Subgroups — C₂×C₁₃×#1543 #15704037

Order 3328.

1274 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#431	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1543	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#333	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#331	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×#41	C₄	1	2	yes	—
7	C₁₃×#532	—	2	4	no	—
8	C₁₃×#532	C₄	1	4	yes	—
9	C₁₃×#532	C₂²	1	4	yes	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#124	C₄	1	2	yes	—
13	C₂×C₁₃×#121	—	2	2	no	—
14	C₂×C₄×D₄×C₁₃	—	2	2	no	—
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#431	—	2	4	no	—
17	C₁₃×#431	C₂²	1	8	yes	complemented
18	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
20	C₂×C₄×C₈×C₁₃	—	2	2	no	—
21	C₁₃×#333	C₂²	1	8	yes	complemented
22	C₁₃×#331	C₂²	1	8	yes	complemented
23	C₂×C₁₃×#41	—	2	4	no	—
24	C₂×C₁₃×#41	—	4	8	no	—
25	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	2	2	no	—
29	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₁₃×C₄⋊Q₈	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
37	C₁₃×#127	—	2	2	no	—
38	C₁₃×#127	C₂×C₄	1	2	yes	—
39	C₁₃×#127	C₂³	1	2	yes	—
40	C₁₃×#124	—	2	4	no	—
41	C₁₃×#124	C₂×C₄	1	8	yes	—
42	C₁₃×#121	—	2	8	no	—
43	C₁₃×#121	—	2	8	no	—
44	C₄×D₄×C₁₃	—	2	8	no	—
45	C₄×D₄×C₁₃	—	2	8	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₃	D₄	1	4	yes	—
51	C₂×C₄²×C₁₃	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₈	—	2	4	no	—
55	C₁₃×C₄⋊C₈	C₂³	1	8	yes	—
56	C₁₃×C₂²⋊C₈	—	2	8	no	—
57	C₄×C₈×C₁₃	—	2	8	no	—
58	C₂×#1543	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	2	2	no	—
60	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₁₃×#41	—	4	16	no	—
62	C₁₃×#41	—	4	16	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	4	4	no	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂×D₄×C₁₃	—	2	8	no	—
67	C₂×D₄×C₁₃	—	2	4	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	2	4	no	—
70	C₂×D₄×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	4	8	no	—
72	C₂×D₄×C₁₃	—	4	8	no	—
73	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	8	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	8	no	—
78	C₂²×C₄×C₁₃	—	4	8	no	—
79	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	8	no	—
88	C₂×C₈×C₁₃	—	4	8	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₂²⋊C₄	—	2	16	no	—
92	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
93	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	4	yes	—
97	C₄²×C₁₃	C₂×D₄	1	2	yes	—
98	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂×#532	C₂×C₁₃	1	1	yes	characteristic
100	C₂×#431	C₂×C₁₃	1	2	yes	complemented
101	#1543	C₂×C₁₃	1	8	yes	direct product factor
102	C₂×#333	C₂×C₁₃	1	2	yes	complemented
103	C₂×#331	C₂×C₁₃	1	2	yes	complemented
104	C₂³×C₁₃	—	2	2	no	—
105	C₂³×C₁₃	—	2	4	no	—
106	C₂³×C₁₃	—	2	4	no	—
107	C₂³×C₁₃	—	4	4	no	—
108	C₂³×C₁₃	—	4	8	no	—
109	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	4	4	no	—
112	Q₈×C₁₃	—	4	4	no	—
113	D₄×C₁₃	—	4	4	no	—
114	D₄×C₁₃	—	4	4	no	—
115	D₄×C₁₃	—	4	16	no	—
116	D₄×C₁₃	—	4	8	no	—
117	D₄×C₁₃	—	4	8	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	8	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	16	no	—
127	C₂×C₄×C₁₃	#100	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₃	#135	1	8	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
135	C₈×C₁₃	—	4	8	no	—
136	C₂²×#41	C₄×C₁₃	1	2	yes	—
137	#532	—	2	4	no	—
138	#532	C₄×C₁₃	1	4	yes	—
139	#532	C₂²×C₁₃	1	4	yes	—
140	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×#124	C₄×C₁₃	1	2	yes	—
143	C₂×#121	—	2	2	no	—
144	C₂×C₄×D₄	—	2	2	no	—
145	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
146	#431	—	2	4	no	—
147	#431	C₂²×C₁₃	1	8	yes	complemented
148	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
149	C₂×C₂²⋊C₈	—	2	2	no	—
150	C₂×C₄×C₈	—	2	2	no	—
151	#333	C₂²×C₁₃	1	8	yes	complemented
152	#331	C₂²×C₁₃	1	8	yes	complemented
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	4	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	—	4	8	no	—
158	C₂²×C₁₃	—	8	8	no	—
159	C₂²×C₁₃	C₂×#100	1	2	yes	—
