Subgroups — C₁₃×#21820 #15708159

Order 3328.

1270 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1985	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1984	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1817	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	—	2	4	no	—
9	C₂×C₄×Q₈×C₁₃	—	2	4	no	—
10	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃	C₄	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
16	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#432	—	2	4	no	—
19	C₁₃×#431	—	2	4	no	—
20	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₁₃×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#336	C₂²	1	16	yes	—
27	C₂²×Q₈×C₁₃	—	2	8	no	—
28	C₂²×Q₈×C₁₃	—	2	4	no	—
29	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₄×Q₈×C₁₃	—	2	8	no	—
39	C₄×Q₈×C₁₃	—	2	8	no	—
40	C₄×Q₈×C₁₃	—	2	4	no	—
41	C₄×Q₈×C₁₃	—	2	4	no	—
42	C₄×Q₈×C₁₃	—	2	4	no	—
43	C₄×Q₈×C₁₃	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₁₃	—	2	4	no	—
51	C₂×C₄²×C₁₃	—	2	4	no	—
52	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	D₄	1	4	yes	—
56	C₂×C₄²×C₁₃	D₄	1	4	yes	—
57	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
60	C₁₃×C₄⋊C₈	—	2	4	no	—
61	C₁₃×C₄⋊C₈	—	2	4	no	—
62	C₁₃×C₂²⋊C₈	—	2	4	no	—
63	C₁₃×C₂²⋊C₈	—	2	4	no	—
64	C₁₃×#97	—	2	4	no	—
65	C₄×C₈×C₁₃	—	2	4	no	—
66	#21820	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×Q₈×C₁₃	—	2	32	no	—
69	C₂×Q₈×C₁₃	—	2	4	no	—
70	C₂×Q₈×C₁₃	—	2	4	no	—
71	C₂×Q₈×C₁₃	—	2	8	no	—
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	M₁₆	1	8	yes	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	4	no	—
76	C₂²×C₄×C₁₃	—	2	8	no	—
77	C₂²×C₄×C₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₃	—	2	8	no	—
81	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₂×C₈×C₁₃	—	2	4	no	—
95	C₂×C₈×C₁₃	—	2	8	no	—
96	C₂×C₈×C₁₃	—	2	8	no	—
97	C₁₃×C₄⋊C₄	—	2	8	no	—
98	C₁₃×C₄⋊C₄	—	2	8	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₄²×C₁₃	—	2	8	no	—
102	C₄²×C₁₃	—	2	4	no	—
103	C₄²×C₁₃	—	2	4	no	—
104	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
105	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
106	C₄²×C₁₃	C₂×D₄	1	2	yes	—
107	C₄²×C₁₃	C₂×D₄	1	2	yes	—
108	C₄²×C₁₃	C₂×D₄	1	2	yes	—
109	C₄²×C₁₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic
111	C₂×#432	C₂×C₁₃	1	1	yes	characteristic
112	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
113	#1985	C₂×C₁₃	1	4	yes	complemented
114	#1984	C₂×C₁₃	1	4	yes	complemented
115	#1817	C₂×C₁₃	1	4	yes	—
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂³×C₁₃	#131	1	4	yes	—
120	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	16	no	—
122	Q₈×C₁₃	—	2	16	no	—
123	C₂×C₄×C₁₃	—	2	16	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	8	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	8	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	8	no	—
136	C₂×C₄×C₁₃	—	4	8	no	—
137	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
143	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
147	C₈×C₁₃	—	4	4	no	—
148	C₈×C₁₃	—	4	4	no	—
149	C₂³×Q₈	C₄×C₁₃	1	1	yes	characteristic
150	C₂×C₄×Q₈	—	2	4	no	—
151	C₂×C₄×Q₈	—	2	4	no	—
152	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
153	C₂×C₄×Q₈	C₄×C₁₃	1	2	yes	—
154	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
156	C₂²×C₄⋊C₄	C₄×C₁₃	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
158	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
160	#432	—	2	4	no	—
161	#431	—	2	4	no	—
162	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
164	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
165	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
166	C₂×#97	C₂²×C₁₃	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
168	#336	C₂²×C₁₃	1	16	yes	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	2	2	no	—
171	C₂²×C₁₃	—	2	8	no	—
172	C₂²×C₁₃	#408	1	4	yes	—
173	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₁₃	#519	1	2	yes	—
175	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
