Subgroups — C₁₃×#13260 #15699599

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×SD₁₆	—	2	1	no	—
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#124	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#495	—	2	2	no	—
15	C₁₃×#495	C₄	1	4	yes	complemented
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#405	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#364	—	2	2	no	—
21	C₁₃×#362	—	2	2	no	—
22	C₁₃×#361	—	2	2	no	—
23	C₁₃×#358	—	2	2	no	—
24	C₁₃×#357	—	2	2	no	—
25	C₁₃×#339	—	2	1	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂×C₁₃×Q₁₆	—	2	1	no	—
29	C₂×C₁₃×Q₁₆	—	2	1	no	—
30	C₂×C₁₃×Q₁₆	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×SD₁₆	—	2	4	no	—
35	C₂×C₁₃×SD₁₆	—	4	1	no	—
36	C₂×C₁₃×SD₁₆	—	4	1	no	—
37	C₂×C₁₃×SD₁₆	—	4	2	no	—
38	C₂×C₁₃×D₈	—	2	1	no	—
39	C₂×C₁₃×D₈	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	1	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
44	C₂×C₁₃×M₁₆	—	2	1	no	—
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#124	—	2	4	no	—
48	C₁₃×#124	—	4	2	no	—
49	C₁₃×#117	—	2	1	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
51	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
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	—	2	1	no	—
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	D₄	1	2	yes	—
60	C₁₃×#107	D₄	1	2	yes	—
61	C₁₃×#106	—	2	1	no	—
62	C₁₃×#97	—	4	1	no	—
63	C₄×C₈×C₁₃	—	4	1	no	—
64	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₃	D₄	1	2	yes	—
66	#13260	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	4	1	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	C₂²×C₄×C₁₃	—	2	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	4	1	no	—
86	C₁₃×Q₁₆	—	4	1	no	—
87	C₁₃×SD₁₆	—	4	4	no	—
88	C₁₃×SD₁₆	—	4	4	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	—	4	1	no	—
92	C₁₃×D₈	—	4	1	no	—
93	C₁₃×M₁₆	—	2	2	no	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₂×C₈×C₁₃	—	4	1	no	—
96	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
98	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
99	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
105	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	8	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₄²×C₁₃	—	2	1	no	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
112	C₁₃×C₁₆	—	4	1	no	—
113	C₁₃×C₁₆	—	4	1	no	—
114	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
115	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂×#362	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂×#361	C₂×C₁₃	1	1	yes	characteristic
118	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#2140	C₂×C₁₃	1	1	yes	characteristic
120	#1772	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	—	4	2	no	—
122	C₂³×C₁₃	—	8	4	no	—
123	C₂³×C₁₃	—	8	1	no	—
124	C₂³×C₁₃	—	8	1	no	—
125	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	8	1	no	—
129	Q₈×C₁₃	—	8	1	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	8	1	no	—
133	D₄×C₁₃	—	8	1	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	8	1	no	—
139	C₂×C₄×C₁₃	—	8	1	no	—
140	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₈×C₁₃	—	4	2	no	—
147	C₈×C₁₃	#102	1	2	yes	—
148	C₈×C₁₃	#102	1	2	yes	—
149	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₈×C₁₃	#124	1	1	yes	characteristic
152	C₈×C₁₃	#127	1	1	yes	characteristic
153	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
154	C₂²×SD₁₆	—	2	1	no	—
155	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#124	—	2	1	no	—
157	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
159	#495	—	2	2	no	—
160	#495	C₄×C₁₃	1	4	yes	complemented
161	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
162	#427	C₂²×C₁₃	1	1	yes	characteristic
163	#405	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
165	#364	—	2	2	no	—
166	#362	—	2	2	no	—
167	#361	—	2	2	no	—
168	#358	—	2	2	no	—
169	#357	—	2	2	no	—
170	#339	—	2	1	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	8	4	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	16	2	no	—
175	C₂²×C₁₃	—	16	2	no	—
176	C₂²×C₁₃	#418	1	1	yes	characteristic
177	C₂²×C₁₃	#459	1	1	yes	characteristic
178	C₂²×C₁₃	#460	1	1	yes	characteristic
179	C₄×C₁₃	—	4	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	#390	1	1	yes	characteristic
183	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
184	C₄×C₁₃	#459	1	1	yes	characteristic
185	C₄×C₁₃	#460	1	1	yes	characteristic
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×D₄	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
194	C₂×SD₁₆	—	2	4	no	—
195	C₂×SD₁₆	—	4	1	no	—
196	C₂×SD₁₆	—	4	1	no	—
197	C₂×SD₁₆	—	4	2	no	—
198	C₂×D₈	—	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	C₂×C₄×C₁₃	1	2	yes	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	#124	—	2	4	no	—
208	#124	—	4	2	no	—
209	#117	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	4	2	no	—
211	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
212	M₃₂	—	2	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	D₄×C₁₃	1	2	yes	—
220	#107	D₄×C₁₃	1	2	yes	—
221	#106	—	2	1	no	—
222	#97	—	4	1	no	—
223	C₄×C₈	—	4	1	no	—
224	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
225	C₄×C₈	D₄×C₁₃	1	2	yes	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	16	2	no	—
229	C₂×C₁₃	#1982	1	1	yes	characteristic
230	C₂×C₁₃	#2212	1	2	yes	—
231	C₂⁴	—	4	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	Q₁₆	—	2	2	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	4	1	no	—
250	Q₁₆	—	4	1	no	—
251	SD₁₆	—	4	4	no	—
252	SD₁₆	—	4	4	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	1	no	—
256	D₈	—	4	1	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
261	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
262	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
263	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
269	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	4	8	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
276	C₁₆	—	4	1	no	—
277	C₁₆	—	4	1	no	—
278	C₁₃	#13260	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	—	4	2	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	1	no	—
287	Q₈	—	8	1	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	8	1	no	—
291	D₄	—	8	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
303	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	C₁₃×#102	1	2	yes	—
306	C₈	C₁₃×#102	1	2	yes	—
307	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
308	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
309	C₈	C₁₃×#124	1	1	yes	characteristic
310	C₈	C₁₃×#127	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	C₁₃×#418	1	1	yes	characteristic
317	C₂²	C₁₃×#459	1	1	yes	characteristic
318	C₂²	C₁₃×#460	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₁₃×#390	1	1	yes	characteristic
323	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
324	C₄	C₁₃×#459	1	1	yes	characteristic
325	C₄	C₁₃×#460	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	16	2	no	—
329	C₂	C₁₃×#1982	1	1	yes	characteristic
330	C₂	C₁₃×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#364	C₂	semidirect
3. C₂×C₁₃×#362	C₂	semidirect
5. C₁₃×#2141	C₂	semidirect
7. C₁₃×#1772	C₂	semidirect
15. C₁₃×#495	C₄	semidirect
66. #13260	C₁₃	direct (N × H)
115. C₂×#364	C₂×C₁₃	semidirect
116. C₂×#362	C₂×C₁₃	semidirect
118. #2141	C₂×C₁₃	semidirect
120. #1772	C₂×C₁₃	semidirect
160. #495	C₄×C₁₃	semidirect
278. C₁₃	#13260	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×#495	2
C₂×C₁₃×#364	2
C₂×C₁₃×#362	2
C₂×C₁₃×#361	2
C₁₃×#2141	2
C₁₃×#2140	2
C₁₃×#1772	2
#13260	13

Subgroups — C13×#13260 #15699599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13260 #15699599