Subgroups — C₂×C₁₃×#2568 #15721283

Order 3328.

2614 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2568	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#400	C₂	1	4	yes	maximal
5	C₂×C₁₃×#391	C₂	1	2	yes	maximal
6	C₂×C₁₃×#391	C₂	1	4	yes	maximal
7	C₂×C₁₃×#382	C₂	1	2	yes	maximal
8	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
10	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	—	2	8	no	—
12	C₂×C₁₃×C₄⋊Q₈	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	1	yes	characteristic
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#400	C₂²	1	32	yes	—
24	C₁₃×#391	C₂²	1	32	yes	—
25	C₁₃×#391	C₂²	1	16	yes	—
26	C₁₃×#382	C₂²	1	16	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	8	yes	—
28	C₂×C₁₃×#95	C₂²	1	2	yes	—
29	C₂×C₁₃×#95	C₂²	1	8	yes	—
30	C₂×C₁₃×#95	C₂²	1	8	yes	—
31	C₂²×Q₈×C₁₃	—	2	4	no	—
32	C₂²×Q₈×C₁₃	—	2	8	no	—
33	C₂²×Q₈×C₁₃	—	2	4	no	—
34	C₂²×Q₈×C₁₃	—	2	16	no	—
35	C₂²×Q₈×C₁₃	Q₈	1	8	yes	complemented
36	C₂²×Q₈×C₁₃	C₂³	1	8	yes	—
37	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊Q₈	—	2	32	no	—
46	C₄×Q₈×C₁₃	—	2	32	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
51	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
56	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
57	C₂×C₄²×C₁₃	—	2	8	no	—
58	C₂×C₄²×C₁₃	—	2	8	no	—
59	C₂×C₄²×C₁₃	D₄	1	8	yes	—
60	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
61	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
62	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
63	C₁₃×#95	C₂³	1	8	yes	—
64	C₁₃×#95	C₂³	1	32	yes	—
65	C₁₃×#95	C₂³	1	32	yes	—
66	C₂×#2568	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂×Q₈×C₁₃	—	2	32	no	—
69	C₂×Q₈×C₁₃	—	2	8	no	—
70	C₂×Q₈×C₁₃	—	2	16	no	—
71	C₂×Q₈×C₁₃	—	2	8	no	—
72	C₂×Q₈×C₁₃	—	2	32	no	—
73	C₂×Q₈×C₁₃	—	4	32	no	—
74	C₂×Q₈×C₁₃	C₂×Q₈	1	16	yes	complemented
75	C₂²×C₄×C₁₃	—	2	4	no	—
76	C₂²×C₄×C₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	—	2	8	no	—
80	C₂²×C₄×C₁₃	—	2	8	no	—
81	C₂²×C₄×C₁₃	—	2	8	no	—
82	C₂²×C₄×C₁₃	—	2	16	no	—
83	C₂²×C₄×C₁₃	—	2	8	no	—
84	C₂²×C₄×C₁₃	—	2	8	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
89	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
90	C₂²×C₄×C₁₃	#41	1	4	yes	—
91	C₂²×C₄×C₁₃	#41	1	8	yes	—
92	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
94	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
96	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
97	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
98	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
99	C₁₃×C₄⋊C₄	—	2	16	no	—
100	C₁₃×C₄⋊C₄	—	2	8	no	—
101	C₁₃×C₄⋊C₄	—	2	16	no	—
102	C₁₃×C₄⋊C₄	—	2	16	no	—
103	C₁₃×C₄⋊C₄	—	2	16	no	—
104	C₄²×C₁₃	—	2	16	no	—
105	C₄²×C₁₃	—	2	16	no	—
106	C₄²×C₁₃	C₂×D₄	1	16	yes	—
107	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
108	C₂²×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic
109	#2568	C₂×C₁₃	1	16	yes	direct product factor
110	C₂×#400	C₂×C₁₃	1	4	yes	—
111	C₂×#391	C₂×C₁₃	1	2	yes	—
112	C₂×#391	C₂×C₁₃	1	4	yes	—
113	C₂×#382	C₂×C₁₃	1	2	yes	—
114	C₂×C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
115	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
116	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
117	C₂³×C₁₃	C₂×#41	1	2	yes	—
118	C₂³×C₁₃	C₂×#41	1	4	yes	—
119	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
