Subgroups — C₁₃×#13036 #15699375

Order 3328.

476 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#509	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2143	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₁₃×#568	—	2	1	no	—
11	C₁₃×#509	—	2	2	no	—
12	C₁₃×#509	—	2	2	no	—
13	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₃×D₁₆	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×#443	—	2	2	no	—
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
21	C₁₃×#421	—	2	1	no	—
22	C₁₃×#418	—	2	1	no	—
23	C₂×C₁₃×#100	—	2	1	no	—
24	C₁₃×#405	C₂²	1	1	yes	characteristic
25	C₁₃×#361	—	2	2	no	—
26	C₁₃×#359	—	2	2	no	—
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×D₄×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	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×SD₁₆	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	1	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
46	C₂×C₁₃×M₁₆	—	4	2	no	—
47	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#117	D₄	1	1	yes	characteristic
50	C₁₃×SD₃₂	—	2	4	no	—
51	C₁₃×D₁₆	—	2	4	no	—
52	C₁₃×M₃₂	—	2	1	no	—
53	C₁₃×M₃₂	—	2	1	no	—
54	C₁₃×M₃₂	—	2	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₂×C₁₃×C₁₆	—	2	1	no	—
57	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	—	2	1	no	—
61	C₁₃×#108	D₄	1	2	yes	—
62	C₁₃×#104	—	2	4	no	—
63	C₁₃×#102	—	4	2	no	—
64	C₁₃×#100	—	4	2	no	—
65	C₁₃×#97	—	2	1	no	—
66	C₄×C₈×C₁₃	—	2	1	no	—
67	#13036	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂⁴×C₁₃	—	4	1	no	—
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×D₄×C₁₃	—	2	1	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	1	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂×D₄×C₁₃	—	4	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	2	1	no	—
85	C₁₃×Q₁₆	C₂²⋊C₄	1	2	yes	—
86	C₁₃×SD₁₆	—	2	2	no	—
87	C₁₃×SD₁₆	—	2	2	no	—
88	C₁₃×D₈	—	2	2	no	—
89	C₁₃×D₈	—	2	1	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	—	4	1	no	—
92	C₁₃×D₈	—	4	1	no	—
93	C₁₃×D₈	C₂²⋊C₄	1	2	yes	—
94	C₁₃×M₁₆	—	2	1	no	—
95	C₁₃×M₁₆	—	2	2	no	—
96	C₁₃×M₁₆	—	4	4	no	—
97	C₁₃×M₁₆	—	4	1	no	—
98	C₁₃×M₁₆	—	4	1	no	—
99	C₁₃×M₁₆	—	4	1	no	—
100	C₁₃×M₁₆	D₈	1	2	yes	—
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	—	4	1	no	—
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	—	2	1	no	—
112	C₁₃×C₂²⋊C₄	—	2	1	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	C₁₃×C₁₆	—	4	1	no	—
116	C₂×#509	C₂×C₁₃	1	1	yes	characteristic
117	#2957	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2143	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#1978	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#1385	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#1378	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	—	4	1	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	—	8	1	no	—
126	C₂³×C₁₃	—	8	1	no	—
127	C₂³×C₁₃	—	16	2	no	—
128	C₂³×C₁₃	#135	1	1	yes	characteristic
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	4	1	no	—
131	D₄×C₁₃	—	2	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	1	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	8	1	no	—
138	D₄×C₁₃	—	8	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
145	C₂×C₄×C₁₃	#121	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	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	4	2	no	—
151	C₈×C₁₃	—	8	1	no	—
152	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₁₃	#121	1	1	yes	characteristic
155	C₈×C₁₃	#123	1	1	yes	characteristic
156	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
157	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
158	#568	—	2	1	no	—
159	#509	—	2	2	no	—
160	#509	—	2	2	no	—
161	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
162	C₂×D₁₆	C₄×C₁₃	1	2	yes	complemented
163	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
165	#444	—	2	2	no	—
166	#443	—	2	2	no	—
167	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
169	#421	—	2	1	no	—
170	#418	—	2	1	no	—
171	C₂×#100	—	2	1	no	—
172	#405	C₂²×C₁₃	1	1	yes	characteristic
173	#361	—	2	2	no	—
174	#359	—	2	2	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	16	1	no	—
179	C₂²×C₁₃	—	16	2	no	—
180	C₂²×C₁₃	—	16	2	no	—
181	C₂²×C₁₃	—	16	2	no	—
182	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
183	C₂²×C₁₃	#449	1	1	yes	characteristic
