Finite Group Databasebeta

← #34056514

Subgroups — #34056514

Order 19072.

462 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 123 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#23750336C211yesmaximal, characteristic, complemented
2#23750260C211yesmaximal, characteristic
3#23749942C212yesmaximal, complemented
4#23749912C211yesmaximal, characteristic, complemented
5#23749853C212yesmaximal, complemented
6#23749822C211yesmaximal, characteristic, complemented
7#23749777C212yesmaximal, complemented
8C4×#14558725C211yesmaximal, characteristic, complemented
9C8×D596C211yesmaximal, characteristic, complemented
10#102×D149C212yesmaximal, complemented
11#486×C149C211yesmaximal, characteristic, complemented
12C2×D4×D149—21no—
13C2×D4×D149C2212yes—
14C2×#14558725C2211yescharacteristic
15C2×#14558724C2212yescomplemented
16C2×C8×D149C2212yescomplemented
17#18011902—21no—
18#18011891C2212yescomplemented
19#18011887—22no—
20#18011882—21no—
21#18011882C2212yes—
22#18011879—22no—
23C4×Dic298C2211yescharacteristic, complemented
24C4×D596C2211yescharacteristic
25C4×D596C2211yescharacteristic
26#18011873—21no—
27C4⋊C4×D149C2212yescomplemented
28#18011843C2211yescharacteristic, complemented
29#18011842C2212yescomplemented
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—22no—
51C2×C4×D149—21no—
52C2×C4×D149C2312yes—
53#14558725—22no—
54#14558724—42no—
55C2×#11924746C2312yes—
56C8×D149—42no—
57C4⋊Dic149—22no—
58C4⋊Dic149D411yescharacteristic
59C4⋊Dic149C2311yescharacteristic
60C4⋊Dic149C2311yescharacteristic
61#14558715—22no—
62#14558715—21no—
63C22⋊Dic149—22no—
64#14558713—22no—
65#14558713—21no—
66#14558713D412yes—
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—21no—
85D596SD1614yescomplemented
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—
96C4×D149—82no—
97#11924746—22no—
98C23×C149—22no—
99Q8×C149—81no—
100D4×C149—22no—
101D4×C149—22no—
102D4×C149—42no—
103D4×C149—81no—
104C2×C4×C149—22no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2411yescharacteristic
108C8×C149—21no—
109C8×C149—21no—
110C22×C149—42no—
111C22×C149—42no—
112C22×C149—42no—
113C22×C149C22×D411yescharacteristic
114C2×D149—22no—
115C2×D149—22no—
116C2×D149—22no—
117C2×D149—41no—
118C2×D149—41no—
119C2×D149—88no—
120C2×D149C2×SD1614yes—
121C4×C149—21no—
122C4×C149—42no—
123C4×C149C2×SD1612yes—
124C4×C149C2×#4111yescharacteristic
125C4×C149H(5,𝔽2)11yescharacteristic
126Dic149—22no—
127Dic149—41no—
128Dic149—41no—
129Dic149—41no—
130C2×C149—82no—
131C2×C149#54611yescharacteristic
132C2×C149C22×SD1611yescharacteristic
133C2×C149#57611yescharacteristic
134D149—24no—
135D149—42no—
136C149#329511yescharacteristic, complemented, Sylow
137#3295—1491nomaximal, Sylow
138#554—1491no—
139D42—1491no—
140#486D14911yescharacteristic, complemented
141#481—1492no—
142#476—1491no—
143#465—1492no—
144#460—1491no—
145#450—1492no—
146C4×SD16—1491no—
147C8×D4—1491no—
148C2×#102—1492no—
149C22×D4—2981no—
150C22×D4—1492no—
151C2×SD16—1492no—
152C2×SD16—1491no—
153C22×C8—1492no—
154C4⋊Q8C2×D14911yescharacteristic, complemented
155#127C2×D14911yescharacteristic
156#125—2981no—
157C22⋊Q8—2982no—
158C22⋊Q8—1492no—
159#121—2981no—
160#121—1492no—
161#120—2982no—
162C4×Q8—1491no—
163C4×D4—2981no—
164C4×D4—1491no—
