Finite Group Databasebeta

← #34078337

Subgroups — #34078337

Order 19328.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#18012765C212yesmaximal, complemented
2#23752723C211yesmaximal, characteristic, complemented
3#23752363C214yesmaximal, complemented
4#23752346C214yesmaximal, complemented
5#23752217C212yesmaximal, complemented
6#23752176C211yesmaximal, characteristic, complemented
7C23×D4×C151C211yesmaximal, characteristic, complemented
8C2×#14559004C2211yescharacteristic, complemented
9#18012765—28no—
10C2×#14558992C2214yescomplemented
11C2×#14558989C2214yescomplemented
12C2×#14558988C2212yescomplemented
13#18012722—21no—
14#18012718C2212yescomplemented
15#18012713C2212yescomplemented
16#18012708—21no—
17C4×C22⋊D151—22no—
18#18012698C2211yescharacteristic, complemented
19#18012660C2218yescomplemented
20C22⋊(C151⋊C8)C2214yescomplemented
21C25×C151—21no—
22C22×D4×C151—28no—
23C22×D4×C151—22no—
24C22×D4×C151—21no—
25C22×D4×C151C2214yes—
26C22×D4×C151C2212yes—
27C23×C4×C151C2211yescharacteristic
28#14559004—24no—
29C2×Dic302C2311yescharacteristic, complemented
30C2×D604C2311yescharacteristic, complemented
31C2×C22⋊D151—22no—
32C2×C22⋊D151—21no—
33C2×C4×D151—21no—
34#14558992—28no—
35#14558989—28no—
36#14558988—24no—
37C2×C151⋊C8C2314yescomplemented
38C4⋊Dic151C2312yescomplemented
39Dic151⋊C4—22no—
40C22⋊Dic151—21no—
41C22⋊Dic151—22no—
42(C2×D151)⋊C4—22no—
43C4×Dic151—21no—
44C24×C151—28no—
45C24×C151—24no—
46C24×C151—22no—
47C24×C151—48no—
48C24×C151D411yescharacteristic
49C2×D4×C151—28no—
50C2×D4×C151—216no—
51C2×D4×C151—216no—
52C2×D4×C151—24no—
53C2×D4×C151—22no—
54C2×D4×C151—24no—
55C2×D4×C151D418yescomplemented
56C2×D4×C151C2314yes—
57C22×C4×C151—24no—
58C22×C4×C151D412yes—
59C22×C4×C151D411yescharacteristic
60C22×C4×C151C2312yes—
61C22×C4×C151C2311yescharacteristic
62C22×D151—21no—
63Dic302—22no—
64D604—22no—
65C22⋊D151—42no—
66C22⋊D151—44no—
67C2×Dic151—22no—
68C2×Dic151—21no—
69C4×D151—42no—
70C151⋊C8—24no—
71C23×C151—22no—
72C23×C151—24no—
73C23×C151—216no—
74C23×C151—28no—
75C23×C151—24no—
76C23×C151—432no—
77C23×C151—48no—
78C23×C151—48no—
79C23×C151—48no—
80C23×C151C2×D411yescharacteristic
81C23×C151C2×D412yes—
82D4×C151—216no—
83D4×C151—28no—
84D4×C151—24no—
85D4×C151C2×D418yes—
86C2×C4×C151—22no—
87C2×C4×C151—24no—
88C2×C4×C151—24no—
89C2×C4×C151C2×D412yes—
90C2×C4×C151C2×D411yescharacteristic
91C2×C4×C151C2×D414yes—
92C2×C4×C151C2411yescharacteristic
93C22×C151—28no—
94C22×C151—22no—
95C22×C151—22no—
96C22×C151—22no—
97C22×C151—28no—
98C22×C151—416no—
99C22×C151—432no—
100C22×C151—48no—
101C22×C151#12012yes—
102C22×C151AGL(1,ℤ/8)14yescomplemented
103C22×C151C22×D411yescharacteristic
104C2×D151—41no—
105C2×D151—42no—
106C4×C151—22no—
107C4×C151#12012yes—
108C4×C151C22×D412yes—
109Dic151—41no—
110Dic151—42no—
111C2×C151—22no—
112C2×C151—24no—
113C2×C151—48no—
114C2×C151C2×#12011yescharacteristic
115C2×C151C2×AGL(1,ℤ/8)12yes—
116D151—81no—
117C151#300311yescharacteristic, complemented, Sylow
118#3003—1511nomaximal, Sylow
