Subgroups — C₁₃×#14276 #15700615

Order 3328.

1406 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#389	C₂	1	2	yes	maximal
2	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1441	C₂	1	8	yes	maximal
6	C₁₃×C₄⋊#95	C₂	1	2	yes	maximal
7	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#389	—	2	8	no	—
15	C₁₃×#384	C₄	1	8	yes	complemented
16	C₁₃×#382	—	2	4	no	—
17	C₁₃×#376	—	2	4	no	—
18	C₂×C₁₃×#95	C₂²	1	4	yes	—
19	C₂×C₁₃×#95	C₂²	1	4	yes	—
20	C₂×C₁₃×#95	C₂²	1	4	yes	—
21	C₂×C₁₃×#95	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	8	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	8	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
38	C₂×C₄²×C₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	—	2	2	no	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	D₄	1	4	yes	—
44	C₂×C₄²×C₁₃	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
47	C₁₃×#95	—	2	8	no	—
48	C₁₃×#95	—	2	8	no	—
49	C₁₃×#95	—	2	8	no	—
50	C₁₃×#95	—	4	8	no	—
51	#14276	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	8	no	—
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	8	no	—
71	C₂²×C₄×C₁₃	—	2	8	no	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₁₃	#41	1	2	yes	—
84	C₂²×C₄×C₁₃	#41	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	16	no	—
86	C₁₃×C₄⋊C₄	—	4	16	no	—
87	C₁₃×C₄⋊C₄	—	4	16	no	—
88	C₁₃×C₄⋊C₄	—	4	8	no	—
89	C₄²×C₁₃	—	2	8	no	—
90	C₄²×C₁₃	—	2	8	no	—
91	C₄²×C₁₃	C₄⋊C₄	1	16	yes	complemented
92	C₂×#389	C₂×C₁₃	1	2	yes	—
93	C₂×#384	C₂×C₁₃	1	1	yes	characteristic
94	C₂×#382	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#376	C₂×C₁₃	1	1	yes	characteristic
96	#1441	C₂×C₁₃	1	8	yes	—
97	C₄⋊#95	C₂×C₁₃	1	2	yes	—
98	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
104	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
105	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
106	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	2	8	no	—
111	C₂×C₄×C₁₃	—	2	8	no	—
112	C₂×C₄×C₁₃	—	2	16	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	8	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	4	no	—
122	C₂×C₄×C₁₃	—	4	8	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	8	no	—
130	C₂×C₄×C₁₃	—	4	8	no	—
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
132	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
135	C₂×C₄×C₁₃	#121	1	4	yes	—
136	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
137	C₂×C₄×C₁₃	#125	1	4	yes	—
138	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
139	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
140	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
141	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
143	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
144	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
145	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
146	#389	—	2	8	no	—
147	#384	C₄×C₁₃	1	8	yes	complemented
148	#382	—	2	4	no	—
149	#376	—	2	4	no	—
150	C₂×#95	C₂²×C₁₃	1	4	yes	—
151	C₂×#95	C₂²×C₁₃	1	4	yes	—
152	C₂×#95	C₂²×C₁₃	1	4	yes	—
153	C₂×#95	C₂²×C₁₃	1	8	yes	—
154	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂²×C₁₃	#518	1	2	yes	—
158	C₂²×C₁₃	#519	1	2	yes	—
159	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₁₃	#535	1	2	yes	—
164	C₂²×C₁₃	#536	1	2	yes	—
165	C₂²×C₁₃	#537	1	4	yes	—
166	C₂²×C₁₃	#540	1	2	yes	—
167	C₂²×C₁₃	#541	1	2	yes	—
168	C₂²×C₁₃	#542	1	4	yes	—
169	C₂²×C₁₃	#544	1	4	yes	—
170	C₂²×C₁₃	#544	1	4	yes	—
171	C₄×C₁₃	—	2	8	no	—
172	C₄×C₁₃	—	4	8	no	—
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	—	4	4	no	—
175	C₄×C₁₃	—	8	4	no	—
176	C₄×C₁₃	#389	1	8	yes	—
177	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
178	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
179	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
180	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
