Subgroups — C₁₃×#20545 #15706884

Order 3328.

934 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1770	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1768	C₂	1	4	yes	maximal, complemented
7	C₁₃×#1752	C₂	1	2	yes	maximal
8	C₈×C₁₃×C₂²⋊C₄	C₂	1	2	yes	maximal
9	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#431	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#97	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₁₃×#383	C₄	1	2	yes	—
23	C₁₃×#382	C₄	1	2	yes	—
24	C₁₃×#381	C₄	1	2	yes	—
25	C₁₃×#380	C₄	1	2	yes	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₁₃×#376	C₂²	1	2	yes	—
29	C₁₃×#336	C₂²	1	4	yes	—
30	C₁₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	4	4	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
54	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
55	C₁₃×#97	—	4	4	no	—
56	C₄×C₈×C₁₃	—	4	4	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
58	C₁₃×#95	C₂×C₄	1	4	yes	—
59	C₁₃×#95	C₂×C₄	1	4	yes	—
60	C₁₃×#95	C₂×C₄	1	4	yes	—
61	C₁₃×#95	C₂×C₄	1	4	yes	—
62	#20545	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
69	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	8	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	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	4	8	no	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
90	C₁₃×C₄⋊C₄	—	2	8	no	—
91	C₁₃×C₂²⋊C₄	—	2	8	no	—
92	C₄²×C₁₃	—	2	4	no	—
93	C₄²×C₁₃	—	2	4	no	—
94	C₄²×C₁₃	C₄²	1	4	yes	—
95	C₄²×C₁₃	C₄²	1	4	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
101	C₂²×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
102	#2283	C₂×C₁₃	1	1	yes	characteristic
103	#1770	C₂×C₁₃	1	2	yes	complemented
104	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	complemented
105	#1768	C₂×C₁₃	1	4	yes	complemented
106	#1752	C₂×C₁₃	1	2	yes	—
107	C₈×C₂²⋊C₄	C₂×C₁₃	1	2	yes	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂³×C₁₃	#131	1	4	yes	—
112	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	8	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	—	2	8	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₁₃	#131	1	4	yes	—
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
134	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
135	C₈×C₁₃	—	2	4	no	—
136	C₈×C₁₃	—	2	4	no	—
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	2	4	no	—
139	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
141	#431	C₄×C₁₃	1	8	yes	complemented
142	C₂×C₄⋊C₈	C₄×C₁₃	1	8	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
147	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#97	C₂²×C₁₃	1	4	yes	—
150	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
152	#383	C₄×C₁₃	1	2	yes	—
153	#382	C₄×C₁₃	1	2	yes	—
154	#381	C₄×C₁₃	1	2	yes	—
155	#380	C₄×C₁₃	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
157	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
158	#376	C₂²×C₁₃	1	2	yes	—
159	#336	C₂²×C₁₃	1	4	yes	—
160	#336	C₂²×C₁₃	1	4	yes	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	8	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	#405	1	4	yes	—
165	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₁₃	#520	1	2	yes	—
167	C₂²×C₁₃	C₂×#131	1	2	yes	—
168	C₂²×C₁₃	C₂×#131	1	2	yes	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	2	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	4	8	no	—
174	C₄×C₁₃	#405	1	4	yes	—
175	C₄×C₁₃	#518	1	1	yes	characteristic
176	C₄×C₁₃	#519	1	1	yes	characteristic
177	C₄×C₁₃	#520	1	2	yes	—
178	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²×C₈	—	4	4	no	—
182	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
183	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
192	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
193	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
194	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
197	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
198	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
199	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
200	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
201	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
202	#97	—	4	4	no	—
203	C₄×C₈	—	4	4	no	—
204	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
205	#95	C₂×C₄×C₁₃	1	4	yes	—
206	#95	C₂×C₄×C₁₃	1	4	yes	—
207	#95	C₂×C₄×C₁₃	1	4	yes	—
208	#95	C₂×C₄×C₁₃	1	4	yes	—
209	C₂×C₁₃	—	2	4	no	—
210	C₂×C₁₃	#2274	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
213	C₂×C₁₃	#2994	1	4	yes	—
214	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
220	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	8	no	—
237	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
238	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
241	C₄⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₄²×C₁₃	1	4	yes	—
246	C₄²	C₄²×C₁₃	1	4	yes	—
247	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
248	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
249	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
250	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
251	C₁₃	#20545	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
255	C₂³	C₁₃×#131	1	4	yes	—
256	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
270	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
274	C₂×C₄	C₁₃×#131	1	4	yes	—
275	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
277	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
278	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	8	no	—
286	C₂²	C₁₃×#405	1	4	yes	—
287	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
288	C₂²	C₁₃×#520	1	2	yes	—
289	C₂²	C₂×C₁₃×#131	1	2	yes	—
290	C₂²	C₂×C₁₃×#131	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	8	no	—
296	C₄	C₁₃×#405	1	4	yes	—
297	C₄	C₁₃×#518	1	1	yes	characteristic
298	C₄	C₁₃×#519	1	1	yes	characteristic
299	C₄	C₁₃×#520	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	C₁₃×#2274	1	1	yes	characteristic
302	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
303	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
304	C₂	C₁₃×#2994	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1770	C₂	semidirect
5. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
6. C₁₃×#1768	C₂	semidirect
11. C₁₃×#431	C₄	semidirect
12. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
62. #20545	C₁₃	direct (N × H)
103. #1770	C₂×C₁₃	semidirect
104. C₈×C₄⋊C₄	C₂×C₁₃	semidirect
105. #1768	C₂×C₁₃	semidirect
141. #431	C₄×C₁₃	semidirect
142. C₂×C₄⋊C₈	C₄×C₁₃	semidirect
251. C₁₃	#20545	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
C₂²×C₁₃×C₄⋊C₈	2
C₁₃×#2283	2
C₁₃×#1770	2
C₈×C₁₃×C₄⋊C₄	2
C₁₃×#1768	2
C₁₃×#1752	2
C₈×C₁₃×C₂²⋊C₄	2
#20545	13

Subgroups — C13×#20545 #15706884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20545 #15706884