Subgroups — C₁₃×#14685 #15701024

Order 3328.

2430 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#381	C₂	1	2	yes	maximal
3	C₂×C₁₃×#380	C₂	1	2	yes	maximal
4	C₁₃×C₄⋊#95	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1434	C₂	1	4	yes	maximal, complemented
6	C₁₃×C₂²⋊#95	C₂	1	2	yes	maximal
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₃	—	2	8	no	—
13	C₂²×C₄²×C₁₃	C₄	1	8	yes	complemented
14	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#381	—	2	8	no	—
17	C₁₃×#380	—	2	8	no	—
18	C₂×C₁₃×#95	C₂²	1	4	yes	—
19	C₂×C₁₃×#95	C₂²	1	8	yes	—
20	C₂×C₁₃×#95	C₂²	1	8	yes	—
21	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	—	2	4	no	—
23	C₂³×C₄×C₁₃	—	2	4	no	—
24	C₂³×C₄×C₁₃	—	2	4	no	—
25	C₂³×C₄×C₁₃	—	2	8	no	—
26	C₂³×C₄×C₁₃	—	2	8	no	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₁₃	D₄	1	8	yes	—
31	C₂³×C₄×C₁₃	D₄	1	4	yes	—
32	C₂³×C₄×C₁₃	Q₈	1	4	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	8	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
40	C₂×C₁₃×C₂²⋊C₄	—	2	8	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	8	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	8	no	—
43	C₂×C₄²×C₁₃	—	2	32	no	—
44	C₂×C₄²×C₁₃	—	2	8	no	—
45	C₂×C₄²×C₁₃	—	2	8	no	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
48	C₁₃×#95	—	2	8	no	—
49	C₁₃×#95	—	2	16	no	—
50	C₁₃×#95	—	2	16	no	—
51	#14685	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	4	no	—
53	C₂⁴×C₁₃	—	2	4	no	—
54	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
56	C₂⁴×C₁₃	C₂×D₄	1	2	yes	—
57	C₂⁴×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂⁴×C₁₃	#41	1	4	yes	—
59	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	32	no	—
61	C₂²×C₄×C₁₃	—	2	8	no	—
62	C₂²×C₄×C₁₃	—	2	8	no	—
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	16	no	—
65	C₂²×C₄×C₁₃	—	2	16	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	16	no	—
72	C₂²×C₄×C₁₃	—	2	16	no	—
73	C₂²×C₄×C₁₃	—	2	32	no	—
74	C₂²×C₄×C₁₃	—	2	8	no	—
75	C₂²×C₄×C₁₃	—	2	8	no	—
76	C₂²×C₄×C₁₃	—	2	16	no	—
77	C₂²×C₄×C₁₃	—	2	8	no	—
78	C₂²×C₄×C₁₃	—	2	16	no	—
79	C₂²×C₄×C₁₃	C₂²⋊C₄	1	16	yes	—
80	C₂²×C₄×C₁₃	C₄⋊C₄	1	16	yes	complemented
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	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	2	yes	—
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	4	yes	—
89	C₂²×C₄×C₁₃	#41	1	8	yes	—
90	C₂²×C₄×C₁₃	#41	1	8	yes	—
91	C₁₃×C₄⋊C₄	—	4	16	no	—
92	C₁₃×C₂²⋊C₄	—	4	16	no	—
93	C₄²×C₁₃	—	2	16	no	—
94	C₄²×C₁₃	—	2	16	no	—
95	C₂³×C₄²	C₂×C₁₃	1	1	yes	characteristic
96	C₂×#381	C₂×C₁₃	1	2	yes	—
97	C₂×#380	C₂×C₁₃	1	2	yes	—
98	C₄⋊#95	C₂×C₁₃	1	4	yes	complemented
99	#1434	C₂×C₁₃	1	4	yes	complemented
100	C₂²⋊#95	C₂×C₁₃	1	2	yes	—
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	—	2	4	no	—
103	C₂³×C₁₃	—	2	8	no	—
104	C₂³×C₁₃	—	2	8	no	—
105	C₂³×C₁₃	—	2	16	no	—
106	C₂³×C₁₃	—	2	8	no	—
107	C₂³×C₁₃	—	2	8	no	—
108	C₂³×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂³×C₁₃	#117	1	4	yes	—
111	C₂³×C₁₃	#121	1	8	yes	—
112	C₂³×C₁₃	C₂²⋊Q₈	1	8	yes	—
113	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
115	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂³×C₁₃	C₂×#41	1	2	yes	—
118	C₂³×C₁₃	C₂×#41	1	4	yes	—
119	C₂³×C₁₃	C₂×#41	1	4	yes	—
120	C₂×C₄×C₁₃	—	2	32	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	2	8	no	—
128	C₂×C₄×C₁₃	—	2	8	no	—
129	C₂×C₄×C₁₃	—	2	16	no	—
130	C₂×C₄×C₁₃	—	2	32	no	—
131	C₂×C₄×C₁₃	—	2	32	no	—
132	C₂×C₄×C₁₃	—	4	8	no	—
