Subgroups — C₁₃×#10840 #15697179

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#339	C₂	1	4	yes	maximal
5	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
6	C₂×C₁₃×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#423	—	2	2	no	—
10	C₁₃×#423	C₄	1	4	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
12	C₁₃×#384	C₄	1	2	yes	complemented
13	C₁₃×#382	C₄	1	2	yes	complemented
14	C₁₃×#380	—	2	2	no	—
15	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
16	C₁₃×#376	—	2	2	no	—
17	C₁₃×#376	C₄	1	4	yes	complemented
18	C₂×C₁₃×#95	—	2	2	no	—
19	C₁₃×#339	—	2	8	no	—
20	C₂³×C₄×C₁₃	—	2	2	no	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₁₃×#117	—	2	4	no	—
29	C₁₃×#117	—	2	4	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₄²×C₁₃	—	2	2	no	—
35	C₂×C₄²×C₁₃	—	2	2	no	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃	D₄	1	4	yes	—
43	C₂×C₄²×C₁₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
47	C₁₃×C₄⋊C₈	—	2	8	no	—
48	C₁₃×#95	—	2	4	no	—
49	C₁₃×#95	—	2	8	no	—
50	#10840	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	4	4	no	—
57	C₂²×C₄×C₁₃	—	4	8	no	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₁₃×M₁₆	—	2	8	no	—
73	C₁₃×M₁₆	—	4	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	2	8	no	—
78	C₁₃×C₄⋊C₄	—	4	8	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₄²×C₁₃	—	2	4	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	4	2	no	—
84	C₄²×C₁₃	—	4	2	no	—
85	C₄²×C₁₃	C₄²	1	2	yes	—
86	C₄²×C₁₃	C₄²	1	2	yes	—
87	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
88	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
89	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
91	C₄²×C₁₃	C₄⋊C₄	1	8	yes	complemented
92	C₂×#423	C₂×C₁₃	1	1	yes	characteristic
93	#2285	C₂×C₁₃	1	1	yes	characteristic
94	C₂×#376	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#339	C₂×C₁₃	1	4	yes	—
96	C₂³×C₁₃	—	2	4	no	—
97	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₃	#121	1	2	yes	—
101	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	8	no	—
108	C₂×C₄×C₁₃	—	4	8	no	—
109	C₂×C₄×C₁₃	—	4	8	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	8	no	—
112	C₂×C₄×C₁₃	#95	1	4	yes	—
113	C₂×C₄×C₁₃	#95	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#117	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₁₃	#121	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#125	1	2	yes	—
129	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
132	C₈×C₁₃	—	4	4	no	—
133	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
134	C₂×#117	C₂²×C₁₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
136	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
137	#423	—	2	2	no	—
138	#423	C₄×C₁₃	1	4	yes	complemented
139	C₂×C₄⋊C₈	—	2	2	no	—
140	#384	C₄×C₁₃	1	2	yes	complemented
141	#382	C₄×C₁₃	1	2	yes	complemented
142	#380	—	2	2	no	—
143	C₄×C₄⋊C₄	—	2	2	no	—
144	#376	—	2	2	no	—
145	#376	C₄×C₁₃	1	4	yes	complemented
146	C₂×#95	—	2	2	no	—
147	#339	—	2	8	no	—
148	C₂²×C₁₃	—	2	4	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
153	C₂²×C₁₃	#380	1	2	yes	—
154	C₂²×C₁₃	#389	1	2	yes	—
155	C₂²×C₁₃	#390	1	1	yes	characteristic
156	C₂²×C₁₃	#391	1	1	yes	characteristic
157	C₂²×C₁₃	#421	1	4	yes	—
158	C₄×C₁₃	—	2	4	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	4	8	no	—
161	C₄×C₁₃	—	8	4	no	—
162	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₁₃	#382	1	1	yes	characteristic
164	C₄×C₁₃	#384	1	1	yes	characteristic
165	C₄×C₁₃	#389	1	2	yes	—
166	C₄×C₁₃	#390	1	1	yes	characteristic
167	C₄×C₁₃	#391	1	1	yes	characteristic
168	C₂³×C₄	—	2	2	no	—
169	C₂³×C₄	D₄×C₁₃	1	2	yes	—
170	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	4	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
