Subgroups — C₉×#2950 #11767671

Order 1152.

684 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₉×#519	C₂	1	2	yes	maximal
3	C₉×#441	C₂	1	4	yes	maximal, complemented
4	C₄×C₉×Q₁₆	C₂	1	4	yes	maximal, complemented
5	C₉×#427	C₂	1	1	yes	maximal, characteristic
6	C₉×#419	C₂	1	2	yes	maximal
7	C₉×#405	C₂	1	1	yes	maximal, characteristic
8	C₃×#2950	C₃	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉	C₂²	1	2	yes	—
10	C₂×C₉×Q₁₆	C₄	1	8	yes	complemented
11	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₉	—	2	4	no	—
16	C₄×Q₈×C₉	C₂²	1	8	yes	—
17	C₉×#117	—	2	4	no	—
18	C₉×#117	C₂²	1	2	yes	—
19	C₉×#117	C₂²	1	1	yes	characteristic
20	C₉×#107	C₂²	1	2	yes	—
21	C₉×#106	C₂²	1	2	yes	—
22	C₉×#103	C₂²	1	8	yes	—
23	C₉×#97	C₂²	1	2	yes	—
24	C₄×C₈×C₉	C₂²	1	2	yes	—
25	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
26	C₃×#519	C₂×C₃	1	2	yes	—
27	C₃×#441	C₂×C₃	1	4	yes	—
28	C₃×C₄×Q₁₆	C₂×C₃	1	4	yes	—
29	C₃×#427	C₂×C₃	1	1	yes	characteristic
30	C₃×#419	C₂×C₃	1	2	yes	—
31	C₃×#405	C₂×C₃	1	1	yes	characteristic
32	C₂×Q₈×C₉	—	2	2	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	C₂×C₄	1	8	yes	—
35	C₂×Q₈×C₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×Q₁₆	C₂×C₄	1	8	yes	—
39	C₉×Q₁₆	C₂×C₄	1	8	yes	—
40	C₉×M₁₆	—	2	2	no	—
41	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
42	C₂×C₈×C₉	C₂³	1	2	yes	—
43	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
45	C₉×C₄⋊C₄	—	2	8	no	—
46	C₉×C₄⋊C₄	D₄	1	2	yes	—
47	C₉×C₄⋊C₄	C₂³	1	4	yes	—
48	C₉×C₂²⋊C₄	—	2	2	no	—
49	C₉×C₂²⋊C₄	D₄	1	2	yes	—
50	C₄²×C₉	—	2	8	no	—
51	C₄²×C₉	—	2	2	no	—
52	C₄²×C₉	C₂³	1	2	yes	—
53	#2950	C₉	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
55	C₂×C₃×Q₁₆	C₃×C₄	1	8	yes	—
56	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
57	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
58	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄×Q₈	—	2	4	no	—
61	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	—
62	C₃×#117	—	2	4	no	—
63	C₃×#117	C₂²×C₃	1	2	yes	—
64	C₃×#117	C₂²×C₃	1	1	yes	characteristic
65	C₃×#107	C₂²×C₃	1	2	yes	—
66	C₃×#106	C₂²×C₃	1	2	yes	—
67	C₃×#103	C₂²×C₃	1	8	yes	—
68	C₃×#97	C₂²×C₃	1	2	yes	—
69	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₉	—	2	8	no	—
72	Q₈×C₉	—	2	4	no	—
73	Q₈×C₉	C₂²×C₄	1	8	yes	—
74	C₂×C₄×C₉	—	2	8	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₉	#41	1	2	yes	—
81	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₉	—	2	2	no	—
83	C₈×C₉	C₂²×C₄	1	2	yes	—
84	C₈×C₉	C₂²×C₄	1	2	yes	—
85	C₂²×Q₁₆	C₂×C₉	1	1	yes	characteristic
86	#519	C₂×C₉	1	2	yes	—
87	#441	C₂×C₉	1	4	yes	complemented
88	C₄×Q₁₆	C₂×C₉	1	4	yes	complemented
89	#427	C₂×C₉	1	1	yes	characteristic
90	#419	C₂×C₉	1	2	yes	—
91	#405	C₂×C₉	1	1	yes	characteristic
92	C₂×C₃×Q₈	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	4	no	—
94	C₂×C₃×Q₈	C₂×C₃×C₄	1	8	yes	—
95	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₃×Q₁₆	C₂×C₃×C₄	1	8	yes	—
99	C₃×Q₁₆	C₂×C₃×C₄	1	8	yes	—
100	C₃×M₁₆	—	2	2	no	—
101	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
102	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
103	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	2	8	no	—
106	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
107	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
110	C₃×C₄²	—	2	8	no	—
111	C₃×C₄²	—	2	2	no	—
112	C₃×C₄²	C₂³×C₃	1	2	yes	—
113	C₂²×C₉	—	2	2	no	—
114	C₂²×C₉	C₄×D₄	1	2	yes	—
115	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₉	—	2	8	no	—
