Subgroups — C₁₃×#17689 #15704028

Order 3328.

876 conjugacy classes of subgroups.

The subgroup lattice has 338 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
2	C₁₃×#2970	C₂	1	2	yes	maximal
3	C₁₃×#1537	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1534	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1531	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1528	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1526	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#330	C₂	1	2	yes	maximal, complemented
9	C₁₃×#557	—	2	2	no	—
10	C₁₃×#554	—	2	2	no	—
11	C₁₃×#554	—	2	4	no	—
12	C₁₃×#544	C₄	1	4	yes	—
13	C₁₃×#544	C₄	1	2	yes	—
14	C₁₃×#533	C₂²	1	1	yes	characteristic
15	C₁₃×#533	C₂²	1	2	yes	—
16	C₂×C₁₃×#125	C₂²	1	2	yes	—
17	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
18	C₁₃×#520	C₄	1	2	yes	—
19	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
20	C₁₃×#446	—	2	2	no	—
21	C₁₃×#435	—	2	4	no	—
22	C₈×D₄×C₁₃	—	2	2	no	—
23	C₁₃×#432	C₂²	1	2	yes	—
24	C₁₃×#432	C₂²	1	4	yes	—
25	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
26	C₂×C₁₃×#97	C₂²	1	4	yes	—
27	C₁₃×#330	C₂²	1	1	yes	characteristic
28	C₁₃×#330	C₂²	1	4	yes	—
29	C₁₃×#330	C₂²	1	2	yes	—
30	C₁₃×#330	C₂²	1	2	yes	—
31	C₁₃×#330	C₂²	1	1	yes	characteristic
32	C₁₃×#330	C₂²	1	4	yes	—
33	C₁₃×#330	C₂²	1	2	yes	—
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×M₁₆	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	—	2	4	no	—
38	C₁₃×C₄⋊Q₈	—	2	2	no	—
39	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
41	C₁₃×#125	—	2	4	no	—
42	C₁₃×#125	—	2	2	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#125	—	2	4	no	—
45	C₁₃×#125	C₂×C₄	1	4	yes	—
46	C₁₃×#125	C₂×C₄	1	2	yes	—
47	C₁₃×#125	C₂³	1	1	yes	characteristic
48	C₁₃×#125	C₂³	1	2	yes	—
49	C₁₃×#125	C₂³	1	1	yes	characteristic
50	C₁₃×C₂²⋊Q₈	—	2	4	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	—	2	4	no	—
53	C₁₃×C₂²⋊Q₈	—	2	4	no	—
54	C₁₃×C₂²⋊Q₈	—	2	4	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×Q₈×C₁₃	—	2	4	no	—
58	C₄×Q₈×C₁₃	D₄	1	2	yes	—
59	C₄×D₄×C₁₃	—	2	4	no	—
60	C₄×D₄×C₁₃	—	2	2	no	—
61	C₄×D₄×C₁₃	D₄	1	2	yes	—
62	C₄×D₄×C₁₃	D₄	1	4	yes	—
63	C₁₃×#117	C₂×C₄	1	2	yes	—
64	C₁₃×#117	C₂×C₄	1	4	yes	—
65	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
66	C₁₃×#117	C₂×C₄	1	2	yes	—
67	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
70	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	C₁₃×C₄⋊C₈	—	2	2	no	—
74	C₁₃×C₄⋊C₈	—	2	4	no	—
75	C₁₃×C₂²⋊C₈	—	2	4	no	—
76	C₁₃×C₂²⋊C₈	—	2	2	no	—
77	C₁₃×#97	—	2	4	no	—
78	C₁₃×#97	C₂³	1	2	yes	—
79	C₁₃×#97	C₂³	1	4	yes	—
80	C₁₃×#97	C₂³	1	2	yes	—
81	C₄×C₈×C₁₃	—	2	2	no	—
82	#17689	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₁₃×#41	—	2	4	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	C₂×Q₈×C₁₃	—	2	4	no	—
86	C₂×Q₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×D₄×C₁₃	C₂²⋊C₄	1	2	yes	—
88	C₂×D₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₃	—	2	4	no	—
91	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₁₃×M₁₆	—	4	4	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	4	no	—
99	C₂×C₈×C₁₃	—	4	8	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₁₃×C₄⋊C₄	—	2	4	no	—
102	C₁₃×C₄⋊C₄	—	2	4	no	—
