Subgroups — C₁₃×#60627 #15746966

Order 3328.

1998 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3499	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#3142	C₂	1	3	yes	maximal, complemented
4	C₁₃×#3139	C₂	1	3	yes	maximal, complemented
5	C₁₃×#3137	C₂	1	12	yes	maximal, complemented
6	C₁₃×#3135	C₂	1	3	yes	maximal, complemented
7	C₂×C₁₃×#491	C₂	1	3	yes	maximal, complemented
8	C₁₃×#2898	C₂	1	4	yes	maximal, complemented
9	C₁₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	2	yes	—
11	C₂×C₁₃×#131	C₂²	1	2	yes	—
12	C₁₃×#557	—	2	2	no	—
13	C₁₃×#554	—	2	6	no	—
14	D₄×Q₈×C₁₃	—	2	2	no	—
15	C₁₃×#547	—	2	6	no	—
16	C₁₃×#544	C₂²	1	6	yes	—
17	C₁₃×#536	C₂²	1	6	yes	—
18	C₁₃×#533	C₂²	1	3	yes	—
19	C₁₃×#532	C₂²	1	3	yes	—
20	C₂×C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
21	C₂×C₁₃×#125	C₂²	1	3	yes	—
22	C₁₃×#520	C₂²	1	2	yes	—
23	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
24	C₁₃×#491	C₂²	1	3	yes	—
25	C₁₃×#491	C₂²	1	12	yes	—
26	C₁₃×#491	C₂²	1	6	yes	—
27	C₁₃×#491	C₂²	1	3	yes	—
28	C₁₃×#490	C₂²	1	1	yes	characteristic
29	C₁₃×#490	C₂²	1	3	yes	—
30	C₁₃×#489	C₂²	1	12	yes	—
31	C₁₃×#488	C₂²	1	12	yes	—
32	C₁₃×#487	C₂²	1	12	yes	—
33	C₁₃×#419	C₂²	1	12	yes	—
34	C₁₃×#417	C₂²	1	12	yes	—
35	C₁₃×#416	C₂²	1	12	yes	—
36	C₂×C₁₃×#103	C₂²	1	12	yes	—
37	C₁₃×#405	C₂²	1	6	yes	—
38	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
39	C₂×C₁₃×#97	C₂²	1	3	yes	—
40	C₁₃×2₋¹⁺⁴	—	2	16	no	—
41	C₂×C₁₃×#41	—	2	6	no	—
42	C₂×C₁₃×#41	C₂³	1	6	yes	—
43	C₂×C₁₃×#41	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₁₃	—	2	2	no	—
45	C₂²×Q₈×C₁₃	C₂³	1	6	yes	—
46	C₁₃×#131	—	2	8	no	—
47	C₂×C₁₃×M₁₆	C₂³	1	6	yes	—
48	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
49	C₁₃×C₄⋊Q₈	—	2	6	no	—
50	C₁₃×C₄⋊Q₈	—	2	6	no	—
51	C₁₃×C₄⋊Q₈	C₂³	1	6	yes	—
52	C₁₃×C₄⋊Q₈	C₂³	1	3	yes	—
53	C₁₃×C₄⋊Q₈	C₂³	1	6	yes	—
54	C₁₃×#127	C₂³	1	1	yes	characteristic
55	C₁₃×#125	—	2	6	no	—
56	C₁₃×#125	—	2	6	no	—
57	C₁₃×#125	C₂³	1	6	yes	—
58	C₁₃×#125	C₂³	1	3	yes	—
59	C₁₃×#125	C₂³	1	1	yes	characteristic
60	C₁₃×#124	C₂³	1	6	yes	—
61	C₁₃×#123	—	2	12	no	—
62	C₁₃×C₂²⋊Q₈	—	2	6	no	—
63	C₁₃×C₂²⋊Q₈	—	2	12	no	—
64	C₁₃×C₂²⋊Q₈	—	2	12	no	—
65	C₁₃×#121	—	2	6	no	—
66	C₄×Q₈×C₁₃	—	2	2	no	—
67	C₄×Q₈×C₁₃	—	2	6	no	—
68	C₄×Q₈×C₁₃	D₄	1	2	yes	—
69	C₄×D₄×C₁₃	—	2	6	no	—
70	C₄×D₄×C₁₃	—	2	6	no	—
71	C₄×D₄×C₁₃	D₄	1	6	yes	—
72	C₁₃×#117	C₂³	1	6	yes	—
73	C₁₃×#117	C₂³	1	3	yes	—
74	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
75	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
76	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
77	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
78	C₁₃×#103	C₂³	1	12	yes	—
79	C₁₃×#103	C₂³	1	24	yes	—
80	C₁₃×#103	C₂³	1	12	yes	—
81	C₁₃×#102	C₂³	1	4	yes	—
82	C₁₃×#102	C₂³	1	12	yes	—
83	C₁₃×#97	C₂³	1	1	yes	characteristic
