C149C2211yescharacteristic
36C4×Q8×C149C2211yescharacteristic, complemented
37C4×D4×C149C2211yescharacteristic
38#106×C149C2211yescharacteristic, complemented
39#103×C149C2211yescharacteristic, complemented
40#102×C149C2211yescharacteristic
41C4×C8×C149C2211yescharacteristic, complemented
42C23×D149—22no—
43#14558737—24no—
44#14558737—42no—
45C2×D596—22no—
46C2×D596—22no—
47C2×D596—22no—
48C2×D596—21no—
49C2×D596—21no—
50C2×D596—21no—
51C2×D596—21no—
52C2×D596—44no—
53C2×D596D411yescharacteristic
54C2×D596C2312yes—
55C2×D596C2312yes—
56C2×C22⋊D149—22no—
57D4×D149—24no—
58D4×D149—42no—
59C2×C4×D149—22no—
60C2×C4×D149C2312yes—
61#14558723—24no—
62D1192—24no—
63D1192—42no—
64#14558721—24no—
65#14558719—24no—
66C2×#11924746C2312yes—
67C4⋊Dic149D411yescharacteristic
68#14558713—22no—
69#14558713—22no—
70#14558713D412yes—
71C2×Q8×C149C2311yescharacteristic
72C2×D4×C149C2311yescharacteristic
73C22×C4×C149—21no—
74SD16×C149D414yescomplemented
75C2×C8×C149C2311yescharacteristic
76C2×C8×C149C2311yescharacteristic
77C4⋊C4×C149—21no—
78C4⋊C4×C149C2311yescharacteristic
79C4⋊C4×C149C2311yescharacteristic
80C22⋊C4×C149—21no—
81C42×C149—21no—
82C42×C149C2311yescharacteristic
83C22×D149—22no—
84C22×D149—22no—
85C22×D149—44no—
86C22×D149—44no—
87C22×D149—82no—
88C22×D149C2×D412yes—
89D596—22no—
90D596—22no—
91D596—22no—
92D596—22no—
93D596—22no—
94D596—44no—
95D596—44no—
96D596—42no—
97D596—42no—
98D596—81no—
99D596—81no—
100D596—82no—
101C22⋊D149—44no—
102C2×Dic149C2×D412yes—
103C4×D149—22no—
104C4×D149—22no—
105C4×D149—44no—
106#11924746—22no—
107C23×C149—21no—
108Q8×C149—21no—
109Q8×C149C2×D412yescomplemented
110D4×C149—21no—
111D4×C149C2×D412yes—
112C2×C4×C149—21no—
113C2×C4×C149—21no—
114C2×C4×C149—21no—
115C2×C4×C149—42no—
116C2×C4×C149C2×D411yescharacteristic
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2411yescharacteristic
119C8×C149—21no—
120C8×C149C2×D412yescomplemented
121C22×C149—22no—
122C22×C149—41no—
123C22×C149—41no—
124C22×C149C22×D411yescharacteristic
125C2×D149—22no—
126C2×D149—22no—
127C2×D149—22no—
128C2×D149—42no—
129C2×D149—42no—
130C2×D149—42no—
131C2×D149—42no—
132C2×D149—42no—
133C2×D149—42no—
134C2×D149—88no—
135C4×C149—22no—
136C4×C149—21no—
137C4×C149—41no—
138C4×C149—41no—
139C4×C149AGL(1,ℤ/8)12yescomplemented
140C4×C149C22×D411yescharacteristic
141C4×C149H(5,𝔽2)11yescharacteristic
142Dic149—22no—
143C2×C149—42no—
144C2×C149D4211yescharacteristic
145C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C149#57611yescharacteristic
147D149—42no—
148D149—82no—
149D149—82no—
150C149#327411yescharacteristic, complemented, Sylow
151#3274—1491nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1492no—
153#550—1491no—
154D42—1491no—
155#493—1491no—
156#469—1492no—
157#460—1491no—
158#459—1491no—
159#449—1492no—
160#447—1492no—
161C4×SD16D14911yescharacteristic, complemented
162#436—1491no—
163C2×#41—1492no—
164C22×D4—2981no—
165C22×D4—1492no—
166AGL(1,ℤ/8)—2988no—
167C2×SD16—1492no—
168C2×SD16C2×D14911yescharacteristic
169C2×D8—1492no—
170C2×D8—1492no—
171C2×D8—1492no—
172C2×M16—1492no—
173#127—2981no—
174#127—1491no—
175#127—1491no—
176#121—2981no—
177#121—2982no—
178#121—1492no—
179#121—1492no—
180#120—2982no—
181C4×Q8C2×D14911yescharacteristic, complemented
182C4×D4—2981no—
183C4×D4—1491no—
184C4×D4C2×D14911yescharacteristic
185#106C2×D14911yescharacteristic, complemented
186C4⋊C8—1491no—
187#103C2×D14911yescharacteristic, complemented
188#102—1492no—
189#102—1492no—
190#102C2×D14911yescharacteristic
191C22⋊C8—1492no—
