Subgroups — C₂×C₁₃×#2634 #15720939

Order 3328.

2286 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	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2634	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#400	C₂	1	2	yes	maximal
5	C₂×C₁₃×#400	C₂	1	2	yes	maximal
6	C₂×C₁₃×#398	C₂	1	2	yes	maximal
7	C₂×C₁₃×#395	C₂	1	2	yes	maximal
8	C₂×C₁₃×#387	C₂	1	4	yes	maximal
9	C₂×C₁₃×#383	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#125	—	2	8	no	—
12	C₂×C₁₃×#125	C₂²	1	8	yes	—
13	C₂×C₄×Q₈×C₁₃	—	2	8	no	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
15	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#400	C₂²	1	16	yes	—
25	C₁₃×#400	C₂²	1	16	yes	—
26	C₁₃×#398	C₂²	1	16	yes	—
27	C₁₃×#395	C₂²	1	16	yes	—
28	C₁₃×#387	C₂²	1	32	yes	—
29	C₁₃×#383	C₂²	1	8	yes	—
30	C₂×C₁₃×#95	C₂²	1	2	yes	—
31	C₂×C₁₃×#95	C₂²	1	2	yes	—
32	C₂×C₁₃×#95	C₂²	1	2	yes	—
33	C₂×C₁₃×#95	C₂²	1	4	yes	—
34	C₂×C₁₃×#95	C₂²	1	4	yes	—
35	C₂×C₁₃×#95	C₂²	1	4	yes	—
36	C₂²×Q₈×C₁₃	—	2	4	no	—
37	C₂²×Q₈×C₁₃	—	2	8	no	—
38	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
43	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
47	C₁₃×#125	—	2	32	no	—
48	C₄×Q₈×C₁₃	—	2	32	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
56	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
57	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
58	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
59	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
60	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
61	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
62	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
63	C₂×C₄²×C₁₃	—	2	8	no	—
64	C₂×C₄²×C₁₃	D₄	1	8	yes	—
65	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
66	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
67	C₁₃×#95	C₂³	1	8	yes	—
68	C₁₃×#95	C₂³	1	8	yes	—
69	C₁₃×#95	C₂³	1	8	yes	—
70	C₁₃×#95	C₂³	1	16	yes	—
71	C₁₃×#95	C₂³	1	16	yes	—
72	C₁₃×#95	C₂³	1	16	yes	—
73	C₂×#2634	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂×Q₈×C₁₃	—	2	32	no	—
76	C₂×Q₈×C₁₃	—	2	8	no	—
77	C₂×Q₈×C₁₃	—	2	16	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₃	—	2	4	no	—
83	C₂²×C₄×C₁₃	—	2	4	no	—
84	C₂²×C₄×C₁₃	—	2	4	no	—
85	C₂²×C₄×C₁₃	—	2	4	no	—
86	C₂²×C₄×C₁₃	—	2	8	no	—
87	C₂²×C₄×C₁₃	—	2	8	no	—
88	C₂²×C₄×C₁₃	—	2	8	no	—
89	C₂²×C₄×C₁₃	—	2	8	no	—
90	C₂²×C₄×C₁₃	—	2	8	no	—
91	C₂²×C₄×C₁₃	—	2	8	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₁₃	#41	1	8	yes	—
96	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
97	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
98	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
99	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
100	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
102	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
103	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
104	C₁₃×C₄⋊C₄	—	2	16	no	—
105	C₁₃×C₄⋊C₄	—	2	8	no	—
106	C₁₃×C₄⋊C₄	—	2	16	no	—
107	C₁₃×C₄⋊C₄	—	2	8	no	—
108	C₁₃×C₄⋊C₄	—	2	8	no	—
109	C₁₃×C₄⋊C₄	—	2	16	no	—
110	C₁₃×C₄⋊C₄	—	2	32	no	—
111	C₄²×C₁₃	—	2	16	no	—
112	C₄²×C₁₃	C₂×D₄	1	16	yes	—
113	C₂²×#125	C₂×C₁₃	1	1	yes	characteristic
114	C₂²×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic
115	#2634	C₂×C₁₃	1	16	yes	direct product factor
116	C₂×#400	C₂×C₁₃	1	2	yes	—
117	C₂×#400	C₂×C₁₃	1	2	yes	—
118	C₂×#398	C₂×C₁₃	1	2	yes	—
119	C₂×#395	C₂×C₁₃	1	2	yes	—
120	C₂×#387	C₂×C₁₃	1	4	yes	—
121	C₂×#383	C₂×C₁₃	1	1	yes	characteristic
122	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
123	C₂³×C₁₃	C₂×#41	1	4	yes	—
124	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
125	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
126	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	32	no	—
128	C₂×C₄×C₁₃	—	2	8	no	—
129	C₂×C₄×C₁₃	—	2	16	no	—
130	C₂×C₄×C₁₃	—	2	8	no	—
131	C₂×C₄×C₁₃	—	2	8	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	4	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	8	no	—
141	C₂×C₄×C₁₃	—	2	8	no	—
142	C₂×C₄×C₁₃	—	2	8	no	—
143	C₂×C₄×C₁₃	—	2	8	no	—
144	C₂×C₄×C₁₃	—	2	8	no	—
145	C₂×C₄×C₁₃	—	2	8	no	—
146	C₂×C₄×C₁₃	—	4	8	no	—
147	C₂×C₄×C₁₃	—	4	8	no	—
148	C₂×C₄×C₁₃	—	4	8	no	—
149	C₂×C₄×C₁₃	—	4	16	no	—
150	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
151	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
153	C₂×C₄×C₁₃	C₂×#41	1	8	yes	—
154	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#125	—	2	8	no	—
156	C₂×#125	C₂²×C₁₃	1	8	yes	—
157	C₂×C₄×Q₈	—	2	8	no	—
158	C₂×C₄×Q₈	C₂²×C₁₃	1	8	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
160	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
161	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
166	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
167	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
168	#400	C₂²×C₁₃	1	16	yes	—
169	#400	C₂²×C₁₃	1	16	yes	—
170	#398	C₂²×C₁₃	1	16	yes	—
171	#395	C₂²×C₁₃	1	16	yes	—
172	#387	C₂²×C₁₃	1	32	yes	—
