Subgroups — C₉×#2379 #11767100

Order 1152.

780 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₉×#391	C₂	1	2	yes	maximal, complemented
3	C₉×#388	C₂	1	4	yes	maximal, complemented
4	C₉×#386	C₂	1	2	yes	maximal
5	C₉×#385	C₂	1	4	yes	maximal, complemented
6	C₉×#382	C₂	1	2	yes	maximal, complemented
7	C₃×#2379	C₃	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉	C₂²	1	2	yes	—
11	C₉×#124	C₄	1	8	yes	complemented
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄²×C₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×#95	C₂²	1	4	yes	—
21	C₉×#95	C₂²	1	4	yes	—
22	C₉×#95	C₂²	1	4	yes	—
23	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
24	C₃×#391	C₂×C₃	1	2	yes	—
25	C₃×#388	C₂×C₃	1	4	yes	—
26	C₃×#386	C₂×C₃	1	2	yes	—
27	C₃×#385	C₂×C₃	1	4	yes	—
28	C₃×#382	C₂×C₃	1	2	yes	—
29	C₂⁴×C₉	C₂³	1	2	yes	—
30	C₂×Q₈×C₉	—	4	1	no	—
31	C₂×Q₈×C₉	—	4	1	no	—
32	C₂×Q₈×C₉	C₂×C₄	1	4	yes	complemented
33	C₂×D₄×C₉	—	4	1	no	—
34	C₂×D₄×C₉	—	4	1	no	—
35	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₉	—	4	4	no	—
37	C₂²×C₄×C₉	—	4	2	no	—
38	C₂²×C₄×C₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₉×C₄⋊C₄	—	4	2	no	—
44	C₉×C₄⋊C₄	—	4	2	no	—
45	C₉×C₄⋊C₄	—	4	2	no	—
46	C₉×C₂²⋊C₄	—	4	2	no	—
47	C₉×C₂²⋊C₄	—	4	4	no	—
48	C₉×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
49	C₄²×C₉	—	4	2	no	—
50	C₄²×C₉	C₂×C₄	1	4	yes	complemented
51	#2379	C₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
54	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
55	C₃×#124	C₃×C₄	1	8	yes	—
56	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
60	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
62	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
63	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
64	C₃×#95	C₂²×C₃	1	4	yes	—
65	C₃×#95	C₂²×C₃	1	4	yes	—
66	C₃×#95	C₂²×C₃	1	4	yes	—
67	C₂³×C₉	—	2	4	no	—
68	C₂³×C₉	—	2	2	no	—
69	C₂³×C₉	—	2	2	no	—
70	C₂³×C₉	—	2	4	no	—
71	C₂³×C₉	C₂²×C₄	1	4	yes	—
72	C₂³×C₉	C₂⁴	1	1	yes	characteristic
73	Q₈×C₉	—	4	4	no	—
74	D₄×C₉	—	4	4	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	2	2	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	—	2	4	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	2	4	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	4	8	no	—
88	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
91	C₂×#124	C₂×C₉	1	1	yes	characteristic
92	#391	C₂×C₉	1	2	yes	complemented
93	#388	C₂×C₉	1	4	yes	complemented
94	#386	C₂×C₉	1	2	yes	—
95	#385	C₂×C₉	1	4	yes	complemented
96	#382	C₂×C₉	1	2	yes	complemented
97	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
98	C₂×C₃×Q₈	—	4	1	no	—
99	C₂×C₃×Q₈	—	4	1	no	—
100	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
101	C₂×C₃×D₄	—	4	1	no	—
102	C₂×C₃×D₄	—	4	1	no	—
103	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
104	C₂²×C₃×C₄	—	4	4	no	—
105	C₂²×C₃×C₄	—	4	2	no	—
106	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
111	C₃×C₄⋊C₄	—	4	2	no	—
112	C₃×C₄⋊C₄	—	4	2	no	—
113	C₃×C₄⋊C₄	—	4	2	no	—
114	C₃×C₂²⋊C₄	—	4	2	no	—
115	C₃×C₂²⋊C₄	—	4	4	no	—
116	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	—
117	C₃×C₄²	—	4	2	no	—
118	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
119	C₂²×C₉	—	2	4	no	—
120	C₂²×C₉	—	2	8	no	—
121	C₂²×C₉	—	4	2	no	—
122	C₂²×C₉	—	4	4	no	—
123	C₂²×C₉	—	4	2	no	—
124	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
128	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	—	2	4	no	—
131	C₄×C₉	—	2	4	no	—
132	C₄×C₉	—	4	4	no	—
133	C₄×C₉	—	4	4	no	—
134	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
135	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