84	C₁₃×#97	C₂³	1	3	yes	—
85	C₁₃×#97	C₂³	1	6	yes	—
86	C₄×C₈×C₁₃	C₂³	1	6	yes	—
87	#60627	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₁₃×#41	—	2	24	no	—
89	C₁₃×#41	—	4	24	no	—
90	C₁₃×#41	#41	1	8	yes	—
91	C₂×Q₈×C₁₃	—	2	6	no	—
92	C₂×Q₈×C₁₃	—	2	6	no	—
93	C₂×Q₈×C₁₃	—	2	24	no	—
94	C₂×Q₈×C₁₃	—	4	8	no	—
95	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₁₃	C₂⁴	1	6	yes	—
97	C₂×D₄×C₁₃	—	2	6	no	—
98	C₂×D₄×C₁₃	—	2	2	no	—
99	C₂×D₄×C₁₃	C₂×D₄	1	3	yes	—
100	C₂×D₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂²×C₄×C₁₃	—	2	6	no	—
102	C₂²×C₄×C₁₃	—	2	6	no	—
103	C₂²×C₄×C₁₃	—	2	6	no	—
104	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
105	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
106	C₁₃×M₁₆	—	2	12	no	—
107	C₂×C₈×C₁₃	—	2	12	no	—
108	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
109	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
110	C₁₃×C₄⋊C₄	—	2	6	no	—
111	C₁₃×C₄⋊C₄	—	2	6	no	—
112	C₁₃×C₄⋊C₄	—	2	12	no	—
113	C₁₃×C₄⋊C₄	—	2	12	no	—
114	C₁₃×C₄⋊C₄	—	2	6	no	—
115	C₁₃×C₄⋊C₄	—	2	2	no	—
116	C₁₃×C₄⋊C₄	—	2	6	no	—
117	C₁₃×C₄⋊C₄	C₂×D₄	1	6	yes	—
118	C₁₃×C₄⋊C₄	C₂⁴	1	6	yes	—
119	C₁₃×C₄⋊C₄	C₂⁴	1	2	yes	—
120	C₁₃×C₂²⋊C₄	—	2	6	no	—
121	C₁₃×C₂²⋊C₄	—	2	12	no	—
122	C₁₃×C₂²⋊C₄	C₂×D₄	1	6	yes	—
123	C₄²×C₁₃	—	2	6	no	—
124	C₄²×C₁₃	C₂×D₄	1	6	yes	—
125	C₄²×C₁₃	C₂⁴	1	3	yes	—
126	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
127	#3504	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#3499	C₂×C₁₃	1	1	yes	characteristic
129	#3142	C₂×C₁₃	1	3	yes	complemented
130	#3139	C₂×C₁₃	1	3	yes	complemented
131	#3137	C₂×C₁₃	1	12	yes	complemented
132	#3135	C₂×C₁₃	1	3	yes	complemented
133	C₂×#491	C₂×C₁₃	1	3	yes	complemented
134	#2898	C₂×C₁₃	1	4	yes	complemented
135	#2875	C₂×C₁₃	1	1	yes	characteristic, complemented
136	C₂³×C₁₃	—	2	2	no	—
137	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
138	Q₈×C₁₃	—	2	12	no	—
139	Q₈×C₁₃	—	4	12	no	—
140	Q₈×C₁₃	—	8	6	no	—
141	Q₈×C₁₃	#124	1	4	yes	—
142	D₄×C₁₃	—	2	4	no	—
143	D₄×C₁₃	—	4	4	no	—
144	D₄×C₁₃	—	4	12	no	—
145	D₄×C₁₃	#124	1	12	yes	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	6	no	—
148	C₂×C₄×C₁₃	—	2	6	no	—
149	C₂×C₄×C₁₃	—	4	6	no	—
150	C₂×C₄×C₁₃	—	4	12	no	—
151	C₂×C₄×C₁₃	—	4	12	no	—
152	C₂×C₄×C₁₃	—	8	6	no	—
153	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
154	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
155	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
156	C₂×C₄×C₁₃	C₂×#41	1	12	yes	—
157	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	—	2	6	no	—
160	C₂×2₋¹⁺⁴	C₂²×C₁₃	1	2	yes	—
161	C₂×#131	C₂²×C₁₃	1	2	yes	—
162	#557	—	2	2	no	—
163	#554	—	2	6	no	—
164	D₄×Q₈	—	2	2	no	—
165	#547	—	2	6	no	—
166	#544	C₂²×C₁₃	1	6	yes	—
167	#536	C₂²×C₁₃	1	6	yes	—
168	#533	C₂²×C₁₃	1	3	yes	—
