Subgroups — C₁₃×#10624901 #23745052

Order 9477.

718 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₃×#737	C₃	1	1	yes	maximal, characteristic
2	C₁₃×#6900	C₃	1	3	yes	maximal, complemented
3	C₁₃×#6899	C₃	1	6	yes	maximal, complemented
4	C₁₃×#6896	C₃	1	6	yes	maximal, complemented
5	C₁₃×#6896	C₃	1	3	yes	maximal, complemented
6	C₁₃×#6895	C₃	1	2	yes	maximal, complemented
7	C₁₃×#6894	C₃	1	2	yes	maximal, complemented
8	C₁₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#6894	C₃	1	6	yes	maximal, complemented
10	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
11	C₁₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
12	C₉×C₁₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
13	C₉×C₁₃×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
14	C₉×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
15	C₃×C₁₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
16	C₁₃×#737	—	3	9	no	—
17	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
18	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
19	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
20	C₃×C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
21	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	—
22	C₃×C₁₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
23	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
25	C₃²×C₉×C₁₃	C₃²	1	1	yes	characteristic
26	C₃²×C₉×C₁₃	C₃²	1	3	yes	—
27	C₃²×C₉×C₁₃	C₃²	1	2	yes	—
28	C₁₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
29	C₁₃×C₉⋊C₉	C₃²	1	2	yes	—
30	C₁₃×C₉⋊C₉	C₃²	1	2	yes	—
31	C₁₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
32	C₁₃×C₉⋊C₉	C₃²	1	2	yes	—
33	C₁₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
34	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
35	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
36	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
37	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
38	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
39	C₁₃×C₉⋊C₉	C₃²	1	3	yes	—
40	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
41	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
42	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
43	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
44	C₁₃×C₉⋊C₉	C₃²	1	6	yes	—
45	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
46	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
47	C₁₃×C₃²⋊C₉	C₃²	1	2	yes	—
48	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	—
49	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	—
50	C₁₃×C₃²⋊C₉	C₃²	1	6	yes	—
51	C₉²×C₁₃	C₃²	1	6	yes	—
52	C₉²×C₁₃	C₃²	1	3	yes	—
53	#10624901	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₃³×C₁₃	C₃³	1	3	yes	—
55	C₃³×C₁₃	C₃³	1	1	yes	characteristic
56	C₁₃×3₋¹⁺²	—	3	3	no	—
57	C₁₃×3₋¹⁺²	—	3	9	no	—
58	C₁₃×3₋¹⁺²	—	3	9	no	—
59	C₁₃×3₋¹⁺²	—	3	9	no	—
60	C₁₃×3₋¹⁺²	—	9	6	no	—
61	C₁₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
62	C₁₃×H(3,𝔽₃)	—	3	3	no	—
63	C₃×C₉×C₁₃	—	3	3	no	—
64	C₃×C₉×C₁₃	—	3	9	no	—
65	C₃×C₉×C₁₃	—	3	3	no	—
66	C₃×C₉×C₁₃	—	9	2	no	—
67	C₃×C₉×C₁₃	C₃³	1	2	yes	—
68	C₃×C₉×C₁₃	C₃³	1	6	yes	—
