Subgroups — C₉×#2375 #11767096

Order 1152.

1032 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#120	C₂	1	1	yes	maximal, characteristic
2	C₉×#390	C₂	1	3	yes	maximal, complemented
3	C₉×#388	C₂	1	6	yes	maximal, complemented
4	C₉×#386	C₂	1	3	yes	maximal, complemented
5	C₉×#379	C₂	1	2	yes	maximal, complemented
6	C₃×#2375	C₃	1	1	yes	maximal, characteristic
7	C₂⁵×C₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₉	C₂²	1	3	yes	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic, complemented
10	C₉×#120	C₄	1	8	yes	complemented
11	C₂×C₉×C₄⋊C₄	C₂²	1	3	yes	complemented
12	C₂×C₉×C₂²⋊C₄	C₂²	1	6	yes	—
13	C₂×C₉×C₂²⋊C₄	C₂²	1	6	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂×C₉×C₂²⋊C₄	C₂²	1	3	yes	complemented
16	C₂×C₄²×C₉	C₂²	1	3	yes	complemented
17	C₉×#95	C₂²	1	6	yes	complemented
18	C₂×C₃×#120	C₂×C₃	1	1	yes	characteristic
19	C₃×#390	C₂×C₃	1	3	yes	—
20	C₃×#388	C₂×C₃	1	6	yes	—
21	C₃×#386	C₂×C₃	1	3	yes	—
22	C₃×#379	C₂×C₃	1	2	yes	—
23	C₂⁴×C₉	—	4	3	no	—
24	C₂⁴×C₉	—	4	3	no	—
25	C₂⁴×C₉	C₂×C₄	1	4	yes	complemented
26	C₂⁴×C₉	C₂³	1	3	yes	—
27	C₂⁴×C₉	C₂³	1	1	yes	characteristic
28	C₂×D₄×C₉	—	4	3	no	—
29	C₂×D₄×C₉	—	4	3	no	—
30	C₂×D₄×C₉	C₂×C₄	1	12	yes	—
31	C₂²×C₄×C₉	—	4	3	no	—
32	C₂²×C₄×C₉	C₂³	1	6	yes	—
33	C₂²×C₄×C₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₉	C₂³	1	3	yes	—
35	C₉×C₄⋊C₄	—	4	3	no	—
36	C₉×C₂²⋊C₄	—	4	6	no	—
37	C₉×C₂²⋊C₄	—	4	6	no	—
38	C₉×C₂²⋊C₄	—	4	3	no	—
39	C₉×C₂²⋊C₄	C₂×C₄	1	12	yes	complemented
40	C₄²×C₉	—	4	3	no	—
41	#2375	C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃×D₄	C₂²×C₃	1	3	yes	—
44	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₃×#120	C₃×C₄	1	8	yes	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	3	yes	—
47	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
48	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
49	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
50	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
51	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
52	C₃×#95	C₂²×C₃	1	6	yes	—
53	C₂³×C₉	—	2	3	no	—
54	C₂³×C₉	—	2	6	no	—
55	C₂³×C₉	—	2	6	no	—
56	C₂³×C₉	—	2	3	no	—
57	C₂³×C₉	—	2	3	no	—
58	C₂³×C₉	—	2	3	no	—
59	C₂³×C₉	—	4	12	no	—
60	C₂³×C₉	—	4	12	no	—
61	C₂³×C₉	—	4	4	no	—
62	C₂³×C₉	C₂²×C₄	1	6	yes	—
63	C₂³×C₉	C₂²×C₄	1	2	yes	—
64	C₂³×C₉	C₂⁴	1	1	yes	characteristic
65	D₄×C₉	—	4	12	no	—
66	C₂×C₄×C₉	—	2	3	no	—
67	C₂×C₄×C₉	—	2	3	no	—
68	C₂×C₄×C₉	—	2	6	no	—
69	C₂×C₄×C₉	—	2	6	no	—
70	C₂×C₄×C₉	—	2	6	no	—
71	C₂×C₄×C₉	—	2	6	no	—
72	C₂×C₄×C₉	—	4	4	no	—
73	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
74	C₂×#120	C₂×C₉	1	1	yes	characteristic
75	#390	C₂×C₉	1	3	yes	complemented
76	#388	C₂×C₉	1	6	yes	complemented
