Subgroups — C₁₃×#13039 #15699378

Order 3328.

472 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#444	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#360	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	—	2	1	no	—
11	C₁₃×#510	—	2	2	no	—
12	C₁₃×#510	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₁₃×#444	—	2	2	no	—
18	C₁₃×#444	—	2	2	no	—
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#104	—	2	1	no	—
22	C₂×C₁₃×#103	—	2	1	no	—
23	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
24	C₁₃×#391	—	2	1	no	—
25	C₁₃×#362	—	2	2	no	—
26	C₁₃×#360	—	2	2	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×Q₈×C₁₃	—	2	1	no	—
29	C₁₃×#135	—	2	1	no	—
30	C₁₃×#135	—	2	1	no	—
31	C₁₃×#135	D₄	1	2	yes	—
32	C₁₃×#135	D₄	1	2	yes	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×SD₁₆	—	2	1	no	—
41	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
42	C₁₃×#131	—	2	4	no	—
43	C₁₃×#131	—	4	2	no	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂×C₁₃×M₁₆	—	4	1	no	—
46	C₂²×C₈×C₁₃	—	4	1	no	—
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×Q₃₂	—	2	4	no	—
52	C₁₃×SD₃₂	—	2	4	no	—
53	C₁₃×M₃₂	—	2	1	no	—
54	C₁₃×M₃₂	—	2	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₁₃×M₃₂	—	2	1	no	—
57	C₂×C₁₃×C₁₆	—	2	1	no	—
58	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	D₄	1	2	yes	—
61	C₁₃×#106	—	2	1	no	—
62	C₁₃×#106	—	2	1	no	—
63	C₁₃×#104	—	2	4	no	—
64	C₁₃×#103	—	4	2	no	—
65	C₁₃×#97	—	2	1	no	—
66	C₁₃×#97	—	2	1	no	—
67	C₁₃×#95	—	4	2	no	—
68	#13039	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₁₃×#41	—	2	2	no	—
70	C₁₃×#41	—	2	2	no	—
71	C₁₃×#41	—	4	1	no	—
72	C₁₃×#41	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	2	1	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	—	4	1	no	—
77	C₂×Q₈×C₁₃	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	2	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	—	4	1	no	—
91	C₁₃×Q₁₆	—	4	1	no	—
92	C₁₃×Q₁₆	C₂²⋊C₄	1	2	yes	—
93	C₁₃×SD₁₆	—	2	2	no	—
94	C₁₃×SD₁₆	—	2	2	no	—
95	C₁₃×D₈	—	2	1	no	—
96	C₁₃×D₈	C₂²⋊C₄	1	2	yes	—
97	C₁₃×M₁₆	—	2	2	no	—
98	C₁₃×M₁₆	—	4	4	no	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₂×C₈×C₁₃	—	4	4	no	—
103	C₂×C₈×C₁₃	—	4	1	no	—
104	C₂×C₈×C₁₃	D₈	1	2	yes	—
105	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₄²×C₁₃	—	2	1	no	—
115	C₁₃×C₁₆	—	2	2	no	—
116	C₁₃×C₁₆	—	4	1	no	—
117	C₂×#510	C₂×C₁₃	1	1	yes	characteristic
118	C₂×#444	C₂×C₁₃	1	1	yes	characteristic
119	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂×#360	C₂×C₁₃	1	1	yes	characteristic
121	#1971	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1383	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#1375	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	1	no	—
125	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	Q₈×C₁₃	—	2	2	no	—
127	Q₈×C₁₃	—	4	1	no	—
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	Q₈×C₁₃	—	8	1	no	—
131	Q₈×C₁₃	—	8	1	no	—
132	D₄×C₁₃	—	2	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	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	1	no	—
142	C₂×C₄×C₁₃	—	8	1	no	—
143	C₂×C₄×C₁₃	—	8	1	no	—
144	C₂×C₄×C₁₃	—	8	2	no	—
145	C₂×C₄×C₁₃	—	8	1	no	—
146	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
151	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	4	2	no	—
154	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
156	C₈×C₁₃	#121	1	1	yes	characteristic
157	C₈×C₁₃	#123	1	1	yes	characteristic
158	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
159	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#131	—	2	1	no	—
161	#510	—	2	2	no	—
162	#510	—	2	2	no	—
163	C₂×Q₃₂	C₄×C₁₃	1	2	yes	complemented
164	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
165	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
167	#444	—	2	2	no	—
168	#444	—	2	2	no	—
169	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×#104	—	2	1	no	—
172	C₂×#103	—	2	1	no	—
173	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
