Subgroups — C₁₃×#20558 #15706897

Order 3328.

1146 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃	C₂	1	2	yes	maximal
2	C₄²×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1770	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1752	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1749	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#336	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₁₃	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₁₃	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
13	C₈×D₄×C₁₃	C₄	1	16	yes	complemented
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₁₃×#97	C₂²	1	4	yes	—
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
22	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₄³×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#336	C₂²	1	1	yes	characteristic
25	C₁₃×#336	C₂²	1	2	yes	—
26	C₁₃×#336	C₂²	1	4	yes	—
27	C₁₃×#336	C₂²	1	1	yes	characteristic
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	—	4	4	no	—
33	C₂²×C₈×C₁₃	—	4	4	no	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₄×D₄×C₁₃	—	2	8	no	—
39	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	—	4	4	no	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
53	C₁₃×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
54	C₁₃×#97	—	4	2	no	—
55	C₁₃×#97	—	4	4	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	4	yes	—
59	#20558	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×D₄×C₁₃	C₄²	1	4	yes	—
62	C₂×D₄×C₁₃	M₁₆	1	8	yes	—
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	8	no	—
67	C₂²×C₄×C₁₃	—	2	8	no	—
68	C₂²×C₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	4	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	—	2	2	no	—
83	C₂×C₈×C₁₃	—	4	16	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
89	C₁₃×C₂²⋊C₄	—	2	8	no	—
90	C₁₃×C₂²⋊C₄	C₄²	1	8	yes	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	4	no	—
93	C₄²×C₁₃	—	2	1	no	—
94	C₄²×C₁₃	—	2	4	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	—	4	4	no	—
98	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×D₄	C₂×C₁₃	1	2	yes	—
100	C₄²×D₄	C₂×C₁₃	1	1	yes	characteristic
101	#1770	C₂×C₁₃	1	2	yes	complemented
102	#1767	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#1752	C₂×C₁₃	1	4	yes	complemented
104	#1749	C₂×C₁₃	1	2	yes	complemented
105	#1744	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₂×#336	C₂×C₁₃	1	2	yes	complemented
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	4	no	—
110	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
111	C₂³×C₁₃	C₂×M₁₆	1	8	yes	—
112	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	D₄×C₁₃	#97	1	16	yes	complemented
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	4	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	2	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	4	16	no	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
132	C₂×C₄×C₁₃	#131	1	4	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
134	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	2	8	no	—
139	C₈×C₁₃	—	2	4	no	—
140	C₂³×C₈	C₂²×C₁₃	1	4	yes	—
141	C₂×C₄×D₄	C₄×C₁₃	1	2	yes	—
142	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
144	C₈×D₄	C₄×C₁₃	1	16	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
146	C₂×C₂²⋊C₈	C₂²×C₁₃	1	4	yes	—
147	C₂×#97	C₂²×C₁₃	1	4	yes	—
148	C₂×#97	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
150	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
151	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×C₂²⋊C₄	C₄×C₁₃	1	4	yes	—
153	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
154	C₄³	C₂²×C₁₃	1	1	yes	characteristic
155	#336	C₂²×C₁₃	1	1	yes	characteristic
156	#336	C₂²×C₁₃	1	2	yes	—
157	#336	C₂²×C₁₃	1	4	yes	—
158	#336	C₂²×C₁₃	1	1	yes	characteristic
159	#336	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	8	no	—
162	C₂²×C₁₃	—	2	8	no	—
163	C₂²×C₁₃	—	2	4	no	—
164	C₂²×C₁₃	C₂×#97	1	8	yes	—
165	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
166	C₂²×C₁₃	#518	1	1	yes	characteristic
167	C₂²×C₁₃	#520	1	1	yes	characteristic
168	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
169	C₂²×C₁₃	C₂×#131	1	2	yes	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	2	4	no	—
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	—	4	2	no	—
