Subgroups — C₁₃×#9674 #15696013

Order 3328.

736 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₁₃	—	2	1	no	—
7	C₂×C₄×D₄×C₁₃	—	2	1	no	—
8	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₁₃×#390	—	2	1	no	—
14	C₁₃×#390	—	2	1	no	—
15	C₁₃×#390	C₄	1	2	yes	—
16	C₁₃×#384	C₄	1	1	yes	characteristic
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#336	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	C₁₃×D₄⋊C₈	—	2	8	no	—
21	C₂²×D₄×C₁₃	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₁₃×#127	—	2	2	no	—
28	C₁₃×#127	C₈	1	4	yes	complemented
29	C₄×D₄×C₁₃	—	2	4	no	—
30	C₄×D₄×C₁₃	—	2	4	no	—
31	C₄×D₄×C₁₃	—	4	4	no	—
32	C₄×D₄×C₁₃	—	4	2	no	—
33	C₄×D₄×C₁₃	—	4	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	—	2	4	no	—
48	C₁₃×C₄⋊C₈	—	2	2	no	—
49	C₁₃×C₄⋊C₈	D₄	1	4	yes	complemented
50	C₄×C₈×C₁₃	—	2	4	no	—
51	C₄×C₈×C₁₃	—	4	2	no	—
52	#9674	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	2	4	no	—
55	C₂×D₄×C₁₃	—	2	4	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	4	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂×D₄×C₁₃	—	4	2	no	—
61	C₂×D₄×C₁₃	—	4	2	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	4	no	—
86	C₁₃×C₂²⋊C₄	—	4	4	no	—
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₁₃×C₂²⋊C₄	—	4	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	4	1	no	—
93	C₄²×C₁₃	C₂×C₈	1	2	yes	—
94	C₄²×C₁₃	M₁₆	1	2	yes	—
95	C₄²×C₁₃	D₈	1	2	yes	—
96	C₄²×C₁₃	SD₁₆	1	2	yes	—
97	C₄²×C₁₃	C₂×D₄	1	2	yes	—
98	C₄²×C₁₃	#41	1	2	yes	—
99	C₄×#127	C₂×C₁₃	1	1	yes	characteristic
100	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂×D₄⋊C₈	C₂×C₁₃	1	4	yes	complemented
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	—	4	2	no	—
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	D₄×C₁₃	—	2	8	no	—
112	D₄×C₁₃	—	4	8	no	—
113	D₄×C₁₃	—	4	8	no	—
114	D₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	—	8	4	no	—
129	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₁₃	#102	1	2	yes	—
132	C₂×C₄×C₁₃	#102	1	2	yes	—
133	C₂×C₄×C₁₃	#104	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	#121	1	2	yes	—
136	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#131	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₁₃	—	4	4	no	—
145	C₈×C₁₃	—	4	4	no	—
146	C₂×#127	C₄×C₁₃	1	1	yes	characteristic
147	C₂×C₄×D₄	—	2	1	no	—
148	C₂×C₄×D₄	—	2	1	no	—
149	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄×C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
154	#390	—	2	1	no	—
155	#390	—	2	1	no	—
156	#390	C₄×C₁₃	1	2	yes	—
157	#384	C₄×C₁₃	1	1	yes	characteristic
158	C₄³	C₂²×C₁₃	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	—	2	1	no	—
161	D₄⋊C₈	—	2	8	no	—
162	C₂²×C₁₃	—	4	4	no	—
163	C₂²×C₁₃	—	4	4	no	—
164	C₂²×C₁₃	—	4	4	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	2	no	—
169	C₂²×C₁₃	#390	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂²×C₁₃	#408	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
173	C₂²×C₁₃	#418	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
175	C₂²×C₁₃	#421	1	1	yes	characteristic
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	4	1	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	D₄⋊C₈	1	4	yes	complemented
183	C₄×C₁₃	C₈×D₄	1	2	yes	—
184	C₄×C₁₃	#436	1	2	yes	—
185	C₄×C₁₃	#459	1	2	yes	—
186	C₄×C₁₃	#460	1	2	yes	—
187	C₂²×D₄	—	2	1	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
190	C₂³×C₄	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	#127	—	2	2	no	—