120	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	32	no	—
122	Q₈×C₁₃	—	4	32	no	—
123	C₂×C₄×C₁₃	—	2	16	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	8	no	—
130	C₂×C₄×C₁₃	—	2	8	no	—
131	C₂×C₄×C₁₃	—	2	8	no	—
132	C₂×C₄×C₁₃	—	2	16	no	—
133	C₂×C₄×C₁₃	—	2	8	no	—
134	C₂×C₄×C₁₃	—	2	8	no	—
135	C₂×C₄×C₁₃	—	4	16	no	—
136	C₂×C₄×C₁₃	—	4	8	no	—
137	C₂×C₄×C₁₃	—	4	16	no	—
138	C₂×C₄×C₁₃	C₂²⋊Q₈	1	16	yes	—
139	C₂×C₄×C₁₃	#124	1	16	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
142	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
143	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
144	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
145	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
146	C₂×C₄×C₁₃	C₂×#41	1	8	yes	—
147	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
148	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×C₄⋊Q₈	—	2	8	no	—
150	C₂×C₄⋊Q₈	C₂²×C₁₃	1	8	yes	—
151	C₂×C₄×Q₈	—	2	8	no	—
152	C₂×C₄×Q₈	C₂²×C₁₃	1	8	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
154	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
155	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
157	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
158	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
160	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
161	#400	C₂²×C₁₃	1	32	yes	—
162	#391	C₂²×C₁₃	1	32	yes	—
163	#391	C₂²×C₁₃	1	16	yes	—
164	#382	C₂²×C₁₃	1	16	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃	1	8	yes	—
166	C₂×#95	C₂²×C₁₃	1	2	yes	—
167	C₂×#95	C₂²×C₁₃	1	8	yes	—
168	C₂×#95	C₂²×C₁₃	1	8	yes	—
169	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	4	yes	—
170	C₂²×C₁₃	C₂×#124	1	4	yes	—
171	C₂²×C₁₃	#536	1	4	yes	—
172	C₂²×C₁₃	#556	1	8	yes	—
173	C₂²×C₁₃	#557	1	8	yes	—
174	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
175	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
176	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
177	C₂²×C₁₃	C₂²×#41	1	2	yes	—
178	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
179	C₄×C₁₃	—	2	8	no	—
180	C₄×C₁₃	—	2	4	no	—
181	C₄×C₁₃	—	4	8	no	—
182	C₄×C₁₃	—	4	4	no	—
183	C₄×C₁₃	—	4	8	no	—
184	C₄×C₁₃	C₂×C₂²⋊Q₈	1	8	yes	—
185	C₄×C₁₃	C₂×#124	1	8	yes	—
186	C₂²×Q₈	—	2	4	no	—
187	C₂²×Q₈	—	2	8	no	—
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	2	16	no	—
190	C₂²×Q₈	Q₈×C₁₃	1	8	yes	complemented
191	C₂²×Q₈	C₂³×C₁₃	1	8	yes	—
192	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
193	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
194	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
195	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
196	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
198	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	32	no	—
201	C₄×Q₈	—	2	32	no	—
202	C₂×C₄⋊C₄	—	2	8	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	8	no	—
206	C₂×C₄⋊C₄	—	2	8	no	—
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
212	C₂×C₄²	—	2	8	no	—
213	C₂×C₄²	—	2	8	no	—
214	C₂×C₄²	D₄×C₁₃	1	8	yes	—
215	C₂×C₄²	C₂³×C₁₃	1	8	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	8	yes	—
217	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
218	#95	C₂³×C₁₃	1	8	yes	—
219	#95	C₂³×C₁₃	1	32	yes	—
220	#95	C₂³×C₁₃	1	32	yes	—
221	C₂×C₁₃	#2568	1	8	yes	direct product factor
222	C₂×C₁₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
