Subgroups — C₂×C₁₃×#2643 #15720904

Order 3328.

2546 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2643	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₁₃×#401	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#400	C₂	1	3	yes	maximal, complemented
4	C₂×C₁₃×#397	C₂	1	3	yes	maximal, complemented
5	C₂×C₁₃×#396	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#386	C₂	1	3	yes	maximal
7	C₂×C₁₃×#385	C₂	1	3	yes	maximal, complemented
8	C₂²×C₁₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₁₃×#401	C₂²	1	8	yes	complemented
17	C₁₃×#400	C₂²	1	24	yes	complemented
18	C₁₃×#397	C₂²	1	24	yes	complemented
19	C₁₃×#396	C₂²	1	8	yes	complemented
20	C₁₃×#386	C₂²	1	24	yes	—
21	C₁₃×#385	C₂²	1	24	yes	complemented
22	C₂×C₁₃×#95	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×#95	C₂²	1	3	yes	—
24	C₂×C₁₃×#95	C₂²	1	3	yes	—
25	C₂×C₁₃×#95	C₂²	1	3	yes	—
26	C₂×C₁₃×#95	C₂²	1	3	yes	—
27	C₂×C₁₃×#95	C₂²	1	3	yes	—
28	C₂×C₁₃×#95	C₂²	1	8	yes	complemented
29	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×D₄×C₁₃	—	2	12	no	—
32	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₃	—	2	12	no	—
34	C₂³×C₄×C₁₃	—	4	6	no	—
35	C₂³×C₄×C₁₃	—	4	6	no	—
36	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
42	C₂×C₁₃×C₄⋊C₄	—	4	6	no	—
43	C₂×C₁₃×C₄⋊C₄	—	4	6	no	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	4	6	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	4	6	no	—
49	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
50	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
51	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
52	C₁₃×#95	C₂³	1	12	yes	—
53	C₁₃×#95	C₂³	1	12	yes	—
54	C₁₃×#95	C₂³	1	12	yes	—
55	C₁₃×#95	C₂³	1	12	yes	—
56	C₁₃×#95	C₂³	1	4	yes	—
57	C₁₃×#95	C₂³	1	12	yes	—
58	C₂×#2643	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	2	6	no	—
60	C₂⁴×C₁₃	—	2	2	no	—
61	C₂⁴×C₁₃	—	2	6	no	—
62	C₂⁴×C₁₃	—	2	6	no	—
63	C₂⁴×C₁₃	—	2	2	no	—
64	C₂⁴×C₁₃	#41	1	12	yes	—
65	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
66	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
67	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂×D₄×C₁₃	—	2	24	no	—
69	C₂×D₄×C₁₃	—	4	16	no	—
70	C₂²×C₄×C₁₃	—	2	6	no	—
71	C₂²×C₄×C₁₃	—	2	6	no	—
72	C₂²×C₄×C₁₃	—	2	24	no	—
73	C₂²×C₄×C₁₃	—	2	6	no	—
74	C₂²×C₄×C₁₃	—	2	6	no	—
75	C₂²×C₄×C₁₃	—	2	6	no	—
76	C₂²×C₄×C₁₃	—	2	6	no	—
77	C₂²×C₄×C₁₃	—	2	6	no	—
78	C₂²×C₄×C₁₃	—	2	6	no	—
79	C₂²×C₄×C₁₃	—	2	6	no	—
80	C₂²×C₄×C₁₃	—	2	6	no	—
81	C₂²×C₄×C₁₃	—	2	6	no	—
82	C₂²×C₄×C₁₃	—	2	6	no	—
83	C₂²×C₄×C₁₃	—	2	6	no	—
84	C₂²×C₄×C₁₃	—	4	12	no	—
85	C₂²×C₄×C₁₃	—	4	12	no	—
86	C₂²×C₄×C₁₃	—	4	48	no	—
87	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
90	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
91	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