192C4×C8C2×D14911yescharacteristic, complemented
193C24—2982no—
194#41—2984no—
195#41—5962no—
196C2×Q8C22×D14911yescharacteristic
197C2×D4—2982no—
198C2×D4—2982no—
199C2×D4—2982no—
200C2×D4—2984no—
201C2×D4—2982no—
202C2×D4—2981no—
203C2×D4—2981no—
204C2×D4—2981no—
205C2×D4—2981no—
206C2×D4—5964no—
207C2×D4—5962no—
208C2×D4—1491no—
209C2×D4—1492no—
210C2×D4—1492no—
211C2×D4C22×D14911yescharacteristic
212C22×C4—21no—
213C22×C4—2982no—
214C22×C4—1492no—
215SD16—2984no—
216SD16D59614yescomplemented
217D8—2984no—
218D8—2984no—
219D8—5962no—
220M16—2984no—
221C2×C8—1492no—
222C2×C8C22×D14911yescharacteristic
223C2×C8C22×D14911yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—1491no—
226C4⋊C4C22×D14911yescharacteristic
227C4⋊C4C22×D14911yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2982no—
230C22⋊C4—2982no—
231C22⋊C4—1492no—
232C42—21no—
233C42C22×D14911yescharacteristic
234C23—21no—
235C23—2982no—
236C23—2982no—
237C23—5964no—
238C23—5964no—
239C23—1492no—
240C23—11922no—
241Q8—21no—
242Q8C2×D59612yescomplemented
243D4—21no—
244D4—2982no—
245D4—2982no—
246D4—2982no—
247D4—2982no—
248D4—2982no—
249D4—5964no—
250D4—5964no—
251D4—5964no—
252D4—5962no—
253D4—5962no—
254D4—11921no—
255D4—11921no—
256D4—11922no—
257D4C2×D59612yes—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—2982no—
263C2×C4—2982no—
264C2×C4—5964no—
265C2×C4—1492no—
266C2×C4D4×D14911yescharacteristic
267C2×C4D4×D14911yescharacteristic
268C2×C4C23×D14911yescharacteristic
269C8—21no—
270C8—2982no—
271C8C2×D59612yescomplemented
272C22—22no—
273C22—41no—
274C22—41no—
275C22—2982no—
276C22—2982no—
277C22—2982no—
278C22—5962no—
279C22—5962no—
280C22—5962no—
281C22—5962no—
282C22—5962no—
283C22—5962no—
284C22—11928no—
285C22C2×D4×D14911yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—2982no—
291C4#1801193812yescomplemented
292C4C22×D59611yescharacteristic
293C4#1801195711yescharacteristic
294C2—42no—
295C2—5962no—
296C2—11922no—
297C2—11922no—
298C2D4×D59611yescharacteristic
299C2C2×#1801193811yescharacteristic
300C2#2375042711yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×SD16×C149C2
socleC22×C149C22×D4
hypercenterC4×SD16D149
hypocenterC149#3274
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#18011938C2semidirect
2. #23750297C2semidirect
3. D4×D596C2semidirect
4. #23750002C2semidirect
5. #23749879C2semidirect
6. #23749825C2semidirect
7. #23749820C2semidirect
8. #23749773C2semidirect
9. #23749760C2semidirect
10. #23749704C2semidirect
11. C4×SD16×C149C2semidirect
12. C2×#14558737C22semidirect
16. C2×#14558723C22semidirect
17. C2×D1192C22semidirect
18. C2×D1192C22semidirect
20. C2×#14558719C22semidirect
23. #18011908C22semidirect
26. #18011886C22semidirect
31. C4⋊#11924746C22semidirect
32. #18011831C22semidirect
33. #18011831C22semidirect
34. #18011823C22semidirect
36. C4×Q8×C149C22semidirect
38. #106×C149C22semidirect
39. #103×C149C22semidirect
41. C4×C8×C149C22semidirect
74. SD16×C149D4semidirect
109. Q8×C149C2×D4semidirect
120. C8×C149C2×D4semidirect
139. C4×C149AGL(1,ℤ/8)semidirect
150. C149#3274semidirect
161. C4×SD16D149semidirect
181. C4×Q8C2×D149semidirect
185. #106C2×D149semidirect
187. #103C2×D149semidirect
192. C4×C8C2×D149semidirect
216. SD16D596semidirect
242. Q8C2×D596semidirect
271. C8C2×D596semidirect
291. C4#18011938semidirect

Maximal subgroups

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

subgroupindex
C2×#180119382
#237502972
D4×D5962
#237500022
#237498792
#237498252
#237498202
#237497732
#237497602
#237497042
C4×SD16×C1492
#3274149