Subgroups — C₁₃×#10817 #15697156

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#344	C₂	1	2	yes	maximal
3	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1735	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₁₃×#405	—	2	4	no	—
12	C₂×C₁₃×#97	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
14	C₁₃×#344	—	2	4	no	—
15	C₁₃×#341	—	2	2	no	—
16	C₁₃×#340	—	2	2	no	—
17	C₁₃×H(3,ℤ/4)	C₄	1	8	yes	complemented
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
37	C₁₃×#117	—	2	2	no	—
38	C₁₃×#117	—	2	2	no	—
39	C₁₃×#117	—	2	2	no	—
40	C₁₃×#117	—	2	2	no	—
41	C₁₃×#117	C₂×C₄	1	4	yes	—
42	C₁₃×#117	C₂×C₄	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₄²×C₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
49	C₁₃×#97	—	2	4	no	—
50	C₄×C₈×C₁₃	—	2	4	no	—
51	#10817	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	#41	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₂²⋊C₄	—	2	4	no	—
89	C₁₃×C₂²⋊C₄	—	2	4	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	2	4	no	—
93	C₄²×C₁₃	—	4	2	no	—
94	C₄²×C₁₃	C₄²	1	4	yes	—
95	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#344	C₂×C₁₃	1	2	yes	—
97	C₂×#341	C₂×C₁₃	1	1	yes	characteristic
98	#1735	C₂×C₁₃	1	1	yes	characteristic
99	C₂×H(3,ℤ/4)	C₂×C₁₃	1	2	yes	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	#117	1	1	yes	characteristic
102	C₂³×C₁₃	C₄×D₄	1	2	yes	—
103	C₂³×C₁₃	#124	1	1	yes	characteristic
104	C₂³×C₁₃	#125	1	1	yes	characteristic
105	C₂³×C₁₃	#126	1	2	yes	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	8	4	no	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	#117	1	2	yes	—
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	#121	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	#123	1	2	yes	—
124	C₂×C₄×C₁₃	#123	1	2	yes	—
125	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#126	1	2	yes	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	4	4	no	—
132	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
134	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×#117	C₂²×C₁₃	1	2	yes	—
136	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
137	#405	—	2	4	no	—
138	C₂×#97	C₄×C₁₃	1	2	yes	complemented
139	C₂×C₄×C₈	C₄×C₁₃	1	2	yes	complemented
140	#344	—	2	4	no	—
141	#341	—	2	2	no	—
142	#340	—	2	2	no	—
143	H(3,ℤ/4)	C₄×C₁₃	1	8	yes	complemented
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	4	2	no	—
149	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	#382	1	1	yes	characteristic
151	C₂²×C₁₃	#383	1	1	yes	characteristic
152	C₂²×C₁₃	#387	1	2	yes	—
153	C₂²×C₁₃	#388	1	2	yes	—
154	C₄×C₁₃	—	4	4	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#376	1	1	yes	characteristic
157	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₁₃	#380	1	1	yes	characteristic
159	C₄×C₁₃	#381	1	1	yes	characteristic
160	C₄×C₁₃	#387	1	2	yes	—
161	C₄×C₁₃	#388	1	2	yes	—
162	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
172	C₂×M₁₆	D₄×C₁₃	1	4	yes	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
177	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₁₃	1	2	yes	—
180	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	C₂×C₄×C₁₃	1	4	yes	—
186	#117	C₂×C₄×C₁₃	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
193	#97	—	2	4	no	—
194	C₄×C₈	—	2	4	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	#1434	1	1	yes	characteristic
200	C₂×C₁₃	#1833	1	2	yes	—
201	C₂⁴	C₁₃×#41	1	1	yes	characteristic
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₁₃×#41	1	2	yes	—
210	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
211	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
212	C₂²×C₄	C₁₃×#41	1	2	yes	—
213	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	C₁₃×C₂²⋊C₄	1	4	yes	—
232	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
233	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₄²×C₁₃	1	4	yes	—
244	C₁₃	#10817	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	4	2	no	—
246	C₂³	C₁₃×#117	1	1	yes	characteristic
247	C₂³	C₄×D₄×C₁₃	1	2	yes	—
248	C₂³	C₁₃×#124	1	1	yes	characteristic
249	C₂³	C₁₃×#125	1	1	yes	characteristic
250	C₂³	C₁₃×#126	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
259	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
260	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
261	C₂×C₄	C₁₃×#117	1	2	yes	—
262	C₂×C₄	C₁₃×#117	1	2	yes	—
263	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
265	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
266	C₂×C₄	C₁₃×#121	1	2	yes	—
267	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
268	C₂×C₄	C₁₃×#123	1	2	yes	—
269	C₂×C₄	C₁₃×#123	1	2	yes	—
270	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
271	C₂×C₄	C₁₃×#125	1	1	yes	characteristic
272	C₂×C₄	C₁₃×#126	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
283	C₂²	C₁₃×#382	1	1	yes	characteristic
284	C₂²	C₁₃×#383	1	1	yes	characteristic
285	C₂²	C₁₃×#387	1	2	yes	—
286	C₂²	C₁₃×#388	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	C₁₃×#376	1	1	yes	characteristic
290	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
291	C₄	C₁₃×#380	1	1	yes	characteristic
292	C₄	C₁₃×#381	1	1	yes	characteristic
293	C₄	C₁₃×#387	1	2	yes	—
294	C₄	C₁₃×#388	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	C₁₃×#1434	1	1	yes	characteristic
300	C₂	C₁₃×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#117
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	#382
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₃×#97	C₄	semidirect
13. C₂×C₄×C₈×C₁₃	C₄	semidirect
17. C₁₃×H(3,ℤ/4)	C₄	semidirect
51. #10817	C₁₃	direct (N × H)
138. C₂×#97	C₄×C₁₃	semidirect
139. C₂×C₄×C₈	C₄×C₁₃	semidirect
143. H(3,ℤ/4)	C₄×C₁₃	semidirect
244. C₁₃	#10817	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×#405	2
C₂×C₁₃×#344	2
C₂×C₁₃×#341	2
C₁₃×#1735	2
C₂×C₁₃×H(3,ℤ/4)	2
#10817	13

Subgroups — C13×#10817 #15697156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10817 #15697156