Subgroups — C₁₃×#18410 #15704749

Order 3328.

906 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3229	C₂	1	2	yes	maximal
3	C₁₃×#2546	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1611	C₂	1	8	yes	maximal
5	C₁₃×#1503	C₂	1	2	yes	maximal
6	C₁₃×#1461	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₁₃	—	2	2	no	—
8	C₁₃×#557	—	2	2	no	—
9	C₁₃×#557	—	2	4	no	—
10	C₁₃×#544	—	2	1	no	—
11	C₁₃×#544	C₂²	1	4	yes	—
12	C₁₃×#533	—	2	2	no	—
13	C₁₃×#533	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
15	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#519	—	2	1	no	—
17	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#498	—	2	2	no	—
19	C₁₃×#487	—	2	2	no	—
20	C₁₃×#483	C₂²	1	8	yes	—
21	C₁₃×#475	—	2	4	no	—
22	C₁₃×#431	C₂²	1	2	yes	—
23	C₁₃×#398	C₂²	1	4	yes	—
24	C₁₃×#391	—	2	1	no	—
25	C₁₃×#389	—	2	2	no	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₁₃×#328	C₂²	1	8	yes	—
28	C₁₃×#324	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	—	2	2	no	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	—	2	4	no	—
34	C₁₃×C₄⋊Q₈	—	2	8	no	—
35	C₁₃×C₄⋊Q₈	—	2	2	no	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	4	4	no	—
38	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
39	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
40	C₁₃×#125	—	2	4	no	—
41	C₁₃×#125	—	2	4	no	—
42	C₁₃×#125	—	2	8	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×C₂²⋊Q₈	—	2	2	no	—
45	C₁₃×C₂²⋊Q₈	—	2	4	no	—
46	C₁₃×C₂²⋊Q₈	C₂³	1	4	yes	—
47	C₄×Q₈×C₁₃	—	2	4	no	—
48	C₄×Q₈×C₁₃	—	2	4	no	—
49	C₄×Q₈×C₁₃	—	2	4	no	—
50	C₄×Q₈×C₁₃	—	2	4	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	D₄	1	4	yes	—
55	C₁₃×#117	—	2	4	no	—
56	C₁₃×#117	—	2	2	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
62	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₂×C₄²×C₁₃	—	2	1	no	—
64	C₂×C₄²×C₁₃	D₄	1	2	yes	—
65	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₁₃×#106	—	2	8	no	—
67	C₁₃×C₄⋊C₈	—	2	2	no	—
68	C₁₃×#103	—	2	8	no	—
69	C₁₃×C₂²⋊C₈	C₂³	1	4	yes	—
70	C₄×C₈×C₁₃	—	2	2	no	—
71	C₁₃×#95	—	2	4	no	—
72	C₁₃×#95	C₂³	1	2	yes	—
73	#18410	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×Q₈×C₁₃	—	2	4	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	2	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	—	4	2	no	—
81	C₂×Q₈×C₁₃	SD₁₆	1	4	yes	—
82	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	2	8	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	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	—	4	1	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	8	no	—
107	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
108	C₁₃×C₂²⋊C₄	—	2	4	no	—
109	C₄²×C₁₃	—	2	8	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	2	4	no	—
113	C₄²×C₁₃	—	4	2	no	—
114	C₄²×C₁₃	C₂×D₄	1	2	yes	—
115	C₄²×C₁₃	C₂×D₄	1	2	yes	—
116	#3502	C₂×C₁₃	1	1	yes	characteristic
117	#3229	C₂×C₁₃	1	2	yes	—
118	#2546	C₂×C₁₃	1	1	yes	characteristic
119	#1611	C₂×C₁₃	1	8	yes	—
120	#1503	C₂×C₁₃	1	2	yes	—
121	#1461	C₂×C₁₃	1	1	yes	characteristic
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	8	no	—
124	Q₈×C₁₃	—	4	4	no	—
125	Q₈×C₁₃	—	4	4	no	—
126	Q₈×C₁₃	—	8	4	no	—
127	Q₈×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#120	1	2	yes	—
144	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
146	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₁₃	#137	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
149	C₈×C₁₃	—	4	4	no	—
150	Q₈²	—	2	2	no	—
