Subgroups — C₂²×C₉×#123 #11768155

Order 1152.

2952 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂⁴×C₃	C₂³×C₃
prime radical	C₂³×C₃×D₄	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×C₉	C₂	semidirect
3. C₂×C₉×#123	C₂	direct (N × H)
4. C₂²×C₉×C₄⋊C₄	C₂	semidirect
5. C₂²×C₉×C₂²⋊C₄	C₂	semidirect
6. C₂²×C₉×C₂²⋊C₄	C₂	semidirect
16. C₂³×C₄×C₉	C₂²	semidirect
17. C₉×#123	C₂²	direct (N × H)
18. C₂×C₉×C₄⋊C₄	C₂²	semidirect
19. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
20. C₂×C₉×C₂²⋊C₄	C₂²	semidirect
43. C₂²×C₄×C₉	C₂³	semidirect
44. C₉×C₄⋊C₄	C₂³	semidirect
45. C₉×C₂²⋊C₄	C₂³	semidirect
46. C₉×C₂²⋊C₄	C₂³	semidirect
47. C₂²×#123	C₉	direct (N × H)
81. C₂⁴×C₄	C₂×C₉	semidirect
82. C₂×#123	C₂×C₉	direct (N × H)
83. C₂²×C₄⋊C₄	C₂×C₉	semidirect
84. C₂²×C₂²⋊C₄	C₂×C₉	semidirect
85. C₂²×C₂²⋊C₄	C₂×C₉	semidirect
111. C₂²×C₉	#123	direct (N × H)
127. C₂³×C₄	C₂²×C₉	semidirect
128. #123	C₂²×C₉	direct (N × H)
129. C₂×C₄⋊C₄	C₂²×C₉	semidirect
130. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
131. C₂×C₂²⋊C₄	C₂²×C₉	semidirect
153. C₂×C₉	C₂×#123	direct (N × H)
172. C₂²×C₄	C₂³×C₉	semidirect
173. C₄⋊C₄	C₂³×C₉	semidirect
174. C₂²⋊C₄	C₂³×C₉	semidirect
175. C₂²⋊C₄	C₂³×C₉	semidirect
189. C₉	C₂²×#123	direct (N × H)
219. C₂²	C₉×#123	direct (N × H)
230. C₂	C₂×C₉×#123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₉	2
C₂⁴×C₄×C₉	2
C₂×C₉×#123	2
C₂²×C₉×C₄⋊C₄	2
C₂²×C₉×C₂²⋊C₄	2
C₂²×C₃×#123	3

Subgroups — C22×C9×#123 #11768155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×#123 #11768155