117	C₄×C₉	—	2	4	no	—
118	C₄×C₉	—	4	4	no	—
119	C₄×C₉	C₄×D₄	1	2	yes	—
120	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₉	C₂×#41	1	1	yes	characteristic
122	C₂²×Q₈	C₂²×C₉	1	2	yes	—
123	C₂×Q₁₆	C₄×C₉	1	8	yes	complemented
124	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
125	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
126	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
128	C₄×Q₈	—	2	4	no	—
129	C₄×Q₈	C₂²×C₉	1	8	yes	—
130	#117	—	2	4	no	—
131	#117	C₂²×C₉	1	2	yes	—
132	#117	C₂²×C₉	1	1	yes	characteristic
133	#107	C₂²×C₉	1	2	yes	—
134	#106	C₂²×C₉	1	2	yes	—
135	#103	C₂²×C₉	1	8	yes	—
136	#97	C₂²×C₉	1	2	yes	—
137	C₄×C₈	C₂²×C₉	1	2	yes	—
138	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
139	C₃×Q₈	—	2	8	no	—
140	C₃×Q₈	—	2	4	no	—
141	C₃×Q₈	C₂²×C₃×C₄	1	8	yes	—
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
147	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₃×#41	1	2	yes	—
149	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
152	C₃×C₈	C₂²×C₃×C₄	1	2	yes	—
153	C₂×C₉	—	2	2	no	—
154	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₉	#578	1	2	yes	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	C₂×C₄×C₉	1	8	yes	—
159	C₂×Q₈	C₂³×C₉	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
162	Q₁₆	C₂×C₄×C₉	1	8	yes	—
163	Q₁₆	C₂×C₄×C₉	1	8	yes	—
164	M₁₆	—	2	2	no	—
165	C₂×C₈	C₂×C₄×C₉	1	4	yes	—
166	C₂×C₈	C₂³×C₉	1	2	yes	—
167	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
169	C₄⋊C₄	—	2	8	no	—
170	C₄⋊C₄	D₄×C₉	1	2	yes	—
171	C₄⋊C₄	C₂³×C₉	1	4	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	D₄×C₉	1	2	yes	—
174	C₄²	—	2	8	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂³×C₉	1	2	yes	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
179	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
180	C₃×C₄	—	2	4	no	—
181	C₃×C₄	—	2	8	no	—
182	C₃×C₄	—	4	4	no	—
183	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
184	C₃×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
185	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
186	C₉	#2950	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
188	Q₈	—	2	8	no	—
189	Q₈	—	2	4	no	—
190	Q₈	C₂²×C₄×C₉	1	8	yes	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
195	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
196	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
197	C₂×C₄	C₉×#41	1	2	yes	—
198	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	C₂²×C₄×C₉	1	2	yes	—
201	C₈	C₂²×C₄×C₉	1	2	yes	—
202	C₂×C₃	—	2	2	no	—
203	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
204	C₂×C₃	C₃×#578	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	C₄×D₄×C₉	1	2	yes	—
207	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	2	8	no	—
210	C₄	—	4	4	no	—
211	C₄	C₄×D₄×C₉	1	2	yes	—
212	C₄	C₂³×C₄×C₉	1	1	yes	characteristic
213	C₄	C₂×C₉×#41	1	1	yes	characteristic
214	C₃	C₃×#2950	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic
217	C₂	C₉×#578	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#441	C₂	semidirect
4. C₄×C₉×Q₁₆	C₂	semidirect
10. C₂×C₉×Q₁₆	C₄	semidirect
53. #2950	C₉	direct (N × H)
87. #441	C₂×C₉	semidirect
88. C₄×Q₁₆	C₂×C₉	semidirect
123. C₂×Q₁₆	C₄×C₉	semidirect
186. C₉	#2950	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×Q₁₆	2
C₉×#519	2
C₉×#441	2
C₄×C₉×Q₁₆	2
C₉×#427	2
C₉×#419	2
C₉×#405	2
C₃×#2950	3

Subgroups — C9×#2950 #11767671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2950 #11767671