165C4×D4—1491no—
166C2×C4⋊C4—1492no—
167#106—1491no—
168#106—1492no—
169C4⋊C8—1491no—
170#103—1491no—
171#102—1492no—
172#102—1491no—
173#102—1492no—
174#102C2×D14912yescomplemented
175#102C2×D14912yescomplemented
176C22⋊C8—1492no—
177C4×C8C2×D14911yescharacteristic, complemented
178C24—2982no—
179C2×Q8—21no—
180C2×Q8—1491no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—2984no—
184C2×D4—2982no—
185C2×D4—2982no—
186C2×D4—5962no—
187C2×D4—5964no—
188C2×D4—1491no—
189C2×D4—1491no—
190C2×D4C22×D14912yes—
191C22×C4—2982no—
192C22×C4—2981no—
193C22×C4—1492no—
194SD16—2982no—
195SD16—5962no—
196C2×C8—5962no—
197C2×C8—1492no—
198C2×C8C22×D14911yescharacteristic
199C2×C8C22×D14911yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—2982no—
202C4⋊C4—2982no—
203C4⋊C4—2981no—
204C4⋊C4—1491no—
205C4⋊C4—1491no—
206C4⋊C4—1491no—
207C4⋊C4C22×D14912yes—
208C22⋊C4—2982no—
209C22⋊C4—2982no—
210C22⋊C4—2981no—
211C22⋊C4—1492no—
212C42—2981no—
213C42C22×D14911yescharacteristic
214C23—22no—
215C23—2981no—
216C23—5964no—
217C23—5964no—
218C23—1492no—
219C23—11922no—
220Q8—81no—
221Q8—2981no—
222Q8—2981no—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—81no—
227D4—2981no—
228D4—5964no—
229D4—5964no—
230D4—1494no—
231D4—1492no—
232C2×C4—22no—
233C2×C4—2982no—
234C2×C4—2982no—
235C2×C4—2981no—
236C2×C4—2981no—
237C2×C4—2981no—
238C2×C4—5962no—
239C2×C4—1492no—
240C2×C4—11922no—
241C2×C4D4×D14911yescharacteristic
242C2×C4D4×D14911yescharacteristic
243C2×C4C23×D14911yescharacteristic
244C8—21no—
245C8—21no—
246C8—2982no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—2982no—
251C22—2982no—
252C22—2982no—
253C22—5961no—
254C22—5961no—
255C22—1494no—
256C22—11928no—
257C22C2×D4×D14911yescharacteristic
258C4—21no—
259C4—42no—
260C4—2982no—
261C4—5961no—
262C4—5961no—
263C4—5961no—
264C4SD16×D14912yes—
265C4#41×D14911yescharacteristic
266C4#1801195711yescharacteristic
267C2—82no—
268C2—2984no—
269C2—5962no—
270C2#2375026811yescharacteristic
271C2C2×SD16×D14911yescharacteristic
272C2#2375042611yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#486×C149C2
socleC22×C149C22×D4
prime radical#23750260C2
hypercenter#486D149
hypocenterC149#3295
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. #23750336C2semidirect
3. #23749942C2semidirect
4. #23749912C2semidirect
5. #23749853C2semidirect
6. #23749822C2semidirect
7. #23749777C2semidirect
8. C4×#14558725C2semidirect
9. C8×D596C2semidirect
10. #102×D149C2semidirect
11. #486×C149C2semidirect
15. C2×#14558724C22semidirect
16. C2×C8×D149C22semidirect
18. #18011891C22semidirect
23. C4×Dic298C22semidirect
27. C4⋊C4×D149C22semidirect
28. #18011843C22semidirect
29. #18011842C22semidirect
30. C4⋊#11924746C22semidirect
31. #18011836C22semidirect
33. #18011832C22semidirect
35. #18011823C22semidirect
36. C4⋊Q8×C149C22semidirect
38. #102×C149C22semidirect
39. #102×C149C22semidirect
40. C4×C8×C149C22semidirect
85. D596SD16semidirect
136. C149#3295semidirect
140. #486D149semidirect
154. C4⋊Q8C2×D149semidirect
174. #102C2×D149semidirect
175. #102C2×D149semidirect
177. C4×C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503362
#237502602
#237499422
#237499122
#237498532
#237498222
#237497772
C4×#145587252
C8×D5962
#102×D1492
#486×C1492
#3295149