173	#383	C₂²×C₁₃	1	8	yes	—
174	C₂×#95	C₂²×C₁₃	1	2	yes	—
175	C₂×#95	C₂²×C₁₃	1	4	yes	—
176	C₂×#95	C₂²×C₁₃	1	2	yes	—
177	C₂×#95	C₂²×C₁₃	1	4	yes	—
178	C₂×#95	C₂²×C₁₃	1	4	yes	—
179	C₂×#95	C₂²×C₁₃	1	2	yes	—
180	C₂²×C₁₃	#525	1	4	yes	—
181	C₂²×C₁₃	#536	1	4	yes	—
182	C₂²×C₁₃	#536	1	4	yes	—
183	C₂²×C₁₃	#541	1	16	yes	—
184	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
185	C₂²×C₁₃	C₂²×#41	1	2	yes	—
186	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
187	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
188	C₄×C₁₃	—	2	4	no	—
189	C₄×C₁₃	—	2	8	no	—
190	C₄×C₁₃	—	2	4	no	—
191	C₄×C₁₃	—	2	4	no	—
192	C₄×C₁₃	—	4	4	no	—
193	C₄×C₁₃	—	4	4	no	—
194	C₄×C₁₃	—	4	4	no	—
195	C₄×C₁₃	—	4	8	no	—
196	C₂²×Q₈	—	2	4	no	—
197	C₂²×Q₈	—	2	8	no	—
198	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
199	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
200	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
201	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
202	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
203	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
205	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
207	#125	—	2	32	no	—
208	C₄×Q₈	—	2	32	no	—
209	C₂×C₄⋊C₄	—	2	8	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	8	no	—
215	C₂×C₄⋊C₄	—	2	16	no	—
216	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
217	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
219	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
220	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
221	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
222	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	—
223	C₂×C₄²	—	2	8	no	—
224	C₂×C₄²	D₄×C₁₃	1	8	yes	—
225	C₂×C₄²	C₂³×C₁₃	1	8	yes	—
226	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
227	#95	C₂³×C₁₃	1	8	yes	—
228	#95	C₂³×C₁₃	1	8	yes	—
229	#95	C₂³×C₁₃	1	8	yes	—
230	#95	C₂³×C₁₃	1	16	yes	—
231	#95	C₂³×C₁₃	1	16	yes	—
232	#95	C₂³×C₁₃	1	16	yes	—
233	C₂×C₁₃	#2634	1	8	yes	direct product factor
234	C₂×C₁₃	C₂×#525	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#536	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#536	1	1	yes	characteristic
237	C₂×C₁₃	C₂×#541	1	4	yes	—
238	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
239	C₂×Q₈	—	2	32	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	16	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
257	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
258	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
259	C₂²×C₄	C₁₃×#41	1	8	yes	—
260	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
261	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
262	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
263	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
264	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
267	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
268	C₄⋊C₄	—	2	16	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	16	no	—
271	C₄⋊C₄	—	2	8	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	16	no	—
274	C₄⋊C₄	—	2	32	no	—
275	C₄²	—	2	16	no	—
276	C₄²	C₂×D₄×C₁₃	1	16	yes	—
277	C₁₃	C₂×#2634	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
279	C₂³	C₂×C₁₃×#41	1	4	yes	—
280	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
281	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
282	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
283	Q₈	—	2	32	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	16	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	16	no	—
306	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
307	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
308	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
309	C₂×C₄	C₂×C₁₃×#41	1	8	yes	—
310	C₂²	C₁₃×#525	1	4	yes	—
311	C₂²	C₁₃×#536	1	4	yes	—
312	C₂²	C₁₃×#536	1	4	yes	—
313	C₂²	C₁₃×#541	1	16	yes	—
314	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
315	C₂²	C₂²×C₁₃×#41	1	2	yes	—
316	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	2	yes	—
317	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	2	8	no	—
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	8	no	—
326	C₂	C₁₃×#2634	1	8	yes	direct product factor
327	C₂	C₂×C₁₃×#525	1	1	yes	characteristic
328	C₂	C₂×C₁₃×#536	1	1	yes	characteristic
329	C₂	C₂×C₁₃×#536	1	1	yes	characteristic
330	C₂	C₂×C₁₃×#541	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
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
3. C₁₃×#2634	C₂	direct (N × H)
73. C₂×#2634	C₁₃	direct (N × H)
115. #2634	C₂×C₁₃	direct (N × H)
233. C₂×C₁₃	#2634	direct (N × H)
277. C₁₃	C₂×#2634	direct (N × H)
326. C₂	C₁₃×#2634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#125	2
C₂²×C₄×Q₈×C₁₃	2
C₁₃×#2634	2
C₂×C₁₃×#400	2
C₂×C₁₃×#398	2
C₂×C₁₃×#395	2
C₂×C₁₃×#387	2
C₂×C₁₃×#383	2
C₂×#2634	13

Subgroups — C2×C13×#2634 #15720939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2634 #15720939