Finite Group Databasebeta

← #510×D181 #37546847

Subgroups — #510×D181 #37546847

Order 23168.

360 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D181C211yesmaximal, characteristic
2C2×Q16×D181C211yesmaximal, characteristic, complemented
3#26041722C211yesmaximal, characteristic, complemented
4#26041721C212yesmaximal, complemented
5#26041720C211yesmaximal, characteristic, complemented
6#26041719C211yesmaximal, characteristic, complemented
7#26041718C212yesmaximal, complemented
8Q32×D181C212yesmaximal, complemented
9SD32×D181C212yesmaximal, complemented
10M32×D181C211yesmaximal, characteristic, complemented
11#510×C181C211yesmaximal, characteristic, complemented
12#41×D181—21no—
13C2×Q8×D181—21no—
14#19153839C2211yescharacteristic
15#19153838C2211yescharacteristic
16#19153837—21no—
17#19153836—21no—
18#19153835C2211yescharacteristic
19C2×Dic724C2211yescharacteristic, complemented
20C2×#15668439—21no—
21Q16×D181—21no—
22Q16×D181—21no—
23Q16×D181C2212yes—
24Q16×D181C2211yescharacteristic
25Q16×D181C2212yescomplemented
26SD16×D181—21no—
27D8×D181C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic
29Dic1448C2212yescomplemented
30#19153762C2212yescomplemented
31#19153762C2212yescomplemented
32#19153761C2212yescomplemented
33#19153760C2212yescomplemented
34#19153759C2212yescomplemented
35#19153756C2211yescharacteristic, complemented
36#19153755C2212yescomplemented
37C16×D181C2212yes—
38#135×C181C2211yescharacteristic
39C2×Q16×C181C2211yescharacteristic, complemented
40Q32×C181C2212yescomplemented
41SD32×C181C2212yescomplemented
42M32×C181C2211yescharacteristic, complemented
43#15668450—21no—
44#15668449—21no—
45#15668449—41no—
46#15668448—21no—
47#15668448—41no—
48C2×Dic362—21no—
49C2×Dic362—41no—
50Q8×D181—21no—
51Q8×D181—22no—
52Q8×D181—22no—
53Q8×D181—41no—
54Q8×D181—41no—
55D4×D181—21no—
56D4×D181—41no—
57C2×C4×D181—41no—
58C2×C4×D181—41no—
59C2×C4×D181D411yescharacteristic
60Dic724—21no—
61Dic724C2312yes—
62Dic724C2311yescharacteristic
63#15668439—22no—
64#15668439—22no—
65#15668439—21no—
66#15668438—21no—
67#15668437—21no—
68#15668436—21no—
69D1448C2311yescharacteristic
70#15668434—21no—
71C2×#13089678D411yescharacteristic
72C8×D181D411yescharacteristic
73C8×D181D411yescharacteristic
74C8×D181C2311yescharacteristic
75C8×D181C2311yescharacteristic
76#15668424C2312yes—
77#41×C181—21no—
78C2×Q8×C181—21no—
79Q16×C181—21no—
80Q16×C181C2312yes—
81Q16×C181C2311yescharacteristic
82SD16×C181—21no—
83D8×C181C2311yescharacteristic
84C2×C8×C181C2311yescharacteristic
85C16×C181C2312yes—
86C22×D181—41no—
87C22×D181D811yescharacteristic
88Dic362—21no—
89Dic362—22no—
90Dic362—41no—
91Dic362—42no—
92Dic362—42no—
93Dic362—41no—
94D724—21no—
95D724—41no—
96C22⋊D181—41no—
97C22⋊D181—41no—
98C22⋊D181—41no—
99C2×Dic181—41no—
100C2×Dic181—41no—
101C2×Dic181D811yescharacteristic
102C4×D181—41no—
103C4×D181—42no—
104C4×D181—42no—
105C4×D181—41no—
106C4×D181—41no—
107C4×D181D811yescharacteristic
108C4×D181D811yescharacteristic
109C4×D181C2×D411yescharacteristic
110C4×D181C2×D411yescharacteristic
111#13089678C2×D411yescharacteristic
112#13089678C2×D411yescharacteristic
113Q8×C181—21no—
114Q8×C181—22no—
115Q8×C181—41no—
116D4×C181—21no—
117D4×C181—41no—
118C2×C4×C181—41no—
119C2×C4×C181—41no—
120C2×C4×C181C2×D411yescharacteristic
121C8×C181C2×D411yescharacteristic
122C8×C181C2411yescharacteristic
123C22×C181—41no—
124C22×C181C2×D811yescharacteristic
125C2×D181—22no—
126C2×D181—41no—
127C2×D181—82no—
128C2×D181C2×D811yescharacteristic
129C2×D181C2×D811yescharacteristic
130C4×C181—41no—
131C4×C181—42no—
132C4×C181C2×D811yescharacteristic
133C4×C181C22×D411yescharacteristic
134Dic181—41no—
135Dic181—42no—
136Dic181C2×D811yescharacteristic
137Dic181C2×D811yescharacteristic
138C2×C181—21no—
139C2×C181—81no—
140C2×C181C22×D811yescharacteristic
141D181—21no—
142D181—81no—
143D181#51012yesdirect product factor
144C181C2×#51011yescharacteristic, complemented, Sylow
