Finite Group Databasebeta

← #34082256

Subgroups — #34082256

Order 19328.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#23753026C211yesmaximal, characteristic, complemented
2#23753013C211yesmaximal, characteristic, complemented
3#23752655C211yesmaximal, characteristic, complemented
4#23752651C212yesmaximal, complemented
5#23752646C212yesmaximal, complemented
6#23752642C211yesmaximal, characteristic, complemented
7#23752641C212yesmaximal, complemented
8C2×#18012683C211yesmaximal, characteristic, complemented
9SD32×D151C212yesmaximal, complemented
10#23752630C211yesmaximal, characteristic, complemented
11#509×C151C211yesmaximal, characteristic, complemented
12#18012786—21no—
13#18012784—21no—
14#18012769—21no—
15#18012768—22no—
16#18012765—21no—
17#18012763—22no—
18#18012761C2211yescharacteristic
19#18012758C2212yes—
20#18012758C2212yescomplemented
21C2×Dic604C2211yescharacteristic, complemented
22Q16×D151C2212yescomplemented
23D8×D151C2212yes—
24#18012745C2211yescharacteristic
25Dic1208C2212yescomplemented
26#18012685C2211yescharacteristic, complemented
27#18012684C2212yescomplemented
28#18012683C2212yescomplemented
29#18012683C2212yescomplemented
30#18012683C2211yescharacteristic, complemented
31#18012682C2211yescharacteristic, complemented
32#18012680C2211yescharacteristic, complemented
33#18012678C2212yescomplemented
34C2×C151⋊C16C2211yescharacteristic, complemented
35C16×D151C2212yescomplemented
36#135×C151C2211yescharacteristic, complemented
37C2×D8×C151C2211yescharacteristic
38SD32×C151C2212yescomplemented
39D16×C151C2212yescomplemented
40M32×C151C2211yescharacteristic, complemented
41#14559004—21no—
42#14559004—41no—
43#14559004D411yescharacteristic
44#14559003—22no—
45#14559003—22no—
46#14559003—41no—
47#14559003—42no—
48C2×Dic302—21no—
49C2×Dic302—41no—
50Q8×D151—22no—
51C2×C22⋊D151—42no—
52D4×D151—22no—
53D4×D151—41no—
54Dic604—21no—
55Dic604C2312yes—
56Dic604C2311yescharacteristic
57#14558994—22no—
58#14558993—21no—
59#14558992—22no—
60#14558992—22no—
61D1208C2311yescharacteristic
62#14558989—22no—
63#14558988D411yescharacteristic
64#14558987D412yes—
65C8×D151C2312yes—
66C151⋊C16C2311yescharacteristic
67C151⋊C16C2311yescharacteristic
68#41×C151—21no—
69C2×D4×C151—21no—
70Q16×C151C2311yescharacteristic
71SD16×C151—21no—
72D8×C151—21no—
73D8×C151C2311yescharacteristic
74D8×C151C2312yes—
75C2×C8×C151C2311yescharacteristic
76C16×C151C2312yes—
77C22×D151—42no—
78Dic302—21no—
79Dic302—22no—
80Dic302—41no—
81Dic302—42no—
82Dic302D811yescharacteristic
83D604—21no—
84D604D811yescharacteristic
85C22⋊D151—41no—
86C22⋊D151—42no—
87C22⋊D151—42no—
88C22⋊D151—42no—
89C22⋊D151D812yes—
90C2×Dic151—41no—
91C2×Dic151—42no—
92C2×Dic151—42no—
93C4×D151—42no—
94C4×D151—41no—
95C4×D151C2×D412yes—
96C151⋊C8C2×D412yes—
97C23×C151—41no—
98Q8×C151—21no—
99D4×C151—21no—
100D4×C151—22no—
101D4×C151—41no—
102D4×C151—41no—
103C2×C4×C151—41no—
104C2×C4×C151C2×D411yescharacteristic
105C8×C151C2×D411yescharacteristic
106C8×C151C2411yescharacteristic
107C22×C151—41no—
108C22×C151—42no—
109C22×C151—161no—
110C22×C151C2×D811yescharacteristic
111C2×D151—41no—
112C2×D151—162no—
113C2×D151C2×D812yes—
114C4×C151—41no—
115C4×C151C2×D811yescharacteristic
116C4×C151C22×D411yescharacteristic
117Dic151—41no—
118Dic151—42no—
119Dic151C2×D812yes—
120C2×C151—21no—
121C2×C151—82no—
122C2×C151—81no—
123C2×C151C22×D811yescharacteristic
124D151—22no—
125D151—81no—
126C151#341611yescharacteristic, complemented, Sylow
127#3416—1511nomaximal, Sylow
128#578—1511no—
129#576—1511no—
