Subgroups — C₁₃×#20469 #15706808

Order 3328.

1002 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×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1768	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1749	C₂	1	2	yes	maximal
6	C₄×C₁₃×C₂²⋊C₈	C₂	1	2	yes	maximal
7	C₁₃×#1746	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1726	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₁₃	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	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
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	C₁₃×#376	C₄	1	2	yes	—
23	C₁₃×#376	C₄	1	2	yes	—
24	C₁₃×#376	C₂²	1	1	yes	characteristic
25	C₁₃×#376	C₂²	1	1	yes	characteristic
26	C₁₃×#376	C₂²	1	2	yes	—
27	C₂×C₁₃×#95	C₄	1	2	yes	—
28	C₂×C₁₃×#95	C₄	1	2	yes	—
29	C₁₃×#336	C₂²	1	4	yes	—
30	C₁₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₃	—	2	8	no	—
39	C₂×C₄²×C₁₃	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	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
49	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
50	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
53	C₁₃×#97	—	2	8	no	—
54	C₁₃×#97	C₂×C₄	1	8	yes	complemented
55	C₄×C₈×C₁₃	—	2	8	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
57	C₁₃×#95	C₂×C₄	1	2	yes	—
58	C₁₃×#95	C₂×C₄	1	2	yes	—
59	C₁₃×#95	C₂×C₄	1	4	yes	—
60	C₁₃×#95	C₂×C₄	1	4	yes	—
61	C₁₃×#95	C₂×C₄	1	2	yes	—
62	C₁₃×#95	C₂×C₄	1	2	yes	—
63	#20469	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	8	no	—
68	C₂²×C₄×C₁₃	—	2	8	no	—
69	C₂²×C₄×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	—	4	4	no	—
71	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
72	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
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	—	2	4	no	—
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	8	yes	—
90	C₄²×C₁₃	—	2	8	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	C₄²	1	4	yes	—
93	C₄²×C₁₃	C₄²	1	4	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	#41	1	8	yes	—
99	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
100	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
101	C₂×#376	C₂×C₁₃	1	1	yes	characteristic
102	#1768	C₂×C₁₃	1	4	yes	complemented
103	#1749	C₂×C₁₃	1	2	yes	—
104	C₄×C₂²⋊C₈	C₂×C₁₃	1	2	yes	—
105	#1746	C₂×C₁₃	1	2	yes	complemented
106	#1726	C₂×C₁₃	1	2	yes	complemented
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂³×C₁₃	#131	1	4	yes	—
111	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	16	no	—
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	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	#117	1	8	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
136	C₈×C₁₃	—	2	4	no	—
137	C₈×C₁₃	—	2	8	no	—
138	C₈×C₁₃	—	2	4	no	—
139	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
141	#432	C₄×C₁₃	1	8	yes	complemented
142	#431	C₄×C₁₃	1	8	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
145	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
148	C₂×#97	C₂²×C₁₃	1	4	yes	—
149	C₂×#97	C₂²×C₁₃	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
152	#376	C₄×C₁₃	1	2	yes	—
153	#376	C₄×C₁₃	1	2	yes	—
154	#376	C₂²×C₁₃	1	1	yes	characteristic
155	#376	C₂²×C₁₃	1	1	yes	characteristic
156	#376	C₂²×C₁₃	1	2	yes	—
157	C₂×#95	C₄×C₁₃	1	2	yes	—
158	C₂×#95	C₄×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	C₂×#117	1	2	yes	—
167	C₂²×C₁₃	C₂×#131	1	2	yes	—
168	C₂²×C₁₃	#568	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	—	2	8	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	C₄×#41	1	2	yes	—
177	C₄×C₁₃	C₄×#41	1	2	yes	—
178	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
179	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂×C₄²	—	2	8	no	—
186	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
187	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
191	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
192	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
195	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	complemented
196	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	complemented
197	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
198	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
199	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
200	#97	—	2	8	no	—
201	#97	C₂×C₄×C₁₃	1	8	yes	complemented
202	C₄×C₈	—	2	8	no	—
203	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	complemented
204	#95	C₂×C₄×C₁₃	1	2	yes	—
205	#95	C₂×C₄×C₁₃	1	2	yes	—
206	#95	C₂×C₄×C₁₃	1	4	yes	—
207	#95	C₂×C₄×C₁₃	1	4	yes	—
208	#95	C₂×C₄×C₁₃	1	2	yes	—
209	#95	C₂×C₄×C₁₃	1	2	yes	—
210	C₂×C₁₃	—	2	4	no	—
211	C₂×C₁₃	C₄×#117	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
213	C₂×C₁₃	#2873	1	1	yes	characteristic
214	C₂×C₁₃	#2924	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
223	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
233	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
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	8	yes	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	C₄²×C₁₃	1	4	yes	—
244	C₄²	C₄²×C₁₃	1	4	yes	—
245	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
246	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
247	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
248	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
249	C₄²	C₁₃×#41	1	8	yes	—
250	C₁₃	#20469	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
254	C₂³	C₁₃×#131	1	4	yes	—
255	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	16	no	—
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	C₂×C₄²×C₁₃	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
276	C₂×C₄	C₁₃×#117	1	8	yes	—
277	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	8	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	C₂×C₁₃×#117	1	2	yes	—
289	C₂²	C₂×C₁₃×#131	1	2	yes	—
290	C₂²	C₁₃×#568	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	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	4	no	—
298	C₄	C₄×C₁₃×#41	1	2	yes	—
299	C₄	C₄×C₁₃×#41	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	C₄×C₁₃×#117	1	1	yes	characteristic
302	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
303	C₂	C₁₃×#2873	1	1	yes	characteristic
304	C₂	C₁₃×#2924	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×#1768	C₂	semidirect
7. C₁₃×#1746	C₂	semidirect
8. C₁₃×#1726	C₂	semidirect
11. C₁₃×#432	C₄	semidirect
12. C₁₃×#431	C₄	semidirect
48. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
49. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
54. C₁₃×#97	C₂×C₄	semidirect
56. C₄×C₈×C₁₃	C₂×C₄	semidirect
63. #20469	C₁₃	direct (N × H)
102. #1768	C₂×C₁₃	semidirect
105. #1746	C₂×C₁₃	semidirect
106. #1726	C₂×C₁₃	semidirect
141. #432	C₄×C₁₃	semidirect
142. #431	C₄×C₁₃	semidirect
195. C₄⋊C₈	C₂×C₄×C₁₃	semidirect
196. C₄⋊C₈	C₂×C₄×C₁₃	semidirect
201. #97	C₂×C₄×C₁₃	semidirect
203. C₄×C₈	C₂×C₄×C₁₃	semidirect
250. C₁₃	#20469	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
C₂×C₁₃×#376	2
C₁₃×#1768	2
C₁₃×#1749	2
C₄×C₁₃×C₂²⋊C₈	2
C₁₃×#1746	2
C₁₃×#1726	2
#20469	13

Subgroups — C13×#20469 #15706808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20469 #15706808