Subgroups — C₂×C₁₃×#1451 #15703656

Order 3328.

966 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1451	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#322	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#322	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×M₁₆	C₄	1	2	yes	—
8	C₂³×C₈×C₁₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₄	1	4	yes	—
11	C₁₃×#431	C₂²	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
14	C₄×C₁₃×M₁₆	C₄	1	4	yes	—
15	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
16	C₂×C₁₃×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#322	C₂²	1	8	yes	complemented
25	C₁₃×#322	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	—	2	8	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₁₃	—	2	8	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
46	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
47	C₁₃×#97	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	4	yes	—
56	C₄×C₈×C₁₃	C₂³	1	4	yes	—
57	C₂×#1451	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
66	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₁₃×M₁₆	—	2	16	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	8	no	—
77	C₂×C₈×C₁₃	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	16	no	—
79	C₂×C₈×C₁₃	M₁₆	1	8	yes	complemented
80	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₁₃	C₄²	1	2	yes	—
85	C₄²×C₁₃	C₄²	1	2	yes	—
86	C₄²×C₁₃	C₄²	1	2	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
95	C₂×C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
96	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
97	#1451	C₂×C₁₃	1	8	yes	direct product factor
98	C₂×#322	C₂×C₁₃	1	2	yes	complemented
99	C₂×#322	C₂×C₁₃	1	2	yes	complemented
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	#97	1	4	yes	—
103	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
104	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	#97	1	8	yes	—
112	C₂×C₄×C₁₃	#97	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
121	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₁₃	#131	1	4	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₁₃	—	2	4	no	—
127	C₈×C₁₃	—	2	8	no	—
128	C₈×C₁₃	—	2	8	no	—
129	C₈×C₁₃	C₂×M₁₆	1	8	yes	complemented
130	C₂²×M₁₆	C₄×C₁₃	1	2	yes	—
131	C₂³×C₈	C₄×C₁₃	1	2	yes	—
132	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
133	#431	C₄×C₁₃	1	4	yes	—
134	#431	C₂²×C₁₃	1	4	yes	—
135	C₂×C₄⋊C₈	C₄×C₁₃	1	2	yes	—
136	C₂×C₂²⋊C₈	C₄×C₁₃	1	2	yes	—
137	C₄×M₁₆	C₄×C₁₃	1	4	yes	—
138	C₄×M₁₆	C₂²×C₁₃	1	4	yes	—
139	C₂×#97	C₄×C₁₃	1	2	yes	—
140	C₂×C₄×C₈	C₄×C₁₃	1	4	yes	—
141	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
145	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
147	#322	C₂²×C₁₃	1	8	yes	complemented
148	#322	C₂²×C₁₃	1	8	yes	complemented
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	8	no	—
152	C₂²×C₁₃	C₂×#97	1	4	yes	—
153	C₂²×C₁₃	C₂×#97	1	2	yes	—
154	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
155	C₂²×C₁₃	#405	1	2	yes	—
156	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	C₂×#97	1	2	yes	—
160	C₄×C₁₃	C₂×#97	1	4	yes	—
161	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
162	C₄×C₁₃	#405	1	2	yes	—
163	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
164	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
165	C₄×C₁₃	C₂×#131	1	2	yes	—
166	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
171	C₂²×C₈	—	2	8	no	—
172	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
174	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
175	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
176	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
184	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
185	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
186	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
187	#97	C₂×C₄×C₁₃	1	8	yes	—
188	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
189	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
190	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
191	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
192	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
193	C₄×C₈	C₂³×C₁₃	1	2	yes	—
194	C₄×C₈	C₂³×C₁₃	1	2	yes	—
195	C₄×C₈	C₂³×C₁₃	1	4	yes	—
196	C₄×C₈	C₂³×C₁₃	1	4	yes	—
197	C₂×C₁₃	—	2	4	no	—
198	C₂×C₁₃	#1451	1	4	yes	direct product factor
199	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
200	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
201	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
202	C₂⁴	C₄²×C₁₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₄²×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₄²×C₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
208	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
209	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
210	C₂²×C₄	C₄²×C₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₁₃×M₁₆	1	8	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	M₁₆	—	2	16	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	16	no	—
223	C₂×C₈	C₁₃×M₁₆	1	8	yes	complemented
224	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
225	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
226	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
227	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
228	C₄²	C₄²×C₁₃	1	2	yes	—
229	C₄²	C₄²×C₁₃	1	2	yes	—
230	C₄²	C₄²×C₁₃	1	2	yes	—
231	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
232	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
233	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
234	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
236	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
237	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
238	C₁₃	C₂×#1451	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₁₃×#97	1	4	yes	—
242	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
243	C₂³	C₂×C₄²×C₁₃	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₁₃×#97	1	8	yes	—
251	C₂×C₄	C₁₃×#97	1	4	yes	—
252	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
253	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
254	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
256	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
259	C₂×C₄	C₂×C₁₃×M₁₆	1	8	yes	—
260	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
261	C₂×C₄	C₁₃×#131	1	4	yes	—
262	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	8	no	—
267	C₈	—	2	8	no	—
268	C₈	C₂×C₁₃×M₁₆	1	8	yes	complemented
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	C₂×C₁₃×#97	1	4	yes	—
273	C₂²	C₂×C₁₃×#97	1	2	yes	—
274	C₂²	C₄×C₁₃×M₁₆	1	2	yes	—
275	C₂²	C₁₃×#405	1	2	yes	—
276	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₂×C₁₃×#97	1	2	yes	—
280	C₄	C₂×C₁₃×#97	1	4	yes	—
281	C₄	C₄×C₁₃×M₁₆	1	2	yes	—
282	C₄	C₁₃×#405	1	2	yes	—
283	C₄	C₂²×C₁₃×M₁₆	1	2	yes	—
284	C₄	C₂²×C₁₃×M₁₆	1	2	yes	—
285	C₄	C₂×C₁₃×#131	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	C₁₃×#1451	1	4	yes	direct product factor
288	C₂	C₂²×C₁₃×#97	1	1	yes	characteristic
289	C₂	C₂×C₄×C₁₃×M₁₆	1	1	yes	characteristic
290	C₂	C₂×C₁₃×#405	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₁₃	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂⁴×C₁₃	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1451	C₂	direct (N × H)
5. C₂×C₁₃×#322	C₂	semidirect
6. C₂×C₁₃×#322	C₂	semidirect
24. C₁₃×#322	C₂²	semidirect
25. C₁₃×#322	C₂²	semidirect
57. C₂×#1451	C₁₃	direct (N × H)
79. C₂×C₈×C₁₃	M₁₆	semidirect
97. #1451	C₂×C₁₃	direct (N × H)
98. C₂×#322	C₂×C₁₃	semidirect
99. C₂×#322	C₂×C₁₃	semidirect
129. C₈×C₁₃	C₂×M₁₆	semidirect
147. #322	C₂²×C₁₃	semidirect
148. #322	C₂²×C₁₃	semidirect
198. C₂×C₁₃	#1451	direct (N × H)
223. C₂×C₈	C₁₃×M₁₆	semidirect
238. C₁₃	C₂×#1451	direct (N × H)
268. C₈	C₂×C₁₃×M₁₆	semidirect
287. C₂	C₁₃×#1451	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×C₁₃×#431	2
C₂×C₄×C₁₃×M₁₆	2
C₂²×C₄×C₈×C₁₃	2
C₁₃×#1451	2
C₂×C₁₃×#322	2
C₂×#1451	13

Subgroups — C2×C13×#1451 #15703656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#1451 #15703656