194	#127	C₈×C₁₃	1	4	yes	complemented
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	—	4	4	no	—
198	C₄×D₄	—	4	2	no	—
199	C₄×D₄	—	4	2	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	D₄×C₁₃	1	4	yes	complemented
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	4	2	no	—
218	C₂×C₁₃	—	8	4	no	—
219	C₂×C₁₃	C₂×D₄⋊C₈	1	1	yes	characteristic
220	C₂×C₁₃	#1478	1	1	yes	characteristic
221	C₂×C₁₃	#1480	1	1	yes	characteristic
222	C₂×C₁₃	#1484	1	1	yes	characteristic
223	C₂×C₁₃	#1954	1	1	yes	characteristic
224	C₂×C₁₃	#1980	1	1	yes	characteristic
225	C₂×C₁₃	#1982	1	1	yes	characteristic
226	C₂⁴	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	C₂×C₈×C₁₃	1	4	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×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	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	1	no	—
266	C₄²	C₂×C₈×C₁₃	1	2	yes	—
267	C₄²	C₁₃×M₁₆	1	2	yes	—
268	C₄²	C₁₃×D₈	1	2	yes	—
269	C₄²	C₁₃×SD₁₆	1	2	yes	—
270	C₄²	C₂×D₄×C₁₃	1	2	yes	—
271	C₄²	C₁₃×#41	1	2	yes	—
272	C₁₃	#9674	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	2	4	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
281	D₄	—	2	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	4	8	no	—
284	D₄	—	4	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
300	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
301	C₂×C₄	C₁₃×#102	1	2	yes	—
302	C₂×C₄	C₁₃×#102	1	2	yes	—
303	C₂×C₄	C₁₃×#104	1	2	yes	—
304	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
305	C₂×C₄	C₁₃×#121	1	2	yes	—
306	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
310	C₂×C₄	C₁₃×#131	1	2	yes	—
311	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
313	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
314	C₈	—	4	4	no	—
315	C₈	—	4	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₁₃×#390	1	1	yes	characteristic
324	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
325	C₂²	C₁₃×#408	1	1	yes	characteristic
326	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
327	C₂²	C₁₃×#418	1	1	yes	characteristic
328	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
329	C₂²	C₁₃×#421	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	8	2	no	—
336	C₄	C₁₃×D₄⋊C₈	1	4	yes	complemented
337	C₄	C₈×D₄×C₁₃	1	2	yes	—
338	C₄	C₁₃×#436	1	2	yes	—
339	C₄	C₁₃×#459	1	2	yes	—
340	C₄	C₁₃×#460	1	2	yes	—
341	C₂	—	8	4	no	—
342	C₂	C₂×C₁₃×D₄⋊C₈	1	1	yes	characteristic
343	C₂	C₁₃×#1478	1	1	yes	characteristic
344	C₂	C₁₃×#1480	1	1	yes	characteristic
345	C₂	C₁₃×#1484	1	1	yes	characteristic
346	C₂	C₁₃×#1954	1	1	yes	characteristic
347	C₂	C₁₃×#1980	1	1	yes	characteristic
348	C₂	C₁₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×C₁₃×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
3. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
4. C₂×C₁₃×D₄⋊C₈	C₂	semidirect
28. C₁₃×#127	C₈	semidirect
49. C₁₃×C₄⋊C₈	D₄	semidirect
52. #9674	C₁₃	direct (N × H)
100. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
101. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
102. C₂×D₄⋊C₈	C₂×C₁₃	semidirect
182. C₄×C₁₃	D₄⋊C₈	semidirect
194. #127	C₈×C₁₃	semidirect
215. C₄⋊C₈	D₄×C₁₃	semidirect
272. C₁₃	#9674	direct (N × H)
336. C₄	C₁₃×D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#127	2
C₄×C₁₃×C₄⋊C₈	2
C₂×C₁₃×D₄⋊C₈	2
#9674	13

Subgroups — C13×#9674 #15696013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9674 #15696013