77	#386	C₂×C₉	1	3	yes	complemented
78	#379	C₂×C₉	1	2	yes	complemented
79	C₂⁴×C₃	—	4	3	no	—
80	C₂⁴×C₃	—	4	3	no	—
81	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
82	C₂⁴×C₃	C₂³×C₃	1	3	yes	—
83	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃×D₄	—	4	3	no	—
85	C₂×C₃×D₄	—	4	3	no	—
86	C₂×C₃×D₄	C₂×C₃×C₄	1	12	yes	—
87	C₂²×C₃×C₄	—	4	3	no	—
88	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
89	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
90	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
91	C₃×C₄⋊C₄	—	4	3	no	—
92	C₃×C₂²⋊C₄	—	4	6	no	—
93	C₃×C₂²⋊C₄	—	4	6	no	—
94	C₃×C₂²⋊C₄	—	4	3	no	—
95	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	12	yes	—
96	C₃×C₄²	—	4	3	no	—
97	C₂²×C₉	—	2	6	no	—
98	C₂²×C₉	—	2	6	no	—
99	C₂²×C₉	—	2	12	no	—
100	C₂²×C₉	—	4	16	no	—
101	C₂²×C₉	—	4	3	no	—
102	C₂²×C₉	—	4	3	no	—
103	C₂²×C₉	—	4	6	no	—
104	C₂²×C₉	—	4	1	no	—
105	C₂²×C₉	—	4	3	no	—
106	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
107	C₂²×C₉	H(5,𝔽₂)	1	3	yes	—
108	C₂²×C₉	H(5,𝔽₂)	1	3	yes	—
109	C₄×C₉	—	2	6	no	—
110	C₄×C₉	—	4	6	no	—
111	C₄×C₉	—	4	2	no	—
112	C₂⁵	C₂²×C₉	1	1	yes	characteristic
113	C₂²×D₄	C₂²×C₉	1	3	yes	—
114	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic, complemented
115	#120	C₄×C₉	1	8	yes	complemented
116	C₂×C₄⋊C₄	C₂²×C₉	1	3	yes	complemented
117	C₂×C₂²⋊C₄	C₂²×C₉	1	6	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₉	1	6	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₉	1	3	yes	complemented
121	C₂×C₄²	C₂²×C₉	1	3	yes	complemented
122	#95	C₂²×C₉	1	6	yes	complemented
123	C₂³×C₃	—	2	3	no	—
124	C₂³×C₃	—	2	6	no	—
125	C₂³×C₃	—	2	6	no	—
126	C₂³×C₃	—	2	3	no	—
127	C₂³×C₃	—	2	3	no	—
128	C₂³×C₃	—	2	3	no	—
129	C₂³×C₃	—	4	12	no	—
130	C₂³×C₃	—	4	12	no	—
131	C₂³×C₃	—	4	4	no	—
132	C₂³×C₃	C₂²×C₃×C₄	1	6	yes	—
133	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
134	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
135	C₃×D₄	—	4	12	no	—
136	C₂×C₃×C₄	—	2	3	no	—
137	C₂×C₃×C₄	—	2	3	no	—
138	C₂×C₃×C₄	—	2	6	no	—
139	C₂×C₃×C₄	—	2	6	no	—
140	C₂×C₃×C₄	—	2	6	no	—
141	C₂×C₃×C₄	—	2	6	no	—
142	C₂×C₃×C₄	—	4	4	no	—
143	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
144	C₂×C₉	—	4	2	no	—
145	C₂×C₉	—	4	6	no	—
146	C₂×C₉	#518	1	3	yes	—
147	C₂×C₉	#560	1	3	yes	—
148	C₂×C₉	H(3,𝔽₄)	1	1	yes	characteristic
149	C₂⁴	—	4	3	no	—
150	C₂⁴	—	4	3	no	—
151	C₂⁴	C₂×C₄×C₉	1	4	yes	complemented
152	C₂⁴	C₂³×C₉	1	3	yes	—
153	C₂⁴	C₂³×C₉	1	1	yes	characteristic
154	C₂×D₄	—	4	3	no	—
155	C₂×D₄	—	4	3	no	—
156	C₂×D₄	C₂×C₄×C₉	1	12	yes	—
157	C₂²×C₄	—	4	3	no	—
158	C₂²×C₄	C₂³×C₉	1	2	yes	—
159	C₂²×C₄	C₂³×C₉	1	6	yes	—
160	C₂²×C₄	C₂³×C₉	1	3	yes	—
161	C₄⋊C₄	—	4	3	no	—
162	C₂²⋊C₄	—	4	6	no	—
