Subgroups — C₁₃×#13004 #15699343

Order 3328.

560 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	—	2	1	no	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#121	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₁₃×#466	—	2	1	no	—
12	C₁₃×#466	—	2	1	no	—
13	C₁₃×#466	C₄	1	2	yes	—
14	C₁₃×#466	C₄	1	2	yes	—
15	C₂×C₁₃×#108	—	2	1	no	—
16	C₂×C₁₃×#108	—	2	4	no	—
17	C₂×C₁₃×#108	C₂²	1	2	yes	—
18	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#102	—	2	1	no	—
20	C₁₃×#361	—	2	4	no	—
21	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
22	C₁₃×C₂²⋊C₁₆	—	2	1	no	—
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	4	1	no	—
25	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₁₃×D₈	—	2	2	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	2	no	—
29	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	—	2	4	no	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	4	2	no	—
33	C₂×C₁₃×M₁₆	—	4	1	no	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	4	1	no	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#121	—	2	2	no	—
44	C₁₃×#121	—	2	2	no	—
45	C₁₃×#121	—	4	1	no	—
46	C₁₃×#121	—	4	1	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
49	C₁₃×M₃₂	—	2	2	no	—
50	C₁₃×M₃₂	—	2	2	no	—
51	C₂×C₁₃×C₁₆	—	2	2	no	—
52	C₁₃×#108	—	2	4	no	—
53	C₁₃×#108	—	2	2	no	—
54	C₁₃×#108	D₄	1	4	yes	complemented
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	C₂×C₄	1	2	yes	—
57	C₁₃×#102	—	2	2	no	—
58	C₁₃×#102	—	2	2	no	—
59	#13004	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₃	—	4	1	no	—
61	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	2	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂×D₄×C₁₃	—	8	4	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×D₈	—	2	4	no	—
83	C₁₃×D₈	—	4	2	no	—
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	—	4	4	no	—
86	C₁₃×M₁₆	—	4	2	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	4	4	no	—
94	C₂×C₈×C₁₃	—	4	2	no	—
95	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	#41	1	2	yes	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₁₆	—	4	2	no	—
106	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
107	C₂²×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#361	C₂×C₁₃	1	2	yes	complemented
109	#2112	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#1387	C₂×C₁₃	1	2	yes	complemented
111	C₂³×C₁₃	—	2	2	no	—
112	C₂³×C₁₃	—	2	1	no	—
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	—	4	1	no	—
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	—	8	2	no	—
117	C₂³×C₁₃	—	8	2	no	—
118	C₂³×C₁₃	—	8	1	no	—
119	C₂³×C₁₃	—	8	1	no	—
120	C₂³×C₁₃	#102	1	1	yes	characteristic
121	C₂³×C₁₃	#102	1	1	yes	characteristic
122	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	8	4	no	—
126	D₄×C₁₃	—	8	4	no	—
127	D₄×C₁₃	—	8	1	no	—
128	D₄×C₁₃	—	8	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	8	2	no	—
141	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	4	4	no	—
152	C₈×C₁₃	C₄×D₄	1	2	yes	—
153	C₈×C₁₃	#121	1	2	yes	—
154	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	1	no	—
156	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#121	—	2	1	no	—
158	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
159	#466	—	2	1	no	—
160	#466	—	2	1	no	—
161	#466	C₄×C₁₃	1	2	yes	—
162	#466	C₄×C₁₃	1	2	yes	—
163	C₂×#108	—	2	1	no	—
164	C₂×#108	—	2	4	no	—
165	C₂×#108	C₂²×C₁₃	1	2	yes	—
166	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×#102	—	2	1	no	—
168	#361	—	2	4	no	—
169	C₂²⋊C₁₆	—	2	1	no	—
170	C₂²⋊C₁₆	—	2	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	4	2	no	—
179	C₂²×C₁₃	—	4	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	16	2	no	—
184	C₂²×C₁₃	—	16	2	no	—
185	C₂²×C₁₃	#361	1	2	yes	complemented
186	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
187	C₂²×C₁₃	#448	1	1	yes	characteristic
