Subgroups — C₂×C₉×#490 #11767854

Order 1152.

732 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#490	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×#102	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#490	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	—	2	2	no	—
8	C₂²×D₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	2	yes	—
10	C₉×#127	—	2	2	no	—
11	C₉×#127	C₂²	1	4	yes	—
12	C₉×#125	—	2	2	no	—
13	C₉×#125	C₂²	1	4	yes	complemented
14	C₂×C₉×C₄⋊C₄	—	2	2	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₉×#102	C₂²	1	16	yes	complemented
18	C₉×#97	C₂²	1	4	yes	complemented
19	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
21	C₃×#490	C₂×C₃	1	8	yes	—
22	C₂×C₃×#102	C₂×C₃	1	4	yes	—
23	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
24	C₂⁴×C₉	—	2	2	no	—
25	C₂×D₄×C₉	—	2	2	no	—
26	C₂×D₄×C₉	—	2	8	no	—
27	C₂×D₄×C₉	—	2	8	no	—
28	C₂×D₄×C₉	—	4	8	no	—
29	C₂×D₄×C₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	D₄	1	2	yes	—
32	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₉	—	2	4	no	—
34	C₂×C₈×C₉	C₂³	1	4	yes	—
35	C₉×C₄⋊C₄	—	2	8	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	C₂³	1	4	yes	—
38	C₄²×C₉	D₄	1	2	yes	—
39	C₄²×C₉	C₂³	1	2	yes	—
40	C₂×#490	C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₃×D₄	—	2	2	no	—
42	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
43	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
44	C₃×#127	—	2	2	no	—
45	C₃×#127	C₂²×C₃	1	4	yes	—
46	C₃×#125	—	2	2	no	—
47	C₃×#125	C₂²×C₃	1	4	yes	—
48	C₂×C₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₃×#102	C₂²×C₃	1	16	yes	—
52	C₃×#97	C₂²×C₃	1	4	yes	—
53	C₂³×C₉	—	2	4	no	—
54	C₂³×C₉	—	4	4	no	—
55	C₂³×C₉	—	4	8	no	—
56	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
57	D₄×C₉	—	2	8	no	—
58	D₄×C₉	—	4	16	no	—
59	D₄×C₉	—	4	4	no	—
60	C₂×C₄×C₉	—	2	4	no	—
61	C₂×C₄×C₉	—	2	4	no	—
62	C₂×C₄×C₉	—	4	4	no	—
63	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	#41	1	4	yes	—
66	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
67	C₈×C₉	—	2	4	no	—
68	C₂×#127	C₂×C₉	1	1	yes	characteristic
69	C₂×#125	C₂×C₉	1	1	yes	characteristic, complemented
70	#490	C₂×C₉	1	8	yes	direct product factor
71	C₂×#102	C₂×C₉	1	4	yes	complemented
72	C₂×#97	C₂×C₉	1	1	yes	characteristic, complemented
73	C₂⁴×C₃	—	2	2	no	—
74	C₂×C₃×D₄	—	2	2	no	—
75	C₂×C₃×D₄	—	2	8	no	—
76	C₂×C₃×D₄	—	2	8	no	—
77	C₂×C₃×D₄	—	4	8	no	—
78	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
79	C₂²×C₃×C₄	—	2	2	no	—
80	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
84	C₃×C₄⋊C₄	—	2	8	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
87	C₃×C₄²	C₃×D₄	1	2	yes	—
88	C₃×C₄²	C₂³×C₃	1	2	yes	—
89	C₂²×C₉	—	4	4	no	—
90	C₂²×C₉	—	4	8	no	—
91	C₂²×C₉	—	8	8	no	—
92	C₂²×C₉	#124	1	2	yes	—
93	C₂²×C₉	AGL(1,ℤ/8)	1	4	yes	—
94	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
95	C₄×C₉	—	2	4	no	—
96	C₄×C₉	—	4	4	no	—
97	C₄×C₉	#124	1	2	yes	—
98	C₄×C₉	C₂×#41	1	2	yes	—
99	C₂²×D₄	—	2	2	no	—
100	C₂²×D₄	C₂²×C₉	1	2	yes	—
101	C₂²×C₈	C₂²×C₉	1	2	yes	—
102	#127	—	2	2	no	—
103	#127	C₂²×C₉	1	4	yes	—
104	#125	—	2	2	no	—
105	#125	C₂²×C₉	1	4	yes	complemented
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
108	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
109	#102	C₂²×C₉	1	16	yes	complemented
