Subgroups — C₉×#3199 #11767920

Order 1152.

663 conjugacy classes of subgroups.

The subgroup lattice has 279 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#535	C₂	1	2	yes	maximal, complemented
2	C₉×#482	C₂	1	4	yes	maximal, complemented
3	C₉×#469	C₂	1	2	yes	maximal, complemented
4	C₉×#466	C₂	1	2	yes	maximal, complemented
5	C₉×#447	C₂	1	4	yes	maximal, complemented
6	C₉×#408	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#3199	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₉	C₂²	1	2	yes	complemented
10	C₂×C₉×D₈	C₂²	1	2	yes	—
11	C₂×C₉×D₈	C₂²	1	2	yes	—
12	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic, complemented
14	C₉×#127	—	2	2	no	—
15	C₉×#124	—	2	2	no	—
16	C₉×#121	C₂²	1	4	yes	—
17	C₉×#121	C₂²	1	4	yes	complemented
18	C₉×#120	—	2	2	no	—
19	C₉×#120	—	2	2	no	—
20	C₉×#117	C₂²	1	2	yes	complemented
21	C₉×#107	C₂²	1	2	yes	complemented
22	C₉×#106	C₂²	1	2	yes	complemented
23	C₉×#102	C₂²	1	4	yes	complemented
24	C₉×#102	C₂²	1	4	yes	complemented
25	C₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
26	C₃×#535	C₂×C₃	1	2	yes	—
27	C₃×#482	C₂×C₃	1	4	yes	—
28	C₃×#469	C₂×C₃	1	2	yes	—
29	C₃×#466	C₂×C₃	1	2	yes	—
30	C₃×#447	C₂×C₃	1	4	yes	—
31	C₃×#408	C₂×C₃	1	1	yes	characteristic
32	C₂⁴×C₉	—	2	2	no	—
33	C₉×#41	—	4	1	no	—
34	C₉×#41	—	4	1	no	—
35	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₉	—	2	4	no	—
37	C₂×D₄×C₉	—	2	4	no	—
38	C₂×D₄×C₉	—	2	2	no	—
39	C₂×D₄×C₉	—	4	2	no	—
40	C₂×D₄×C₉	—	4	2	no	—
41	C₂×D₄×C₉	D₄	1	1	yes	characteristic
42	C₂×D₄×C₉	D₄	1	2	yes	—
43	C₂×D₄×C₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₉×D₈	—	4	2	no	—
47	C₉×D₈	—	4	2	no	—
48	C₉×M₁₆	—	4	1	no	—
49	C₂×C₈×C₉	—	4	1	no	—
50	C₂×C₈×C₉	C₂³	1	2	yes	—
51	C₂×C₈×C₉	C₂³	1	2	yes	—
52	C₉×C₄⋊C₄	C₂³	1	4	yes	—
53	C₉×C₂²⋊C₄	—	2	2	no	—
54	C₉×C₂²⋊C₄	—	2	4	no	—
55	C₉×C₂²⋊C₄	—	2	4	no	—
56	C₄²×C₉	—	2	2	no	—
57	#3199	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
59	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
60	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
61	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
62	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₃×#127	—	2	2	no	—
65	C₃×#124	—	2	2	no	—
66	C₃×#121	C₂²×C₃	1	4	yes	—
67	C₃×#121	C₂²×C₃	1	4	yes	—
68	C₃×#120	—	2	2	no	—
69	C₃×#120	—	2	2	no	—
70	C₃×#117	C₂²×C₃	1	2	yes	—
71	C₃×#107	C₂²×C₃	1	2	yes	—
72	C₃×#106	C₂²×C₃	1	2	yes	—
73	C₃×#102	C₂²×C₃	1	4	yes	—
74	C₃×#102	C₂²×C₃	1	4	yes	—
75	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
76	C₂³×C₉	—	2	4	no	—
77	C₂³×C₉	—	4	4	no	—
78	C₂³×C₉	—	8	4	no	—
79	C₂³×C₉	C₂×D₄	1	2	yes	—
80	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
81	Q₈×C₉	—	4	1	no	—
82	D₄×C₉	—	2	4	no	—
83	D₄×C₉	—	2	4	no	—
84	D₄×C₉	—	4	2	no	—
85	D₄×C₉	—	4	1	no	—
86	D₄×C₉	—	4	4	no	—
87	D₄×C₉	—	8	4	no	—
88	D₄×C₉	—	8	4	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	4	no	—
94	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
97	C₈×C₉	—	2	2	no	—
98	C₈×C₉	—	2	2	no	—
99	#535	C₂×C₉	1	2	yes	complemented
100	#482	C₂×C₉	1	4	yes	complemented
101	#469	C₂×C₉	1	2	yes	complemented
102	#466	C₂×C₉	1	2	yes	complemented
103	#447	C₂×C₉	1	4	yes	complemented
104	#408	C₂×C₉	1	1	yes	characteristic, complemented
105	C₂⁴×C₃	—	2	2	no	—
106	C₃×#41	—	4	1	no	—
107	C₃×#41	—	4	1	no	—
108	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
109	C₂×C₃×D₄	—	2	4	no	—
110	C₂×C₃×D₄	—	2	4	no	—
111	C₂×C₃×D₄	—	2	2	no	—
112	C₂×C₃×D₄	—	4	2	no	—
113	C₂×C₃×D₄	—	4	2	no	—
114	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
115	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
116	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
117	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
118	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₃×D₈	—	4	2	no	—
120	C₃×D₈	—	4	2	no	—
121	C₃×M₁₆	—	4	1	no	—
122	C₂×C₃×C₈	—	4	1	no	—
123	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
124	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
125	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
126	C₃×C₂²⋊C₄	—	2	2	no	—
127	C₃×C₂²⋊C₄	—	2	4	no	—
128	C₃×C₂²⋊C₄	—	2	4	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	2	1	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	4	no	—
134	C₂²×C₉	—	4	4	no	—
135	C₂²×C₉	—	4	4	no	—
136	C₂²×C₉	—	4	4	no	—
137	C₂²×C₉	—	8	8	no	—
138	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	4	4	no	—
