Subgroups — C₉×#1910 #11766631

Order 1152.

516 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#535	C₂	1	1	yes	maximal, characteristic
3	C₉×#418	C₂	1	2	yes	maximal, complemented
4	C₉×#410	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
6	C₃×#1910	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₉×#131	—	2	4	no	—
10	C₂×C₉×M₁₆	—	2	2	no	—
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₉×#127	C₄	1	2	yes	complemented
13	C₉×#124	C₄	1	2	yes	complemented
14	C₉×#121	—	2	2	no	—
15	C₉×#120	—	2	2	no	—
16	C₉×#117	C₂²	1	2	yes	—
17	C₉×#117	C₂²	1	1	yes	characteristic
18	C₉×#102	—	2	4	no	—
19	C₉×#99	—	2	2	no	—
20	C₃×#568	C₂×C₃	1	1	yes	characteristic
21	C₃×#535	C₂×C₃	1	1	yes	characteristic
22	C₃×#418	C₂×C₃	1	2	yes	—
23	C₃×#410	C₂×C₃	1	1	yes	characteristic
24	C₃×H(3,ℤ/4)	C₂×C₃	1	2	yes	—
25	C₂⁴×C₉	—	2	1	no	—
26	C₉×#41	—	2	2	no	—
27	C₉×#41	—	2	2	no	—
28	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₉	—	2	4	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	2	no	—
33	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₉	D₄	1	2	yes	—
36	C₂²×C₄×C₉	D₄	1	2	yes	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×M₁₆	—	2	4	no	—
39	C₉×M₁₆	—	2	2	no	—
40	C₉×M₁₆	—	2	2	no	—
41	C₂×C₈×C₉	—	2	4	no	—
42	C₂×C₈×C₉	D₄	1	4	yes	complemented
43	C₉×C₄⋊C₄	—	2	1	no	—
44	C₉×C₄⋊C₄	—	2	2	no	—
45	C₉×C₂²⋊C₄	—	2	1	no	—
46	C₉×C₂²⋊C₄	—	2	4	no	—
47	C₉×C₂²⋊C₄	—	2	2	no	—
48	C₄²×C₉	—	2	2	no	—
49	C₄²×C₉	C₂×C₄	1	2	yes	—
50	#1910	C₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
53	C₃×#131	—	2	4	no	—
54	C₂×C₃×M₁₆	—	2	2	no	—
55	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
56	C₃×#127	C₃×C₄	1	2	yes	—
57	C₃×#124	C₃×C₄	1	2	yes	—
58	C₃×#121	—	2	2	no	—
59	C₃×#120	—	2	2	no	—
60	C₃×#117	C₂²×C₃	1	2	yes	—
61	C₃×#117	C₂²×C₃	1	1	yes	characteristic
62	C₃×#102	—	2	4	no	—
63	C₃×#99	—	2	2	no	—
64	C₂³×C₉	—	2	2	no	—
65	C₂³×C₉	—	4	2	no	—
66	C₂³×C₉	—	8	2	no	—
67	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₉	—	2	2	no	—
70	D₄×C₉	—	2	4	no	—
71	D₄×C₉	—	2	2	no	—
72	D₄×C₉	—	2	4	no	—
73	D₄×C₉	—	4	1	no	—
74	D₄×C₉	—	4	1	no	—
75	D₄×C₉	—	8	4	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	4	2	no	—
80	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₉	#41	1	2	yes	—
85	C₈×C₉	—	2	4	no	—
86	#568	C₂×C₉	1	1	yes	characteristic, complemented
87	#535	C₂×C₉	1	1	yes	characteristic
88	#418	C₂×C₉	1	2	yes	complemented
89	#410	C₂×C₉	1	1	yes	characteristic, complemented
90	H(3,ℤ/4)	C₂×C₉	1	2	yes	complemented
91	C₂⁴×C₃	—	2	1	no	—
92	C₃×#41	—	2	2	no	—
93	C₃×#41	—	2	2	no	—
94	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₃×D₄	—	2	4	no	—
96	C₂×C₃×D₄	—	2	2	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	—	2	2	no	—
99	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
100	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
102	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
103	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₃×M₁₆	—	2	4	no	—
105	C₃×M₁₆	—	2	2	no	—
106	C₃×M₁₆	—	2	2	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
109	C₃×C₄⋊C₄	—	2	1	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₂²⋊C₄	—	2	1	no	—
112	C₃×C₂²⋊C₄	—	2	4	no	—
113	C₃×C₂²⋊C₄	—	2	2	no	—
114	C₃×C₄²	—	2	2	no	—
115	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
116	C₂²×C₉	—	2	2	no	—
117	C₂²×C₉	—	4	1	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	—	4	4	no	—
120	C₂²×C₉	—	4	2	no	—
121	C₂²×C₉	—	8	4	no	—
122	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₉	#121	1	2	yes	—
124	C₄×C₉	—	2	2	no	—
125	C₄×C₉	—	4	2	no	—
126	C₄×C₉	—	8	2	no	—
127	C₄×C₉	C₄×D₄	1	2	yes	—
128	C₄×C₉	#120	1	1	yes	characteristic