103	C₁₃×C₄⋊C₄	—	2	4	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	4	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	2	4	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	4	yes	—
111	C₁₃×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
112	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
114	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
115	C₁₃×C₂²⋊C₄	—	2	2	no	—
116	C₁₃×C₂²⋊C₄	—	2	4	no	—
117	C₁₃×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
118	C₁₃×C₂²⋊C₄	C₂²⋊C₄	1	4	yes	—
119	C₄²×C₁₃	—	2	4	no	—
120	C₄²×C₁₃	—	2	2	no	—
121	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
122	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₁₃	C₂×D₄	1	2	yes	—
124	C₄²×C₁₃	C₂×D₄	1	4	yes	—
125	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
126	#3504	C₂×C₁₃	1	1	yes	characteristic
127	#2970	C₂×C₁₃	1	2	yes	—
128	#1537	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#1534	C₂×C₁₃	1	2	yes	complemented
130	#1531	C₂×C₁₃	1	4	yes	complemented
131	#1528	C₂×C₁₃	1	2	yes	complemented
132	#1526	C₂×C₁₃	1	1	yes	characteristic, complemented
133	C₂×#330	C₂×C₁₃	1	2	yes	complemented
134	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
136	Q₈×C₁₃	—	2	2	no	—
137	Q₈×C₁₃	—	8	4	no	—
138	Q₈×C₁₃	—	8	2	no	—
139	D₄×C₁₃	—	2	4	no	—
140	D₄×C₁₃	#101	1	4	yes	—
141	C₂×C₄×C₁₃	—	2	4	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	4	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	4	4	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	8	4	no	—
149	C₂×C₄×C₁₃	—	8	2	no	—
150	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
153	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
156	C₈×C₁₃	—	2	4	no	—
157	C₈×C₁₃	—	4	4	no	—
158	C₈×C₁₃	—	4	2	no	—
159	#557	—	2	2	no	—
160	#554	—	2	2	no	—
161	#554	—	2	4	no	—
162	#544	C₄×C₁₃	1	4	yes	—
163	#544	C₄×C₁₃	1	2	yes	—
164	#533	C₂²×C₁₃	1	1	yes	characteristic
165	#533	C₂²×C₁₃	1	2	yes	—
166	C₂×#125	C₂²×C₁₃	1	2	yes	—
167	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
168	#520	C₄×C₁₃	1	2	yes	—
169	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
170	#446	—	2	2	no	—
171	#435	—	2	4	no	—
172	C₈×D₄	—	2	2	no	—
173	#432	C₂²×C₁₃	1	2	yes	—
174	#432	C₂²×C₁₃	1	4	yes	—
175	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
176	C₂×#97	C₂²×C₁₃	1	4	yes	—
177	#330	C₂²×C₁₃	1	2	yes	—
178	#330	C₂²×C₁₃	1	2	yes	—
179	#330	C₂²×C₁₃	1	1	yes	characteristic
180	#330	C₂²×C₁₃	1	4	yes	—
181	#330	C₂²×C₁₃	1	2	yes	—
182	#330	C₂²×C₁₃	1	1	yes	characteristic
183	#330	C₂²×C₁₃	1	4	yes	—
184	C₂²×C₁₃	—	2	4	no	—
185	C₂²×C₁₃	—	2	1	no	—
186	C₂²×C₁₃	—	2	2	no	—
187	C₂²×C₁₃	C₂×#101	1	4	yes	—
188	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	4	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	4	2	no	—
195	C₄×C₁₃	—	4	4	no	—
196	C₄×C₁₃	—	4	2	no	—
197	C₄×C₁₃	C₂×#101	1	2	yes	—
198	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
199	C₂×M₁₆	—	2	2	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
204	C₄⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
205	#125	—	2	4	no	—
206	#125	—	2	2	no	—
207	#125	—	2	2	no	—
208	#125	—	2	4	no	—
209	#125	C₂×C₄×C₁₃	1	4	yes	—
210	#125	C₂×C₄×C₁₃	1	2	yes	—
211	#125	C₂³×C₁₃	1	1	yes	characteristic
212	#125	C₂³×C₁₃	1	2	yes	—
213	#125	C₂³×C₁₃	1	1	yes	characteristic
214	C₂²⋊Q₈	—	2	4	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	2	4	no	—