145C2×#510—1811nomaximal, Sylow
146C2×#135—1811no—
147C22×Q16—1811no—
148#510—1811no—
149#510—1811no—
150#510—1812no—
151#510—1812no—
152#510—1811no—
153#510D18111yescharacteristic, direct product factor
154C2×Q32—1812no—
155C2×SD32—1812no—
156C2×M32—1811no—
157C2×#41—3621no—
158C22×Q8—3621no—
159#135—3621no—
160#135—3621no—
161#135—1811no—
162#135—1811no—
163#135—1811no—
164#135C2×D18111yescharacteristic
165C2×Q16—3621no—
166C2×Q16—3621no—
167C2×Q16—3621no—
168C2×Q16—1812no—
169C2×Q16—1812no—
170C2×Q16—1811no—
171C2×Q16—1811no—
172C2×Q16C2×D18111yescharacteristic, complemented
173C2×SD16—3621no—
174C2×D8—1811no—
175C22×C8—1811no—
176Q32—1812no—
177Q32—1812no—
178Q32—1812no—
179Q32C2×D18112yescomplemented
180SD32—1812no—
181SD32—1812no—
182SD32—1812no—
183SD32C2×D18112yescomplemented
184M32—1811no—
185M32—1812no—
186M32C2×D18111yescharacteristic, complemented
187C2×C16—1812no—
188#41—21no—
189#41—3621no—
190#41—3621no—
191#41—3621no—
192#41—7241no—
193#41—7241no—
194C2×Q8—21no—
195C2×Q8—3621no—
196C2×Q8—3622no—
197C2×Q8—3622no—
198C2×Q8—3621no—
199C2×Q8—7241no—
200C2×Q8—7241no—
201C2×Q8—7241no—
202C2×D4—3621no—
203C2×D4—7241no—
204C22×C4—1811no—
205C22×C4—7241no—
206C22×C4—7241no—
207Q16—21no—
208Q16—3622no—
209Q16—3622no—
210Q16—3621no—
211Q16—3621no—
212Q16—1812no—
213Q16—1811no—
214Q16C22×D18112yes—
215Q16C22×D18111yescharacteristic
216SD16—21no—
217SD16—3621no—
218SD16—3621no—
219SD16—3621no—
220D8—3621no—
221D8—1811no—
222D8C22×D18111yescharacteristic
223C2×C8—1811no—
224C2×C8—1811no—
225C2×C8—1811no—
226C2×C8—1811no—
227C2×C8—1811no—
228C2×C8C22×D18111yescharacteristic
229C16—1812no—
230C16C22×D18112yes—
231C23—1811no—
232C23—7241no—
233Q8—21no—
234Q8—22no—
235Q8—41no—
236Q8—3621no—
237Q8—3622no—
238Q8—7242no—
239Q8—7242no—
240Q8—7241no—
241Q8—7241no—
242D4—21no—
243D4—41no—
244D4—3621no—
245D4—7241no—
246D4—7241no—
247D4—7241no—
248D4—7241no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—1811no—
252C2×C4—1811no—
253C2×C4—1811no—
254C2×C4—1811no—
255C2×C4—1811no—
256C2×C4—7241no—
257C2×C4—7241no—
258C2×C4—7242no—
259C2×C4—7242no—
260C2×C4—7241no—
261C2×C4—7241no—
262C2×C4—7241no—
263C2×C4D4×D18111yescharacteristic
264C8—1811no—
265C8—1811no—
266C8D4×D18111yescharacteristic
267C8C23×D18111yescharacteristic
268C22—41no—
269C22—3622no—
270C22—14482no—
271C22—1811no—
272C22—1811no—
273C22—7241no—
274C22D8×D18111yescharacteristic
275C4—41no—
276C4—42no—
277C4—1811no—
278C4—1811no—
279C4—7241no—
280C4—7242no—
281C4D8×D18111yescharacteristic
282C4C2×D4×D18111yescharacteristic
283C2—21no—
284C2—81no—
285C2—3621no—
286C2—14481no—
287C2—1812no—
288C2C2×D8×D18111yescharacteristic
subgroupG/H
centerC2C2×D8×D181
commutator, cosocleC8×C181C24
FrattiniC8C23×D181
Fitting#510×C181C2
socleC2×C181C22×D8
prime radical#135×D181C2
hypercenter#510D181
hypocenterC181C2×#510
LCSG ▷ C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D181C2semidirect
3. #26041722C2semidirect
4. #26041721C2semidirect
5. #26041720C2semidirect
6. #26041719C2semidirect
7. #26041718C2semidirect
8. Q32×D181C2semidirect
9. SD32×D181C2semidirect
10. M32×D181C2semidirect
11. #510×C181C2semidirect
19. C2×Dic724C22semidirect
25. Q16×D181C22semidirect
29. Dic1448C22semidirect
30. #19153762C22semidirect
31. #19153762C22semidirect
32. #19153761C22semidirect
33. #19153760C22semidirect
34. #19153759C22semidirect
35. #19153756C22semidirect
36. #19153755C22semidirect
39. C2×Q16×C181C22semidirect
40. Q32×C181C22semidirect
41. SD32×C181C22semidirect
42. M32×C181C22semidirect
143. D181#510direct (N × H)
144. C181C2×#510semidirect
153. #510D181direct (N × H)
172. C2×Q16C2×D181semidirect
179. Q32C2×D181semidirect
183. SD32C2×D181semidirect
186. M32C2×D181semidirect

Maximal subgroups

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

subgroupindex
#135×D1812
C2×Q16×D1812
#260417222
#260417212
#260417202
#260417192
#260417182
Q32×D1812
SD32×D1812
M32×D1812
#510×C1812
C2×#510181