176	C₂²×C₁₃	C₂×#131	1	2	yes	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	8	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	4	4	no	—
183	C₄×C₁₃	#407	1	4	yes	—
184	C₄×C₁₃	#536	1	2	yes	—
185	C₄×C₁₃	#536	1	2	yes	—
186	C₂²×Q₈	—	2	8	no	—
187	C₂²×Q₈	—	2	4	no	—
188	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
189	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
190	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
191	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
192	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
193	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
194	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
197	C₄×Q₈	—	2	8	no	—
198	C₄×Q₈	—	2	8	no	—
199	C₄×Q₈	—	2	4	no	—
200	C₄×Q₈	—	2	4	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₄×Q₈	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	8	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
212	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄²	D₄×C₁₃	1	4	yes	—
215	C₂×C₄²	D₄×C₁₃	1	4	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
217	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²⋊C₈	—	2	4	no	—
223	#97	—	2	4	no	—
224	C₄×C₈	—	2	4	no	—
225	C₂×C₁₃	—	2	4	no	—
226	C₂×C₁₃	#2310	1	1	yes	characteristic
227	C₂×C₁₃	C₂×#407	1	1	yes	characteristic
228	C₂×C₁₃	C₂×#408	1	1	yes	characteristic
229	C₂×C₁₃	#2984	1	4	yes	—
230	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
231	C₂×Q₈	—	2	32	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	8	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	C₁₃×M₁₆	1	8	yes	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
245	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
247	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
252	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	2	8	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄²	—	2	8	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	4	no	—
267	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
268	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
269	C₄²	C₂×D₄×C₁₃	1	2	yes	—
270	C₄²	C₂×D₄×C₁₃	1	2	yes	—
271	C₄²	C₂×D₄×C₁₃	1	2	yes	—
272	C₄²	C₂×D₄×C₁₃	1	2	yes	—
273	C₁₃	#21820	1	1	yes	characteristic, direct product factor, Sylow
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
277	C₂³	C₁₃×#131	1	4	yes	—
278	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
279	Q₈	—	2	16	no	—
280	Q₈	—	2	16	no	—
281	C₂×C₄	—	2	16	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
296	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
297	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
298	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
299	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
300	C₂×C₄	C₂×C₁₃×M₁₆	1	8	yes	—
301	C₂×C₄	C₂×C₁₃×M₁₆	1	4	yes	—
302	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
304	C₂×C₄	C₁₃×2₋¹⁺⁴	1	4	yes	—
305	C₈	—	4	4	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	8	no	—
310	C₂²	C₁₃×#408	1	4	yes	—
311	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
312	C₂²	C₁₃×#519	1	2	yes	—
313	C₂²	C₂²×C₁₃×M₁₆	1	2	yes	—
314	C₂²	C₂×C₁₃×#131	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	8	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	4	no	—
321	C₄	C₁₃×#407	1	4	yes	—
322	C₄	C₁₃×#536	1	2	yes	—
323	C₄	C₁₃×#536	1	2	yes	—
324	C₂	—	2	4	no	—
325	C₂	C₁₃×#2310	1	1	yes	characteristic
326	C₂	C₂×C₁₃×#407	1	1	yes	characteristic
327	C₂	C₂×C₁₃×#408	1	1	yes	characteristic
328	C₂	C₁₃×#2984	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₁₃×#1985	C₂	semidirect
5. C₁₃×#1984	C₂	semidirect
66. #21820	C₁₃	direct (N × H)
113. #1985	C₂×C₁₃	semidirect
114. #1984	C₂×C₁₃	semidirect
273. C₁₃	#21820	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×C₄×Q₈×C₁₃	2
C₂×C₁₃×#432	2
C₂×C₁₃×#431	2
C₁₃×#1985	2
C₁₃×#1984	2
C₁₃×#1817	2
#21820	13

Subgroups — C13×#21820 #15708159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#21820 #15708159