136	C₂³×C₄	C₂²×C₉	1	2	yes	—
137	#124	C₄×C₉	1	8	yes	complemented
138	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
144	C₂×C₄²	C₂²×C₉	1	2	yes	—
145	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
146	#95	C₂²×C₉	1	4	yes	—
147	#95	C₂²×C₉	1	4	yes	—
148	#95	C₂²×C₉	1	4	yes	—
149	C₂³×C₃	—	2	4	no	—
150	C₂³×C₃	—	2	2	no	—
151	C₂³×C₃	—	2	2	no	—
152	C₂³×C₃	—	2	4	no	—
153	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
154	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
155	C₃×Q₈	—	4	4	no	—
156	C₃×D₄	—	4	4	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	4	8	no	—
170	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
171	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
173	C₂×C₉	—	4	4	no	—
174	C₂×C₉	#518	1	1	yes	characteristic
175	C₂×C₉	#520	1	2	yes	—
176	C₂×C₉	H(3,𝔽₄)	1	1	yes	characteristic
177	C₂×C₉	#562	1	2	yes	—
178	C₂×C₉	#563	1	1	yes	characteristic
179	C₂⁴	C₂³×C₉	1	2	yes	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	C₂×C₄×C₉	1	4	yes	complemented
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂³×C₉	1	4	yes	—
189	C₂²×C₄	C₂³×C₉	1	4	yes	—
190	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₉	1	2	yes	—
192	C₂²×C₄	C₂³×C₉	1	2	yes	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₂²⋊C₄	C₂×C₄×C₉	1	16	yes	complemented
199	C₄²	—	4	2	no	—
200	C₄²	C₂×C₄×C₉	1	4	yes	complemented
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	2	8	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	4	4	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
207	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
208	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
209	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	4	4	no	—
215	C₃×C₄	—	4	4	no	—
216	C₉	#2379	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	2	4	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	C₂²×C₄×C₉	1	4	yes	—
222	C₂³	C₂⁴×C₉	1	1	yes	characteristic
223	Q₈	—	4	4	no	—
224	D₄	—	4	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
239	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
240	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
241	C₂×C₃	—	4	4	no	—
242	C₂×C₃	C₃×#518	1	1	yes	characteristic
243	C₂×C₃	C₃×#520	1	2	yes	—
244	C₂×C₃	C₃×H(3,𝔽₄)	1	1	yes	characteristic
245	C₂×C₃	C₃×#562	1	2	yes	—
246	C₂×C₃	C₃×#563	1	1	yes	characteristic
247	C₂²	—	2	8	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
253	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
254	C₂²	C₉×H(5,𝔽₂)	1	1	yes	characteristic
255	C₂²	C₉×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	4	no	—
262	C₃	C₃×#2379	1	1	yes	characteristic
263	C₂	—	4	4	no	—
264	C₂	C₉×#518	1	1	yes	characteristic
265	C₂	C₉×#520	1	2	yes	—
266	C₂	C₉×H(3,𝔽₄)	1	1	yes	characteristic
267	C₂	C₉×#562	1	2	yes	—
268	C₂	C₉×#563	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	C₂²×C₃×D₄	C₂²×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×#391	C₂	semidirect
3. C₉×#388	C₂	semidirect
5. C₉×#385	C₂	semidirect
6. C₉×#382	C₂	semidirect
11. C₉×#124	C₄	semidirect
32. C₂×Q₈×C₉	C₂×C₄	semidirect
48. C₉×C₂²⋊C₄	C₂×C₄	semidirect
50. C₄²×C₉	C₂×C₄	semidirect
51. #2379	C₉	direct (N × H)
92. #391	C₂×C₉	semidirect
93. #388	C₂×C₉	semidirect
95. #385	C₂×C₉	semidirect
96. #382	C₂×C₉	semidirect
137. #124	C₄×C₉	semidirect
182. C₂×Q₈	C₂×C₄×C₉	semidirect
198. C₂²⋊C₄	C₂×C₄×C₉	semidirect
200. C₄²	C₂×C₄×C₉	semidirect
216. C₉	#2379	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
C₉×#391	2
C₉×#388	2
C₉×#386	2
C₉×#385	2
C₉×#382	2
C₃×#2379	3

Subgroups — C9×#2379 #11767100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2379 #11767100