169	#532	C₂²×C₁₃	1	3	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₁₃	1	3	yes	—
171	C₂×#125	C₂²×C₁₃	1	3	yes	—
172	#520	C₂²×C₁₃	1	2	yes	—
173	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
174	#491	C₂²×C₁₃	1	3	yes	—
175	#491	C₂²×C₁₃	1	12	yes	—
176	#491	C₂²×C₁₃	1	6	yes	—
177	#491	C₂²×C₁₃	1	3	yes	—
178	#490	C₂²×C₁₃	1	1	yes	characteristic
179	#490	C₂²×C₁₃	1	3	yes	—
180	#489	C₂²×C₁₃	1	12	yes	—
181	#488	C₂²×C₁₃	1	12	yes	—
182	#487	C₂²×C₁₃	1	12	yes	—
183	#419	C₂²×C₁₃	1	12	yes	—
184	#417	C₂²×C₁₃	1	12	yes	—
185	#416	C₂²×C₁₃	1	12	yes	—
186	C₂×#103	C₂²×C₁₃	1	12	yes	—
187	#405	C₂²×C₁₃	1	6	yes	—
188	C₄×M₁₆	C₂²×C₁₃	1	3	yes	—
189	C₂×#97	C₂²×C₁₃	1	3	yes	—
190	C₂²×C₁₃	—	2	6	no	—
191	C₂²×C₁₃	—	4	4	no	—
192	C₂²×C₁₃	—	4	2	no	—
193	C₂²×C₁₃	C₂×#124	1	6	yes	—
194	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	2	6	no	—
197	C₄×C₁₃	—	4	6	no	—
198	C₄×C₁₃	—	4	2	no	—
199	C₄×C₁₃	—	4	6	no	—
200	C₄×C₁₃	C₂×#124	1	6	yes	—
201	C₄×C₁₃	C₂²×#41	1	2	yes	—
202	2₋¹⁺⁴	—	2	16	no	—
203	C₂×#41	—	2	6	no	—
204	C₂×#41	C₂³×C₁₃	1	6	yes	—
205	C₂×#41	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×Q₈	—	2	2	no	—
207	C₂²×Q₈	C₂³×C₁₃	1	6	yes	—
208	#131	—	2	8	no	—
209	C₂×M₁₆	C₂³×C₁₃	1	6	yes	—
210	C₂²×C₈	C₂³×C₁₃	1	6	yes	—
211	C₄⋊Q₈	—	2	6	no	—
212	C₄⋊Q₈	—	2	6	no	—
213	C₄⋊Q₈	C₂³×C₁₃	1	3	yes	—
214	C₄⋊Q₈	C₂³×C₁₃	1	6	yes	—
215	C₄⋊Q₈	C₂³×C₁₃	1	6	yes	—
216	#127	C₂³×C₁₃	1	1	yes	characteristic
217	#125	—	2	6	no	—
218	#125	—	2	6	no	—
219	#125	C₂³×C₁₃	1	3	yes	—
220	#125	C₂³×C₁₃	1	6	yes	—
221	#125	C₂³×C₁₃	1	1	yes	characteristic
222	#124	C₂³×C₁₃	1	6	yes	—
223	#123	—	2	12	no	—
224	C₂²⋊Q₈	—	2	6	no	—
225	C₂²⋊Q₈	—	2	12	no	—
226	C₂²⋊Q₈	—	2	12	no	—
227	#121	—	2	6	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	6	no	—
230	C₄×Q₈	D₄×C₁₃	1	2	yes	—
231	C₄×D₄	—	2	6	no	—
232	C₄×D₄	—	2	6	no	—
233	C₄×D₄	D₄×C₁₃	1	6	yes	—
234	#117	C₂³×C₁₃	1	6	yes	—
235	#117	C₂³×C₁₃	1	3	yes	—
236	C₂×C₄⋊C₄	—	2	6	no	—
237	C₂×C₄⋊C₄	—	2	6	no	—
238	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
239	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
240	#103	C₂³×C₁₃	1	12	yes	—
241	#103	C₂³×C₁₃	1	24	yes	—
242	#103	C₂³×C₁₃	1	12	yes	—
243	#102	C₂³×C₁₃	1	4	yes	—
244	#102	C₂³×C₁₃	1	12	yes	—
245	#97	C₂³×C₁₃	1	1	yes	characteristic
246	#97	C₂³×C₁₃	1	3	yes	—
247	#97	C₂³×C₁₃	1	6	yes	—
248	C₄×C₈	C₂³×C₁₃	1	6	yes	—
249	C₂×C₁₃	—	2	6	no	—
250	C₂×C₁₃	—	8	2	no	—
251	C₂×C₁₃	C₂²×#124	1	1	yes	characteristic
252	C₂×C₁₃	#3586	1	2	yes	—
253	#41	—	2	24	no	—
254	#41	—	4	24	no	—
255	#41	C₁₃×#41	1	8	yes	—
256	C₂×Q₈	—	2	6	no	—
257	C₂×Q₈	—	2	6	no	—
258	C₂×Q₈	—	2	24	no	—
259	C₂×Q₈	—	4	8	no	—
260	C₂×Q₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×Q₈	C₂⁴×C₁₃	1	6	yes	—
262	C₂×D₄	—	2	6	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	C₂×D₄×C₁₃	1	3	yes	—
