Subgroups — C₁₃×#10827 #15697166

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#343	C₂	1	2	yes	maximal
3	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1736	C₂	1	2	yes	maximal
5	C₂×C₁₃×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₁₃×#405	—	2	2	no	—
12	C₁₃×#405	C₄	1	4	yes	complemented
13	C₂×C₁₃×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	—	2	1	no	—
15	C₁₃×#343	C₄	1	8	yes	complemented
16	C₁₃×#341	—	2	2	no	—
17	C₁₃×#340	—	2	1	no	—
18	C₁₃×#340	—	2	1	no	—
19	C₁₃×#340	—	2	2	no	—
20	C₁₃×#340	—	2	2	no	—
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	—	2	1	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	2	yes	—
41	C₁₃×#117	—	2	1	no	—
42	C₁₃×#117	—	2	1	no	—
43	C₁₃×#117	C₂×C₄	1	2	yes	—
44	C₁₃×#117	Q₈	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
54	C₂×C₄²×C₁₃	—	2	1	no	—
55	C₁₃×#97	—	2	4	no	—
56	C₄×C₈×C₁₃	—	2	4	no	—
57	#10827	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	8	2	no	—
65	C₂²×C₄×C₁₃	—	8	2	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂²×C₄×C₁₃	#41	1	2	yes	—
75	C₂²×C₄×C₁₃	#41	1	2	yes	—
76	C₁₃×M₁₆	—	2	2	no	—
77	C₁₃×M₁₆	—	2	4	no	—
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	—	2	4	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₁₃×M₁₆	—	4	2	no	—
82	C₁₃×M₁₆	C₄²	1	4	yes	—
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	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	—	4	2	no	—
92	C₂×C₈×C₁₃	—	4	2	no	—
93	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
95	C₁₃×C₄⋊C₄	—	2	1	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	2	4	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₁₃×C₂²⋊C₄	—	2	2	no	—
104	C₁₃×C₂²⋊C₄	—	2	2	no	—
105	C₄²×C₁₃	—	2	2	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
108	C₂×#343	C₂×C₁₃	1	2	yes	—
109	C₂×#341	C₂×C₁₃	1	1	yes	characteristic
110	#1736	C₂×C₁₃	1	2	yes	—
111	C₂×#340	C₂×C₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	—	2	1	no	—
115	C₂³×C₁₃	#117	1	1	yes	characteristic
116	C₂³×C₁₃	#117	1	2	yes	—
117	C₂³×C₁₃	#121	1	1	yes	characteristic
118	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂³×C₁₃	#126	1	2	yes	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	—	8	2	no	—
129	C₂×C₄×C₁₃	—	8	4	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#117	1	2	yes	—
137	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#123	1	2	yes	—
144	C₂×C₄×C₁₃	#123	1	2	yes	—
145	C₂×C₄×C₁₃	#124	1	2	yes	—
146	C₂×C₄×C₁₃	#125	1	2	yes	—
147	C₂×C₄×C₁₃	#126	1	2	yes	—
148	C₈×C₁₃	—	2	4	no	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	2	2	no	—
151	C₈×C₁₃	—	4	4	no	—
152	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
153	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
157	#405	—	2	2	no	—
158	#405	C₄×C₁₃	1	4	yes	complemented
159	C₂×#97	—	2	1	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	#343	C₄×C₁₃	1	8	yes	complemented
162	#341	—	2	2	no	—
163	#340	—	2	1	no	—
164	#340	—	2	1	no	—
165	#340	—	2	2	no	—
166	#340	—	2	2	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	#376	1	1	yes	characteristic
176	C₂²×C₁₃	#380	1	1	yes	characteristic
177	C₂²×C₁₃	#383	1	1	yes	characteristic
178	C₂²×C₁₃	#387	1	2	yes	—
179	C₂²×C₁₃	#388	1	2	yes	—
180	C₂²×C₁₃	#471	1	2	yes	—
181	C₂²×C₁₃	#472	1	1	yes	characteristic
182	C₂²×C₁₃	#473	1	1	yes	characteristic
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	—	8	4	no	—
186	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
187	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₁₃	#381	1	1	yes	characteristic
189	C₄×C₁₃	#382	1	1	yes	characteristic
190	C₄×C₁₃	#387	1	2	yes	—
191	C₄×C₁₃	#388	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
202	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