188	C₂²×C₁₃	#481	1	1	yes	characteristic
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	#386	1	1	yes	characteristic
193	C₄×C₁₃	#417	1	1	yes	characteristic
194	C₄×C₁₃	#447	1	1	yes	characteristic
195	C₄×C₁₃	#480	1	1	yes	characteristic
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	4	1	no	—
198	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
199	C₂×D₈	—	2	2	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	4	2	no	—
206	C₂×M₁₆	—	4	1	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	4	1	no	—
210	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	—	4	1	no	—
219	#121	—	4	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	4	1	no	—
222	M₃₂	—	2	2	no	—
223	M₃₂	—	2	2	no	—
224	C₂×C₁₆	—	2	2	no	—
225	#108	—	2	4	no	—
226	#108	—	2	2	no	—
227	#108	D₄×C₁₃	1	4	yes	complemented
228	#107	—	2	1	no	—
229	#107	C₂×C₄×C₁₃	1	2	yes	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	2	2	no	—
235	C₂×C₁₃	—	4	1	no	—
236	C₂×C₁₃	—	16	2	no	—
237	C₂×C₁₃	#1874	1	1	yes	characteristic
238	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
239	C₂×C₁₃	#2148	1	1	yes	characteristic
240	C₂⁴	—	4	1	no	—
241	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	8	4	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
257	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
258	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
259	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
260	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
262	D₈	—	2	4	no	—
263	D₈	—	4	2	no	—
264	M₁₆	—	2	4	no	—
265	M₁₆	—	4	4	no	—
266	M₁₆	—	4	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	4	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₂×C₈	C₁₃×#41	1	2	yes	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₁₆	—	4	2	no	—
286	C₁₃	#13004	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³	—	2	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	1	no	—
295	C₂³	—	8	1	no	—
296	C₂³	C₁₃×#102	1	1	yes	characteristic
297	C₂³	C₁₃×#102	1	1	yes	characteristic
298	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	8	1	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
318	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
319	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
321	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
322	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
323	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	4	4	no	—
328	C₈	C₄×D₄×C₁₃	1	2	yes	—
329	C₈	C₁₃×#121	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	16	2	no	—
343	C₂²	—	16	2	no	—
344	C₂²	C₁₃×#361	1	2	yes	complemented
345	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
346	C₂²	C₁₃×#448	1	1	yes	characteristic
347	C₂²	C₁₃×#481	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	C₁₃×#386	1	1	yes	characteristic
352	C₄	C₁₃×#417	1	1	yes	characteristic
353	C₄	C₁₃×#447	1	1	yes	characteristic
354	C₄	C₁₃×#480	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	2	no	—
358	C₂	—	4	1	no	—
359	C₂	—	16	2	no	—
360	C₂	C₁₃×#1874	1	1	yes	characteristic
361	C₂	C₂×C₁₃×#361	1	1	yes	characteristic
362	C₂	C₁₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×#108	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₁₃×#2112	C₂	semidirect
5. C₁₃×#1387	C₂	semidirect
54. C₁₃×#108	D₄	semidirect
59. #13004	C₁₃	direct (N × H)
107. C₂²×#108	C₂×C₁₃	semidirect
108. C₂×#361	C₂×C₁₃	semidirect
109. #2112	C₂×C₁₃	semidirect
110. #1387	C₂×C₁₃	semidirect
185. C₂²×C₁₃	#361	semidirect
227. #108	D₄×C₁₃	semidirect
286. C₁₃	#13004	direct (N × H)
344. C₂²	C₁₃×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#466	2
C₂²×C₁₃×#108	2
C₂×C₁₃×#361	2
C₁₃×#2112	2
C₁₃×#1387	2
#13004	13

Subgroups — C13×#13004 #15699343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13004 #15699343