175	C₄×C₁₃	C₂×#97	1	4	yes	—
176	C₄×C₁₃	#518	1	2	yes	—
177	C₄×C₁₃	#520	1	2	yes	—
178	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
180	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
181	C₂²×C₈	—	4	4	no	—
182	C₂²×C₈	—	4	4	no	—
183	C₂²×C₈	C₂×C₄×C₁₃	1	16	yes	complemented
184	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
186	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
187	C₄×D₄	—	2	8	no	—
188	C₄×D₄	C₂×C₄×C₁₃	1	8	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
193	C₂×C₄²	—	4	4	no	—
194	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
197	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	complemented
202	C₂²⋊C₈	C₂×C₄×C₁₃	1	16	yes	complemented
203	#97	—	4	2	no	—
204	#97	—	4	4	no	—
205	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	complemented
206	#95	C₂×C₄×C₁₃	1	2	yes	—
207	#95	C₂×C₄×C₁₃	1	4	yes	—
208	C₂×C₁₃	—	2	8	no	—
209	C₂×C₁₃	#2275	1	1	yes	characteristic
210	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
211	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
212	C₂×C₁₃	#2974	1	2	yes	—
213	C₂×C₁₃	#2990	1	2	yes	—
214	C₂⁴	C₂²×C₄×C₁₃	1	2	yes	—
215	C₂×D₄	C₄²×C₁₃	1	4	yes	—
216	C₂×D₄	C₁₃×M₁₆	1	8	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₂²×C₄×C₁₃	1	2	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	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	16	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	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₄²×C₁₃	1	4	yes	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₂²⋊C₄	C₄²×C₁₃	1	8	yes	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	4	4	no	—
252	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
253	C₁₃	#20558	1	1	yes	characteristic, direct product factor, Sylow
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	C₂×C₄²×C₁₃	1	4	yes	—
258	C₂³	C₂×C₁₃×M₁₆	1	8	yes	—
259	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
260	D₄	C₁₃×#97	1	16	yes	complemented
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	16	no	—
275	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
277	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
278	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
279	C₂×C₄	C₁₃×#131	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
281	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
283	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	8	no	—
286	C₈	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	8	no	—
291	C₂²	C₂×C₁₃×#97	1	8	yes	—
292	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
293	C₂²	C₁₃×#518	1	1	yes	characteristic
294	C₂²	C₁₃×#520	1	1	yes	characteristic
295	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
296	C₂²	C₂×C₁₃×#131	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	2	no	—
302	C₄	C₂×C₁₃×#97	1	4	yes	—
303	C₄	C₁₃×#518	1	2	yes	—
304	C₄	C₁₃×#520	1	2	yes	—
305	C₂	—	2	8	no	—
306	C₂	C₁₃×#2275	1	1	yes	characteristic
307	C₂	C₂²×C₁₃×#97	1	1	yes	characteristic
308	C₂	C₄×C₁₃×#131	1	1	yes	characteristic
309	C₂	C₁₃×#2974	1	2	yes	—
310	C₂	C₁₃×#2990	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	C₂²×D₄×C₁₃	C₂×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
3. C₁₃×#1770	C₂	semidirect
4. C₁₃×#1767	C₂	semidirect
5. C₁₃×#1752	C₂	semidirect
6. C₁₃×#1749	C₂	semidirect
7. C₁₃×#1744	C₂	semidirect
8. C₂×C₁₃×#336	C₂	semidirect
13. C₈×D₄×C₁₃	C₄	semidirect
34. C₂²×C₈×C₁₃	C₂×C₄	semidirect
52. C₁₃×C₄⋊C₈	C₂×C₄	semidirect
53. C₁₃×C₂²⋊C₈	C₂×C₄	semidirect
56. C₄×C₈×C₁₃	C₂×C₄	semidirect
59. #20558	C₁₃	direct (N × H)
101. #1770	C₂×C₁₃	semidirect
102. #1767	C₂×C₁₃	semidirect
103. #1752	C₂×C₁₃	semidirect
104. #1749	C₂×C₁₃	semidirect
105. #1744	C₂×C₁₃	semidirect
106. C₂×#336	C₂×C₁₃	semidirect
113. D₄×C₁₃	#97	semidirect
144. C₈×D₄	C₄×C₁₃	semidirect
183. C₂²×C₈	C₂×C₄×C₁₃	semidirect
201. C₄⋊C₈	C₂×C₄×C₁₃	semidirect
202. C₂²⋊C₈	C₂×C₄×C₁₃	semidirect
205. C₄×C₈	C₂×C₄×C₁₃	semidirect
253. C₁₃	#20558	direct (N × H)
260. D₄	C₁₃×#97	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×D₄×C₁₃	2
C₄²×D₄×C₁₃	2
C₁₃×#1770	2
C₁₃×#1767	2
C₁₃×#1752	2
C₁₃×#1749	2
C₁₃×#1744	2
C₂×C₁₃×#336	2
#20558	13

Subgroups — C13×#20558 #15706897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20558 #15706897