174	#391	—	2	1	no	—
175	#362	—	2	2	no	—
176	#360	—	2	2	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	—	4	2	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	#442	1	1	yes	characteristic
181	C₂²×C₁₃	#449	1	1	yes	characteristic
182	C₂²×C₁₃	#459	1	1	yes	characteristic
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	—	8	1	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	16	1	no	—
187	C₄×C₁₃	#386	1	1	yes	characteristic
188	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
189	C₄×C₁₃	#447	1	1	yes	characteristic
190	C₄×C₁₃	#463	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	D₄×C₁₃	1	2	yes	—
196	#135	D₄×C₁₃	1	2	yes	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
203	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
204	C₂×SD₁₆	—	2	1	no	—
205	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
206	#131	—	2	4	no	—
207	#131	—	4	2	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	4	1	no	—
210	C₂²×C₈	—	4	1	no	—
211	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
215	Q₃₂	—	2	4	no	—
216	SD₃₂	—	2	4	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
223	#108	—	2	1	no	—
224	#108	D₄×C₁₃	1	2	yes	—
225	#106	—	2	1	no	—
226	#106	—	2	1	no	—
227	#104	—	2	4	no	—
228	#103	—	4	2	no	—
229	#97	—	2	1	no	—
230	#97	—	2	1	no	—
231	#95	—	4	2	no	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	#1879	1	1	yes	characteristic
236	C₂×C₁₃	#2180	1	2	yes	—
237	#41	—	2	2	no	—
238	#41	—	2	2	no	—
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	4	1	no	—
259	Q₁₆	—	4	1	no	—
260	Q₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
261	SD₁₆	—	2	2	no	—
262	SD₁₆	—	2	2	no	—
263	D₈	—	2	1	no	—
264	D₈	C₁₃×C₂²⋊C₄	1	2	yes	—
265	M₁₆	—	2	2	no	—
266	M₁₆	—	4	4	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	C₁₃×D₈	1	2	yes	—
273	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
278	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	4	1	no	—
285	C₁₃	#13039	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	4	1	no	—
287	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	1	no	—
293	Q₈	—	8	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
309	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
310	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
311	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
312	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
313	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	4	2	no	—
316	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
317	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
318	C₈	C₁₃×#121	1	1	yes	characteristic
319	C₈	C₁₃×#123	1	1	yes	characteristic
320	C₂²	—	2	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₁₃×#442	1	1	yes	characteristic
324	C₂²	C₁₃×#449	1	1	yes	characteristic
325	C₂²	C₁₃×#459	1	1	yes	characteristic
326	C₄	—	4	2	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	16	1	no	—
330	C₄	C₁₃×#386	1	1	yes	characteristic
331	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
332	C₄	C₁₃×#447	1	1	yes	characteristic
333	C₄	C₁₃×#463	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	8	2	no	—
337	C₂	C₁₃×#1879	1	1	yes	characteristic
338	C₂	C₁₃×#2180	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#442
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#362	C₂	semidirect
5. C₁₃×#1971	C₂	semidirect
6. C₁₃×#1383	C₂	semidirect
7. C₁₃×#1375	C₂	semidirect
13. C₂×C₁₃×Q₃₂	C₄	semidirect
14. C₂×C₁₃×SD₃₂	C₄	semidirect
68. #13039	C₁₃	direct (N × H)
119. C₂×#362	C₂×C₁₃	semidirect
121. #1971	C₂×C₁₃	semidirect
122. #1383	C₂×C₁₃	semidirect
123. #1375	C₂×C₁₃	semidirect
163. C₂×Q₃₂	C₄×C₁₃	semidirect
164. C₂×SD₃₂	C₄×C₁₃	semidirect
285. C₁₃	#13039	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×#510	2
C₂×C₁₃×#444	2
C₂×C₁₃×#362	2
C₂×C₁₃×#360	2
C₁₃×#1971	2
C₁₃×#1383	2
C₁₃×#1375	2
#13039	13

Subgroups — C13×#13039 #15699378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13039 #15699378