Finite Group Databasebeta

← #34054065

Subgroups — #34054065

Order 19072.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#23750461C211yesmaximal, characteristic, complemented
2C2×#18011939C212yesmaximal, complemented
3#23750189C211yesmaximal, characteristic, complemented
4#23749800C214yesmaximal, complemented
5#23749795C214yesmaximal, complemented
6#23749641C212yesmaximal, complemented
7#413×C149C211yesmaximal, characteristic, complemented
8#18011959C2212yescomplemented
9#18011958—22no—
10#18011957—22no—
11#18011956—22no—
12#18011956C2212yescomplemented
13C2×#14558736—22no—
14C2×#14558736C2212yes—
15#41×D149—24no—
16C22×D596C2211yescharacteristic, complemented
17#18011939—24no—
18#18011939C2218yescomplemented
19C2×#14558725—22no—
20C2×D1192—22no—
21#18011907C2212yescomplemented
22#18011900C2212yescomplemented
23#18011885—22no—
24#18011879—22no—
25#18011870C2211yescharacteristic, complemented
26#18011837C2218yescomplemented
27C2×#41×C149C2211yescharacteristic
28C2×M16×C149C2212yescomplemented
29#101×C149C2212yescomplemented
30#100×C149C2212yescomplemented
31C23×D149—21no—
32#14558737—24no—
33#14558736—24no—
34#14558736—24no—
35#14558736—22no—
36#14558736—24no—
37#14558736—22no—
38#14558736D414yescomplemented
39#14558736C2314yes—
40#14558735—28no—
41#14558735—24no—
42C2×Dic298—22no—
43C2×Dic298D412yescomplemented
44Q8×D149—24no—
45C2×D596—22no—
46C2×D596—22no—
47C2×D596—22no—
48C2×D596—21no—
49C2×D596D412yescomplemented
50C2×D596C2312yes—
51C2×C22⋊D149—24no—
52C2×C22⋊D149—22no—
53C2×C22⋊D149—24no—
54D4×D149—28no—
55D4×D149—24no—
56C2×C4×D149—24no—
57C2×C4×D149—22no—
58#14558725—28no—
59D1192—28no—
60C4⋊Dic149—21no—
61C22⋊Dic149—21no—
62#14558713—24no—
63C4×Dic149C2312yes—
64#41×C149—22no—
65#41×C149—22no—
66C2×Q8×C149C2311yescharacteristic
67C2×D4×C149D412yes—
68C2×D4×C149C2311yescharacteristic
69C22×C4×C149D412yes—
70C22×C4×C149C2311yescharacteristic
71M16×C149—22no—
72M16×C149C2314yes—
73C2×C8×C149—22no—
74C22×D149—24no—
75C22×D149—22no—
76C22×D149—22no—
77C22×D149—42no—
78C22×D149—82no—
79Dic298—24no—
80Dic298—22no—
81Dic298C2×D414yes—
82D596—24no—
83D596—22no—
84D596—24no—
85D596—41no—
86D596—41no—
87D596C2×D414yes—
88C22⋊D149—28no—
89C22⋊D149—24no—
90C22⋊D149—24no—
91C22⋊D149—84no—
92C2×Dic149—24no—
93C2×Dic149—22no—
94C2×Dic149—22no—
95C4×D149—28no—
96C4×D149—24no—
97C4×D149—24no—
98C23×C149C2×D412yes—
99C23×C149C2×D411yescharacteristic
100Q8×C149—22no—
101D4×C149—24no—
102D4×C149—22no—
103C2×C4×C149—22no—
104C2×C4×C149—22no—
105C2×C4×C149C2×D412yes—
106C2×C4×C149C2×D412yes—
107C2×C4×C149C2×D412yes—
108C2×C4×C149C2×D411yescharacteristic
109C2×C4×C149C2411yescharacteristic
110C8×C149—24no—
111C22×C149—22no—
112C22×C149—41no—
113C22×C149—42no—
114C22×C149#12012yes—
115C22×C149C22×D411yescharacteristic
116C2×D149—24no—
117C2×D149—44no—
118C2×D149—42no—
119C2×D149—82no—
120C2×D149—84no—
121C2×D149—84no—
122C4×C149—22no—
123C4×C149#12011yescharacteristic
124C4×C149#12011yescharacteristic
125C4×C149C22×D412yes—
126Dic149—24no—
127Dic149—42no—
128C2×C149—21no—
129C2×C149—22no—
130C2×C149—42no—
131C2×C149C2×#12011yescharacteristic
132D149—44no—
133D149—82no—
134C149#302111yescharacteristic, complemented, Sylow
135#3021—1491nomaximal, Sylow
136#585—1491no—
137C2×AGL(1,ℤ/8)—1492no—
138#535—1491no—
139#454—1494no—
140#453—1494no—
141#421—1492no—
