Subgroups — C₉×#2980 #11767701

Order 1152.

591 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₉×#435	C₂	1	8	yes	maximal, complemented
3	C₉×#432	C₂	1	2	yes	maximal, complemented
4	C₉×#408	C₂	1	4	yes	maximal, complemented
5	C₃×#2980	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×#41	C₂²	1	2	yes	—
7	C₂×C₉×M₁₆	C₂²	1	4	yes	—
8	C₂²×C₈×C₉	C₂²	1	4	yes	—
9	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₉×#127	C₄	1	1	yes	characteristic
11	C₉×#124	C₄	1	2	yes	—
12	C₉×#121	C₄	1	4	yes	—
13	C₄×D₄×C₉	C₂²	1	4	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊C₈	C₂²	1	4	yes	—
16	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
17	C₉×C₂²⋊C₈	C₂²	1	8	yes	—
18	C₉×#97	C₂²	1	4	yes	—
19	C₃×#532	C₂×C₃	1	1	yes	characteristic
20	C₃×#435	C₂×C₃	1	8	yes	—
21	C₃×#432	C₂×C₃	1	2	yes	—
22	C₃×#408	C₂×C₃	1	4	yes	—
23	C₉×#41	—	4	2	no	—
24	C₉×#41	—	4	2	no	—
25	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
26	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
27	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₉	C₂³	1	4	yes	—
32	C₉×M₁₆	—	4	4	no	—
33	C₂×C₈×C₉	—	4	4	no	—
34	C₂×C₈×C₉	C₂³	1	8	yes	—
35	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
37	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₉	C₂³	1	2	yes	—
40	#2980	C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
42	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
43	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
44	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
45	C₃×#127	C₃×C₄	1	1	yes	characteristic
46	C₃×#124	C₃×C₄	1	2	yes	—
47	C₃×#121	C₃×C₄	1	4	yes	—
48	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
51	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
52	C₃×C₂²⋊C₈	C₂²×C₃	1	8	yes	—
53	C₃×#97	C₂²×C₃	1	4	yes	—
54	C₂³×C₉	C₂²×C₄	1	4	yes	—
55	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
56	Q₈×C₉	—	4	2	no	—
57	D₄×C₉	—	4	4	no	—
58	D₄×C₉	—	4	2	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	—	2	2	no	—
63	C₂×C₄×C₉	—	4	2	no	—
64	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
69	C₈×C₉	—	2	8	no	—
70	#532	C₂×C₉	1	1	yes	characteristic
71	#435	C₂×C₉	1	8	yes	complemented
72	#432	C₂×C₉	1	2	yes	complemented
73	#408	C₂×C₉	1	4	yes	complemented
74	C₃×#41	—	4	2	no	—
75	C₃×#41	—	4	2	no	—
76	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
78	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
79	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
83	C₃×M₁₆	—	4	4	no	—
84	C₂×C₃×C₈	—	4	4	no	—
85	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
86	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
87	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
88	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
89	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
90	C₃×C₄²	C₂³×C₃	1	2	yes	—
91	C₂²×C₉	—	2	1	no	—
92	C₂²×C₉	—	2	4	no	—
93	C₂²×C₉	—	2	2	no	—
94	C₂²×C₉	—	2	8	no	—
95	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₉	—	2	1	no	—
97	C₄×C₉	—	2	4	no	—
98	C₄×C₉	—	2	2	no	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
101	C₂×#41	C₂²×C₉	1	2	yes	—
102	C₂×M₁₆	C₂²×C₉	1	4	yes	—
103	C₂²×C₈	C₂²×C₉	1	4	yes	—
104	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
105	#127	C₄×C₉	1	1	yes	characteristic
106	#124	C₄×C₉	1	2	yes	—
107	#121	C₄×C₉	1	4	yes	—
108	C₄×D₄	C₂²×C₉	1	4	yes	—
109	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
110	C₄⋊C₈	C₂²×C₉	1	4	yes	—
111	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
112	C₂²⋊C₈	C₂²×C₉	1	8	yes	—
113	#97	C₂²×C₉	1	4	yes	—
114	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
115	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₃×Q₈	—	4	2	no	—
117	C₃×D₄	—	4	4	no	—
118	C₃×D₄	—	4	2	no	—
119	C₃×D₄	—	4	4	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	4	2	no	—
124	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
126	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
128	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
129	C₃×C₈	—	2	8	no	—
130	C₂×C₉	—	4	1	no	—
131	C₂×C₉	—	4	4	no	—
132	C₂×C₉	#518	1	1	yes	characteristic
133	C₂×C₉	#568	1	2	yes	—
134	#41	—	4	2	no	—
135	#41	—	4	2	no	—
136	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
137	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
138	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
139	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
140	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
141	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₉	1	4	yes	—
143	M₁₆	—	4	4	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	C₂³×C₉	1	8	yes	—
146	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
147	C₂²⋊C₄	C₂×C₄×C₉	1	8	yes	—
148	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
149	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
150	C₄²	C₂³×C₉	1	2	yes	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	2	8	no	—
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
161	C₉	#2980	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³	C₂²×C₄×C₉	1	4	yes	—
163	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
164	Q₈	—	4	2	no	—
165	D₄	—	4	2	no	—
166	D₄	—	4	4	no	—
167	D₄	—	4	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
176	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
177	C₈	—	2	8	no	—
178	C₂×C₃	—	4	1	no	—
179	C₂×C₃	—	4	4	no	—
180	C₂×C₃	C₃×#518	1	1	yes	characteristic
181	C₂×C₃	C₃×#568	1	2	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	8	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	4	no	—
186	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	4	no	—
191	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
192	C₃	C₃×#2980	1	1	yes	characteristic
193	C₂	—	4	1	no	—
194	C₂	—	4	4	no	—
195	C₂	C₉×#518	1	1	yes	characteristic
196	C₂	C₉×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#435	C₂	semidirect
3. C₉×#432	C₂	semidirect
4. C₉×#408	C₂	semidirect
40. #2980	C₉	direct (N × H)
71. #435	C₂×C₉	semidirect
72. #432	C₂×C₉	semidirect
73. #408	C₂×C₉	semidirect
161. C₉	#2980	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#532	2
C₉×#435	2
C₉×#432	2
C₉×#408	2
C₃×#2980	3

Subgroups — C9×#2980 #11767701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2980 #11767701