110	#97	C₂²×C₉	1	4	yes	complemented
111	C₂³×C₃	—	2	4	no	—
112	C₂³×C₃	—	4	4	no	—
113	C₂³×C₃	—	4	8	no	—
114	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
115	C₃×D₄	—	2	8	no	—
116	C₃×D₄	—	4	16	no	—
117	C₃×D₄	—	4	4	no	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	—	4	4	no	—
121	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
122	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄	C₃×#41	1	4	yes	—
124	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
125	C₃×C₈	—	2	4	no	—
126	C₂×C₉	—	8	4	no	—
127	C₂×C₉	#490	1	4	yes	direct product factor
128	C₂×C₉	C₂×#124	1	1	yes	characteristic
129	C₂×C₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₂⁴	—	2	2	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	2	8	no	—
133	C₂×D₄	—	2	8	no	—
134	C₂×D₄	—	4	8	no	—
135	C₂×D₄	C₂³×C₉	1	4	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	D₄×C₉	1	2	yes	—
138	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₂³×C₉	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	C₂³×C₉	1	4	yes	—
144	C₄²	D₄×C₉	1	2	yes	—
145	C₄²	C₂³×C₉	1	2	yes	—
146	C₂²×C₃	—	4	8	no	—
147	C₂²×C₃	—	4	4	no	—
148	C₂²×C₃	—	8	8	no	—
149	C₂²×C₃	C₃×#124	1	2	yes	—
150	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	4	yes	—
151	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
152	C₃×C₄	—	2	4	no	—
153	C₃×C₄	—	4	4	no	—
154	C₃×C₄	C₃×#124	1	2	yes	—
155	C₃×C₄	C₂×C₃×#41	1	2	yes	—
156	C₉	C₂×#490	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³	—	2	4	no	—
158	C₂³	—	4	4	no	—
159	C₂³	—	4	8	no	—
160	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
161	D₄	—	2	8	no	—
162	D₄	—	4	16	no	—
163	D₄	—	4	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
168	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
169	C₂×C₄	C₉×#41	1	4	yes	—
170	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₂×C₃	—	8	4	no	—
173	C₂×C₃	C₃×#490	1	4	yes	—
174	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
175	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
176	C₂²	—	4	4	no	—
177	C₂²	—	4	8	no	—
178	C₂²	—	8	8	no	—
179	C₂²	C₉×#124	1	2	yes	—
180	C₂²	C₉×AGL(1,ℤ/8)	1	4	yes	—
181	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	4	4	no	—
184	C₄	C₉×#124	1	2	yes	—
185	C₄	C₂×C₉×#41	1	2	yes	—
186	C₃	C₂×C₃×#490	1	1	yes	characteristic
187	C₂	—	8	4	no	—
188	C₂	C₉×#490	1	4	yes	direct product factor
189	C₂	C₂×C₉×#124	1	1	yes	characteristic
190	C₂	C₂×C₉×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×#127	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#125	C₂	semidirect
3. C₉×#490	C₂	direct (N × H)
4. C₂×C₉×#102	C₂	semidirect
5. C₂×C₉×#97	C₂	semidirect
13. C₉×#125	C₂²	semidirect
17. C₉×#102	C₂²	semidirect
18. C₉×#97	C₂²	semidirect
40. C₂×#490	C₉	direct (N × H)
69. C₂×#125	C₂×C₉	semidirect
70. #490	C₂×C₉	direct (N × H)
71. C₂×#102	C₂×C₉	semidirect
72. C₂×#97	C₂×C₉	semidirect
105. #125	C₂²×C₉	semidirect
109. #102	C₂²×C₉	semidirect
110. #97	C₂²×C₉	semidirect
127. C₂×C₉	#490	direct (N × H)
156. C₉	C₂×#490	direct (N × H)
188. C₂	C₉×#490	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#127	2
C₂×C₉×#125	2
C₉×#490	2
C₂×C₉×#102	2
C₂×C₉×#97	2
C₂×C₃×#490	3

Subgroups — C2×C9×#490 #11767854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#490 #11767854