142#413D14911yescharacteristic, complemented
1432−1+4—2982no—
1442−1+4—1492no—
145H(5,𝔽2)—2982no—
146H(5,𝔽2)—2982no—
147H(5,𝔽2)—1492no—
148C2×#41—2984no—
149C2×#41—2982no—
150C2×#41—1492no—
151C2×#41C2×D14911yescharacteristic
152C22×D4—1491no—
153AGL(1,ℤ/8)—2984no—
154AGL(1,ℤ/8)—1498no—
155C2×SD16—2982no—
156C2×D8—2982no—
157C2×M16C2×D14912yescomplemented
158#127—1492no—
159#124—1492no—
160#121—2982no—
161#120—2982no—
162#117—1491no—
163#104—1498no—
164#101C2×D14912yescomplemented
165#100C2×D14912yescomplemented
166C24—2981no—
167#41—22no—
168#41—22no—
169#41—2984no—
170#41—2984no—
171#41—2988no—
172#41—2984no—
173#41—2984no—
174#41—2982no—
175#41—2982no—
176#41—2984no—
177#41—1494no—
178#41—1494no—
179C2×Q8—2982no—
180C2×Q8—2984no—
181C2×Q8—1492no—
182C2×Q8C22×D14911yescharacteristic
183C2×D4—2982no—
184C2×D4—2988no—
185C2×D4—2984no—
186C2×D4—2984no—
187C2×D4—2982no—
188C2×D4—2984no—
189C2×D4—2982no—
190C2×D4—2982no—
191C2×D4—2981no—
192C2×D4—1492no—
193C2×D4—1492no—
194C2×D4D59612yes—
195C2×D4C22×D14911yescharacteristic
196C22×C4—2984no—
197C22×C4—2982no—
198C22×C4D59612yes—
199C22×C4C22×D14911yescharacteristic
200SD16—2988no—
201D8—2988no—
202M16—22no—
203M16C22×D14914yes—
204C2×C8—22no—
205C4⋊C4—2981no—
206C22⋊C4—2981no—
207C22⋊C4—2984no—
208C42—1492no—
209C23—2984no—
210C23—2982no—
211C23—2982no—
212C23—5962no—
213C23—11922no—
214C23C2×D59612yes—
215C23C2×D59611yescharacteristic
216Q8—22no—
217Q8—2984no—
218Q8—2982no—
219Q8—1494no—
220D4—24no—
221D4—22no—
222D4—2984no—
223D4—2988no—
224D4—2984no—
225D4—2984no—
226D4—2984no—
227D4—2982no—
228D4—5961no—
229D4—5961no—
230D4—1494no—
231D4—11924no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2988no—
235C2×C4—2984no—
236C2×C4—2982no—
237C2×C4—2984no—
238C2×C4—2984no—
239C2×C4—2982no—
240C2×C4D4×D14912yes—
241C2×C4D4×D14912yes—
242C2×C4C2×D59612yes—
243C2×C4C2×D59611yescharacteristic
244C2×C4C23×D14911yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—2984no—
250C22—5964no—
251C22—5962no—
252C22—11922no—
253C22—11924no—
254C22—11924no—
255C22#1801188112yes—
256C22C22×D59611yescharacteristic
257C4—22no—
258C4—2984no—
259C4—5962no—
260C4#1801188111yescharacteristic
261C4#1801188111yescharacteristic
262C4C2×D4×D14912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—5964no—
267C2—11922no—
268C2C2×#1801188111yescharacteristic
subgroupG/H
centerC2C2×#18011881
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#413×C149C2
socleC2×C149C2×#120
hypercenter#413D149
hypocenterC149#3021
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #23750461C2semidirect
2. C2×#18011939C2semidirect
3. #23750189C2semidirect
4. #23749800C2semidirect
5. #23749795C2semidirect
6. #23749641C2semidirect
7. #413×C149C2semidirect
8. #18011959C22semidirect
12. #18011956C22semidirect
16. C22×D596C22semidirect
18. #18011939C22semidirect
21. #18011907C22semidirect
22. #18011900C22semidirect
25. #18011870C22semidirect
26. #18011837C22semidirect
28. C2×M16×C149C22semidirect
29. #101×C149C22semidirect
30. #100×C149C22semidirect
38. #14558736D4semidirect
43. C2×Dic298D4semidirect
49. C2×D596D4semidirect
134. C149#3021semidirect
142. #413D149semidirect
157. C2×M16C2×D149semidirect
164. #101C2×D149semidirect
165. #100C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504612
C2×#180119392
#237501892
#237498002
#237497952
#237496412
#413×C1492
#3021149