163	C₂²⋊C₄	—	4	6	no	—
164	C₂²⋊C₄	—	4	3	no	—
165	C₂²⋊C₄	C₂×C₄×C₉	1	12	yes	complemented
166	C₄²	—	4	3	no	—
167	C₂²×C₃	—	2	6	no	—
168	C₂²×C₃	—	2	6	no	—
169	C₂²×C₃	—	2	12	no	—
170	C₂²×C₃	—	4	16	no	—
171	C₂²×C₃	—	4	3	no	—
172	C₂²×C₃	—	4	6	no	—
173	C₂²×C₃	—	4	3	no	—
174	C₂²×C₃	—	4	1	no	—
175	C₂²×C₃	—	4	3	no	—
176	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
177	C₂²×C₃	C₃×H(5,𝔽₂)	1	3	yes	—
178	C₂²×C₃	C₃×H(5,𝔽₂)	1	3	yes	—
179	C₃×C₄	—	2	6	no	—
180	C₃×C₄	—	4	6	no	—
181	C₃×C₄	—	4	2	no	—
182	C₉	#2375	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³	—	2	3	no	—
184	C₂³	—	2	6	no	—
185	C₂³	—	2	6	no	—
186	C₂³	—	2	3	no	—
187	C₂³	—	2	3	no	—
188	C₂³	—	2	3	no	—
189	C₂³	—	4	12	no	—
190	C₂³	—	4	12	no	—
191	C₂³	—	4	4	no	—
192	C₂³	C₂²×C₄×C₉	1	6	yes	—
193	C₂³	C₂²×C₄×C₉	1	2	yes	—
194	C₂³	C₂⁴×C₉	1	1	yes	characteristic
195	D₄	—	4	12	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂²×C₄×C₉	1	6	yes	—
204	C₂×C₃	—	4	6	no	—
205	C₂×C₃	—	4	2	no	—
206	C₂×C₃	C₃×#518	1	3	yes	—
207	C₂×C₃	C₃×#560	1	3	yes	—
208	C₂×C₃	C₃×H(3,𝔽₄)	1	1	yes	characteristic
209	C₂²	—	2	6	no	—
210	C₂²	—	2	6	no	—
211	C₂²	—	2	12	no	—
212	C₂²	—	4	16	no	—
213	C₂²	—	4	3	no	—
214	C₂²	—	4	6	no	—
215	C₂²	—	4	3	no	—
216	C₂²	—	4	1	no	—
217	C₂²	—	4	3	no	—
218	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
219	C₂²	C₉×H(5,𝔽₂)	1	3	yes	—
220	C₂²	C₉×H(5,𝔽₂)	1	3	yes	—
221	C₄	—	2	6	no	—
222	C₄	—	4	6	no	—
223	C₄	—	4	2	no	—
224	C₃	C₃×#2375	1	1	yes	characteristic
225	C₂	—	4	6	no	—
226	C₂	—	4	2	no	—
227	C₂	C₉×#518	1	3	yes	—
228	C₂	C₉×#560	1	3	yes	—
229	C₂	C₉×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#120	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. C₉×#390	C₂	semidirect
3. C₉×#388	C₂	semidirect
4. C₉×#386	C₂	semidirect
5. C₉×#379	C₂	semidirect
9. C₂³×C₄×C₉	C₂²	semidirect
10. C₉×#120	C₄	semidirect
11. C₂×C₉×C₄⋊C₄	C₂²	semidirect
15. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
16. C₂×C₄²×C₉	C₂²	semidirect
17. C₉×#95	C₂²	semidirect
25. C₂⁴×C₉	C₂×C₄	semidirect
39. C₉×C₂²⋊C₄	C₂×C₄	semidirect
41. #2375	C₉	direct (N × H)
75. #390	C₂×C₉	semidirect
76. #388	C₂×C₉	semidirect
77. #386	C₂×C₉	semidirect
78. #379	C₂×C₉	semidirect
114. C₂³×C₄	C₂²×C₉	semidirect
115. #120	C₄×C₉	semidirect
116. C₂×C₄⋊C₄	C₂²×C₉	semidirect
120. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
121. C₂×C₄²	C₂²×C₉	semidirect
122. #95	C₂²×C₉	semidirect
151. C₂⁴	C₂×C₄×C₉	semidirect
165. C₂²⋊C₄	C₂×C₄×C₉	semidirect
182. C₉	#2375	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#120	2
C₉×#390	2
C₉×#388	2
C₉×#386	2
C₉×#379	2
C₃×#2375	3

Subgroups — C9×#2375 #11767096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2375 #11767096