Subgroups — C₁₃×#20548 #15706887

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 294 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×C₁₃×#423	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×#1753	C₂	1	4	yes	maximal
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₄	1	8	yes	complemented
11	C₁₃×#423	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#383	C₄	1	2	yes	—
21	C₁₃×#382	C₄	1	2	yes	—
22	C₁₃×#381	C₄	1	2	yes	—
23	C₁₃×#380	C₄	1	2	yes	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₁₃×#376	C₂²	1	2	yes	—
27	C₁₃×#336	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	—	4	4	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
47	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
49	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
50	C₁₃×#97	—	4	4	no	—
51	C₄×C₈×C₁₃	—	4	4	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
54	C₁₃×#95	C₂×C₄	1	4	yes	—
55	C₁₃×#95	C₂×C₄	1	4	yes	—
56	C₁₃×#95	C₂×C₄	1	4	yes	—
57	C₁₃×#95	C₂×C₄	1	4	yes	—
58	#20548	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
65	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×M₁₆	—	4	8	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
85	C₁₃×C₄⋊C₄	—	2	8	no	—
86	C₁₃×C₂²⋊C₄	—	4	4	no	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	C₄²	1	4	yes	—
90	C₄²×C₁₃	C₄²	1	4	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
97	#2283	C₂×C₁₃	1	1	yes	characteristic
98	#1770	C₂×C₁₃	1	2	yes	complemented
99	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	complemented
100	#1768	C₂×C₁₃	1	4	yes	complemented
101	#1753	C₂×C₁₃	1	4	yes	—
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	—	2	2	no	—
104	C₂³×C₁₃	—	2	2	no	—
105	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
106	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	8	no	—
115	C₂×C₄×C₁₃	—	2	8	no	—
116	C₂×C₄×C₁₃	—	2	8	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₁₃	#131	1	4	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
129	C₈×C₁₃	—	2	8	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	2	4	no	—
132	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
134	#431	C₄×C₁₃	1	8	yes	complemented
135	#423	C₄×C₁₃	1	8	yes	complemented
136	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
138	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
139	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
140	C₂×#97	C₂²×C₁₃	1	4	yes	—
141	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
142	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
144	#383	C₄×C₁₃	1	2	yes	—
145	#382	C₄×C₁₃	1	2	yes	—
146	#381	C₄×C₁₃	1	2	yes	—
147	#380	C₄×C₁₃	1	2	yes	—
148	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
149	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
150	#376	C₂²×C₁₃	1	2	yes	—
151	#336	C₂²×C₁₃	1	4	yes	—
152	#336	C₂²×C₁₃	1	4	yes	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
158	C₂²×C₁₃	#520	1	2	yes	—
159	C₂²×C₁₃	C₂×#131	1	2	yes	—
160	C₂²×C₁₃	#568	1	2	yes	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	4	8	no	—
166	C₄×C₁₃	#405	1	4	yes	—
167	C₄×C₁₃	#518	1	1	yes	characteristic
168	C₄×C₁₃	#519	1	1	yes	characteristic
169	C₄×C₁₃	#520	1	2	yes	—
170	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	4	4	no	—
174	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
175	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
176	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
177	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
180	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
183	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
184	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
187	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
188	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
189	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
190	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
191	#97	—	4	4	no	—
192	C₄×C₈	—	4	4	no	—
193	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
194	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
195	#95	C₂×C₄×C₁₃	1	4	yes	—
196	#95	C₂×C₄×C₁₃	1	4	yes	—
197	#95	C₂×C₄×C₁₃	1	4	yes	—
198	#95	C₂×C₄×C₁₃	1	4	yes	—
199	C₂×C₁₃	—	4	2	no	—
200	C₂×C₁₃	#2274	1	1	yes	characteristic
201	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
202	C₂×C₁₃	#2873	1	1	yes	characteristic
203	C₂×C₁₃	#2993	1	4	yes	—
204	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
210	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
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	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
220	M₁₆	—	4	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
228	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
229	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
230	C₄⋊C₄	—	2	8	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	C₄²×C₁₃	1	4	yes	—
235	C₄²	C₄²×C₁₃	1	4	yes	—
236	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
237	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
238	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
239	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
240	C₁₃	#20548	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
245	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
259	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
260	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
262	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
263	C₂×C₄	C₁₃×#131	1	4	yes	—
264	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
267	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
268	C₈	—	2	8	no	—
269	C₈	—	2	4	no	—
270	C₈	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	4	no	—
275	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
276	C₂²	C₁₃×#520	1	2	yes	—
277	C₂²	C₂×C₁₃×#131	1	2	yes	—
278	C₂²	C₁₃×#568	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	8	no	—
284	C₄	C₁₃×#405	1	4	yes	—
285	C₄	C₁₃×#518	1	1	yes	characteristic
286	C₄	C₁₃×#519	1	1	yes	characteristic
287	C₄	C₁₃×#520	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	C₁₃×#2274	1	1	yes	characteristic
290	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
291	C₂	C₁₃×#2873	1	1	yes	characteristic
292	C₂	C₁₃×#2993	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
10. C₁₃×#431	C₄	semidirect
11. C₁₃×#423	C₄	semidirect
58. #20548	C₁₃	direct (N × H)
98. #1770	C₂×C₁₃	semidirect
99. C₈×C₄⋊C₄	C₂×C₁₃	semidirect
100. #1768	C₂×C₁₃	semidirect
134. #431	C₄×C₁₃	semidirect
135. #423	C₄×C₁₃	semidirect
240. C₁₃	#20548	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×C₁₃×#423	2
C₁₃×#2283	2
C₁₃×#1770	2
C₈×C₁₃×C₄⋊C₄	2
C₁₃×#1768	2
C₁₃×#1753	2
#20548	13

Subgroups — C13×#20548 #15706887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20548 #15706887