184	C₂²×C₁₃	#463	1	1	yes	characteristic
185	C₄×C₁₃	—	4	2	no	—
186	C₄×C₁₃	—	8	1	no	—
187	C₄×C₁₃	#386	1	1	yes	characteristic
188	C₄×C₁₃	#442	1	1	yes	characteristic
189	C₄×C₁₃	#447	1	1	yes	characteristic
190	C₄×C₁₃	#459	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	C₂²×D₄	—	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	C₂×C₄×C₁₃	1	1	yes	characteristic
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
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	—	2	1	no	—
210	C₂×M₁₆	—	4	2	no	—
211	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
213	#117	D₄×C₁₃	1	1	yes	characteristic
214	SD₃₂	—	2	4	no	—
215	D₁₆	—	2	4	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
222	#108	—	2	1	no	—
223	#108	—	2	1	no	—
224	#108	—	2	1	no	—
225	#108	D₄×C₁₃	1	2	yes	—
226	#104	—	2	4	no	—
227	#102	—	4	2	no	—
228	#100	—	4	2	no	—
229	#97	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₂×C₁₃	—	2	1	no	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	2	1	no	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	#1879	1	1	yes	characteristic
237	C₂⁴	—	4	1	no	—
238	#41	—	2	2	no	—
239	#41	—	2	2	no	—
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	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	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	Q₁₆	—	2	1	no	—
254	Q₁₆	C₁₃×C₂²⋊C₄	1	2	yes	—
255	SD₁₆	—	2	2	no	—
256	SD₁₆	—	2	2	no	—
257	D₈	—	2	2	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	2	no	—
260	D₈	—	4	1	no	—
261	D₈	—	4	1	no	—
262	D₈	C₁₃×C₂²⋊C₄	1	2	yes	—
263	M₁₆	—	2	1	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	4	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	—	4	1	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	C₁₃×D₈	1	2	yes	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	C₂²×C₄×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	C₂×D₄×C₁₃	1	1	yes	characteristic
278	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
279	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	4	1	no	—
285	C₁₃	#13036	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	4	1	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	8	1	no	—
290	C₂³	—	16	2	no	—
291	C₂³	C₁₃×#135	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	Q₈	—	4	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	8	1	no	—
301	D₄	—	8	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
307	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
308	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
309	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
310	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	8	1	no	—
315	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
316	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
317	C₈	C₁₃×#121	1	1	yes	characteristic
318	C₈	C₁₃×#123	1	1	yes	characteristic
319	C₂²	—	4	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	1	no	—
323	C₂²	—	16	2	no	—
324	C₂²	—	16	2	no	—
325	C₂²	—	16	2	no	—
326	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
327	C₂²	C₁₃×#449	1	1	yes	characteristic
328	C₂²	C₁₃×#463	1	1	yes	characteristic
329	C₄	—	4	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₁₃×#386	1	1	yes	characteristic
332	C₄	C₁₃×#442	1	1	yes	characteristic
333	C₄	C₁₃×#447	1	1	yes	characteristic
334	C₄	C₁₃×#459	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	8	2	no	—
339	C₂	—	16	2	no	—
340	C₂	C₁₃×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#1879
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2957	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₁₃×#2143	C₂	semidirect
5. C₁₃×#1978	C₂	semidirect
6. C₁₃×#1385	C₂	semidirect
7. C₁₃×#1378	C₂	semidirect
13. C₂×C₁₃×SD₃₂	C₄	semidirect
14. C₂×C₁₃×D₁₆	C₄	semidirect
67. #13036	C₁₃	direct (N × H)
117. #2957	C₂×C₁₃	semidirect
118. C₂×#361	C₂×C₁₃	semidirect
119. #2143	C₂×C₁₃	semidirect
120. #1978	C₂×C₁₃	semidirect
121. #1385	C₂×C₁₃	semidirect
122. #1378	C₂×C₁₃	semidirect
161. C₂×SD₃₂	C₄×C₁₃	semidirect
162. C₂×D₁₆	C₄×C₁₃	semidirect
285. C₁₃	#13036	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×#509	2
C₁₃×#2957	2
C₂×C₁₃×#361	2
C₁₃×#2143	2
C₁₃×#1978	2
C₁₃×#1385	2
C₁₃×#1378	2
#13036	13

Subgroups — C13×#13036 #15699375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13036 #15699375