Subgroups — C₁₃×#20552 #15706891

Order 3328.

890 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#433	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2298	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1771	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1768	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1753	C₂	1	2	yes	maximal
9	C₈×C₁₃×C₂²⋊C₄	C₂	1	2	yes	maximal
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
13	C₁₃×#433	C₄	1	8	yes	complemented
14	C₁₃×#424	C₄	1	8	yes	complemented
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#388	C₄	1	4	yes	—
26	C₁₃×#387	C₄	1	4	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#376	C₂²	1	1	yes	characteristic
30	C₄³×C₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×#336	C₂²	1	2	yes	—
32	C₁₃×#336	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	—	4	2	no	—
37	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₁₃	—	4	2	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×#117	C₂×C₄	1	4	yes	—
45	C₁₃×#117	C₂×C₄	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
56	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
57	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
58	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
59	C₁₃×#97	—	4	2	no	—
60	C₁₃×#97	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₁₃	—	4	2	no	—
62	C₄×C₈×C₁₃	—	4	2	no	—
63	C₄×C₈×C₁₃	—	4	2	no	—
64	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
65	C₁₃×#95	C₂×C₄	1	4	yes	—
66	C₁₃×#95	C₂×C₄	1	4	yes	—
67	C₁₃×#95	C₂×C₄	1	4	yes	—
68	#20552	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
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	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×M₁₆	—	4	4	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	4	4	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
94	C₁₃×C₄⋊C₄	—	2	8	no	—
95	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
96	C₁₃×C₂²⋊C₄	—	2	8	no	—
97	C₁₃×C₂²⋊C₄	C₄²	1	8	yes	—
98	C₄²×C₁₃	—	2	8	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	2	2	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
107	C₂×#433	C₂×C₁₃	1	1	yes	characteristic
108	C₂×#424	C₂×C₁₃	1	1	yes	characteristic
109	#2298	C₂×C₁₃	1	1	yes	characteristic
110	#1771	C₂×C₁₃	1	2	yes	complemented
111	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	complemented
112	#1768	C₂×C₁₃	1	2	yes	complemented
113	C₄×C₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
114	#1753	C₂×C₁₃	1	2	yes	—
115	C₈×C₂²⋊C₄	C₂×C₁₃	1	2	yes	—
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	2	4	no	—
118	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	—	2	4	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	4	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
133	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₁₃	#131	1	8	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
139	C₈×C₁₃	—	2	4	no	—
140	C₈×C₁₃	—	2	4	no	—
141	C₈×C₁₃	—	2	4	no	—
142	C₈×C₁₃	—	2	4	no	—
143	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
146	#433	C₄×C₁₃	1	8	yes	complemented
147	#424	C₄×C₁₃	1	8	yes	complemented
148	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
151	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₂×#97	C₂²×C₁₃	1	2	yes	—
153	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	#388	C₄×C₁₃	1	4	yes	—
159	#387	C₄×C₁₃	1	4	yes	—
160	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
161	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
162	#376	C₂²×C₁₃	1	1	yes	characteristic
163	C₄³	C₂²×C₁₃	1	1	yes	characteristic
164	#336	C₂²×C₁₃	1	2	yes	—
165	#336	C₂²×C₁₃	1	2	yes	—
166	#336	C₂²×C₁₃	1	2	yes	—
167	#336	C₂²×C₁₃	1	2	yes	—
168	C₂²×C₁₃	—	2	4	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
172	C₂²×C₁₃	#520	1	2	yes	—
173	C₂²×C₁₃	C₂×#131	1	4	yes	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	2	4	no	—
176	C₄×C₁₃	—	2	8	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	#518	1	1	yes	characteristic
180	C₄×C₁₃	#519	1	1	yes	characteristic
181	C₄×C₁₃	#520	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×M₁₆	—	4	2	no	—
184	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
185	C₂²×C₈	—	4	2	no	—
186	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
191	#117	C₂×C₄×C₁₃	1	4	yes	—
192	#117	C₂×C₄×C₁₃	1	4	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
198	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
203	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
204	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
205	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
206	#97	—	4	2	no	—
207	#97	C₂×C₄×C₁₃	1	4	yes	—
208	C₄×C₈	—	4	2	no	—
209	C₄×C₈	—	4	2	no	—
210	C₄×C₈	—	4	2	no	—
211	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
212	#95	C₂×C₄×C₁₃	1	4	yes	—
213	#95	C₂×C₄×C₁₃	1	4	yes	—
214	#95	C₂×C₄×C₁₃	1	4	yes	—
215	C₂×C₁₃	—	4	2	no	—
216	C₂×C₁₃	#2274	1	1	yes	characteristic
217	C₂×C₁₃	C₄×#131	1	2	yes	—
218	C₂×C₁₃	#2994	1	4	yes	—
219	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
223	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
229	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
242	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	C₄²×C₁₃	1	8	yes	—
246	C₂²⋊C₄	—	2	8	no	—
247	C₂²⋊C₄	C₄²×C₁₃	1	8	yes	—
248	C₄²	—	2	8	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	C₂²×C₄×C₁₃	1	2	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	C₁₃	#20552	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	—	2	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
261	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
274	C₂×C₄	C₂×C₄²×C₁₃	1	4	yes	—
275	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
276	C₂×C₄	C₁₃×#131	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	C₁₃×H(5,𝔽₂)	1	2	yes	—
280	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
289	C₂²	C₁₃×#520	1	2	yes	—
290	C₂²	C₂×C₁₃×#131	1	4	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	8	no	—
295	C₄	—	4	4	no	—
296	C₄	C₁₃×#518	1	1	yes	characteristic
297	C₄	C₁₃×#519	1	1	yes	characteristic
298	C₄	C₁₃×#520	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₁₃×#2274	1	1	yes	characteristic
301	C₂	C₄×C₁₃×#131	1	2	yes	—
302	C₂	C₁₃×#2994	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×#1771	C₂	semidirect
5. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
6. C₁₃×#1768	C₂	semidirect
7. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
13. C₁₃×#433	C₄	semidirect
14. C₁₃×#424	C₄	semidirect
68. #20552	C₁₃	direct (N × H)
110. #1771	C₂×C₁₃	semidirect
111. C₈×C₄⋊C₄	C₂×C₁₃	semidirect
112. #1768	C₂×C₁₃	semidirect
113. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
146. #433	C₄×C₁₃	semidirect
147. #424	C₄×C₁₃	semidirect
257. C₁₃	#20552	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×#433	2
C₂×C₁₃×#424	2
C₁₃×#2298	2
C₁₃×#1771	2
C₈×C₁₃×C₄⋊C₄	2
C₁₃×#1768	2
C₄×C₁₃×C₄⋊C₈	2
C₁₃×#1753	2
C₈×C₁₃×C₂²⋊C₄	2
#20552	13

Subgroups — C13×#20552 #15706891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20552 #15706891