92	C₁₃×C₄⋊C₄	—	4	12	no	—
93	C₁₃×C₄⋊C₄	—	4	12	no	—
94	C₁₃×C₂²⋊C₄	—	2	24	no	—
95	C₁₃×C₂²⋊C₄	—	4	12	no	—
96	C₁₃×C₂²⋊C₄	—	4	12	no	—
97	#2643	C₂×C₁₃	1	16	yes	direct product factor
98	C₂×#401	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂×#400	C₂×C₁₃	1	3	yes	complemented
100	C₂×#397	C₂×C₁₃	1	3	yes	complemented
101	C₂×#396	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂×#386	C₂×C₁₃	1	3	yes	—
103	C₂×#385	C₂×C₁₃	1	3	yes	complemented
104	C₂²×#95	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂³×C₁₃	—	2	24	no	—
106	C₂³×C₁₃	—	2	24	no	—
107	C₂³×C₁₃	—	2	6	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	6	no	—
110	C₂³×C₁₃	—	2	6	no	—
111	C₂³×C₁₃	—	2	2	no	—
112	C₂³×C₁₃	—	4	4	no	—
113	C₂³×C₁₃	—	4	12	no	—
114	C₂³×C₁₃	—	4	12	no	—
115	C₂³×C₁₃	#126	1	16	yes	—
116	C₂³×C₁₃	C₂×#41	1	12	yes	—
117	C₂³×C₁₃	C₂×#41	1	6	yes	—
118	C₂³×C₁₃	H(5,𝔽₂)	1	6	yes	—
119	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
120	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
121	D₄×C₁₃	—	4	16	no	—
122	C₂×C₄×C₁₃	—	2	6	no	—
123	C₂×C₄×C₁₃	—	2	6	no	—
124	C₂×C₄×C₁₃	—	2	6	no	—
125	C₂×C₄×C₁₃	—	2	6	no	—
126	C₂×C₄×C₁₃	—	2	6	no	—
127	C₂×C₄×C₁₃	—	2	6	no	—
128	C₂×C₄×C₁₃	—	2	6	no	—
129	C₂×C₄×C₁₃	—	2	6	no	—
130	C₂×C₄×C₁₃	—	2	6	no	—
131	C₂×C₄×C₁₃	—	2	6	no	—
132	C₂×C₄×C₁₃	—	2	6	no	—
133	C₂×C₄×C₁₃	—	2	6	no	—
134	C₂×C₄×C₁₃	—	2	6	no	—
135	C₂×C₄×C₁₃	—	4	48	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	4	12	no	—
138	C₂×C₄×C₁₃	—	4	12	no	—
139	C₂×C₄×C₁₃	—	4	12	no	—
140	C₂×C₄×C₁₃	—	4	12	no	—
141	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
142	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
145	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	3	yes	—
146	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	3	yes	—
147	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	3	yes	—
148	#401	C₂²×C₁₃	1	8	yes	complemented
149	#400	C₂²×C₁₃	1	24	yes	complemented
150	#397	C₂²×C₁₃	1	24	yes	complemented
151	#396	C₂²×C₁₃	1	8	yes	complemented
152	#386	C₂²×C₁₃	1	24	yes	—
153	#385	C₂²×C₁₃	1	24	yes	complemented
154	C₂×#95	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#95	C₂²×C₁₃	1	3	yes	—
156	C₂×#95	C₂²×C₁₃	1	3	yes	—
157	C₂×#95	C₂²×C₁₃	1	3	yes	—
158	C₂×#95	C₂²×C₁₃	1	3	yes	—
159	C₂×#95	C₂²×C₁₃	1	3	yes	—
160	C₂×#95	C₂²×C₁₃	1	8	yes	complemented
161	C₂²×C₁₃	—	2	32	no	—
162	C₂²×C₁₃	—	2	12	no	—
163	C₂²×C₁₃	—	2	12	no	—
164	C₂²×C₁₃	—	4	2	no	—
165	C₂²×C₁₃	—	4	6	no	—
166	C₂²×C₁₃	—	4	6	no	—
167	C₂²×C₁₃	—	8	8	no	—
168	C₂²×C₁₃	C₂×#126	1	8	yes	—
169	C₂²×C₁₃	C₂×#126	1	4	yes	—
170	C₂²×C₁₃	#538	1	12	yes	—
171	C₂²×C₁₃	#539	1	12	yes	—
172	C₂²×C₁₃	C₂²×#41	1	3	yes	—
173	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	3	yes	—
174	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