133	C₂×C₄×C₁₃	—	4	8	no	—
134	C₂×C₄×C₁₃	—	4	8	no	—
135	C₂×C₄×C₁₃	—	4	16	no	—
136	C₂×C₄×C₁₃	—	4	16	no	—
137	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
138	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
140	C₂×C₄×C₁₃	#117	1	8	yes	—
141	C₂×C₄×C₁₃	#117	1	8	yes	—
142	C₂×C₄×C₁₃	#123	1	16	yes	—
143	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
144	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
145	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
146	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
147	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
148	C₂²×C₄²	—	2	8	no	—
149	C₂²×C₄²	C₄×C₁₃	1	8	yes	complemented
150	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
152	#381	—	2	8	no	—
153	#380	—	2	8	no	—
154	C₂×#95	C₂²×C₁₃	1	8	yes	—
155	C₂×#95	C₂²×C₁₃	1	4	yes	—
156	C₂×#95	C₂²×C₁₃	1	8	yes	—
157	C₂²×C₁₃	—	2	16	no	—
158	C₂²×C₁₃	—	2	16	no	—
159	C₂²×C₁₃	—	2	4	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	8	no	—
162	C₂²×C₁₃	—	2	8	no	—
163	C₂²×C₁₃	#380	1	8	yes	—
164	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₂²×C₁₃	C₂×#117	1	2	yes	—
168	C₂²×C₁₃	C₂×#117	1	2	yes	—
169	C₂²×C₁₃	C₂×#121	1	2	yes	—
170	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
171	C₂²×C₁₃	C₂×#123	1	4	yes	—
172	C₂²×C₁₃	#525	1	4	yes	—
173	C₂²×C₁₃	#529	1	8	yes	—
174	C₂²×C₁₃	#532	1	4	yes	—
175	C₂²×C₁₃	#533	1	4	yes	—
176	C₄×C₁₃	—	2	8	no	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	—	2	8	no	—
179	C₄×C₁₃	—	8	8	no	—
180	C₄×C₁₃	#381	1	8	yes	complemented
181	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
182	C₂³×C₄	—	2	4	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	—	2	4	no	—
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	—	2	8	no	—
187	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
188	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
189	C₂³×C₄	C₂×C₄×C₁₃	1	4	yes	—
190	C₂³×C₄	D₄×C₁₃	1	8	yes	—
191	C₂³×C₄	D₄×C₁₃	1	4	yes	—
192	C₂³×C₄	Q₈×C₁₃	1	4	yes	—
193	C₂³×C₄	C₂³×C₁₃	1	8	yes	—
194	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
197	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₄⋊C₄	—	2	8	no	—
199	C₂×C₄⋊C₄	—	2	16	no	—
200	C₂×C₂²⋊C₄	—	2	8	no	—
201	C₂×C₂²⋊C₄	—	2	8	no	—
202	C₂×C₂²⋊C₄	—	2	8	no	—
203	C₂×C₄²	—	2	32	no	—
204	C₂×C₄²	—	2	8	no	—
205	C₂×C₄²	—	2	8	no	—
206	C₂×C₄²	C₂×C₄×C₁₃	1	8	yes	—
207	C₂×C₄²	C₂×C₄×C₁₃	1	8	yes	—
208	#95	—	2	8	no	—
209	#95	—	2	16	no	—
210	#95	—	2	16	no	—
211	C₂×C₁₃	—	2	8	no	—
212	C₂×C₁₃	C₂×#380	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#381	1	1	yes	characteristic
214	C₂×C₁₃	#2280	1	1	yes	characteristic
215	C₂×C₁₃	#2283	1	2	yes	—
216	C₂×C₁₃	#2285	1	2	yes	—
217	C₂×C₁₃	#2854	1	4	yes	—
218	C₂×C₁₃	#2855	1	4	yes	—
219	C₂⁴	—	2	4	no	—
220	C₂⁴	—	2	4	no	—
221	C₂⁴	C₂²×C₄×C₁₃	1	2	yes	—
222	C₂⁴	C₂×D₄×C₁₃	1	4	yes	—
223	C₂⁴	C₂×D₄×C₁₃	1	2	yes	—
224	C₂⁴	C₂×Q₈×C₁₃	1	2	yes	—
225	C₂⁴	C₁₃×#41	1	4	yes	—
226	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
227	C₂²×C₄	—	2	32	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	16	no	—
232	C₂²×C₄	—	2	16	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	16	no	—
239	C₂²×C₄	—	2	16	no	—
240	C₂²×C₄	—	2	32	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	16	no	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	2	16	no	—
246	C₂²×C₄	C₁₃×C₂²⋊C₄	1	16	yes	—
247	C₂²×C₄	C₁₃×C₄⋊C₄	1	16	yes	complemented
248	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
