Subgroups — C₁₃×#20537 #15706876

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1771	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1770	C₂	1	2	yes	maximal, complemented
6	C₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1750	C₂	1	4	yes	maximal
8	C₂×C₁₃×#117	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#432	C₄	1	8	yes	complemented
11	C₁₃×#431	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×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×#97	C₂²	1	4	yes	—
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#384	C₄	1	2	yes	—
22	C₁₃×#382	C₄	1	2	yes	—
23	C₁₃×#380	C₄	1	2	yes	—
24	C₁₃×#380	C₄	1	2	yes	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₁₃×C₄⋊C₄	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	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₁₃×#117	—	4	4	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
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×C₂²⋊C₈	C₂×C₄	1	8	yes	—
51	C₁₃×#97	—	4	4	no	—
52	C₁₃×#97	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₁₃	—	4	4	no	—
54	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
56	C₁₃×#95	C₂×C₄	1	4	yes	—
57	C₁₃×#95	C₂×C₄	1	4	yes	—
58	#20537	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	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	2	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	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	4	4	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₂²⋊C₄	—	4	4	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	C₄²	1	4	yes	—
91	C₄²×C₁₃	C₄²	1	4	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	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
98	#2285	C₂×C₁₃	1	1	yes	characteristic
99	#1771	C₂×C₁₃	1	4	yes	complemented
100	#1770	C₂×C₁₃	1	2	yes	complemented
101	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	complemented
102	#1750	C₂×C₁₃	1	4	yes	—
103	C₂³×C₁₃	—	2	1	no	—
104	C₂³×C₁₃	—	2	1	no	—
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	—	2	2	no	—
107	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
108	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	8	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
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	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₁₃	#131	1	4	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
130	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
131	C₈×C₁₃	—	2	4	no	—
132	C₈×C₁₃	—	2	8	no	—
133	C₈×C₁₃	—	2	4	no	—
134	C₂×#117	C₂²×C₁₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
136	#432	C₄×C₁₃	1	8	yes	complemented
137	#431	C₄×C₁₃	1	8	yes	complemented
138	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
140	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
141	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
142	C₂×#97	C₂²×C₁₃	1	4	yes	—
143	C₂×#97	C₂²×C₁₃	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
145	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	#384	C₄×C₁₃	1	2	yes	—
148	#382	C₄×C₁₃	1	2	yes	—
149	#380	C₄×C₁₃	1	2	yes	—
150	#380	C₄×C₁₃	1	2	yes	—
151	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
152	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
153	#336	C₂²×C₁₃	1	4	yes	—
154	#336	C₂²×C₁₃	1	4	yes	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	4	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
161	C₂²×C₁₃	#518	1	1	yes	characteristic
162	C₂²×C₁₃	#519	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#131	1	2	yes	—
164	C₂²×C₁₃	#568	1	2	yes	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	4	8	no	—
170	C₄×C₁₃	#405	1	4	yes	—
171	C₄×C₁₃	#520	1	2	yes	—
172	C₄×C₁₃	#520	1	2	yes	—
173	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
177	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
178	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
179	#117	—	4	4	no	—
180	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
184	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
187	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
188	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
189	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
191	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
192	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
193	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
194	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
195	#97	—	4	4	no	—
196	#97	C₂×C₄×C₁₃	1	8	yes	—
197	C₄×C₈	—	4	4	no	—
198	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
199	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
200	#95	C₂×C₄×C₁₃	1	4	yes	—
201	#95	C₂×C₄×C₁₃	1	4	yes	—
202	C₂×C₁₃	—	4	2	no	—
203	C₂×C₁₃	#2274	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
205	C₂×C₁₃	#2873	1	1	yes	characteristic
206	C₂×C₁₃	#2981	1	2	yes	—
207	C₂×C₁₃	#2985	1	2	yes	—
208	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
214	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
231	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
232	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₄²×C₁₃	1	4	yes	—
240	C₄²	C₄²×C₁₃	1	4	yes	—
241	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
242	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
243	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
244	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
245	C₁₃	#20537	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
251	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
265	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
266	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
267	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
268	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
269	C₂×C₄	C₁₃×#131	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
272	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
273	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	2	8	no	—
276	C₈	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
283	C₂²	C₁₃×#518	1	1	yes	characteristic
284	C₂²	C₁₃×#519	1	1	yes	characteristic
285	C₂²	C₂×C₁₃×#131	1	2	yes	—
286	C₂²	C₁₃×#568	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	8	no	—
292	C₄	C₁₃×#405	1	4	yes	—
293	C₄	C₁₃×#520	1	2	yes	—
294	C₄	C₁₃×#520	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₁₃×#2274	1	1	yes	characteristic
297	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
298	C₂	C₁₃×#2873	1	1	yes	characteristic
299	C₂	C₁₃×#2981	1	2	yes	—
300	C₂	C₁₃×#2985	1	2	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×#1771	C₂	semidirect
5. C₁₃×#1770	C₂	semidirect
6. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
10. C₁₃×#432	C₄	semidirect
11. C₁₃×#431	C₄	semidirect
58. #20537	C₁₃	direct (N × H)
99. #1771	C₂×C₁₃	semidirect
100. #1770	C₂×C₁₃	semidirect
101. C₈×C₄⋊C₄	C₂×C₁₃	semidirect
136. #432	C₄×C₁₃	semidirect
137. #431	C₄×C₁₃	semidirect
245. C₁₃	#20537	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×#432	2
C₂×C₁₃×#431	2
C₁₃×#2285	2
C₁₃×#1771	2
C₁₃×#1770	2
C₈×C₁₃×C₄⋊C₄	2
C₁₃×#1750	2
#20537	13

Subgroups — C13×#20537 #15706876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20537 #15706876