Subgroups — C₁₃×#20549 #15706888

Order 3328.

966 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×C₁₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1771	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1767	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1753	C₂	1	4	yes	maximal
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₁₃×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#432	C₄	1	8	yes	complemented
12	C₁₃×#423	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×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	4	yes	—
18	C₂×C₁₃×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
20	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
23	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₁₃×#376	C₄	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#336	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	4	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	C₂×C₁₃×M₁₆	—	4	4	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
36	C₁₃×#117	—	2	8	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
53	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
55	C₁₃×#97	—	4	4	no	—
56	C₁₃×#97	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₁₃	—	4	4	no	—
58	C₁₃×#95	C₂×C₄	1	4	yes	—
59	C₁₃×#95	C₂×C₄	1	4	yes	—
60	#20549	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
67	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×M₁₆	—	4	8	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	—	2	4	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
85	C₁₃×C₄⋊C₄	—	2	8	no	—
86	C₁₃×C₂²⋊C₄	—	2	8	no	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	4	2	no	—
93	C₄²×C₁₃	—	4	2	no	—
94	C₄²×C₁₃	C₄²	1	4	yes	—
95	C₄²×C₁₃	C₄²	1	4	yes	—
96	C₄²×C₁₃	M₁₆	1	8	yes	complemented
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	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
102	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
103	C₄×#117	C₂×C₁₃	1	1	yes	characteristic
104	#1771	C₂×C₁₃	1	4	yes	complemented
105	#1767	C₂×C₁₃	1	2	yes	complemented
106	C₄×C₄⋊C₈	C₂×C₁₃	1	2	yes	complemented
107	#1753	C₂×C₁₃	1	4	yes	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	—	2	2	no	—
111	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	4	no	—
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	4	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	4	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	4	4	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	C₂×M₁₆	1	8	yes	—
133	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
138	C₈×C₁₃	—	2	8	no	—
139	C₈×C₁₃	—	2	8	no	—
140	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
141	C₂×#117	C₄×C₁₃	1	2	yes	—
142	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
143	#432	C₄×C₁₃	1	8	yes	complemented
144	#423	C₄×C₁₃	1	8	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
147	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
148	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
149	C₂×#97	C₂²×C₁₃	1	4	yes	—
150	C₂×#97	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
152	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
153	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
154	C₄×C₂²⋊C₄	C₄×C₁₃	1	2	yes	—
155	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
156	#376	C₄×C₁₃	1	2	yes	—
157	C₄³	C₂²×C₁₃	1	1	yes	characteristic
158	C₄³	C₂²×C₁₃	1	1	yes	characteristic
159	#336	C₂²×C₁₃	1	4	yes	—
160	#336	C₂²×C₁₃	1	4	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×C₄²	1	1	yes	characteristic
166	C₂²×C₁₃	#520	1	2	yes	—
167	C₂²×C₁₃	C₂²×M₁₆	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	8	no	—
173	C₄×C₁₃	—	2	2	no	—
174	C₄×C₁₃	—	4	4	no	—
175	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
176	C₄×C₁₃	#518	1	1	yes	characteristic
177	C₄×C₁₃	#519	1	1	yes	characteristic
178	C₄×C₁₃	#520	1	2	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	C₂×M₁₆	—	4	4	no	—
183	C₂×M₁₆	C₂×C₄×C₁₃	1	8	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
185	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
186	#117	—	2	8	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
190	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
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	2	yes	—
197	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
198	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
202	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
203	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
204	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
205	#97	—	4	4	no	—
206	#97	C₂×C₄×C₁₃	1	8	yes	—
207	C₄×C₈	—	4	4	no	—
208	#95	C₂×C₄×C₁₃	1	4	yes	—
209	#95	C₂×C₄×C₁₃	1	4	yes	—
210	C₂×C₁₃	—	4	2	no	—
211	C₂×C₁₃	#2274	1	1	yes	characteristic
212	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
213	C₂×C₁₃	#2873	1	1	yes	characteristic
214	C₂×C₁₃	#2989	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
221	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
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	1	yes	characteristic
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	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
232	M₁₆	—	4	8	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
238	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	2	8	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₄²×C₁₃	1	4	yes	—
249	C₄²	C₄²×C₁₃	1	4	yes	—
250	C₄²	C₁₃×M₁₆	1	8	yes	complemented
251	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
252	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
253	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
254	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
255	C₁₃	#20549	1	1	yes	characteristic, direct product factor, Sylow
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
260	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
277	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
278	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
279	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
280	C₂×C₄	C₂×C₁₃×M₁₆	1	8	yes	—
281	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
282	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
283	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
285	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
286	C₈	—	2	8	no	—
287	C₈	—	2	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
293	C₂²	C₁₃×#520	1	2	yes	—
294	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
295	C₂²	C₁₃×#568	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	8	no	—
301	C₄	—	4	4	no	—
302	C₄	C₄×C₁₃×M₁₆	1	4	yes	—
303	C₄	C₁₃×#518	1	1	yes	characteristic
304	C₄	C₁₃×#519	1	1	yes	characteristic
305	C₄	C₁₃×#520	1	2	yes	—
306	C₂	—	4	2	no	—
307	C₂	C₁₃×#2274	1	1	yes	characteristic
308	C₂	C₂×C₄×C₁₃×M₁₆	1	1	yes	characteristic
309	C₂	C₁₃×#2873	1	1	yes	characteristic
310	C₂	C₁₃×#2989	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. C₁₃×#1767	C₂	semidirect
6. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
11. C₁₃×#432	C₄	semidirect
12. C₁₃×#423	C₄	semidirect
60. #20549	C₁₃	direct (N × H)
96. C₄²×C₁₃	M₁₆	semidirect
104. #1771	C₂×C₁₃	semidirect
105. #1767	C₂×C₁₃	semidirect
106. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
143. #432	C₄×C₁₃	semidirect
144. #423	C₄×C₁₃	semidirect
250. C₄²	C₁₃×M₁₆	semidirect
255. C₁₃	#20549	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×#423	2
C₄×C₁₃×#117	2
C₁₃×#1771	2
C₁₃×#1767	2
C₄×C₁₃×C₄⋊C₈	2
C₁₃×#1753	2
#20549	13

Subgroups — C13×#20549 #15706888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#20549 #15706888