Subgroups — C₉×#2315 #11767036

Order 1152.

1224 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal
2	C₂²×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₉×#385	C₂	1	8	yes	maximal, complemented
4	C₉×#380	C₂	1	4	yes	maximal, complemented
5	C₃×#2315	C₃	1	1	yes	maximal, characteristic
6	C₂⁵×C₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₂²	1	4	yes	—
8	C₂³×C₄×C₉	C₂²	1	2	yes	—
9	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	complemented
10	C₂×C₉×C₂²⋊C₄	—	2	8	no	—
11	C₂×C₉×C₂²⋊C₄	C₄	1	8	yes	complemented
12	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₉×#95	C₂²	1	8	yes	complemented
16	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	2	yes	—
17	C₂²×C₃×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₃×#385	C₂×C₃	1	8	yes	—
19	C₃×#380	C₂×C₃	1	4	yes	—
20	C₂⁴×C₉	—	4	4	no	—
21	C₂⁴×C₉	C₂×C₄	1	4	yes	—
22	C₂⁴×C₉	D₄	1	4	yes	—
23	C₂⁴×C₉	Q₈	1	4	yes	—
24	C₂⁴×C₉	C₂³	1	2	yes	—
25	C₂⁴×C₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₉	—	2	8	no	—
27	C₂²×C₄×C₉	—	4	4	no	—
28	C₂²×C₄×C₉	—	4	8	no	—
29	C₂²×C₄×C₉	C₂×C₄	1	8	yes	complemented
30	C₂²×C₄×C₉	C₂³	1	8	yes	—
31	C₂²×C₄×C₉	C₂³	1	4	yes	—
32	C₉×C₄⋊C₄	—	4	8	no	—
33	C₉×C₂²⋊C₄	—	2	8	no	—
34	C₉×C₂²⋊C₄	—	2	8	no	—
35	C₉×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
36	#2315	C₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₂⁵×C₃	C₂²×C₃	1	1	yes	characteristic
38	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
39	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
40	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
41	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
42	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	8	yes	—
43	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
44	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
45	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
46	C₃×#95	C₂²×C₃	1	8	yes	—
47	C₂³×C₉	—	2	4	no	—
48	C₂³×C₉	—	2	8	no	—
49	C₂³×C₉	—	4	8	no	—
50	C₂³×C₉	—	4	8	no	—
51	C₂³×C₉	—	4	8	no	—
52	C₂³×C₉	C₄⋊C₄	1	16	yes	complemented
53	C₂³×C₉	C₂²×C₄	1	4	yes	—
54	C₂³×C₉	C₂²×C₄	1	2	yes	—
55	C₂³×C₉	C₂×D₄	1	2	yes	—
56	C₂³×C₉	C₂×D₄	1	4	yes	—
57	C₂³×C₉	C₂×Q₈	1	4	yes	—
58	C₂³×C₉	C₂×Q₈	1	2	yes	—
59	C₂³×C₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₉	—	2	8	no	—
61	C₂×C₄×C₉	—	2	8	no	—
62	C₂×C₄×C₉	—	2	16	no	—
63	C₂×C₄×C₉	—	4	8	no	—
64	C₂×C₄×C₉	—	4	16	no	—
65	C₂×C₄×C₉	C₂²×C₄	1	8	yes	—
66	C₂²×C₂²⋊C₄	C₂×C₉	1	2	yes	—
67	C₂²×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
68	#385	C₂×C₉	1	8	yes	complemented
69	#380	C₂×C₉	1	4	yes	complemented
70	C₂⁴×C₃	—	4	4	no	—
71	C₂⁴×C₃	C₂×C₃×C₄	1	4	yes	—
72	C₂⁴×C₃	C₃×D₄	1	4	yes	—
73	C₂⁴×C₃	C₃×Q₈	1	4	yes	—
74	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
75	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	—	2	8	no	—
77	C₂²×C₃×C₄	—	4	4	no	—
78	C₂²×C₃×C₄	—	4	8	no	—
79	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
80	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
82	C₃×C₄⋊C₄	—	4	8	no	—
83	C₃×C₂²⋊C₄	—	2	8	no	—
84	C₃×C₂²⋊C₄	—	2	8	no	—
85	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	—
86	C₂²×C₉	—	2	4	no	—
87	C₂²×C₉	—	2	8	no	—
88	C₂²×C₉	—	2	8	no	—
89	C₂²×C₉	—	2	4	no	—
90	C₂²×C₉	—	2	4	no	—
91	C₂²×C₉	—	2	8	no	—
92	C₂²×C₉	—	4	16	no	—
93	C₂²×C₉	C₂×C₄⋊C₄	1	8	yes	—
94	C₂²×C₉	C₂×C₄⋊C₄	1	4	yes	—
95	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
99	C₄×C₉	—	2	8	no	—
100	C₄×C₉	—	4	8	no	—
101	C₂⁵	C₂²×C₉	1	1	yes	characteristic
102	C₂³×C₄	C₂²×C₉	1	4	yes	—
103	C₂³×C₄	C₂²×C₉	1	2	yes	—
104	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	complemented
105	C₂×C₂²⋊C₄	—	2	8	no	—
106	C₂×C₂²⋊C₄	C₄×C₉	1	8	yes	complemented
107	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
110	#95	C₂²×C₉	1	8	yes	complemented
111	C₂³×C₃	—	2	4	no	—
112	C₂³×C₃	—	2	8	no	—
113	C₂³×C₃	—	4	8	no	—
114	C₂³×C₃	—	4	8	no	—
115	C₂³×C₃	—	4	8	no	—
116	C₂³×C₃	C₃×C₄⋊C₄	1	16	yes	—