130#510—1512no—
131#510—1511no—
132#509—1512no—
133#509D15111yescharacteristic, complemented
134#508—1511no—
135#508—1512no—
136C2×SD32—1511no—
137C2×SD32—1512no—
138#504—1511no—
1392−1+4—3021no—
140H(5,𝔽2)—3021no—
141#137—3022no—
142#137—3021no—
143AGL(1,ℤ/8)—3022no—
144AGL(1,ℤ/8)—3021no—
145#135—1512no—
146#135—1512no—
147#135—1511no—
148#135C2×D15111yescharacteristic, complemented
149C2×Q16—1512no—
150C2×Q16—1511no—
151C2×D8—1512no—
152C2×D8C2×D15111yescharacteristic
153#131—1511no—
154Q32—1511no—
155Q32—1512no—
156SD32—1511no—
157SD32—1512no—
158SD32—1512no—
159SD32—1511no—
160SD32—1512no—
161SD32C2×D15112yescomplemented
162D16—1511no—
163D16C2×D15112yescomplemented
164M32—1512no—
165M32C2×D15111yescharacteristic, complemented
166C2×C16—1511no—
167C2×C16—1512no—
168#41—21no—
169#41—3022no—
170#41—3022no—
171#41—3021no—
172#41—6041no—
173#41—6041no—
174#41—6042no—
175#41—1511no—
176C2×Q8—3022no—
177C2×Q8—3021no—
178C2×Q8—6041no—
179C2×D4—21no—
180C2×D4—3022no—
181C2×D4—6041no—
182C2×D4—6042no—
183Q16—3022no—
184Q16—3021no—
185Q16—1512no—
186Q16—1511no—
187Q16C22×D15111yescharacteristic
188SD16—21no—
189SD16—3022no—
190SD16—3022no—
191SD16—3021no—
192D8—21no—
193D8—3022no—
194D8—1511no—
195D8C22×D15111yescharacteristic
196D8C22×D15112yes—
197M16—1511no—
198M16—1512no—
199C2×C8—1512no—
200C2×C8C22×D15111yescharacteristic
201C16—1511no—
202C16—1511no—
203C16C22×D15112yes—
204C23—41no—
205C23—6042no—
206Q8—21no—
207Q8—3021no—
208Q8—3022no—
209Q8—6042no—
210Q8—6041no—
211Q8—1511no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3021no—
217D4—6042no—
218D4—6042no—
219D4—6041no—
220D4—6042no—
221D4—1512no—
222D4—1511no—
223C2×C4—41no—
224C2×C4—6042no—
225C2×C4—6041no—
226C2×C4—6042no—
227C2×C4—6042no—
228C2×C4—6041no—
229C2×C4—1512no—
230C2×C4D4×D15111yescharacteristic
231C8—1512no—
232C8D4×D15111yescharacteristic
233C8C23×D15111yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—6041no—
238C22—24162no—
239C22—1512no—
240C22D8×D15111yescharacteristic
241C4—41no—
242C4—6041no—
243C4—6042no—
244C4—1512no—
245C4D8×D15111yescharacteristic
246C4C2×D4×D15111yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—3022no—
251C2—12081no—
252C2C2×D8×D15111yescharacteristic
subgroupG/H
centerC2C2×D8×D151
commutator, cosocleC8×C151C24
FrattiniC8C23×D151
Fitting#509×C151C2
socleC2×C151C22×D8
hypercenter#509D151
hypocenterC151#3416
LCSG ▷ C8×C151 ▷ C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #23753026C2semidirect
2. #23753013C2semidirect
3. #23752655C2semidirect
4. #23752651C2semidirect
5. #23752646C2semidirect
6. #23752642C2semidirect
7. #23752641C2semidirect
8. C2×#18012683C2semidirect
9. SD32×D151C2semidirect
10. #23752630C2semidirect
11. #509×C151C2semidirect
20. #18012758C22semidirect
21. C2×Dic604C22semidirect
22. Q16×D151C22semidirect
25. Dic1208C22semidirect
26. #18012685C22semidirect
27. #18012684C22semidirect
28. #18012683C22semidirect
29. #18012683C22semidirect
30. #18012683C22semidirect
31. #18012682C22semidirect
32. #18012680C22semidirect
33. #18012678C22semidirect
34. C2×C151⋊C16C22semidirect
35. C16×D151C22semidirect
36. #135×C151C22semidirect
38. SD32×C151C22semidirect
39. D16×C151C22semidirect
40. M32×C151C22semidirect
126. C151#3416semidirect
133. #509D151semidirect
148. #135C2×D151semidirect
161. SD32C2×D151semidirect
163. D16C2×D151semidirect
165. M32C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530262
#237530132
#237526552
#237526512
#237526462
#237526422
#237526412
C2×#180126832
SD32×D1512
#237526302
#509×C1512
#3416151