175	C₄×C₁₃	—	4	2	no	—
176	C₄×C₁₃	—	4	6	no	—
177	C₄×C₁₃	—	4	6	no	—
178	C₄×C₁₃	—	4	6	no	—
179	C₄×C₁₃	—	4	6	no	—
180	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
181	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
182	C₂²×D₄	—	2	12	no	—
183	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
184	C₂³×C₄	—	2	12	no	—
185	C₂³×C₄	—	4	6	no	—
186	C₂³×C₄	—	4	6	no	—
187	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
188	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
189	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
190	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
191	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
192	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
193	C₂×C₄⋊C₄	—	4	6	no	—
194	C₂×C₄⋊C₄	—	4	6	no	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	12	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	12	yes	—
197	C₂×C₂²⋊C₄	—	2	12	no	—
198	C₂×C₂²⋊C₄	—	4	6	no	—
199	C₂×C₂²⋊C₄	—	4	6	no	—
200	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
201	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
202	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
203	#95	C₂³×C₁₃	1	4	yes	—
204	#95	C₂³×C₁₃	1	12	yes	—
205	#95	C₂³×C₁₃	1	12	yes	—
206	#95	C₂³×C₁₃	1	12	yes	—
207	#95	C₂³×C₁₃	1	12	yes	—
208	#95	C₂³×C₁₃	1	12	yes	—
209	C₂×C₁₃	—	2	8	no	—
210	C₂×C₁₃	—	8	2	no	—
211	C₂×C₁₃	#2643	1	8	yes	direct product factor
212	C₂×C₁₃	C₂²×#126	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#538	1	3	yes	—
214	C₂×C₁₃	C₂×#539	1	3	yes	—
215	C₂⁴	—	2	6	no	—
216	C₂⁴	—	2	2	no	—
217	C₂⁴	—	2	6	no	—
218	C₂⁴	—	2	6	no	—
219	C₂⁴	—	2	2	no	—
220	C₂⁴	C₁₃×#41	1	12	yes	—
221	C₂⁴	C₂⁴×C₁₃	1	2	yes	—
222	C₂⁴	C₂⁴×C₁₃	1	2	yes	—
223	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₂×D₄	—	2	24	no	—
225	C₂×D₄	—	4	16	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	2	24	no	—
229	C₂²×C₄	—	2	6	no	—
230	C₂²×C₄	—	2	6	no	—
231	C₂²×C₄	—	2	6	no	—
232	C₂²×C₄	—	2	6	no	—
233	C₂²×C₄	—	2	6	no	—
234	C₂²×C₄	—	2	6	no	—
235	C₂²×C₄	—	2	6	no	—
236	C₂²×C₄	—	2	6	no	—
237	C₂²×C₄	—	2	6	no	—
238	C₂²×C₄	—	2	6	no	—
239	C₂²×C₄	—	2	6	no	—
240	C₂²×C₄	—	4	12	no	—
241	C₂²×C₄	—	4	12	no	—
242	C₂²×C₄	—	4	48	no	—
243	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
244	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
245	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
246	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
247	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
248	C₄⋊C₄	—	4	12	no	—
249	C₄⋊C₄	—	4	12	no	—
250	C₂²⋊C₄	—	2	24	no	—
251	C₂²⋊C₄	—	4	12	no	—
252	C₂²⋊C₄	—	4	12	no	—
253	C₁₃	C₂×#2643	1	1	yes	characteristic, direct product factor, Sylow
254	C₂³	—	2	24	no	—
255	C₂³	—	2	24	no	—
256	C₂³	—	2	6	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	6	no	—
