Subgroups — C₁₃×#10816 #15697155

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1742	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1740	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#339	C₂	1	2	yes	maximal
5	C₂×C₁₃×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×#117	C₂²	1	2	yes	—
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
12	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
13	C₁₃×#376	C₄	1	2	yes	complemented
14	C₄³×C₁₃	C₄	1	2	yes	complemented
15	C₁₃×#343	—	2	2	no	—
16	C₁₃×#342	—	2	2	no	—
17	C₁₃×#339	—	2	4	no	—
18	C₁₃×H(3,ℤ/4)	C₄	1	8	yes	complemented
19	C₂³×C₄×C₁₃	—	2	1	no	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₁₃×#117	—	2	2	no	—
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	—	2	2	no	—
31	C₁₃×#117	—	2	2	no	—
32	C₁₃×#117	—	2	2	no	—
33	C₁₃×#117	—	2	2	no	—
34	C₁₃×#117	C₂×C₄	1	4	yes	—
35	C₁₃×#117	C₂×C₄	1	4	yes	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
39	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃	—	2	1	no	—
43	C₂×C₄²×C₁₃	—	2	1	no	—
44	C₂×C₄²×C₁₃	—	2	1	no	—
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
53	C₁₃×#95	—	2	4	no	—
54	#10816	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	2	4	no	—
86	C₁₃×C₂²⋊C₄	—	2	4	no	—
87	C₁₃×C₂²⋊C₄	—	2	4	no	—
88	C₁₃×C₂²⋊C₄	—	4	2	no	—
89	C₄²×C₁₃	—	2	1	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	4	no	—
97	C₄²×C₁₃	—	4	8	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	C₄²×C₁₃	C₄²	1	4	yes	—
100	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
101	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
102	C₄×#117	C₂×C₁₃	1	1	yes	characteristic
103	#1742	C₂×C₁₃	1	1	yes	characteristic
104	#1740	C₂×C₁₃	1	1	yes	characteristic
105	C₂×#339	C₂×C₁₃	1	2	yes	—
106	C₂×H(3,ℤ/4)	C₂×C₁₃	1	2	yes	—
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₁₃	C₄×D₄	1	2	yes	—
110	C₂³×C₁₃	#121	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂³×C₁₃	#123	1	2	yes	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	8	4	no	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#117	1	2	yes	—
129	C₂×C₄×C₁₃	#117	1	2	yes	—
130	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	#123	1	2	yes	—
139	C₂×C₄×C₁₃	#124	1	2	yes	—
140	C₂×C₄×C₁₃	#125	1	2	yes	—
141	C₂×C₄×C₁₃	#126	1	2	yes	—
142	C₂×C₄×C₁₃	#126	1	2	yes	—
143	C₈×C₁₃	—	4	4	no	—
144	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
145	C₂×#117	C₂²×C₁₃	1	2	yes	—
146	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
148	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₄×C₂²⋊C₄	—	2	2	no	—
151	#376	C₄×C₁₃	1	2	yes	complemented
152	C₄³	C₄×C₁₃	1	2	yes	complemented
153	#343	—	2	2	no	—
154	#342	—	2	2	no	—
155	#339	—	2	4	no	—
156	H(3,ℤ/4)	C₄×C₁₃	1	8	yes	complemented
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	2	4	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	—	4	2	no	—
162	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₁₃	#380	1	1	yes	characteristic
164	C₂²×C₁₃	#381	1	1	yes	characteristic
165	C₂²×C₁₃	#387	1	2	yes	—
166	C₂²×C₁₃	#388	1	2	yes	—
167	C₄×C₁₃	—	2	4	no	—
168	C₄×C₁₃	—	4	4	no	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	—	4	4	no	—
171	C₄×C₁₃	—	8	4	no	—
172	C₄×C₁₃	#376	1	1	yes	characteristic
173	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₁₃	#382	1	1	yes	characteristic
175	C₄×C₁₃	#383	1	1	yes	characteristic
176	C₄×C₁₃	#387	1	2	yes	—
177	C₄×C₁₃	#388	1	2	yes	—
178	C₂³×C₄	—	2	1	no	—
179	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	C₂×C₄×C₁₃	1	4	yes	—
194	#117	C₂×C₄×C₁₃	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
207	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₁₃	1	2	yes	—
211	C₂×C₄²	Q₈×C₁₃	1	2	yes	—
212	#95	—	2	4	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	#1434	1	1	yes	characteristic
218	C₂×C₁₃	#1827	1	2	yes	—
219	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
234	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
235	C₂²×C₄	C₁₃×#41	1	2	yes	—
236	C₂²×C₄	C₁₃×#41	1	2	yes	—
237	C₂²×C₄	C₁₃×#41	1	2	yes	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	4	8	no	—
262	C₄²	—	4	2	no	—
263	C₄²	C₄²×C₁₃	1	4	yes	—
264	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
265	C₄²	C₁₃×C₄⋊C₄	1	2	yes	—
266	C₁₃	#10816	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	4	2	no	—
268	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
269	C₂³	C₄×D₄×C₁₃	1	2	yes	—
270	C₂³	C₁₃×#121	1	1	yes	characteristic
271	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
272	C₂³	C₁₃×#123	1	2	yes	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
289	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
290	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
291	C₂×C₄	C₁₃×#117	1	2	yes	—
292	C₂×C₄	C₁₃×#117	1	2	yes	—
293	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
294	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
295	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
296	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
297	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#123	1	2	yes	—
299	C₂×C₄	C₁₃×#124	1	2	yes	—
300	C₂×C₄	C₁₃×#125	1	2	yes	—
301	C₂×C₄	C₁₃×#126	1	2	yes	—
302	C₂×C₄	C₁₃×#126	1	2	yes	—
303	C₈	—	4	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
310	C₂²	C₁₃×#380	1	1	yes	characteristic
311	C₂²	C₁₃×#381	1	1	yes	characteristic
312	C₂²	C₁₃×#387	1	2	yes	—
313	C₂²	C₁₃×#388	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	4	no	—
319	C₄	C₁₃×#376	1	1	yes	characteristic
320	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
321	C₄	C₁₃×#382	1	1	yes	characteristic
322	C₄	C₁₃×#383	1	1	yes	characteristic
323	C₄	C₁₃×#387	1	2	yes	—
324	C₄	C₁₃×#388	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	C₁₃×#1434	1	1	yes	characteristic
330	C₂	C₁₃×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	#381
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
13. C₁₃×#376	C₄	semidirect
14. C₄³×C₁₃	C₄	semidirect
18. C₁₃×H(3,ℤ/4)	C₄	semidirect
54. #10816	C₁₃	direct (N × H)
151. #376	C₄×C₁₃	semidirect
152. C₄³	C₄×C₁₃	semidirect
156. H(3,ℤ/4)	C₄×C₁₃	semidirect
266. C₁₃	#10816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#117	2
C₁₃×#1742	2
C₁₃×#1740	2
C₂×C₁₃×#339	2
C₂×C₁₃×H(3,ℤ/4)	2
#10816	13

Subgroups — C13×#10816 #15697155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10816 #15697155