Subgroups — C₁₃×#22685 #15709024

Order 3328.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×Q₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2351	C₂	1	2	yes	maximal
4	C₁₃×#1942	C₂	1	2	yes	maximal
5	C₁₃×#1865	C₂	1	4	yes	maximal
6	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1805	C₂	1	2	yes	maximal
8	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₄×C₁₃×Q₁₆	—	2	8	no	—
13	C₄×C₁₃×Q₁₆	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#106	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#103	C₂²	1	4	yes	—
19	C₂×C₁₃×#103	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#389	—	2	4	no	—
23	C₁₃×#389	C₂²	1	4	yes	—
24	C₁₃×#382	—	2	2	no	—
25	C₁₃×#382	C₂²	1	2	yes	—
26	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×Q₁₆	D₄	1	4	yes	—
33	C₂×C₁₃×Q₁₆	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	—	2	4	no	—
38	C₄×Q₈×C₁₃	—	2	8	no	—
39	C₄×Q₈×C₁₃	—	4	4	no	—
40	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₃	—	2	4	no	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#107	—	2	4	no	—
52	C₁₃×#107	C₂×C₄	1	4	yes	—
53	C₁₃×#106	—	4	2	no	—
54	C₁₃×C₄⋊C₈	—	2	2	no	—
55	C₁₃×#103	—	2	8	no	—
56	C₁₃×#103	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₁₃	—	2	4	no	—
58	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
59	C₁₃×#95	—	2	2	no	—
60	C₁₃×#95	—	2	4	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	D₄	1	1	yes	characteristic
63	C₁₃×#95	D₄	1	1	yes	characteristic
64	#22685	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
70	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×Q₁₆	C₄⋊C₄	1	16	yes	complemented
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	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	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	8	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	4	8	no	—
95	C₁₃×C₄⋊C₄	—	4	8	no	—
96	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
97	C₄²×C₁₃	—	2	8	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	4	8	no	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	#41	1	2	yes	—
103	C₂×C₄×Q₁₆	C₂×C₁₃	1	2	yes	—
104	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
105	#2351	C₂×C₁₃	1	2	yes	—
106	#1942	C₂×C₁₃	1	2	yes	—
107	#1865	C₂×C₁₃	1	4	yes	—
108	#1846	C₂×C₁₃	1	1	yes	characteristic
109	#1805	C₂×C₁₃	1	2	yes	—
110	C₈×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	—	2	4	no	—
114	Q₈×C₁₃	—	2	4	no	—
115	Q₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	#135	1	4	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	—	4	2	no	—
142	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
143	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
145	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
146	C₄×Q₁₆	—	2	8	no	—
147	C₄×Q₁₆	C₄×C₁₃	1	8	yes	complemented
148	C₂×#107	C₂²×C₁₃	1	2	yes	—
149	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×#106	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×#103	C₂²×C₁₃	1	4	yes	—
153	C₂×#103	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
156	#389	—	2	4	no	—
157	#389	C₂²×C₁₃	1	4	yes	—
158	#382	—	2	2	no	—
159	#382	C₂²×C₁₃	1	2	yes	—
160	C₄×C₄⋊C₄	—	2	2	no	—
161	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
162	#340	C₂²×C₁₃	1	4	yes	—
163	#336	C₂²×C₁₃	1	2	yes	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₁₃	D₄²	1	1	yes	characteristic
166	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#135	1	2	yes	—
168	C₂²×C₁₃	#578	1	2	yes	—
169	C₄×C₁₃	—	2	8	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	2	no	—
172	C₄×C₁₃	—	2	4	no	—
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	—	4	2	no	—
175	C₄×C₁₃	—	8	4	no	—
176	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₁₃	#520	1	1	yes	characteristic
178	C₄×C₁₃	#547	1	1	yes	characteristic
179	C₄×C₁₃	#554	1	1	yes	characteristic
180	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
181	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
182	C₂×Q₁₆	D₄×C₁₃	1	4	yes	—
183	C₂×Q₁₆	Q₈×C₁₃	1	4	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	2	8	no	—
189	C₄×Q₈	—	4	4	no	—
190	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	C₂×C₄×C₁₃	1	4	yes	—
203	#106	—	4	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	8	no	—
206	#103	C₂×C₄×C₁₃	1	8	yes	—
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	—	2	2	no	—
212	#95	D₄×C₁₃	1	1	yes	characteristic
213	#95	D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
215	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
216	C₂×C₁₃	#2950	1	1	yes	characteristic
217	C₂×C₁₃	#3285	1	2	yes	—
218	C₂×C₁₃	#3383	1	2	yes	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
223	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
224	C₂×Q₈	C₂×Q₈×C₁₃	1	4	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
233	Q₁₆	C₁₃×C₄⋊C₄	1	16	yes	complemented
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
242	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
251	C₄²	—	2	8	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	8	no	—
255	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
256	C₄²	C₁₃×#41	1	2	yes	—
257	C₁₃	#22685	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
259	Q₈	—	2	4	no	—
260	Q₈	—	2	4	no	—
261	Q₈	—	2	4	no	—
262	Q₈	C₂×C₁₃×C₄⋊C₄	1	8	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
280	C₂×C₄	C₁₃×#135	1	4	yes	—
281	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
285	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
290	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
291	C₂²	D₄²×C₁₃	1	1	yes	characteristic
292	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
293	C₂²	C₂×C₁₃×#135	1	2	yes	—
294	C₂²	C₁₃×#578	1	2	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	2	8	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
303	C₄	C₁₃×#520	1	1	yes	characteristic
304	C₄	C₁₃×#547	1	1	yes	characteristic
305	C₄	C₁₃×#554	1	1	yes	characteristic
306	C₂	D₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
307	C₂	C₄×C₁₃×#135	1	1	yes	characteristic
308	C₂	C₁₃×#2950	1	1	yes	characteristic
309	C₂	C₁₃×#3285	1	2	yes	—
310	C₂	C₁₃×#3383	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 ◁ C₄×C₁₃×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₁₃×Q₁₆	C₄	semidirect
64. #22685	C₁₃	direct (N × H)
79. C₁₃×Q₁₆	C₄⋊C₄	semidirect
147. C₄×Q₁₆	C₄×C₁₃	semidirect
233. Q₁₆	C₁₃×C₄⋊C₄	semidirect
257. C₁₃	#22685	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×C₄×C₁₃×Q₁₆	2
C₁₃×#2351	2
C₁₃×#1942	2
C₁₃×#1865	2
C₁₃×#1846	2
C₁₃×#1805	2
C₈×C₁₃×C₄⋊C₄	2
#22685	13

Subgroups — C13×#22685 #15709024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22685 #15709024