Subgroups — C₉×#1983 #11766704

Order 1152.

516 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#103	C₂	1	2	yes	maximal, complemented
4	C₂×C₉×#102	C₂	1	2	yes	maximal
5	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1983	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	2	yes	—
10	C₉×#124	—	2	2	no	—
11	C₉×#124	C₄	1	4	yes	complemented
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
14	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₂×C₄²×C₉	—	2	1	no	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₉×C₄⋊C₈	—	2	2	no	—
18	C₉×#103	—	2	4	no	—
19	C₉×#102	—	2	4	no	—
20	C₉×#95	—	2	1	no	—
21	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#103	C₂×C₃	1	2	yes	—
24	C₂×C₃×#102	C₂×C₃	1	2	yes	—
25	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
26	C₂⁴×C₉	—	2	1	no	—
27	C₂×Q₈×C₉	—	2	1	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	—	2	2	no	—
34	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₉	—	2	2	no	—
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₉	—	2	2	no	—
41	C₂×C₈×C₉	—	2	2	no	—
42	C₂×C₈×C₉	—	2	2	no	—
43	C₉×C₄⋊C₄	—	2	2	no	—
44	C₉×C₄⋊C₄	D₄	1	4	yes	—
45	C₉×C₂²⋊C₄	—	2	4	no	—
46	C₉×C₂²⋊C₄	—	2	4	no	—
47	C₄²×C₉	—	4	2	no	—
48	C₄²×C₉	C₂×C₄	1	2	yes	—
49	C₄²×C₉	D₄	1	2	yes	—
50	#1983	C₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
54	C₃×#124	—	2	2	no	—
55	C₃×#124	C₃×C₄	1	4	yes	—
56	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
58	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
59	C₂×C₃×C₄²	—	2	1	no	—
60	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₃×C₄⋊C₈	—	2	2	no	—
62	C₃×#103	—	2	4	no	—
63	C₃×#102	—	2	4	no	—
64	C₃×#95	—	2	1	no	—
65	C₂³×C₉	—	2	2	no	—
66	C₂³×C₉	—	4	2	no	—
67	C₂³×C₉	—	4	2	no	—
68	C₂³×C₉	—	4	1	no	—
69	C₂³×C₉	—	4	1	no	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₉	—	2	4	no	—
72	Q₈×C₉	—	4	1	no	—
73	Q₈×C₉	—	4	1	no	—
74	D₄×C₉	—	2	4	no	—
75	D₄×C₉	—	4	1	no	—
76	D₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	4	2	no	—
83	C₂×C₄×C₉	—	4	2	no	—
84	C₂×C₄×C₉	—	4	2	no	—
85	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₉	#41	1	2	yes	—
91	C₈×C₉	—	4	2	no	—
92	C₂×#124	C₂×C₉	1	1	yes	characteristic
93	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
94	C₂×#103	C₂×C₉	1	2	yes	complemented
95	C₂×#102	C₂×C₉	1	2	yes	—
96	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
97	C₂⁴×C₃	—	2	1	no	—
98	C₂×C₃×Q₈	—	2	1	no	—
99	C₂×C₃×Q₈	—	2	2	no	—
100	C₂×C₃×Q₈	—	2	2	no	—
101	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
102	C₂×C₃×D₄	—	2	2	no	—
103	C₂×C₃×D₄	—	2	1	no	—
104	C₂×C₃×D₄	—	2	2	no	—
105	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₃×C₄⋊C₄	—	2	2	no	—
115	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
116	C₃×C₂²⋊C₄	—	2	4	no	—
117	C₃×C₂²⋊C₄	—	2	4	no	—
118	C₃×C₄²	—	4	2	no	—
119	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
120	C₃×C₄²	C₃×D₄	1	2	yes	—
121	C₂²×C₉	—	4	2	no	—
122	C₂²×C₉	—	4	2	no	—
123	C₂²×C₉	—	4	2	no	—
124	C₂²×C₉	—	8	2	no	—
125	C₂²×C₉	—	8	2	no	—
126	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₉	#121	1	2	yes	—
128	C₂²×C₉	#135	1	2	yes	—
129	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂²×C₉	#137	1	1	yes	characteristic
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	4	2	no	—
134	C₄×C₉	—	4	4	no	—
135	C₄×C₉	C₄×D₄	1	2	yes	—
136	C₄×C₉	#124	1	1	yes	characteristic
137	C₄×C₉	#127	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