259	C₂³	—	2	6	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	12	no	—
263	C₂³	—	4	12	no	—
264	C₂³	C₁₃×#126	1	16	yes	—
265	C₂³	C₂×C₁₃×#41	1	12	yes	—
266	C₂³	C₂×C₁₃×#41	1	6	yes	—
267	C₂³	C₁₃×H(5,𝔽₂)	1	6	yes	—
268	C₂³	C₁₃×2₋¹⁺⁴	1	2	yes	—
269	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
270	D₄	—	4	16	no	—
271	C₂×C₄	—	2	6	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	6	no	—
275	C₂×C₄	—	2	6	no	—
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	6	no	—
279	C₂×C₄	—	2	6	no	—
280	C₂×C₄	—	2	6	no	—
281	C₂×C₄	—	2	6	no	—
282	C₂×C₄	—	2	6	no	—
283	C₂×C₄	—	2	6	no	—
284	C₂×C₄	—	4	48	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	12	no	—
287	C₂×C₄	—	4	12	no	—
288	C₂×C₄	—	4	12	no	—
289	C₂×C₄	—	4	12	no	—
290	C₂²	—	2	32	no	—
291	C₂²	—	2	12	no	—
292	C₂²	—	2	12	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	6	no	—
295	C₂²	—	4	6	no	—
296	C₂²	—	8	8	no	—
297	C₂²	C₂×C₁₃×#126	1	8	yes	—
298	C₂²	C₂×C₁₃×#126	1	4	yes	—
299	C₂²	C₁₃×#538	1	12	yes	—
300	C₂²	C₁₃×#539	1	12	yes	—
301	C₂²	C₂²×C₁₃×#41	1	3	yes	—
302	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	3	yes	—
303	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
304	C₄	—	4	2	no	—
305	C₄	—	4	6	no	—
306	C₄	—	4	6	no	—
307	C₄	—	4	6	no	—
308	C₄	—	4	6	no	—
309	C₂	—	2	8	no	—
310	C₂	—	8	2	no	—
311	C₂	C₁₃×#2643	1	8	yes	direct product factor
312	C₂	C₂²×C₁₃×#126	1	1	yes	characteristic
313	C₂	C₂×C₁₃×#538	1	3	yes	—
314	C₂	C₂×C₁₃×#539	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
prime radical	C₂³×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2643	C₂	direct (N × H)
2. C₂×C₁₃×#401	C₂	semidirect
3. C₂×C₁₃×#400	C₂	semidirect
4. C₂×C₁₃×#397	C₂	semidirect
5. C₂×C₁₃×#396	C₂	semidirect
7. C₂×C₁₃×#385	C₂	semidirect
8. C₂²×C₁₃×#95	C₂	semidirect
16. C₁₃×#401	C₂²	semidirect
17. C₁₃×#400	C₂²	semidirect
18. C₁₃×#397	C₂²	semidirect
19. C₁₃×#396	C₂²	semidirect
21. C₁₃×#385	C₂²	semidirect
28. C₂×C₁₃×#95	C₂²	semidirect
58. C₂×#2643	C₁₃	direct (N × H)
97. #2643	C₂×C₁₃	direct (N × H)
98. C₂×#401	C₂×C₁₃	semidirect
99. C₂×#400	C₂×C₁₃	semidirect
100. C₂×#397	C₂×C₁₃	semidirect
101. C₂×#396	C₂×C₁₃	semidirect
103. C₂×#385	C₂×C₁₃	semidirect
104. C₂²×#95	C₂×C₁₃	semidirect
148. #401	C₂²×C₁₃	semidirect
149. #400	C₂²×C₁₃	semidirect
150. #397	C₂²×C₁₃	semidirect
151. #396	C₂²×C₁₃	semidirect
153. #385	C₂²×C₁₃	semidirect
160. C₂×#95	C₂²×C₁₃	semidirect
211. C₂×C₁₃	#2643	direct (N × H)
253. C₁₃	C₂×#2643	direct (N × H)
311. C₂	C₁₃×#2643	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×#2643	2
C₂×C₁₃×#401	2
C₂×C₁₃×#400	2
C₂×C₁₃×#397	2
C₂×C₁₃×#396	2
C₂×C₁₃×#386	2
C₂×C₁₃×#385	2
C₂²×C₁₃×#95	2
C₂×#2643	13

Subgroups — C2×C13×#2643 #15720904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2643 #15720904