Subgroups — C₁₃×#13259 #15699598

Order 3328.

472 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2145	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2143	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#135	—	2	1	no	—
7	C₂×C₁₃×#135	C₂²	1	2	yes	—
8	C₁₃×#532	—	2	1	no	—
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₁₃×#495	—	2	1	no	—
11	C₁₃×#495	C₄	1	2	yes	complemented
12	C₁₃×#494	C₄	1	1	yes	characteristic, complemented
13	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
14	C₁₃×#492	—	2	1	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#405	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#364	—	2	1	no	—
20	C₁₃×#364	—	2	1	no	—
21	C₁₃×#360	—	2	4	no	—
22	C₁₃×#359	—	2	4	no	—
23	C₁₃×#339	—	2	1	no	—
24	C₂×C₁₃×#41	—	2	2	no	—
25	C₁₃×#135	—	2	2	no	—
26	C₁₃×#135	—	2	2	no	—
27	C₁₃×#135	—	2	2	no	—
28	C₁₃×#135	—	2	2	no	—
29	C₁₃×#135	—	4	1	no	—
30	C₁₃×#135	—	4	1	no	—
31	C₁₃×#135	—	4	1	no	—
32	C₁₃×#135	—	4	1	no	—
33	C₂×C₁₃×Q₁₆	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×SD₁₆	—	2	2	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×M₁₆	—	2	1	no	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×#127	—	2	1	no	—
44	C₁₃×#124	—	2	2	no	—
45	C₁₃×#121	—	4	2	no	—
46	C₄×D₄×C₁₃	—	4	2	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
49	C₁₃×M₃₂	—	2	2	no	—
50	C₁₃×M₃₂	—	2	2	no	—
51	C₂×C₁₃×C₁₆	—	2	2	no	—
52	C₁₃×#108	—	2	1	no	—
53	C₁₃×#108	D₄	1	2	yes	complemented
54	C₁₃×#107	—	2	1	no	—
55	C₁₃×#106	D₄	1	2	yes	complemented
56	C₁₃×#97	—	4	1	no	—
57	C₄×C₈×C₁₃	—	4	1	no	—
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	#13259	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₃×#41	—	4	2	no	—
64	C₁₃×#41	—	4	2	no	—
65	C₁₃×#41	—	4	2	no	—
66	C₁₃×#41	—	4	2	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₁₃×Q₁₆	—	2	2	no	—
75	C₁₃×Q₁₆	—	2	2	no	—
76	C₁₃×Q₁₆	—	4	1	no	—
77	C₁₃×Q₁₆	—	4	1	no	—
78	C₁₃×SD₁₆	—	4	2	no	—
79	C₁₃×SD₁₆	—	4	2	no	—
80	C₁₃×SD₁₆	—	4	2	no	—
81	C₁₃×SD₁₆	—	4	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	4	1	no	—
85	C₁₃×D₈	—	4	1	no	—
86	C₁₃×M₁₆	—	2	2	no	—
87	C₁₃×M₁₆	—	4	1	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
92	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
93	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₂²⋊C₄	—	4	4	no	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	2	1	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₁₃×C₁₆	—	4	2	no	—
110	#3147	C₂×C₁₃	1	1	yes	characteristic
111	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#2145	C₂×C₁₃	1	2	yes	complemented
113	#2143	C₂×C₁₃	1	2	yes	complemented
114	#1772	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	4	2	no	—
116	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	8	4	no	—
122	D₄×C₁₃	—	8	2	no	—
123	D₄×C₁₃	—	8	2	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	8	2	no	—
130	C₂×C₄×C₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₈×C₁₃	—	4	2	no	—
138	C₈×C₁₃	#102	1	1	yes	characteristic
139	C₈×C₁₃	#102	1	1	yes	characteristic
140	C₈×C₁₃	#102	1	1	yes	characteristic
141	C₈×C₁₃	#102	1	1	yes	characteristic
142	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
143	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
144	C₈×C₁₃	#124	1	1	yes	characteristic
145	C₈×C₁₃	#127	1	1	yes	characteristic
146	C₂×#135	—	2	1	no	—
147	C₂×#135	C₂²×C₁₃	1	2	yes	—
148	#532	—	2	1	no	—
149	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
150	#495	—	2	1	no	—
151	#495	C₄×C₁₃	1	2	yes	complemented
152	#494	C₄×C₁₃	1	1	yes	characteristic, complemented
153	#493	C₄×C₁₃	1	1	yes	characteristic, complemented
154	#492	—	2	1	no	—
155	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
156	#427	C₂²×C₁₃	1	1	yes	characteristic
157	#405	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	#364	—	2	1	no	—
160	#364	—	2	1	no	—
161	#360	—	2	4	no	—
162	#359	—	2	4	no	—
163	#339	—	2	1	no	—
164	C₂²×C₁₃	—	4	1	no	—
165	C₂²×C₁₃	—	8	4	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	#418	1	1	yes	characteristic
168	C₂²×C₁₃	#459	1	1	yes	characteristic