217	C₂²⋊Q₈	—	2	4	no	—
218	C₂²⋊Q₈	—	2	4	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	4	no	—
222	C₄×Q₈	D₄×C₁₃	1	2	yes	—
223	C₄×D₄	—	2	4	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	D₄×C₁₃	1	2	yes	—
226	C₄×D₄	D₄×C₁₃	1	4	yes	—
227	#117	C₂×C₄×C₁₃	1	2	yes	—
228	#117	C₂×C₄×C₁₃	1	4	yes	—
229	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
230	#117	C₂×C₄×C₁₃	1	2	yes	—
231	C₂×C₄⋊C₄	—	2	4	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
234	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
235	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
236	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	4	no	—
239	C₂²⋊C₈	—	2	4	no	—
240	C₂²⋊C₈	—	2	2	no	—
241	#97	—	2	4	no	—
242	#97	C₂³×C₁₃	1	2	yes	—
243	#97	C₂³×C₁₃	1	4	yes	—
244	#97	C₂³×C₁₃	1	2	yes	—
245	C₄×C₈	—	2	2	no	—
246	C₂×C₁₃	—	2	4	no	—
247	C₂×C₁₃	—	4	1	no	—
248	C₂×C₁₃	C₂²×#101	1	1	yes	characteristic
249	C₂×C₁₃	#2901	1	2	yes	—
250	#41	—	2	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×D₄	C₁₃×C₂²⋊C₄	1	2	yes	—
255	C₂×D₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
261	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
262	M₁₆	—	4	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
278	C₄⋊C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
279	C₄⋊C₄	C₂²×C₄×C₁₃	1	2	yes	—
280	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
281	C₄⋊C₄	C₂²×C₄×C₁₃	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
285	C₂²⋊C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
289	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₁₃	1	2	yes	—
291	C₄²	C₂×D₄×C₁₃	1	4	yes	—
292	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
293	C₁₃	#17689	1	1	yes	characteristic, direct product factor, Sylow
294	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
295	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
296	Q₈	—	2	2	no	—
297	Q₈	—	8	4	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	2	4	no	—
300	D₄	C₁₃×#101	1	4	yes	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
311	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
312	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
313	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
316	C₈	—	2	4	no	—
317	C₈	—	4	4	no	—
318	C₈	—	4	2	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	C₂×C₁₃×#101	1	4	yes	—
323	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	2	no	—
332	C₄	C₂×C₁₃×#101	1	2	yes	—
333	C₂	—	2	4	no	—
334	C₂	—	4	1	no	—
335	C₂	C₂²×C₁₃×#101	1	1	yes	characteristic
336	C₂	C₁₃×#2901	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
3. C₁₃×#1537	C₂	semidirect
4. C₁₃×#1534	C₂	semidirect
5. C₁₃×#1531	C₂	semidirect
6. C₁₃×#1528	C₂	semidirect
7. C₁₃×#1526	C₂	semidirect
8. C₂×C₁₃×#330	C₂	semidirect
82. #17689	C₁₃	direct (N × H)
128. #1537	C₂×C₁₃	semidirect
129. #1534	C₂×C₁₃	semidirect
130. #1531	C₂×C₁₃	semidirect
131. #1528	C₂×C₁₃	semidirect
132. #1526	C₂×C₁₃	semidirect
133. C₂×#330	C₂×C₁₃	semidirect
293. C₁₃	#17689	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×#2970	2
C₁₃×#1537	2
C₁₃×#1534	2
C₁₃×#1531	2
C₁₃×#1528	2
C₁₃×#1526	2
C₂×C₁₃×#330	2
#17689	13

Subgroups — C13×#17689 #15704028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17689 #15704028