69	C₃×C₉×C₁₃	C₃³	1	6	yes	—
70	C₃×C₉×C₁₃	C₃³	1	6	yes	—
71	C₃×C₉×C₁₃	C₃³	1	2	yes	—
72	C₃×C₉×C₁₃	C₃³	1	2	yes	—
73	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
74	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
75	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
76	C₃×C₉×C₁₃	C₃³	1	3	yes	—
77	C₃×C₉×C₁₃	C₃³	1	2	yes	—
78	C₃×C₉×C₁₃	C₃³	1	3	yes	—
79	C₃×C₉×C₁₃	C₃³	1	1	yes	characteristic
80	C₃×#737	C₃×C₁₃	1	1	yes	characteristic
81	#6900	C₃×C₁₃	1	3	yes	complemented
82	#6899	C₃×C₁₃	1	6	yes	complemented
83	#6896	C₃×C₁₃	1	6	yes	complemented
84	#6896	C₃×C₁₃	1	3	yes	complemented
85	#6895	C₃×C₁₃	1	2	yes	complemented
86	#6894	C₃×C₁₃	1	2	yes	complemented
87	#6894	C₃×C₁₃	1	1	yes	characteristic, complemented
88	#6894	C₃×C₁₃	1	6	yes	complemented
89	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	2	yes	—
90	C₉⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic
91	C₉×3₋¹⁺²	C₃×C₁₃	1	2	yes	complemented
92	C₉×3₋¹⁺²	C₃×C₁₃	1	3	yes	complemented
93	C₉×3₋¹⁺²	C₃×C₁₃	1	1	yes	characteristic, complemented
94	C₃×C₉⋊C₉	C₃×C₁₃	1	1	yes	characteristic, complemented
95	C₃²×C₁₃	—	3	1	no	—
96	C₃²×C₁₃	—	3	2	no	—
97	C₃²×C₁₃	—	3	3	no	—
98	C₃²×C₁₃	—	3	3	no	—
99	C₃²×C₁₃	—	3	6	no	—
100	C₃²×C₁₃	C₃×3₋¹⁺²	1	3	yes	—
101	C₃²×C₁₃	C₃⁴	1	1	yes	characteristic
102	C₉×C₁₃	—	3	1	no	—
103	C₉×C₁₃	—	3	1	no	—
104	C₉×C₁₃	—	3	1	no	—
105	C₉×C₁₃	—	3	3	no	—
106	C₉×C₁₃	—	3	2	no	—
107	C₉×C₁₃	—	3	6	no	—
108	C₉×C₁₃	—	3	2	no	—
109	C₉×C₁₃	—	3	2	no	—
110	C₉×C₁₃	—	3	2	no	—
111	C₉×C₁₃	—	3	6	no	—
112	C₉×C₁₃	—	3	6	no	—
113	C₉×C₁₃	C₃×3₋¹⁺²	1	9	yes	—
114	C₉×C₁₃	#737	1	3	yes	complemented
115	#737	—	3	9	no	—
116	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
117	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
118	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
119	C₃×3₋¹⁺²	C₃²×C₁₃	1	3	yes	—
120	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	—
121	C₃×3₋¹⁺²	C₃²×C₁₃	1	1	yes	characteristic
122	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	1	yes	characteristic
123	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
124	C₃²×C₉	C₃²×C₁₃	1	3	yes	—
125	C₃²×C₉	C₃²×C₁₃	1	1	yes	characteristic
126	C₃²×C₉	C₃²×C₁₃	1	2	yes	—
127	C₉⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
128	C₉⋊C₉	C₃²×C₁₃	1	2	yes	—
129	C₉⋊C₉	C₃²×C₁₃	1	2	yes	—
130	C₉⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
131	C₉⋊C₉	C₃²×C₁₃	1	2	yes	—
132	C₉⋊C₉	C₃²×C₁₃	1	1	yes	characteristic
133	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
134	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
135	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
136	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
137	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
138	C₉⋊C₉	C₃²×C₁₃	1	3	yes	—
139	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
140	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
141	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
142	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
143	C₉⋊C₉	C₃²×C₁₃	1	6	yes	—
144	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
145	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
