Subgroups — C₁₃×#20470 #15706809

Order 3328.

958 conjugacy classes of subgroups.

The subgroup lattice has 318 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	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1767	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1750	C₂	1	4	yes	maximal
7	C₁₃×#1746	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#432	C₄	1	8	yes	complemented
12	C₁₃×#431	C₄	1	8	yes	complemented
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	4	yes	—
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
23	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
24	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
25	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
26	C₁₃×#376	C₂²	1	1	yes	characteristic
27	C₁₃×#376	C₂²	1	1	yes	characteristic
28	C₄³×C₁₃	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	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₁₃×#117	—	2	4	no	—
38	C₁₃×#117	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
55	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₁₃×#97	—	2	8	no	—
57	C₁₃×#97	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₃	—	2	8	no	—
59	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
60	C₁₃×#95	C₂×C₄	1	2	yes	—
61	C₁₃×#95	C₂×C₄	1	2	yes	—
62	C₁₃×#95	C₂×C₄	1	2	yes	—
63	C₁₃×#95	C₂×C₄	1	2	yes	—
64	#20470	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
71	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	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	8	yes	—
89	C₁₃×C₄⋊C₄	—	2	8	no	—
90	C₁₃×C₂²⋊C₄	—	2	8	no	—
91	C₄²×C₁₃	—	2	8	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	4	4	no	—
95	C₄²×C₁₃	C₄²	1	4	yes	—
96	C₄²×C₁₃	C₄²	1	4	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	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃	#41	1	4	yes	—
102	C₄²×C₁₃	#41	1	4	yes	—
103	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
104	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
105	C₄×#117	C₂×C₁₃	1	1	yes	characteristic
106	#1767	C₂×C₁₃	1	2	yes	complemented
107	C₄×C₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
108	#1750	C₂×C₁₃	1	4	yes	—
109	#1746	C₂×C₁₃	1	2	yes	complemented
110	#1726	C₂×C₁₃	1	2	yes	complemented
111	C₂³×C₁₃	—	2	2	no	—
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	4	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	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	8	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₁₃	#117	1	8	yes	—
136	C₂×C₄×C₁₃	#131	1	4	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₈×C₁₃	—	2	4	no	—
142	C₈×C₁₃	—	2	8	no	—
143	C₈×C₁₃	—	2	4	no	—
144	C₂×#117	C₂²×C₁₃	1	2	yes	—
145	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
146	#432	C₄×C₁₃	1	8	yes	complemented
147	#431	C₄×C₁₃	1	8	yes	complemented
148	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
151	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
153	C₂×#97	C₂²×C₁₃	1	4	yes	—
154	C₂×#97	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
156	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
157	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
158	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
159	C₄×C₂²⋊C₄	C₄×C₁₃	1	2	yes	—
160	C₄×C₂²⋊C₄	C₄×C₁₃	1	2	yes	—
161	#376	C₂²×C₁₃	1	1	yes	characteristic
162	#376	C₂²×C₁₃	1	1	yes	characteristic
163	C₄³	C₂²×C₁₃	1	2	yes	—
164	#336	C₂²×C₁₃	1	4	yes	—
165	#336	C₂²×C₁₃	1	4	yes	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	4	4	no	—
170	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×#117	1	2	yes	—
172	C₂²×C₁₃	C₂×#131	1	2	yes	—
173	C₂²×C₁₃	#568	1	2	yes	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	2	8	no	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	#405	1	4	yes	—
181	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
182	C₄×C₁₃	C₄×#41	1	2	yes	—
183	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	#117	—	2	4	no	—
191	#117	—	2	4	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
194	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
199	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
204	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
205	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
206	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
207	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
208	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
209	#97	—	2	8	no	—
210	#97	C₂×C₄×C₁₃	1	8	yes	—
211	C₄×C₈	—	2	8	no	—
212	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
213	#95	C₂×C₄×C₁₃	1	2	yes	—
214	#95	C₂×C₄×C₁₃	1	2	yes	—
215	#95	C₂×C₄×C₁₃	1	2	yes	—
216	#95	C₂×C₄×C₁₃	1	2	yes	—
217	C₂×C₁₃	—	4	2	no	—
218	C₂×C₁₃	C₄×#117	1	1	yes	characteristic
219	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
220	C₂×C₁₃	#2873	1	1	yes	characteristic
221	C₂×C₁₃	#2925	1	4	yes	—
222	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
228	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
245	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
246	C₄⋊C₄	—	2	8	no	—
247	C₂²⋊C₄	—	2	8	no	—
248	C₄²	—	2	8	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	4	no	—
252	C₄²	C₄²×C₁₃	1	4	yes	—
253	C₄²	C₄²×C₁₃	1	4	yes	—
254	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
255	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
256	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
257	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
258	C₄²	C₁₃×#41	1	4	yes	—
259	C₄²	C₁₃×#41	1	4	yes	—
260	C₁₃	#20470	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
265	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
281	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
282	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
283	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
284	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
285	C₂×C₄	C₁₃×#117	1	8	yes	—
286	C₂×C₄	C₁₃×#131	1	4	yes	—
287	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	8	no	—
293	C₈	—	2	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	4	no	—
298	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
299	C₂²	C₂×C₁₃×#117	1	2	yes	—
300	C₂²	C₂×C₁₃×#131	1	2	yes	—
301	C₂²	C₁₃×#568	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	8	no	—
307	C₄	—	4	4	no	—
308	C₄	C₁₃×#405	1	4	yes	—
309	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
310	C₄	C₄×C₁₃×#41	1	2	yes	—
311	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
312	C₂	—	4	2	no	—
313	C₂	C₄×C₁₃×#117	1	1	yes	characteristic
314	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
315	C₂	C₁₃×#2873	1	1	yes	characteristic
316	C₂	C₁₃×#2925	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×#1767	C₂	semidirect
5. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
7. C₁₃×#1746	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
11. C₁₃×#432	C₄	semidirect
12. C₁₃×#431	C₄	semidirect
64. #20470	C₁₃	direct (N × H)
106. #1767	C₂×C₁₃	semidirect
107. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
109. #1746	C₂×C₁₃	semidirect
110. #1726	C₂×C₁₃	semidirect
146. #432	C₄×C₁₃	semidirect
147. #431	C₄×C₁₃	semidirect
260. C₁₃	#20470	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
C₄×C₁₃×#117	2
C₁₃×#1767	2
C₄×C₁₃×C₄⋊C₈	2
C₁₃×#1750	2
C₁₃×#1746	2
C₁₃×#1726	2
#20470	13

Subgroups — C13×#20470 #15706809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20470 #15706809