151	#557	—	2	2	no	—
152	#557	—	2	4	no	—
153	#544	—	2	1	no	—
154	#544	C₂²×C₁₃	1	4	yes	—
155	#533	—	2	2	no	—
156	#533	C₂²×C₁₃	1	2	yes	—
157	C₂×C₄⋊Q₈	—	2	1	no	—
158	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
159	#519	—	2	1	no	—
160	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
161	#498	—	2	2	no	—
162	#487	—	2	2	no	—
163	#483	C₂²×C₁₃	1	8	yes	—
164	#475	—	2	4	no	—
165	#431	C₂²×C₁₃	1	2	yes	—
166	#398	C₂²×C₁₃	1	4	yes	—
167	#391	—	2	1	no	—
168	#389	—	2	2	no	—
169	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
170	#328	C₂²×C₁₃	1	8	yes	—
171	#324	C₂²×C₁₃	1	4	yes	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
175	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	4	4	no	—
181	C₄×C₁₃	—	4	4	no	—
182	C₄×C₁₃	—	4	2	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	#448	1	4	yes	—
186	C₄×C₁₃	#456	1	2	yes	—
187	C₂²×Q₈	—	2	1	no	—
188	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	2	8	no	—
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	4	4	no	—
196	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
197	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
198	#125	—	2	4	no	—
199	#125	—	2	4	no	—
200	#125	—	2	8	no	—
201	#125	—	2	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	2	4	no	—
204	C₂²⋊Q₈	C₂³×C₁₃	1	4	yes	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	C₄×Q₈	—	2	4	no	—
208	C₄×Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	D₄×C₁₃	1	4	yes	—
213	#117	—	2	4	no	—
214	#117	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	D₄×C₁₃	1	2	yes	—
223	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
224	#106	—	2	8	no	—
225	C₄⋊C₈	—	2	2	no	—
226	#103	—	2	8	no	—
227	C₂²⋊C₈	C₂³×C₁₃	1	4	yes	—
228	C₄×C₈	—	2	2	no	—
229	#95	—	2	4	no	—
230	#95	C₂³×C₁₃	1	2	yes	—
231	C₂×C₁₃	—	2	1	no	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	C₂×#448	1	1	yes	characteristic
234	C₂×C₁₃	C₂×#456	1	1	yes	characteristic
235	C₂×C₁₃	#3028	1	1	yes	characteristic
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	C₁₃×SD₁₆	1	4	yes	—
244	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	8	no	—
269	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₄²	—	2	8	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	4	2	no	—
276	C₄²	C₂×D₄×C₁₃	1	2	yes	—
277	C₄²	C₂×D₄×C₁₃	1	2	yes	—
278	C₁₃	#18410	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
280	Q₈	—	2	8	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	Q₈	—	8	4	no	—
284	Q₈	—	8	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#120	1	2	yes	—
301	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×SD₁₆	1	4	yes	—
303	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
304	C₂×C₄	C₁₃×#137	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
310	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
311	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	4	no	—
319	C₄	—	8	2	no	—
320	C₄	C₁₃×#448	1	4	yes	—
321	C₄	C₁₃×#456	1	2	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	C₂×C₁₃×#448	1	1	yes	characteristic
325	C₂	C₂×C₁₃×#456	1	1	yes	characteristic
326	C₂	C₁₃×#3028	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
73. #18410	C₁₃	direct (N × H)
278. C₁₃	#18410	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×#3502	2
C₁₃×#3229	2
C₁₃×#2546	2
C₁₃×#1611	2
C₁₃×#1503	2
C₁₃×#1461	2
#18410	13

Subgroups — C13×#18410 #15704749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18410 #15704749