119C23×D4D15111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1512no—
121#525—1511no—
122#450—1514no—
123#447—1514no—
124#418—1512no—
125#407—1511no—
126C25—21no—
127C2×#41—1511no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D15114yes—
132C22×D4C2×D15112yes—
133C23×C4C2×D15111yescharacteristic
134AGL(1,ℤ/8)—3028no—
135C2×SD16—1514no—
136C2×D8—1514no—
137C2×M16—1512no—
138#123—3021no—
139C22⋊Q8—1512no—
140#121—1512no—
141#120—3021no—
142C4×D4—3022no—
143#117—1511no—
144#102—1518no—
145C22⋊C8—1514no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D15111yescharacteristic
151#41—3024no—
152C2×Q8—1511no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—3022no—
160C2×D4—3021no—
161C2×D4—1511no—
162C2×D4C22⋊D15118yescomplemented
163C2×D4C22×D15114yes—
164C22×C4—24no—
165C22×C4—3021no—
166C22×C4C22⋊D15112yes—
167C22×C4C22⋊D15111yescharacteristic
168C22×C4C22×D15112yes—
169C22×C4C22×D15111yescharacteristic
170SD16—3028no—
171D8—3028no—
172M16—3024no—
173C2×C8—1514no—
174C4⋊C4—3022no—
175C4⋊C4—1512no—
176C22⋊C4—3021no—
177C22⋊C4—3022no—
178C22⋊C4—3022no—
179C42—3021no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—3021no—
190C23C2×C22⋊D15111yescharacteristic
191C23C2×C22⋊D15112yes—
192Q8—3022no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—3022no—
197D4—6042no—
198D4—6044no—
199D4C2×C22⋊D15118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—3022no—
204C2×C4—3021no—
205C2×C4—6042no—
206C2×C4C2×C22⋊D15112yes—
207C2×C4C2×C22⋊D15111yescharacteristic
208C2×C4C2×C22⋊D15114yes—
209C2×C4C23×D15111yescharacteristic
210C8—3024no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—6042no—
220C22—6041no—
221C22#1801270812yes—
222C22#1801276514yescomplemented
223C22C22×C22⋊D15111yescharacteristic
224C4—22no—
225C4—6042no—
226C4—6041no—
227C4#1801270812yes—
228C4C22×C22⋊D15112yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—12081no—
233C2C2×#1801270811yescharacteristic
234C2C2×#1801276512yes—
subgroupG/H
centerC22C22×C22⋊D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC23×D4×C151C2
socleC22×C151C22×D4
hypercenterC23×D4D151
hypocenterC151#3003
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#18012765C2semidirect
2. #23752723C2semidirect
3. #23752363C2semidirect
4. #23752346C2semidirect
5. #23752217C2semidirect
6. #23752176C2semidirect
7. C23×D4×C151C2semidirect
8. C2×#14559004C22semidirect
10. C2×#14558992C22semidirect
11. C2×#14558989C22semidirect
12. C2×#14558988C22semidirect
14. #18012718C22semidirect
15. #18012713C22semidirect
18. #18012698C22semidirect
19. #18012660C22semidirect
20. C22⋊(C151⋊C8)C22semidirect
29. C2×Dic302C23semidirect
30. C2×D604C23semidirect
37. C2×C151⋊C8C23semidirect
38. C4⋊Dic151C23semidirect
55. C2×D4×C151D4semidirect
102. C22×C151AGL(1,ℤ/8)semidirect
117. C151#3003semidirect
119. C23×D4D151semidirect
162. C2×D4C22⋊D151semidirect
222. C22#18012765semidirect

Maximal subgroups

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

subgroupindex
C2×#180127652
#237527232
#237523632
#237523462
#237522172
#237521762
C23×D4×C1512
#3003151