Subgroups — C₉×#2309 #11767030

Order 1152.

1656 conjugacy classes of subgroups.

The subgroup lattice has 201 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#117	C₂	1	2	yes	maximal, complemented
3	C₉×#390	C₂	1	8	yes	maximal, complemented
4	C₉×#379	C₂	1	4	yes	maximal, complemented
5	C₃×#2309	C₃	1	1	yes	maximal, characteristic
6	C₂⁵×C₉	C₂²	1	2	yes	—
7	C₂²×D₄×C₉	—	2	8	no	—
8	C₂²×D₄×C₉	C₄	1	8	yes	complemented
9	C₂²×D₄×C₉	C₂²	1	4	yes	—
10	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₉×#117	—	2	8	no	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	complemented
13	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	16	yes	complemented
15	C₂×C₄²×C₉	C₂²	1	4	yes	complemented
16	C₂³×C₃×D₄	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×#117	C₂×C₃	1	2	yes	—
18	C₃×#390	C₂×C₃	1	8	yes	—
19	C₃×#379	C₂×C₃	1	4	yes	—
20	C₂⁴×C₉	—	2	8	no	—
21	C₂⁴×C₉	—	4	8	no	—
22	C₂⁴×C₉	C₂×C₄	1	8	yes	—
23	C₂⁴×C₉	C₂³	1	4	yes	—
24	C₂⁴×C₉	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₉	—	2	32	no	—
26	C₂×D₄×C₉	—	2	8	no	—
27	C₂×D₄×C₉	—	2	8	no	—
28	C₂×D₄×C₉	C₂×C₄	1	16	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₉	D₄	1	8	yes	—
31	C₂²×C₄×C₉	C₂³	1	8	yes	—
32	C₂²×C₄×C₉	C₂³	1	2	yes	—
33	C₉×C₄⋊C₄	—	2	8	no	—
34	C₉×C₂²⋊C₄	—	2	8	no	—
35	C₉×C₂²⋊C₄	—	4	16	no	—
36	C₄²×C₉	—	2	8	no	—
37	#2309	C₉	1	1	yes	characteristic, direct product factor, Sylow
38	C₂⁵×C₃	C₂²×C₃	1	2	yes	—
39	C₂²×C₃×D₄	—	2	8	no	—
40	C₂²×C₃×D₄	C₃×C₄	1	8	yes	—
41	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
42	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₃×#117	—	2	8	no	—
44	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	16	yes	—
47	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
48	C₂³×C₉	—	2	4	no	—
49	C₂³×C₉	—	2	16	no	—
50	C₂³×C₉	—	2	8	no	—
51	C₂³×C₉	—	2	16	no	—
52	C₂³×C₉	—	4	16	no	—
53	C₂³×C₉	—	4	32	no	—
54	C₂³×C₉	C₂²×C₄	1	8	yes	—
55	C₂³×C₉	C₂²×C₄	1	2	yes	—
56	C₂³×C₉	C₂×D₄	1	4	yes	—
57	C₂³×C₉	C₂⁴	1	1	yes	characteristic
58	D₄×C₉	—	2	32	no	—
59	C₂×C₄×C₉	—	2	8	no	—
60	C₂×C₄×C₉	—	2	16	no	—
61	C₂×C₄×C₉	C₂²⋊C₄	1	16	yes	complemented
62	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
65	C₂³×D₄	C₂×C₉	1	1	yes	characteristic
66	C₂×#117	C₂×C₉	1	2	yes	complemented
67	#390	C₂×C₉	1	8	yes	complemented
68	#379	C₂×C₉	1	4	yes	complemented
69	C₂⁴×C₃	—	2	8	no	—
70	C₂⁴×C₃	—	4	8	no	—
71	C₂⁴×C₃	C₂×C₃×C₄	1	8	yes	—
72	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
73	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₃×D₄	—	2	32	no	—
75	C₂×C₃×D₄	—	2	8	no	—
76	C₂×C₃×D₄	—	2	8	no	—
77	C₂×C₃×D₄	C₂×C₃×C₄	1	16	yes	—
78	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
79	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
80	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
82	C₃×C₄⋊C₄	—	2	8	no	—
83	C₃×C₂²⋊C₄	—	2	8	no	—
84	C₃×C₂²⋊C₄	—	4	16	no	—
85	C₃×C₄²	—	2	8	no	—
86	C₂²×C₉	—	2	4	no	—
87	C₂²×C₉	—	2	8	no	—
88	C₂²×C₉	—	2	8	no	—
89	C₂²×C₉	—	2	16	no	—
90	C₂²×C₉	—	4	32	no	—
91	C₂²×C₉	—	4	16	no	—
92	C₂²×C₉	C₂×C₂²⋊C₄	1	4	yes	—
93	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₉	C₂²×D₄	1	2	yes	—
95	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
96	C₄×C₉	—	4	8	no	—
97	C₄×C₉	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂⁵	C₂²×C₉	1	2	yes	—
99	C₂²×D₄	—	2	8	no	—
100	C₂²×D₄	C₄×C₉	1	8	yes	complemented
101	C₂²×D₄	C₂²×C₉	1	4	yes	—
102	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
103	#117	—	2	8	no	—
104	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	complemented
105	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₉	1	16	yes	complemented
107	C₂×C₄²	C₂²×C₉	1	4	yes	complemented
108	C₂³×C₃	—	2	4	no	—
109	C₂³×C₃	—	2	16	no	—
110	C₂³×C₃	—	2	8	no	—
111	C₂³×C₃	—	2	16	no	—
112	C₂³×C₃	—	4	16	no	—
113	C₂³×C₃	—	4	32	no	—