203	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
209	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
210	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₈	D₄×C₁₃	1	2	yes	—
212	#117	—	2	1	no	—
213	#117	—	2	1	no	—
214	#117	C₂×C₄×C₁₃	1	2	yes	—
215	#117	Q₈×C₁₃	1	2	yes	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
222	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₄²	—	2	1	no	—
226	#97	—	2	4	no	—
227	C₄×C₈	—	2	4	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	2	2	no	—
231	C₂×C₁₃	#1434	1	1	yes	characteristic
232	C₂×C₁₃	#1815	1	1	yes	characteristic
233	C₂×C₁₃	#1816	1	1	yes	characteristic
234	C₂×C₁₃	#1930	1	2	yes	—
235	C₂×C₁₃	#1931	1	2	yes	—
236	C₂⁴	C₁₃×#41	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	8	2	no	—
243	C₂²×C₄	—	8	2	no	—
244	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
251	C₂²×C₄	C₁₃×#41	1	2	yes	—
252	C₂²×C₄	C₁₃×#41	1	2	yes	—
253	C₂²×C₄	C₁₃×#41	1	2	yes	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	2	4	no	—
258	M₁₆	—	4	2	no	—
259	M₁₆	—	4	2	no	—
260	M₁₆	C₄²×C₁₃	1	4	yes	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
272	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	C₁₃×C₄⋊C₄	1	2	yes	—
280	C₄⋊C₄	C₁₃×C₄⋊C₄	1	2	yes	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₁₃	#10827	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₁₃×#117	1	1	yes	characteristic
290	C₂³	C₁₃×#117	1	2	yes	—
291	C₂³	C₁₃×#121	1	1	yes	characteristic
292	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
293	C₂³	C₁₃×#126	1	2	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
308	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
309	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
310	C₂×C₄	C₁₃×#117	1	2	yes	—
311	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
312	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
313	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
314	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
315	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
316	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
317	C₂×C₄	C₁₃×#123	1	2	yes	—
318	C₂×C₄	C₁₃×#123	1	2	yes	—
319	C₂×C₄	C₁₃×#124	1	2	yes	—
320	C₂×C₄	C₁₃×#125	1	2	yes	—
321	C₂×C₄	C₁₃×#126	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	—	4	4	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	C₁₃×#376	1	1	yes	characteristic
335	C₂²	C₁₃×#380	1	1	yes	characteristic
336	C₂²	C₁₃×#383	1	1	yes	characteristic
337	C₂²	C₁₃×#387	1	2	yes	—
338	C₂²	C₁₃×#388	1	2	yes	—
339	C₂²	C₁₃×#471	1	2	yes	—
340	C₂²	C₁₃×#472	1	1	yes	characteristic
341	C₂²	C₁₃×#473	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	8	4	no	—
345	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
346	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
347	C₄	C₁₃×#381	1	1	yes	characteristic
348	C₄	C₁₃×#382	1	1	yes	characteristic
349	C₄	C₁₃×#387	1	2	yes	—
350	C₄	C₁₃×#388	1	2	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	2	no	—
353	C₂	—	2	1	no	—
354	C₂	C₁₃×#1434	1	1	yes	characteristic
355	C₂	C₁₃×#1815	1	1	yes	characteristic
356	C₂	C₁₃×#1816	1	1	yes	characteristic
357	C₂	C₁₃×#1930	1	2	yes	—
358	C₂	C₁₃×#1931	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	#121
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#405	C₄	semidirect
15. C₁₃×#343	C₄	semidirect
57. #10827	C₁₃	direct (N × H)
158. #405	C₄×C₁₃	semidirect
161. #343	C₄×C₁₃	semidirect
285. C₁₃	#10827	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×#405	2
C₂×C₁₃×#343	2
C₂×C₁₃×#341	2
C₁₃×#1736	2
C₂×C₁₃×#340	2
#10827	13

Subgroups — C13×#10827 #15697166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10827 #15697166