181	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	4	8	no	—
190	C₂×C₄⋊C₄	—	4	8	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	16	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
198	C₂×C₄²	D₄×C₁₃	1	4	yes	—
199	C₂×C₄²	D₄×C₁₃	1	4	yes	—
200	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
201	C₂×C₄²	Q₈×C₁₃	1	4	yes	—
202	#95	—	2	8	no	—
203	#95	—	2	8	no	—
204	#95	—	2	8	no	—
205	#95	—	4	8	no	—
206	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
207	C₂×C₁₃	#2316	1	2	yes	—
208	C₂×C₁₃	#2327	1	1	yes	characteristic
209	C₂×C₁₃	#2328	1	1	yes	characteristic
210	C₂×C₁₃	#2331	1	2	yes	—
211	C₂×C₁₃	#2620	1	2	yes	—
212	C₂×C₁₃	#2629	1	2	yes	—
213	C₂×C₁₃	#2647	1	2	yes	—
214	C₂×C₁₃	#2660	1	2	yes	—
215	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
236	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
242	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
243	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
244	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
245	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
246	C₂²×C₄	C₁₃×#41	1	2	yes	—
247	C₂²×C₄	C₁₃×#41	1	2	yes	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	4	16	no	—
250	C₄⋊C₄	—	4	16	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₄²	—	2	8	no	—
253	C₄²	—	2	8	no	—
254	C₄²	C₁₃×C₄⋊C₄	1	16	yes	complemented
255	C₁₃	#14276	1	1	yes	characteristic, direct product factor, Sylow
256	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
257	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
258	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
259	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
260	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
261	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
262	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
263	C₂³	C₁₃×H(5,𝔽₂)	1	4	yes	—
264	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	16	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	8	yes	—
290	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
291	C₂×C₄	C₄×D₄×C₁₃	1	4	yes	—
292	C₂×C₄	C₄×Q₈×C₁₃	1	4	yes	—
293	C₂×C₄	C₁₃×#121	1	4	yes	—
294	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
295	C₂×C₄	C₁₃×#125	1	4	yes	—
296	C₂×C₄	C₁₃×C₄⋊Q₈	1	4	yes	—
297	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
298	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
299	C₂²	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
300	C₂²	C₁₃×#518	1	2	yes	—
301	C₂²	C₁₃×#519	1	2	yes	—
302	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
303	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂²	C₂×C₁₃×#125	1	1	yes	characteristic
305	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
306	C₂²	C₁₃×#535	1	2	yes	—
307	C₂²	C₁₃×#536	1	2	yes	—
308	C₂²	C₁₃×#537	1	4	yes	—
309	C₂²	C₁₃×#540	1	2	yes	—
310	C₂²	C₁₃×#541	1	2	yes	—
311	C₂²	C₁₃×#542	1	4	yes	—
312	C₂²	C₁₃×#544	1	4	yes	—
313	C₂²	C₁₃×#544	1	4	yes	—
314	C₄	—	2	8	no	—
315	C₄	—	4	8	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	4	no	—
319	C₄	C₁₃×#389	1	8	yes	—
320	C₂	C₂×C₁₃×#389	1	1	yes	characteristic
321	C₂	C₁₃×#2316	1	2	yes	—
322	C₂	C₁₃×#2327	1	1	yes	characteristic
323	C₂	C₁₃×#2328	1	1	yes	characteristic
324	C₂	C₁₃×#2331	1	2	yes	—
325	C₂	C₁₃×#2620	1	2	yes	—
326	C₂	C₁₃×#2629	1	2	yes	—
327	C₂	C₁₃×#2647	1	2	yes	—
328	C₂	C₁₃×#2660	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
15. C₁₃×#384	C₄	semidirect
51. #14276	C₁₃	direct (N × H)
91. C₄²×C₁₃	C₄⋊C₄	semidirect
147. #384	C₄×C₁₃	semidirect
254. C₄²	C₁₃×C₄⋊C₄	semidirect
255. C₁₃	#14276	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×#389	2
C₂×C₁₃×#384	2
C₂×C₁₃×#382	2
C₂×C₁₃×#376	2
C₁₃×#1441	2
C₁₃×C₄⋊#95	2
#14276	13

Subgroups — C13×#14276 #15700615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#14276 #15700615