139	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²×C₉	1	2	yes	—
141	#124	—	2	2	no	—
142	#124	C₄×C₉	1	4	yes	complemented
143	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
144	C₂×C₂²⋊C₄	—	2	1	no	—
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	#103	—	2	4	no	—
150	#102	—	2	4	no	—
151	#95	—	2	1	no	—
152	C₂³×C₃	—	2	2	no	—
153	C₂³×C₃	—	4	2	no	—
154	C₂³×C₃	—	4	2	no	—
155	C₂³×C₃	—	4	1	no	—
156	C₂³×C₃	—	4	1	no	—
157	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
158	C₃×Q₈	—	2	4	no	—
159	C₃×Q₈	—	4	1	no	—
160	C₃×Q₈	—	4	1	no	—
161	C₃×D₄	—	2	4	no	—
162	C₃×D₄	—	4	1	no	—
163	C₃×D₄	—	4	1	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	4	2	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
173	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₃×#41	1	2	yes	—
178	C₃×C₈	—	4	2	no	—
179	C₂×C₉	—	8	2	no	—
180	C₂×C₉	#390	1	1	yes	characteristic
181	C₂×C₉	#416	1	1	yes	characteristic
182	C₂×C₉	#417	1	1	yes	characteristic
183	C₂×C₉	#463	1	2	yes	—
184	C₂×C₉	#464	1	2	yes	—
185	C₂⁴	—	2	1	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	D₄×C₉	1	4	yes	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	4	2	no	—
207	C₄²	C₂×C₄×C₉	1	2	yes	—
208	C₄²	D₄×C₉	1	2	yes	—
209	C₂²×C₃	—	4	2	no	—
210	C₂²×C₃	—	4	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	8	2	no	—
213	C₂²×C₃	—	8	2	no	—
214	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	C₃×#121	1	2	yes	—
216	C₂²×C₃	C₃×#135	1	2	yes	—
217	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂²×C₃	C₃×#137	1	1	yes	characteristic
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	—	4	4	no	—
223	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
224	C₃×C₄	C₃×#124	1	1	yes	characteristic
225	C₃×C₄	C₃×#127	1	1	yes	characteristic
226	C₉	#1983	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³	—	2	2	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	4	1	no	—
235	Q₈	—	4	1	no	—
236	D₄	—	2	4	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
252	C₂×C₄	C₉×#41	1	2	yes	—
253	C₈	—	4	2	no	—
254	C₂×C₃	—	8	2	no	—
255	C₂×C₃	C₃×#390	1	1	yes	characteristic
256	C₂×C₃	C₃×#416	1	1	yes	characteristic
257	C₂×C₃	C₃×#417	1	1	yes	characteristic
258	C₂×C₃	C₃×#463	1	2	yes	—
259	C₂×C₃	C₃×#464	1	2	yes	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
266	C₂²	C₉×#121	1	2	yes	—
267	C₂²	C₉×#135	1	2	yes	—
268	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
269	C₂²	C₉×#137	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	4	no	—
274	C₄	C₄×D₄×C₉	1	2	yes	—
275	C₄	C₉×#124	1	1	yes	characteristic
276	C₄	C₉×#127	1	1	yes	characteristic
277	C₃	C₃×#1983	1	1	yes	characteristic
278	C₂	—	8	2	no	—
279	C₂	C₉×#390	1	1	yes	characteristic
280	C₂	C₉×#416	1	1	yes	characteristic
281	C₂	C₉×#417	1	1	yes	characteristic
282	C₂	C₉×#463	1	2	yes	—
283	C₂	C₉×#464	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
3. C₂×C₉×#103	C₂	semidirect
5. C₄×C₉×C₄⋊C₄	C₂	semidirect
11. C₉×#124	C₄	semidirect
50. #1983	C₉	direct (N × H)
93. C₂×C₄⋊C₈	C₂×C₉	semidirect
94. C₂×#103	C₂×C₉	semidirect
96. C₄×C₄⋊C₄	C₂×C₉	semidirect
142. #124	C₄×C₉	semidirect
226. C₉	#1983	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×#124	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#103	2
C₂×C₉×#102	2
C₄×C₉×C₄⋊C₄	2
C₃×#1983	3

Subgroups — C9×#1983 #11766704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1983 #11766704