129	C₄×C₉	#123	1	1	yes	characteristic
130	C₂×#41	C₂²×C₉	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
132	#131	—	2	4	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₂²×C₉	1	2	yes	—
135	#127	C₄×C₉	1	2	yes	complemented
136	#124	C₄×C₉	1	2	yes	complemented
137	#121	—	2	2	no	—
138	#120	—	2	2	no	—
139	#117	C₂²×C₉	1	2	yes	—
140	#117	C₂²×C₉	1	1	yes	characteristic
141	#102	—	2	4	no	—
142	#99	—	2	2	no	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	—	4	2	no	—
145	C₂³×C₃	—	8	2	no	—
146	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
147	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
148	C₃×Q₈	—	2	2	no	—
149	C₃×D₄	—	2	4	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	2	4	no	—
152	C₃×D₄	—	4	1	no	—
153	C₃×D₄	—	4	1	no	—
154	C₃×D₄	—	8	4	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
160	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₃×#41	1	2	yes	—
164	C₃×C₈	—	2	4	no	—
165	C₂×C₉	—	2	1	no	—
166	C₂×C₉	—	2	2	no	—
167	C₂×C₉	—	4	2	no	—
168	C₂×C₉	—	8	2	no	—
169	C₂×C₉	#386	1	1	yes	characteristic
170	C₂⁴	—	2	1	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
179	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
180	C₂×D₄	D₄×C₉	1	2	yes	—
181	C₂²×C₄	D₄×C₉	1	2	yes	—
182	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
183	M₁₆	—	2	4	no	—
184	M₁₆	—	2	2	no	—
185	M₁₆	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	D₄×C₉	1	4	yes	complemented
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×C₄×C₉	1	2	yes	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	4	4	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	8	4	no	—
201	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₃	C₃×#121	1	2	yes	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	—	8	2	no	—
206	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
207	C₃×C₄	C₃×#120	1	1	yes	characteristic
208	C₃×C₄	C₃×#123	1	1	yes	characteristic
209	C₉	#1910	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³	—	2	2	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
214	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	D₄	—	2	4	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	4	no	—
219	D₄	—	4	1	no	—
220	D₄	—	4	1	no	—
221	D₄	—	8	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
229	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
230	C₂×C₄	C₉×#41	1	2	yes	—
231	C₈	—	2	4	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	—	8	2	no	—
236	C₂×C₃	C₃×#386	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	8	4	no	—
243	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
244	C₂²	C₉×#121	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	8	2	no	—
248	C₄	C₄×D₄×C₉	1	2	yes	—
249	C₄	C₉×#120	1	1	yes	characteristic
250	C₄	C₉×#123	1	1	yes	characteristic
251	C₃	C₃×#1910	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	2	no	—
254	C₂	—	4	2	no	—
255	C₂	—	8	2	no	—
256	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₃×#535	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#568	C₂	semidirect
3. C₉×#418	C₂	semidirect
4. C₉×#410	C₂	semidirect
5. C₉×H(3,ℤ/4)	C₂	semidirect
12. C₉×#127	C₄	semidirect
13. C₉×#124	C₄	semidirect
42. C₂×C₈×C₉	D₄	semidirect
50. #1910	C₉	direct (N × H)
86. #568	C₂×C₉	semidirect
88. #418	C₂×C₉	semidirect
89. #410	C₂×C₉	semidirect
90. H(3,ℤ/4)	C₂×C₉	semidirect
135. #127	C₄×C₉	semidirect
136. #124	C₄×C₉	semidirect
187. C₂×C₈	D₄×C₉	semidirect
209. C₉	#1910	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×#568	2
C₉×#535	2
C₉×#418	2
C₉×#410	2
C₉×H(3,ℤ/4)	2
C₃×#1910	3

Subgroups — C9×#1910 #11766631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1910 #11766631