265	C₂×D₄	C₂⁴×C₁₃	1	2	yes	—
266	C₂²×C₄	—	2	6	no	—
267	C₂²×C₄	—	2	6	no	—
268	C₂²×C₄	—	2	6	no	—
269	C₂²×C₄	C₂×D₄×C₁₃	1	6	yes	—
270	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
271	M₁₆	—	2	12	no	—
272	C₂×C₈	—	2	12	no	—
273	C₂×C₈	C₂⁴×C₁₃	1	6	yes	—
274	C₂×C₈	C₂⁴×C₁₃	1	2	yes	—
275	C₄⋊C₄	—	2	6	no	—
276	C₄⋊C₄	—	2	6	no	—
277	C₄⋊C₄	—	2	12	no	—
278	C₄⋊C₄	—	2	12	no	—
279	C₄⋊C₄	—	2	6	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	6	no	—
282	C₄⋊C₄	C₂×D₄×C₁₃	1	6	yes	—
283	C₄⋊C₄	C₂⁴×C₁₃	1	6	yes	—
284	C₄⋊C₄	C₂⁴×C₁₃	1	2	yes	—
285	C₂²⋊C₄	—	2	6	no	—
286	C₂²⋊C₄	—	2	12	no	—
287	C₂²⋊C₄	C₂×D₄×C₁₃	1	6	yes	—
288	C₄²	—	2	6	no	—
289	C₄²	C₂×D₄×C₁₃	1	6	yes	—
290	C₄²	C₂⁴×C₁₃	1	3	yes	—
291	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
292	C₁₃	#60627	1	1	yes	characteristic, direct product factor, Sylow
293	C₂³	—	2	2	no	—
294	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
295	Q₈	—	2	12	no	—
296	Q₈	—	4	12	no	—
297	Q₈	—	8	6	no	—
298	Q₈	C₁₃×#124	1	4	yes	—
299	D₄	—	2	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	12	no	—
302	D₄	C₁₃×#124	1	12	yes	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	6	no	—
305	C₂×C₄	—	2	6	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	12	no	—
309	C₂×C₄	—	8	6	no	—
310	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
311	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
312	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
313	C₂×C₄	C₂×C₁₃×#41	1	12	yes	—
314	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	6	no	—
317	C₂²	—	2	6	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₂×C₁₃×#124	1	6	yes	—
321	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	6	no	—
324	C₄	—	4	6	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂×C₁₃×#124	1	6	yes	—
328	C₄	C₂²×C₁₃×#41	1	2	yes	—
329	C₂	—	2	6	no	—
330	C₂	—	8	2	no	—
331	C₂	C₂²×C₁₃×#124	1	1	yes	characteristic
332	C₂	C₁₃×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₁₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3504	C₂	semidirect
3. C₁₃×#3142	C₂	semidirect
4. C₁₃×#3139	C₂	semidirect
5. C₁₃×#3137	C₂	semidirect
6. C₁₃×#3135	C₂	semidirect
7. C₂×C₁₃×#491	C₂	semidirect
8. C₁₃×#2898	C₂	semidirect
9. C₁₃×#2875	C₂	semidirect
87. #60627	C₁₃	direct (N × H)
127. #3504	C₂×C₁₃	semidirect
129. #3142	C₂×C₁₃	semidirect
130. #3139	C₂×C₁₃	semidirect
131. #3137	C₂×C₁₃	semidirect
132. #3135	C₂×C₁₃	semidirect
133. C₂×#491	C₂×C₁₃	semidirect
134. #2898	C₂×C₁₃	semidirect
135. #2875	C₂×C₁₃	semidirect
292. C₁₃	#60627	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3504	2
C₁₃×#3499	2
C₁₃×#3142	2
C₁₃×#3139	2
C₁₃×#3137	2
C₁₃×#3135	2
C₂×C₁₃×#491	2
C₁₃×#2898	2
C₁₃×#2875	2
#60627	13

Subgroups — C13×#60627 #15746966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#60627 #15746966