160	C₂²×C₁₃	#416	1	4	yes	—
161	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	4	no	—
165	C₄×C₁₃	—	4	4	no	—
166	C₄×C₁₃	C₂×#100	1	4	yes	—
167	C₄×C₁₃	#416	1	4	yes	—
168	C₄×C₁₃	C₂×#135	1	4	yes	—
169	C₂×#41	—	2	4	no	—
170	C₂×#41	—	4	8	no	—
171	C₂×#41	C₂×C₄×C₁₃	1	8	yes	—
172	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	—	2	2	no	—
175	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
178	C₂³×C₄	D₄×C₁₃	1	2	yes	—
179	C₂²×C₈	—	2	4	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
182	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
183	#127	—	2	2	no	—
184	#127	C₂×C₄×C₁₃	1	2	yes	—
185	#127	C₂³×C₁₃	1	2	yes	—
186	#124	—	2	4	no	—
187	#124	C₂×C₄×C₁₃	1	8	yes	—
188	#121	—	2	8	no	—
189	#121	—	2	8	no	—
190	C₄×D₄	—	2	8	no	—
191	C₄×D₄	—	2	8	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
196	C₂×C₄²	D₄×C₁₃	1	2	yes	—
197	C₂×C₄²	D₄×C₁₃	1	4	yes	—
198	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	C₂³×C₁₃	1	8	yes	—
202	C₂²⋊C₈	—	2	8	no	—
203	C₄×C₈	—	2	8	no	—
204	C₂×C₁₃	—	4	2	no	—
205	C₂×C₁₃	—	8	4	no	—
206	C₂×C₁₃	#1543	1	4	yes	direct product factor
207	C₂×C₁₃	C₂²×#100	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#416	1	2	yes	—
209	C₂⁴	—	2	2	no	—
210	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
211	#41	—	4	16	no	—
212	#41	—	4	16	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	4	4	no	—
215	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
216	C₂×D₄	—	2	8	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	4	8	no	—
222	C₂×D₄	—	4	8	no	—
223	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
230	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
231	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
232	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
236	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	4	8	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	16	no	—
242	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
243	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
244	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
245	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₁₃	1	4	yes	—
247	C₄²	C₂×D₄×C₁₃	1	2	yes	—
248	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
249	C₁₃	C₂×#1543	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	4	8	no	—
255	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
256	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	16	no	—
262	D₄	—	4	8	no	—
263	D₄	—	4	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	C₁₃×#100	1	4	yes	—
274	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
275	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄	C₁₃×#135	1	8	yes	—
279	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
281	C₈	—	4	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	8	no	—
287	C₂²	—	8	8	no	—
288	C₂²	C₂×C₁₃×#100	1	2	yes	—
289	C₂²	C₁₃×#416	1	4	yes	—
290	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	4	no	—
295	C₄	C₂×C₁₃×#100	1	4	yes	—
296	C₄	C₁₃×#416	1	4	yes	—
297	C₄	C₂×C₁₃×#135	1	4	yes	—
298	C₂	—	4	2	no	—
299	C₂	—	8	4	no	—
300	C₂	C₁₃×#1543	1	4	yes	direct product factor
301	C₂	C₂²×C₁₃×#100	1	1	yes	characteristic
302	C₂	C₂×C₁₃×#416	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×C₁₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#431	C₂	semidirect
3. C₁₃×#1543	C₂	direct (N × H)
4. C₂×C₁₃×#333	C₂	semidirect
5. C₂×C₁₃×#331	C₂	semidirect
17. C₁₃×#431	C₂²	semidirect
21. C₁₃×#333	C₂²	semidirect
22. C₁₃×#331	C₂²	semidirect
58. C₂×#1543	C₁₃	direct (N × H)
100. C₂×#431	C₂×C₁₃	semidirect
101. #1543	C₂×C₁₃	direct (N × H)
102. C₂×#333	C₂×C₁₃	semidirect
103. C₂×#331	C₂×C₁₃	semidirect
147. #431	C₂²×C₁₃	semidirect
151. #333	C₂²×C₁₃	semidirect
152. #331	C₂²×C₁₃	semidirect
206. C₂×C₁₃	#1543	direct (N × H)
249. C₁₃	C₂×#1543	direct (N × H)
300. C₂	C₁₃×#1543	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×#532	2
C₂×C₁₃×#431	2
C₁₃×#1543	2
C₂×C₁₃×#333	2
C₂×C₁₃×#331	2
C₂×#1543	13

Subgroups — C2×C13×#1543 #15704037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#1543 #15704037