142	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×#41	C₂²×C₉	1	1	yes	characteristic
145	C₂²×D₄	C₂²×C₉	1	2	yes	complemented
146	C₂×D₈	C₂²×C₉	1	2	yes	—
147	C₂×D₈	C₂²×C₉	1	2	yes	—
148	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic, complemented
149	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic, complemented
150	#127	—	2	2	no	—
151	#124	—	2	2	no	—
152	#121	C₂²×C₉	1	4	yes	—
153	#121	C₂²×C₉	1	4	yes	complemented
154	#120	—	2	2	no	—
155	#120	—	2	2	no	—
156	#117	C₂²×C₉	1	2	yes	complemented
157	#107	C₂²×C₉	1	2	yes	complemented
158	#106	C₂²×C₉	1	2	yes	complemented
159	#102	C₂²×C₉	1	4	yes	complemented
160	#102	C₂²×C₉	1	4	yes	complemented
161	C₂²⋊C₈	C₂²×C₉	1	4	yes	complemented
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	—	4	4	no	—
164	C₂³×C₃	—	8	4	no	—
165	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
166	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
167	C₃×Q₈	—	4	1	no	—
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	4	2	no	—
171	C₃×D₄	—	4	1	no	—
172	C₃×D₄	—	4	4	no	—
173	C₃×D₄	—	8	4	no	—
174	C₃×D₄	—	8	4	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	4	no	—
180	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	2	no	—
185	C₂×C₉	—	4	1	no	—
186	C₂×C₉	—	4	2	no	—
187	C₂×C₉	—	8	4	no	—
188	C₂×C₉	#534	1	1	yes	characteristic
189	C₂×C₉	#576	1	2	yes	—
190	C₂⁴	—	2	2	no	—
191	#41	—	4	1	no	—
192	#41	—	4	1	no	—
193	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	D₄×C₉	1	1	yes	characteristic
200	C₂×D₄	D₄×C₉	1	2	yes	—
201	C₂×D₄	C₂³×C₉	1	4	yes	—
202	C₂²×C₄	C₂³×C₉	1	2	yes	—
203	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
204	D₈	—	4	2	no	—
205	D₈	—	4	2	no	—
206	M₁₆	—	4	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	C₂³×C₉	1	2	yes	—
209	C₂×C₈	C₂³×C₉	1	2	yes	—
210	C₄⋊C₄	C₂³×C₉	1	4	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₄²	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	4	4	no	—
220	C₂²×C₃	—	4	4	no	—
221	C₂²×C₃	—	4	4	no	—
222	C₂²×C₃	—	8	8	no	—
223	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	4	4	no	—
227	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
228	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₉	#3199	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	2	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	8	4	no	—
233	C₂³	C₂×D₄×C₉	1	2	yes	—
234	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	1	no	—
240	D₄	—	4	4	no	—
241	D₄	—	8	4	no	—
242	D₄	—	8	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
249	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
250	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₂×C₃	—	4	2	no	—
254	C₂×C₃	—	4	1	no	—
255	C₂×C₃	—	8	4	no	—
256	C₂×C₃	C₃×#534	1	1	yes	characteristic
257	C₂×C₃	C₃×#576	1	2	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	8	8	no	—
266	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
271	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
272	C₃	C₃×#3199	1	1	yes	characteristic
273	C₂	—	4	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	8	4	no	—
276	C₂	C₉×#534	1	1	yes	characteristic
277	C₂	C₉×#576	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3199	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₉×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#535	C₂	semidirect
2. C₉×#482	C₂	semidirect
3. C₉×#469	C₂	semidirect
4. C₉×#466	C₂	semidirect
5. C₉×#447	C₂	semidirect
6. C₉×#408	C₂	semidirect
9. C₂²×D₄×C₉	C₂²	semidirect
12. C₂×C₉×M₁₆	C₂²	semidirect
13. C₂²×C₈×C₉	C₂²	semidirect
17. C₉×#121	C₂²	semidirect
20. C₉×#117	C₂²	semidirect
21. C₉×#107	C₂²	semidirect
22. C₉×#106	C₂²	semidirect
23. C₉×#102	C₂²	semidirect
24. C₉×#102	C₂²	semidirect
25. C₉×C₂²⋊C₈	C₂²	semidirect
57. #3199	C₉	direct (N × H)
99. #535	C₂×C₉	semidirect
100. #482	C₂×C₉	semidirect
101. #469	C₂×C₉	semidirect
102. #466	C₂×C₉	semidirect
103. #447	C₂×C₉	semidirect
104. #408	C₂×C₉	semidirect
145. C₂²×D₄	C₂²×C₉	semidirect
148. C₂×M₁₆	C₂²×C₉	semidirect
149. C₂²×C₈	C₂²×C₉	semidirect
153. #121	C₂²×C₉	semidirect
156. #117	C₂²×C₉	semidirect
157. #107	C₂²×C₉	semidirect
158. #106	C₂²×C₉	semidirect
159. #102	C₂²×C₉	semidirect
160. #102	C₂²×C₉	semidirect
161. C₂²⋊C₈	C₂²×C₉	semidirect
229. C₉	#3199	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×#535	2
C₉×#482	2
C₉×#469	2
C₉×#466	2
C₉×#447	2
C₉×#408	2
C₃×#3199	3

Subgroups — C9×#3199 #11767920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3199 #11767920