Subgroups — C₁₃×#21821 #15708160

Order 3328.

1014 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1985	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1984	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1818	C₂	1	4	yes	maximal
7	C₁₃×#533	—	2	4	no	—
8	C₁₃×#533	—	2	4	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₁₃×#125	C₄	1	2	yes	—
12	C₂×C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₂×C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
16	C₂×C₁₃×#117	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#432	—	2	4	no	—
19	C₁₃×#431	—	2	4	no	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₁₃×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#336	C₂²	1	16	yes	—
27	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	2	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	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₁₃×C₄⋊Q₈	—	2	4	no	—
35	C₁₃×C₄⋊Q₈	—	2	4	no	—
36	C₁₃×#125	—	2	8	no	—
37	C₁₃×C₂²⋊Q₈	—	2	8	no	—
38	C₁₃×C₂²⋊Q₈	—	2	8	no	—
39	C₄×Q₈×C₁₃	—	2	4	no	—
40	C₄×Q₈×C₁₃	—	2	4	no	—
41	C₄×Q₈×C₁₃	—	2	4	no	—
42	C₄×Q₈×C₁₃	—	2	4	no	—
43	C₁₃×#117	—	2	4	no	—
44	C₁₃×#117	—	2	4	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	D₄	1	4	yes	—
55	C₂×C₄²×C₁₃	D₄	1	4	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×C₄⋊C₈	—	2	4	no	—
60	C₁₃×C₄⋊C₈	—	2	4	no	—
61	C₁₃×C₂²⋊C₈	—	2	4	no	—
62	C₁₃×C₂²⋊C₈	—	2	4	no	—
63	C₁₃×#97	—	2	4	no	—
64	C₄×C₈×C₁₃	—	2	4	no	—
65	#21821	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×Q₈×C₁₃	—	2	4	no	—
68	C₂×Q₈×C₁₃	—	2	4	no	—
69	C₂×Q₈×C₁₃	—	2	8	no	—
70	C₂×Q₈×C₁₃	—	4	4	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	8	no	—
89	C₂×C₈×C₁₃	—	2	8	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	8	no	—
93	C₁₃×C₄⋊C₄	—	2	8	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	2	4	no	—
96	C₁₃×C₂²⋊C₄	—	2	8	no	—
97	C₄²×C₁₃	—	2	4	no	—
98	C₄²×C₁₃	—	2	4	no	—
99	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
101	C₄²×C₁₃	C₂×D₄	1	2	yes	—
102	C₄²×C₁₃	C₂×D₄	1	2	yes	—
103	C₄²×C₁₃	C₂×D₄	1	2	yes	—
104	C₄²×C₁₃	C₂×D₄	1	2	yes	—
105	C₂×#533	C₂×C₁₃	1	1	yes	characteristic
106	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
107	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
108	#1985	C₂×C₁₃	1	4	yes	complemented
109	#1984	C₂×C₁₃	1	4	yes	complemented
110	#1818	C₂×C₁₃	1	4	yes	—
111	C₂³×C₁₃	—	2	2	no	—
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₁₃	—	4	8	no	—
117	Q₈×C₁₃	—	4	8	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	8	no	—
128	C₂×C₄×C₁₃	—	2	8	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₃	#131	1	4	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
139	C₈×C₁₃	—	4	4	no	—
140	C₈×C₁₃	—	4	4	no	—
141	#533	—	2	4	no	—
142	#533	—	2	4	no	—
143	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
145	C₂×#125	C₄×C₁₃	1	2	yes	—
146	C₂×C₂²⋊Q₈	C₄×C₁₃	1	2	yes	—
147	C₂×C₂²⋊Q₈	C₄×C₁₃	1	2	yes	—
148	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
150	C₂×#117	C₂²×C₁₃	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
152	#432	—	2	4	no	—
153	#431	—	2	4	no	—
154	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
156	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
158	C₂×#97	C₂²×C₁₃	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
160	#336	C₂²×C₁₃	1	16	yes	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	4	4	no	—
165	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₁₃	#519	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	—	4	4	no	—
174	C₄×C₁₃	—	4	4	no	—
175	C₄×C₁₃	#408	1	4	yes	—
176	C₄×C₁₃	#536	1	2	yes	—
177	C₄×C₁₃	#536	1	2	yes	—
178	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
179	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
180	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	—	2	4	no	—
187	#125	—	2	8	no	—
188	C₂²⋊Q₈	—	2	8	no	—
189	C₂²⋊Q₈	—	2	8	no	—
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	2	4	no	—
192	C₄×Q₈	—	2	4	no	—
193	C₄×Q₈	—	2	4	no	—
194	#117	—	2	4	no	—
195	#117	—	2	4	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
203	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
205	C₂×C₄²	D₄×C₁₃	1	4	yes	—
206	C₂×C₄²	D₄×C₁₃	1	4	yes	—
207	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
208	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	C₄⋊C₈	—	2	4	no	—
212	C₂²⋊C₈	—	2	4	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	#97	—	2	4	no	—
215	C₄×C₈	—	2	4	no	—
216	C₂×C₁₃	—	4	2	no	—
217	C₂×C₁₃	#2310	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#408	1	1	yes	characteristic
219	C₂×C₁₃	#2879	1	1	yes	characteristic
220	C₂×C₁₃	#2985	1	4	yes	—
221	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
230	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
240	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	8	no	—
244	C₂×C₈	—	2	8	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	8	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
255	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
256	C₄²	C₂×D₄×C₁₃	1	2	yes	—
257	C₄²	C₂×D₄×C₁₃	1	2	yes	—
258	C₄²	C₂×D₄×C₁₃	1	2	yes	—
259	C₄²	C₂×D₄×C₁₃	1	2	yes	—
260	C₁₃	#21821	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×C₁₃×C₂²⋊C₄	1	2	yes	—
265	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
266	Q₈	—	4	8	no	—
267	Q₈	—	4	8	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
281	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
285	C₂×C₄	C₁₃×#131	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	C₂×C₄	C₁₃×2₋¹⁺⁴	1	4	yes	—
289	C₈	—	4	4	no	—
290	C₈	—	4	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	4	no	—
295	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
296	C₂²	C₁₃×#519	1	2	yes	—
297	C₂²	C₂×C₁₃×#131	1	2	yes	—
298	C₂²	C₁₃×#568	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	C₁₃×#408	1	4	yes	—
306	C₄	C₁₃×#536	1	2	yes	—
307	C₄	C₁₃×#536	1	2	yes	—
308	C₂	—	4	2	no	—
309	C₂	C₁₃×#2310	1	1	yes	characteristic
310	C₂	C₂×C₁₃×#408	1	1	yes	characteristic
311	C₂	C₁₃×#2879	1	1	yes	characteristic
312	C₂	C₁₃×#2985	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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×#1985	C₂	semidirect
5. C₁₃×#1984	C₂	semidirect
65. #21821	C₁₃	direct (N × H)
108. #1985	C₂×C₁₃	semidirect
109. #1984	C₂×C₁₃	semidirect
260. C₁₃	#21821	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×#533	2
C₂×C₁₃×#432	2
C₂×C₁₃×#431	2
C₁₃×#1985	2
C₁₃×#1984	2
C₁₃×#1818	2
#21821	13

Subgroups — C13×#21821 #15708160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21821 #15708160