117	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
118	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
119	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
120	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
121	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
122	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
123	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
124	C₂×C₃×C₄	—	2	8	no	—
125	C₂×C₃×C₄	—	2	8	no	—
126	C₂×C₃×C₄	—	2	16	no	—
127	C₂×C₃×C₄	—	4	8	no	—
128	C₂×C₃×C₄	—	4	16	no	—
129	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
130	C₂×C₉	—	2	4	no	—
131	C₂×C₉	—	2	8	no	—
132	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₉	#518	1	2	yes	—
134	C₂×C₉	#534	1	2	yes	—
135	C₂×C₉	#543	1	2	yes	—
136	C₂⁴	—	4	4	no	—
137	C₂⁴	C₂×C₄×C₉	1	4	yes	—
138	C₂⁴	D₄×C₉	1	4	yes	—
139	C₂⁴	Q₈×C₉	1	4	yes	—
140	C₂⁴	C₂³×C₉	1	2	yes	—
141	C₂⁴	C₂³×C₉	1	1	yes	characteristic
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	4	4	no	—
144	C₂²×C₄	—	4	8	no	—
145	C₂²×C₄	C₂×C₄×C₉	1	8	yes	complemented
146	C₂²×C₄	C₂³×C₉	1	8	yes	—
147	C₂²×C₄	C₂³×C₉	1	4	yes	—
148	C₄⋊C₄	—	4	8	no	—
149	C₂²⋊C₄	—	2	8	no	—
150	C₂²⋊C₄	—	2	8	no	—
151	C₂²⋊C₄	C₂×C₄×C₉	1	16	yes	complemented
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	2	8	no	—
154	C₂²×C₃	—	2	8	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	4	no	—
157	C₂²×C₃	—	2	8	no	—
158	C₂²×C₃	—	4	16	no	—
159	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	8	yes	—
160	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
161	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
162	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
163	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
164	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
165	C₃×C₄	—	2	8	no	—
166	C₃×C₄	—	4	8	no	—
167	C₉	#2315	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	—	2	8	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	4	8	no	—
171	C₂³	—	4	8	no	—
172	C₂³	—	4	8	no	—
173	C₂³	C₉×C₄⋊C₄	1	16	yes	complemented
174	C₂³	C₂²×C₄×C₉	1	4	yes	—
175	C₂³	C₂²×C₄×C₉	1	2	yes	—
176	C₂³	C₂×D₄×C₉	1	2	yes	—
177	C₂³	C₂×D₄×C₉	1	4	yes	—
178	C₂³	C₂×Q₈×C₉	1	4	yes	—
179	C₂³	C₂×Q₈×C₉	1	2	yes	—
180	C₂³	C₂⁴×C₉	1	1	yes	characteristic
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	C₂²×C₄×C₉	1	8	yes	—
187	C₂×C₃	—	2	4	no	—
188	C₂×C₃	—	2	8	no	—
189	C₂×C₃	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₃	C₃×#518	1	2	yes	—
191	C₂×C₃	C₃×#534	1	2	yes	—
192	C₂×C₃	C₃×#543	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	8	no	—
199	C₂²	—	4	16	no	—
200	C₂²	C₂×C₉×C₄⋊C₄	1	8	yes	—
201	C₂²	C₂×C₉×C₄⋊C₄	1	4	yes	—
202	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
203	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
204	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
205	C₂²	C₉×H(5,𝔽₂)	1	4	yes	—
206	C₄	—	2	8	no	—
207	C₄	—	4	8	no	—
208	C₃	C₃×#2315	1	1	yes	characteristic
209	C₂	—	2	8	no	—
210	C₂	—	2	4	no	—
211	C₂	C₂²×C₉×C₄⋊C₄	1	1	yes	characteristic
212	C₂	C₉×#518	1	2	yes	—
213	C₂	C₉×#534	1	2	yes	—
214	C₂	C₉×#543	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁵×C₃	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
3. C₉×#385	C₂	semidirect
4. C₉×#380	C₂	semidirect
9. C₂×C₉×C₄⋊C₄	C₂²	semidirect
11. C₂×C₉×C₂²⋊C₄	C₄	semidirect
15. C₉×#95	C₂²	semidirect
29. C₂²×C₄×C₉	C₂×C₄	semidirect
35. C₉×C₂²⋊C₄	C₂×C₄	semidirect
36. #2315	C₉	direct (N × H)
52. C₂³×C₉	C₄⋊C₄	semidirect
68. #385	C₂×C₉	semidirect
69. #380	C₂×C₉	semidirect
104. C₂×C₄⋊C₄	C₂²×C₉	semidirect
106. C₂×C₂²⋊C₄	C₄×C₉	semidirect
110. #95	C₂²×C₉	semidirect
145. C₂²×C₄	C₂×C₄×C₉	semidirect
151. C₂²⋊C₄	C₂×C₄×C₉	semidirect
167. C₉	#2315	direct (N × H)
173. C₂³	C₉×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₂²⋊C₄	2
C₉×#385	2
C₉×#380	2
C₃×#2315	3

Subgroups — C9×#2315 #11767036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2315 #11767036