Subgroups — C₁₃×#22744 #15709083

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2355	C₂	1	2	yes	maximal
3	C₁₃×#1949	C₂	1	2	yes	maximal
4	C₁₃×#1949	C₂	1	2	yes	maximal
5	C₁₃×#1947	C₂	1	2	yes	maximal
6	C₁₃×#1836	C₂	1	2	yes	maximal
7	C₁₃×#1835	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1776	C₂	1	2	yes	maximal
9	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#391	—	2	2	no	—
20	C₁₃×#387	—	2	2	no	—
21	C₁₃×#387	—	2	2	no	—
22	C₁₃×#384	C₂²	1	2	yes	—
23	C₁₃×#382	C₂²	1	2	yes	—
24	C₁₃×#382	C₂²	1	2	yes	—
25	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	1	yes	characteristic
33	C₁₃×#340	C₂²	1	1	yes	characteristic
34	C₁₃×#340	C₂²	1	2	yes	—
35	C₁₃×#340	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₄²×C₁₃	—	2	2	no	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
55	C₁₃×#107	—	2	4	no	—
56	C₁₃×#107	—	2	4	no	—
57	C₁₃×#106	—	2	4	no	—
58	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
59	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
60	C₁₃×#97	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	#22744	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×Q₈×C₁₃	—	4	2	no	—
71	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₄⋊C₄	—	8	2	no	—
97	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
98	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
99	C₁₃×C₄⋊C₄	#41	1	4	yes	—
100	C₄²×C₁₃	—	4	4	no	—
101	C₄²×C₁₃	—	8	2	no	—
102	C₄²×C₁₃	—	8	2	no	—
103	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
105	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
106	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
107	C₂×#446	C₂×C₁₃	1	1	yes	characteristic
108	#2355	C₂×C₁₃	1	2	yes	—
109	#1949	C₂×C₁₃	1	2	yes	—
110	#1949	C₂×C₁₃	1	2	yes	—
111	#1947	C₂×C₁₃	1	2	yes	—
112	#1836	C₂×C₁₃	1	2	yes	—
113	#1835	C₂×C₁₃	1	1	yes	characteristic
114	#1776	C₂×C₁₃	1	2	yes	—
115	C₄×#107	C₂×C₁₃	1	1	yes	characteristic
116	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
145	C₈×C₁₃	—	2	4	no	—
146	C₈×C₁₃	—	2	2	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
149	#446	C₄×C₁₃	1	8	yes	complemented
150	C₂×#107	C₂²×C₁₃	1	2	yes	—
151	C₂×#107	C₂²×C₁₃	1	2	yes	—
152	C₂×#106	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×#97	C₂²×C₁₃	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
157	#391	—	2	2	no	—
158	#387	—	2	2	no	—
159	#387	—	2	2	no	—
160	#384	C₂²×C₁₃	1	2	yes	—
161	#382	C₂²×C₁₃	1	2	yes	—
162	#382	C₂²×C₁₃	1	2	yes	—
163	C₄×C₄⋊C₄	—	2	2	no	—
164	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
166	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
167	#340	C₂²×C₁₃	1	2	yes	—
168	#340	C₂²×C₁₃	1	1	yes	characteristic
169	#340	C₂²×C₁₃	1	1	yes	characteristic
170	#340	C₂²×C₁₃	1	1	yes	characteristic
171	#340	C₂²×C₁₃	1	2	yes	—
172	#340	C₂²×C₁₃	1	2	yes	—
173	#340	C₂²×C₁₃	1	1	yes	characteristic
174	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₁₃	#548	1	2	yes	—
176	C₂²×C₁₃	#575	1	2	yes	—
177	C₂²×C₁₃	#576	1	1	yes	characteristic
178	C₂²×C₁₃	#578	1	1	yes	characteristic
179	C₄×C₁₃	—	2	4	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	4	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
188	C₄×C₁₃	#519	1	1	yes	characteristic
189	C₄×C₁₃	#557	1	2	yes	—
190	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
194	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
195	C₄×Q₈	Q₈×C₁₃	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
199	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
209	#107	—	2	4	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
213	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
214	#97	C₂×C₄×C₁₃	1	8	yes	—
215	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	C₂×C₁₃	#2351	1	1	yes	characteristic
224	C₂×C₁₃	#2911	1	1	yes	characteristic
225	C₂×C₁₃	#2913	1	1	yes	characteristic
226	C₂×C₁₃	#3370	1	2	yes	—
227	C₂×C₁₃	#3377	1	2	yes	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	C₁₃×C₄⋊C₄	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
244	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	8	2	no	—
255	C₄⋊C₄	C₁₃×C₄⋊C₄	1	8	yes	—
256	C₄⋊C₄	C₁₃×C₄⋊C₄	1	4	yes	—
257	C₄⋊C₄	C₁₃×#41	1	4	yes	—
258	C₄²	—	4	4	no	—
259	C₄²	—	8	2	no	—
260	C₄²	—	8	2	no	—
261	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
262	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
263	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
264	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
265	C₁₃	#22744	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
289	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
291	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
294	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
299	C₂²	C₁₃×#548	1	2	yes	—
300	C₂²	C₁₃×#575	1	2	yes	—
301	C₂²	C₁₃×#576	1	1	yes	characteristic
302	C₂²	C₁₃×#578	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
312	C₄	C₁₃×#519	1	1	yes	characteristic
313	C₄	C₁₃×#557	1	2	yes	—
314	C₂	C₁₃×#2351	1	1	yes	characteristic
315	C₂	C₁₃×#2911	1	1	yes	characteristic
316	C₂	C₁₃×#2913	1	1	yes	characteristic
317	C₂	C₁₃×#3370	1	2	yes	—
318	C₂	C₁₃×#3377	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₃×#446	C₄	semidirect
69. #22744	C₁₃	direct (N × H)
149. #446	C₄×C₁₃	semidirect
265. C₁₃	#22744	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×#446	2
C₁₃×#2355	2
C₁₃×#1949	2
C₁₃×#1947	2
C₁₃×#1836	2
C₁₃×#1835	2
C₁₃×#1776	2
C₄×C₁₃×#107	2
#22744	13

Subgroups — C13×#22744 #15709083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22744 #15709083