Subgroups — C₉×#1857 #11766578

Order 1152.

408 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#519	C₂	1	1	yes	maximal, characteristic
2	C₉×#428	C₂	1	2	yes	maximal
3	C₂×C₉×#101	C₂	1	1	yes	maximal, characteristic
4	C₉×#391	C₂	1	1	yes	maximal, characteristic
5	C₉×#339	C₂	1	2	yes	maximal
6	C₃×#1857	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₄×Q₈×C₉	—	2	2	no	—
10	C₄×Q₈×C₉	—	2	2	no	—
11	C₉×#117	—	2	2	no	—
12	C₉×#117	C₂²	1	2	yes	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×#107	—	2	2	no	—
16	C₉×#106	—	2	2	no	—
17	C₉×#101	C₄	1	4	yes	complemented
18	C₉×#95	—	2	2	no	—
19	C₃×#519	C₂×C₃	1	1	yes	characteristic
20	C₃×#428	C₂×C₃	1	2	yes	—
21	C₂×C₃×#101	C₂×C₃	1	1	yes	characteristic
22	C₃×#391	C₂×C₃	1	1	yes	characteristic
23	C₃×#339	C₂×C₃	1	2	yes	—
24	C₂×Q₈×C₉	—	2	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₉	Q₈	1	2	yes	—
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	—	2	1	no	—
30	C₂²×C₄×C₉	D₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₉×M₁₆	—	2	2	no	—
33	C₉×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	—	2	2	no	—
37	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	D₄	1	2	yes	—
39	C₉×C₄⋊C₄	D₄	1	2	yes	—
40	C₉×C₂²⋊C₄	—	2	2	no	—
41	C₄²×C₉	—	2	4	no	—
42	C₄²×C₉	—	2	2	no	—
43	C₄²×C₉	—	2	2	no	—
44	#1857	C₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
47	C₃×C₄×Q₈	—	2	2	no	—
48	C₃×C₄×Q₈	—	2	2	no	—
49	C₃×#117	—	2	2	no	—
50	C₃×#117	C₂²×C₃	1	2	yes	—
51	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₃×#107	—	2	2	no	—
54	C₃×#106	—	2	2	no	—
55	C₃×#101	C₃×C₄	1	4	yes	—
56	C₃×#95	—	2	2	no	—
57	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
58	Q₈×C₉	—	2	4	no	—
59	Q₈×C₉	—	2	4	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	4	1	no	—
66	C₂×C₄×C₉	—	4	1	no	—
67	C₂×C₄×C₉	—	4	2	no	—
68	C₂×C₄×C₉	—	4	1	no	—
69	C₂×C₄×C₉	C₄⋊C₄	1	4	yes	—
70	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	2	yes	—
74	C₈×C₉	—	2	4	no	—
75	#519	C₂×C₉	1	1	yes	characteristic
76	#428	C₂×C₉	1	2	yes	—
77	C₂×#101	C₂×C₉	1	1	yes	characteristic
78	#391	C₂×C₉	1	1	yes	characteristic
79	#339	C₂×C₉	1	2	yes	—
80	C₂×C₃×Q₈	—	2	2	no	—
81	C₂×C₃×Q₈	—	2	2	no	—
82	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₃×Q₈	C₃×Q₈	1	2	yes	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	1	no	—
86	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₃×M₁₆	—	2	2	no	—
89	C₃×M₁₆	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₃×C₄⋊C₄	—	2	2	no	—
92	C₃×C₄⋊C₄	—	2	2	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
95	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
96	C₃×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₄²	—	2	4	no	—
98	C₃×C₄²	—	2	2	no	—
99	C₃×C₄²	—	2	2	no	—
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	1	no	—
102	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₉	#120	1	1	yes	characteristic
104	C₂²×C₉	#123	1	1	yes	characteristic
105	C₄×C₉	—	2	4	no	—
106	C₄×C₉	—	4	2	no	—
107	C₄×C₉	—	4	2	no	—
108	C₄×C₉	—	4	2	no	—
109	C₄×C₉	—	8	1	no	—
110	C₄×C₉	C₄×D₄	1	2	yes	—
111	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
112	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
113	C₂×M₁₆	C₂²×C₉	1	2	yes	—
114	C₄×Q₈	—	2	2	no	—
115	C₄×Q₈	—	2	2	no	—
116	#117	—	2	2	no	—
117	#117	C₂²×C₉	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
120	#107	—	2	2	no	—
121	#106	—	2	2	no	—
122	#101	C₄×C₉	1	4	yes	complemented
123	#95	—	2	2	no	—
124	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
125	C₃×Q₈	—	2	4	no	—
126	C₃×Q₈	—	2	4	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	—	4	1	no	—
134	C₂×C₃×C₄	—	4	2	no	—
135	C₂×C₃×C₄	—	4	1	no	—
136	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
137	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
139	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
140	C₂×C₃×C₄	C₃×#41	1	2	yes	—
141	C₃×C₈	—	2	4	no	—
142	C₂×C₉	—	2	1	no	—
143	C₂×C₉	—	2	1	no	—
144	C₂×C₉	#385	1	1	yes	characteristic
145	C₂×C₉	#454	1	1	yes	characteristic
146	C₂×C₉	#456	1	1	yes	characteristic
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
150	C₂×Q₈	Q₈×C₉	1	2	yes	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	D₄×C₉	1	2	yes	—
154	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	C₂×C₄×C₉	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	D₄×C₉	1	2	yes	—
162	C₄⋊C₄	D₄×C₉	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₃	C₃×#120	1	1	yes	characteristic
171	C₂²×C₃	C₃×#123	1	1	yes	characteristic
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	4	2	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	8	1	no	—
177	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
178	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
179	C₉	#1857	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	C₉×C₄⋊C₄	1	4	yes	—
193	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
195	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
196	C₂×C₄	C₉×#41	1	2	yes	—
197	C₈	—	2	4	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	C₃×#385	1	1	yes	characteristic
201	C₂×C₃	C₃×#454	1	1	yes	characteristic
202	C₂×C₃	C₃×#456	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
206	C₂²	C₉×#120	1	1	yes	characteristic
207	C₂²	C₉×#123	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	8	1	no	—
213	C₄	C₄×D₄×C₉	1	2	yes	—
214	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
215	C₃	C₃×#1857	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	C₉×#385	1	1	yes	characteristic
219	C₂	C₉×#454	1	1	yes	characteristic
220	C₂	C₉×#456	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₉×#101	C₄	semidirect
44. #1857	C₉	direct (N × H)
122. #101	C₄×C₉	semidirect
179. C₉	#1857	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×#519	2
C₉×#428	2
C₂×C₉×#101	2
C₉×#391	2
C₉×#339	2
C₃×#1857	3

Subgroups — C9×#1857 #11766578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1857 #11766578