146	C₃²⋊C₉	C₃²×C₁₃	1	2	yes	—
147	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	—
148	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	—
149	C₃²⋊C₉	C₃²×C₁₃	1	6	yes	—
150	C₉²	C₃²×C₁₃	1	6	yes	—
151	C₉²	C₃²×C₁₃	1	3	yes	—
152	C₃×C₁₃	—	3	3	no	—
153	C₃×C₁₃	—	9	3	no	—
154	C₃×C₁₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
155	C₃×C₁₃	C₃×#737	1	1	yes	characteristic
156	C₃×C₁₃	3₋¹⁺⁴	1	2	yes	—
157	C₃³	C₃³×C₁₃	1	3	yes	—
158	C₃³	C₃³×C₁₃	1	1	yes	characteristic
159	3₋¹⁺²	—	3	3	no	—
160	3₋¹⁺²	—	3	9	no	—
161	3₋¹⁺²	—	3	9	no	—
162	3₋¹⁺²	—	3	9	no	—
163	3₋¹⁺²	—	9	6	no	—
164	3₋¹⁺²	C₁₃×3₋¹⁺²	1	9	yes	complemented
165	H(3,𝔽₃)	—	3	3	no	—
166	C₃×C₉	—	3	3	no	—
167	C₃×C₉	—	3	9	no	—
168	C₃×C₉	—	3	3	no	—
169	C₃×C₉	—	9	2	no	—
170	C₃×C₉	C₃³×C₁₃	1	2	yes	—
171	C₃×C₉	C₃³×C₁₃	1	6	yes	—
172	C₃×C₉	C₃³×C₁₃	1	6	yes	—
173	C₃×C₉	C₃³×C₁₃	1	6	yes	—
174	C₃×C₉	C₃³×C₁₃	1	2	yes	—
175	C₃×C₉	C₃³×C₁₃	1	2	yes	—
176	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
177	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
178	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
179	C₃×C₉	C₃³×C₁₃	1	3	yes	—
180	C₃×C₉	C₃³×C₁₃	1	2	yes	—
181	C₃×C₉	C₃³×C₁₃	1	3	yes	—
182	C₃×C₉	C₃³×C₁₃	1	1	yes	characteristic
183	C₁₃	#10624901	1	1	yes	characteristic, direct product factor, Sylow
184	C₃²	—	3	1	no	—
185	C₃²	—	3	2	no	—
186	C₃²	—	3	3	no	—
187	C₃²	—	3	3	no	—
188	C₃²	—	3	6	no	—
189	C₃²	C₃×C₁₃×3₋¹⁺²	1	3	yes	—
190	C₃²	C₃⁴×C₁₃	1	1	yes	characteristic
191	C₉	—	3	1	no	—
192	C₉	—	3	1	no	—
193	C₉	—	3	1	no	—
194	C₉	—	3	3	no	—
195	C₉	—	3	2	no	—
196	C₉	—	3	6	no	—
197	C₉	—	3	2	no	—
198	C₉	—	3	2	no	—
199	C₉	—	3	2	no	—
200	C₉	—	3	6	no	—
201	C₉	—	3	6	no	—
202	C₉	C₃×C₁₃×3₋¹⁺²	1	9	yes	—
203	C₉	C₁₃×#737	1	3	yes	complemented
204	C₃	—	3	3	no	—
205	C₃	—	9	3	no	—
206	C₃	C₃²×C₁₃×3₋¹⁺²	1	1	yes	characteristic
207	C₃	C₃×C₁₃×#737	1	1	yes	characteristic
208	C₃	C₁₃×3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₁₃
prime radical	C₃×C₁₃×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#6900	C₃	semidirect
3. C₁₃×#6899	C₃	semidirect
4. C₁₃×#6896	C₃	semidirect
5. C₁₃×#6896	C₃	semidirect
6. C₁₃×#6895	C₃	semidirect
7. C₁₃×#6894	C₃	semidirect
8. C₁₃×#6894	C₃	semidirect
9. C₁₃×#6894	C₃	semidirect
12. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
13. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
14. C₉×C₁₃×3₋¹⁺²	C₃	semidirect
15. C₃×C₁₃×C₉⋊C₉	C₃	semidirect
53. #10624901	C₁₃	direct (N × H)
61. C₁₃×3₋¹⁺²	3₋¹⁺²	semidirect
81. #6900	C₃×C₁₃	semidirect
82. #6899	C₃×C₁₃	semidirect
83. #6896	C₃×C₁₃	semidirect
84. #6896	C₃×C₁₃	semidirect
85. #6895	C₃×C₁₃	semidirect
86. #6894	C₃×C₁₃	semidirect
87. #6894	C₃×C₁₃	semidirect
88. #6894	C₃×C₁₃	semidirect
91. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
92. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
93. C₉×3₋¹⁺²	C₃×C₁₃	semidirect
94. C₃×C₉⋊C₉	C₃×C₁₃	semidirect
114. C₉×C₁₃	#737	semidirect
164. 3₋¹⁺²	C₁₃×3₋¹⁺²	semidirect
183. C₁₃	#10624901	direct (N × H)
203. C₉	C₁₃×#737	semidirect

Subgroups — C13×#10624901 #23745052

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624901 #23745052