250	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
251	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
253	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
254	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
255	C₂²×C₄	C₂×Q₈×C₁₃	1	4	yes	—
256	C₂²×C₄	C₁₃×#41	1	8	yes	—
257	C₂²×C₄	C₁₃×#41	1	8	yes	—
258	C₄⋊C₄	—	4	16	no	—
259	C₂²⋊C₄	—	4	16	no	—
260	C₄²	—	2	16	no	—
261	C₄²	—	2	16	no	—
262	C₁₃	#14685	1	1	yes	characteristic, direct product factor, Sylow
263	C₂³	—	2	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	8	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	16	no	—
268	C₂³	—	2	8	no	—
269	C₂³	—	2	8	no	—
270	C₂³	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
271	C₂³	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
272	C₂³	C₁₃×#117	1	4	yes	—
273	C₂³	C₁₃×#121	1	8	yes	—
274	C₂³	C₁₃×C₂²⋊Q₈	1	8	yes	—
275	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
276	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
277	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
278	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
279	C₂³	C₂×C₁₃×#41	1	2	yes	—
280	C₂³	C₂×C₁₃×#41	1	4	yes	—
281	C₂³	C₂×C₁₃×#41	1	4	yes	—
282	C₂×C₄	—	2	32	no	—
283	C₂×C₄	—	2	32	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	16	no	—
293	C₂×C₄	—	2	32	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	16	no	—
298	C₂×C₄	—	4	16	no	—
299	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
300	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
301	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	8	yes	—
302	C₂×C₄	C₁₃×#117	1	8	yes	—
303	C₂×C₄	C₁₃×#117	1	8	yes	—
304	C₂×C₄	C₁₃×#123	1	16	yes	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	8	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	2	16	no	—
310	C₂²	—	2	16	no	—
311	C₂²	C₁₃×#380	1	8	yes	—
312	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
313	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
314	C₂²	C₂×C₁₃×#117	1	1	yes	characteristic
315	C₂²	C₂×C₁₃×#117	1	2	yes	—
316	C₂²	C₂×C₁₃×#117	1	2	yes	—
317	C₂²	C₂×C₁₃×#121	1	2	yes	—
318	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	2	yes	—
319	C₂²	C₂×C₁₃×#123	1	4	yes	—
320	C₂²	C₁₃×#525	1	4	yes	—
321	C₂²	C₁₃×#529	1	8	yes	—
322	C₂²	C₁₃×#532	1	4	yes	—
323	C₂²	C₁₃×#533	1	4	yes	—
324	C₄	—	2	8	no	—
325	C₄	—	2	4	no	—
326	C₄	—	2	8	no	—
327	C₄	—	8	8	no	—
328	C₄	C₁₃×#381	1	8	yes	complemented
329	C₂	—	2	8	no	—
330	C₂	C₂×C₁₃×#380	1	1	yes	characteristic
331	C₂	C₂×C₁₃×#381	1	1	yes	characteristic
332	C₂	C₁₃×#2280	1	1	yes	characteristic
333	C₂	C₁₃×#2283	1	2	yes	—
334	C₂	C₁₃×#2285	1	2	yes	—
335	C₂	C₁₃×#2854	1	4	yes	—
336	C₂	C₁₃×#2855	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃
prime radical	C₂⁵×C₁₃	C₂³
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
4. C₁₃×C₄⋊#95	C₂	semidirect
5. C₁₃×#1434	C₂	semidirect
13. C₂²×C₄²×C₁₃	C₄	semidirect
51. #14685	C₁₃	direct (N × H)
80. C₂²×C₄×C₁₃	C₄⋊C₄	semidirect
98. C₄⋊#95	C₂×C₁₃	semidirect
99. #1434	C₂×C₁₃	semidirect
149. C₂²×C₄²	C₄×C₁₃	semidirect
180. C₄×C₁₃	#381	semidirect
247. C₂²×C₄	C₁₃×C₄⋊C₄	semidirect
262. C₁₃	#14685	direct (N × H)
328. C₄	C₁₃×#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₁₃	2
C₂×C₁₃×#381	2
C₂×C₁₃×#380	2
C₁₃×C₄⋊#95	2
C₁₃×#1434	2
C₁₃×C₂²⋊#95	2
#14685	13

Subgroups — C13×#14685 #15701024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#14685 #15701024