175	C₂²×C₈	—	2	2	no	—
176	#117	—	2	4	no	—
177	#117	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
186	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
188	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₁₃	1	4	yes	—
191	C₂×C₄²	D₄×C₁₃	1	2	yes	—
192	C₂×C₄²	D₄×C₁₃	1	2	yes	—
193	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
194	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
195	C₄⋊C₈	—	2	8	no	—
196	#95	—	2	8	no	—
197	#95	—	2	4	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	2	2	no	—
200	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
201	C₂×C₁₃	#1733	1	2	yes	—
202	C₂×C₁₃	#1963	1	2	yes	—
203	C₂×C₁₃	#1995	1	2	yes	—
204	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
212	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
213	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
214	C₂²×C₄	C₁₃×C₄⋊C₄	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
221	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
223	C₂²×C₄	C₁₃×#41	1	2	yes	—
224	C₂²×C₄	C₁₃×#41	1	2	yes	—
225	M₁₆	—	2	8	no	—
226	M₁₆	—	4	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₄²×C₁₃	1	2	yes	—
239	C₄²	C₄²×C₁₃	1	2	yes	—
240	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
241	C₄²	C₁₃×C₄⋊C₄	1	4	yes	—
242	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
243	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
244	C₄²	C₁₃×C₄⋊C₄	1	8	yes	complemented
245	C₁₃	#10840	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	—	2	4	no	—
247	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₂³	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
250	C₂³	C₁₃×#121	1	2	yes	—
251	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	C₁₃×#95	1	4	yes	—
263	C₂×C₄	C₁₃×#95	1	4	yes	—
264	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
267	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₁₃×#117	1	2	yes	—
270	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
271	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
274	C₂×C₄	C₁₃×#121	1	2	yes	—
275	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
276	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
277	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
278	C₂×C₄	C₁₃×#125	1	2	yes	—
279	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
280	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
281	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	C₂×C₁₃×#95	1	1	yes	characteristic
288	C₂²	C₁₃×#380	1	2	yes	—
289	C₂²	C₁₃×#389	1	2	yes	—
290	C₂²	C₁₃×#390	1	1	yes	characteristic
291	C₂²	C₁₃×#391	1	1	yes	characteristic
292	C₂²	C₁₃×#421	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	8	4	no	—
297	C₄	C₄×C₁₃×C₄⋊C₄	1	2	yes	—
298	C₄	C₁₃×#382	1	1	yes	characteristic
299	C₄	C₁₃×#384	1	1	yes	characteristic
300	C₄	C₁₃×#389	1	2	yes	—
301	C₄	C₁₃×#390	1	1	yes	characteristic
302	C₄	C₁₃×#391	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	C₁₃×C₄⋊#95	1	1	yes	characteristic
306	C₂	C₁₃×#1733	1	2	yes	—
307	C₂	C₁₃×#1963	1	2	yes	—
308	C₂	C₁₃×#1995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. C₁₃×#423	C₄	semidirect
12. C₁₃×#384	C₄	semidirect
13. C₁₃×#382	C₄	semidirect
17. C₁₃×#376	C₄	semidirect
50. #10840	C₁₃	direct (N × H)
91. C₄²×C₁₃	C₄⋊C₄	semidirect
138. #423	C₄×C₁₃	semidirect
140. #384	C₄×C₁₃	semidirect
141. #382	C₄×C₁₃	semidirect
145. #376	C₄×C₁₃	semidirect
244. C₄²	C₁₃×C₄⋊C₄	semidirect
245. C₁₃	#10840	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×#423	2
C₁₃×#2285	2
C₂×C₁₃×#376	2
C₂×C₁₃×#339	2
#10840	13

Subgroups — C13×#10840 #15697179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10840 #15697179