114	C₂³×C₃	C₂²×C₃×C₄	1	8	yes	—
115	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
116	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
117	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
118	C₃×D₄	—	2	32	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	2	16	no	—
121	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	16	yes	—
122	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
123	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
124	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
125	C₂×C₉	—	2	4	no	—
126	C₂×C₉	—	4	8	no	—
127	C₂×C₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₉	#518	1	2	yes	—
129	C₂×C₉	#535	1	4	yes	—
130	C₂⁴	—	2	8	no	—
131	C₂⁴	—	4	8	no	—
132	C₂⁴	C₂×C₄×C₉	1	8	yes	—
133	C₂⁴	C₂³×C₉	1	4	yes	—
134	C₂⁴	C₂³×C₉	1	1	yes	characteristic
135	C₂×D₄	—	2	32	no	—
136	C₂×D₄	—	2	8	no	—
137	C₂×D₄	—	2	8	no	—
138	C₂×D₄	C₂×C₄×C₉	1	16	yes	—
139	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
140	C₂²×C₄	D₄×C₉	1	8	yes	—
141	C₂²×C₄	C₂³×C₉	1	8	yes	—
142	C₂²×C₄	C₂³×C₉	1	2	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₂²⋊C₄	—	2	8	no	—
145	C₂²⋊C₄	—	4	16	no	—
146	C₄²	—	2	8	no	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	2	8	no	—
149	C₂²×C₃	—	2	8	no	—
150	C₂²×C₃	—	2	16	no	—
151	C₂²×C₃	—	4	32	no	—
152	C₂²×C₃	—	4	16	no	—
153	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	4	yes	—
154	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
155	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
156	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
157	C₃×C₄	—	4	8	no	—
158	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	8	yes	—
159	C₉	#2309	1	1	yes	characteristic, direct product factor, Sylow
160	C₂³	—	2	4	no	—
161	C₂³	—	2	16	no	—
162	C₂³	—	2	8	no	—
163	C₂³	—	2	16	no	—
164	C₂³	—	4	16	no	—
165	C₂³	—	4	32	no	—
166	C₂³	C₂²×C₄×C₉	1	8	yes	—
167	C₂³	C₂²×C₄×C₉	1	2	yes	—
168	C₂³	C₂×D₄×C₉	1	4	yes	—
169	C₂³	C₂⁴×C₉	1	1	yes	characteristic
170	D₄	—	2	32	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	C₉×C₂²⋊C₄	1	16	yes	complemented
174	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
175	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
176	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
177	C₂×C₃	—	2	4	no	—
178	C₂×C₃	—	4	8	no	—
179	C₂×C₃	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₃	C₃×#518	1	2	yes	—
181	C₂×C₃	C₃×#535	1	4	yes	—
182	C₂²	—	2	8	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	8	no	—
185	C₂²	—	2	16	no	—
186	C₂²	—	4	32	no	—
187	C₂²	—	4	16	no	—
188	C₂²	C₂×C₉×C₂²⋊C₄	1	4	yes	—
189	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
190	C₂²	C₂²×D₄×C₉	1	2	yes	—
191	C₂²	C₉×H(5,𝔽₂)	1	4	yes	—
192	C₄	—	4	8	no	—
193	C₄	C₂×C₉×C₂²⋊C₄	1	8	yes	—
194	C₃	C₃×#2309	1	1	yes	characteristic
195	C₂	—	2	4	no	—
196	C₂	—	4	8	no	—
197	C₂	C₂²×C₉×C₂²⋊C₄	1	1	yes	characteristic
198	C₂	C₉×#518	1	2	yes	—
199	C₂	C₉×#535	1	4	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂³×C₃×D₄	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#117	C₂	semidirect
3. C₉×#390	C₂	semidirect
4. C₉×#379	C₂	semidirect
8. C₂²×D₄×C₉	C₄	semidirect
12. C₂×C₉×C₄⋊C₄	C₂²	semidirect
14. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
15. C₂×C₄²×C₉	C₂²	semidirect
37. #2309	C₉	direct (N × H)
61. C₂×C₄×C₉	C₂²⋊C₄	semidirect
66. C₂×#117	C₂×C₉	semidirect
67. #390	C₂×C₉	semidirect
68. #379	C₂×C₉	semidirect
100. C₂²×D₄	C₄×C₉	semidirect
104. C₂×C₄⋊C₄	C₂²×C₉	semidirect
106. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
107. C₂×C₄²	C₂²×C₉	semidirect
159. C₉	#2309	direct (N × H)
173. C₂×C₄	C₉×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₉	2
C₂×C₉×#117	2
C₉×#390	2
C₉×#379	2
C₃×#2309	3

Subgroups — C9×#2309 #11767030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2309 #11767030