223	C₂×C₁₃	C₂²×#124	1	1	yes	characteristic
224	C₂×C₁₃	C₂×#536	1	1	yes	characteristic
225	C₂×C₁₃	C₂×#556	1	2	yes	—
226	C₂×C₁₃	C₂×#557	1	2	yes	—
227	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
228	C₂×Q₈	—	2	32	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	2	16	no	—
231	C₂×Q₈	—	2	8	no	—
232	C₂×Q₈	—	2	32	no	—
233	C₂×Q₈	—	4	32	no	—
234	C₂×Q₈	C₂×Q₈×C₁₃	1	16	yes	complemented
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	16	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
248	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
249	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
250	C₂²×C₄	C₁₃×#41	1	4	yes	—
251	C₂²×C₄	C₁₃×#41	1	8	yes	—
252	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
253	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
254	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
255	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
257	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	16	no	—
262	C₄⋊C₄	—	2	16	no	—
263	C₄⋊C₄	—	2	16	no	—
264	C₄²	—	2	16	no	—
265	C₄²	—	2	16	no	—
266	C₄²	C₂×D₄×C₁₃	1	16	yes	—
267	C₁₃	C₂×#2568	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
269	C₂³	C₂²×Q₈×C₁₃	1	2	yes	—
270	C₂³	C₂×C₁₃×#41	1	2	yes	—
271	C₂³	C₂×C₁₃×#41	1	4	yes	—
272	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
273	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
274	Q₈	—	2	32	no	—
275	Q₈	—	4	32	no	—
276	C₂×C₄	—	2	16	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	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	—	4	16	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	C₁₃×C₂²⋊Q₈	1	16	yes	—
292	C₂×C₄	C₁₃×#124	1	16	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
296	C₂×C₄	C₂²×Q₈×C₁₃	1	8	yes	—
297	C₂×C₄	C₂²×Q₈×C₁₃	1	4	yes	—
298	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
299	C₂×C₄	C₂×C₁₃×#41	1	8	yes	—
300	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	4	yes	—
301	C₂²	C₂×C₁₃×#124	1	4	yes	—
302	C₂²	C₁₃×#536	1	4	yes	—
303	C₂²	C₁₃×#556	1	8	yes	—
304	C₂²	C₁₃×#557	1	8	yes	—
305	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
306	C₂²	C₂³×Q₈×C₁₃	1	1	yes	characteristic
307	C₂²	C₂²×C₁₃×#41	1	1	yes	characteristic
308	C₂²	C₂²×C₁₃×#41	1	2	yes	—
309	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	2	yes	—
310	C₄	—	2	8	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	8	no	—
313	C₄	—	4	4	no	—
314	C₄	—	4	8	no	—
315	C₄	C₂×C₁₃×C₂²⋊Q₈	1	8	yes	—
316	C₄	C₂×C₁₃×#124	1	8	yes	—
317	C₂	C₁₃×#2568	1	8	yes	direct product factor
318	C₂	C₂²×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
319	C₂	C₂²×C₁₃×#124	1	1	yes	characteristic
320	C₂	C₂×C₁₃×#536	1	1	yes	characteristic
321	C₂	C₂×C₁₃×#556	1	2	yes	—
322	C₂	C₂×C₁₃×#557	1	2	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×#2568	C₂	direct (N × H)
35. C₂²×Q₈×C₁₃	Q₈	semidirect
66. C₂×#2568	C₁₃	direct (N × H)
74. C₂×Q₈×C₁₃	C₂×Q₈	semidirect
109. #2568	C₂×C₁₃	direct (N × H)
190. C₂²×Q₈	Q₈×C₁₃	semidirect
221. C₂×C₁₃	#2568	direct (N × H)
234. C₂×Q₈	C₂×Q₈×C₁₃	semidirect
267. C₁₃	C₂×#2568	direct (N × H)
317. C₂	C₁₃×#2568	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×C₄⋊Q₈	2
C₂²×C₄×Q₈×C₁₃	2
C₁₃×#2568	2
C₂×C₁₃×#400	2
C₂×C₁₃×#391	2
C₂×C₁₃×#382	2
C₂×C₄×C₁₃×C₄⋊C₄	2
C₂×#2568	13

Subgroups — C2×C13×#2568 #15721283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2568 #15721283