169	C₂²×C₁₃	#460	1	1	yes	characteristic
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	#390	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
176	C₄×C₁₃	#459	1	1	yes	characteristic
177	C₄×C₁₃	#460	1	1	yes	characteristic
178	C₂×#41	—	2	2	no	—
179	#135	—	2	2	no	—
180	#135	—	2	2	no	—
181	#135	—	2	2	no	—
182	#135	—	2	2	no	—
183	#135	—	4	1	no	—
184	#135	—	4	1	no	—
185	#135	—	4	1	no	—
186	#135	—	4	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×SD₁₆	—	2	2	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	1	no	—
197	#127	—	2	1	no	—
198	#124	—	2	2	no	—
199	#121	—	4	2	no	—
200	C₄×D₄	—	4	2	no	—
201	#117	—	2	1	no	—
202	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
203	M₃₂	—	2	2	no	—
204	M₃₂	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	#108	—	2	1	no	—
207	#108	D₄×C₁₃	1	2	yes	complemented
208	#107	—	2	1	no	—
209	#106	D₄×C₁₃	1	2	yes	complemented
210	#97	—	4	1	no	—
211	C₄×C₈	—	4	1	no	—
212	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
214	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
215	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₁₃	—	2	1	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	16	2	no	—
220	C₂×C₁₃	#1982	1	1	yes	characteristic
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	#41	—	4	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	4	1	no	—
235	Q₁₆	—	4	1	no	—
236	SD₁₆	—	4	2	no	—
237	SD₁₆	—	4	2	no	—
238	SD₁₆	—	4	2	no	—
239	SD₁₆	—	4	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	2	2	no	—
242	D₈	—	4	1	no	—
243	D₈	—	4	1	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
250	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
251	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
252	C₂×C₈	C₁₃×SD₁₆	1	1	yes	characteristic
253	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
254	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
258	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
266	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
267	C₁₆	—	4	2	no	—
268	C₁₃	#13259	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	—	4	2	no	—
270	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	8	4	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
288	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
290	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	C₁₃×#102	1	1	yes	characteristic
293	C₈	C₁₃×#102	1	1	yes	characteristic
294	C₈	C₁₃×#102	1	1	yes	characteristic
295	C₈	C₁₃×#102	1	1	yes	characteristic
296	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
297	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
298	C₈	C₁₃×#124	1	1	yes	characteristic
299	C₈	C₁₃×#127	1	1	yes	characteristic
300	C₂²	—	4	1	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₁₃×#418	1	1	yes	characteristic
304	C₂²	C₁₃×#459	1	1	yes	characteristic
305	C₂²	C₁₃×#460	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₁₃×#390	1	1	yes	characteristic
311	C₄	C₂×C₁₃×#102	1	1	yes	characteristic
312	C₄	C₁₃×#459	1	1	yes	characteristic
313	C₄	C₁₃×#460	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	C₁₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1982
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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. C₁₃×#2145	C₂	semidirect
4. C₁₃×#2143	C₂	semidirect
5. C₁₃×#1772	C₂	semidirect
11. C₁₃×#495	C₄	semidirect
12. C₁₃×#494	C₄	semidirect
13. C₁₃×#493	C₄	semidirect
53. C₁₃×#108	D₄	semidirect
55. C₁₃×#106	D₄	semidirect
62. #13259	C₁₃	direct (N × H)
111. C₂×#364	C₂×C₁₃	semidirect
112. #2145	C₂×C₁₃	semidirect
113. #2143	C₂×C₁₃	semidirect
114. #1772	C₂×C₁₃	semidirect
151. #495	C₄×C₁₃	semidirect
152. #494	C₄×C₁₃	semidirect
153. #493	C₄×C₁₃	semidirect
207. #108	D₄×C₁₃	semidirect
209. #106	D₄×C₁₃	semidirect
268. C₁₃	#13259	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3147	2
C₂×C₁₃×#364	2
C₁₃×#2145	2
C₁₃×#2143	2
C₁₃×#1772	2
#13259	13

Subgroups — C13×#13259 #15699598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13259 #15699598