Subgroups — C₁₃×#29142 #15715481

Order 3328.

770 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2024	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2023	C₂	1	2	yes	maximal
5	C₁₃×#1958	C₂	1	2	yes	maximal
6	C₁₃×#1835	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1805	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1804	C₂	1	2	yes	maximal
9	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#124	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#103	C₂²	1	4	yes	—
16	C₂×C₁₃×#102	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
18	C₁₃×#391	—	2	2	no	—
19	C₁₃×#391	—	2	2	no	—
20	C₁₃×#391	C₂²	1	1	yes	characteristic
21	C₁₃×#390	C₂²	1	1	yes	characteristic
22	C₁₃×#388	—	2	2	no	—
23	C₁₃×#382	C₂²	1	1	yes	characteristic
24	C₁₃×#382	C₂²	1	2	yes	—
25	C₁₃×#382	C₂²	1	2	yes	—
26	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
27	C₄³×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#340	C₂²	1	4	yes	—
30	C₁₃×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₃	—	2	2	no	—
32	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₃	—	2	1	no	—
35	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
36	C₁₃×C₄⋊Q₈	—	4	4	no	—
37	C₁₃×#124	—	2	2	no	—
38	C₁₃×#124	Q₈	1	4	yes	complemented
39	C₄×Q₈×C₁₃	—	2	4	no	—
40	C₄×D₄×C₁₃	—	2	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
49	C₂×C₄²×C₁₃	—	2	1	no	—
50	C₂×C₄²×C₁₃	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₈	—	2	4	no	—
56	C₁₃×#103	—	4	4	no	—
57	C₁₃×#102	—	4	4	no	—
58	C₄×C₈×C₁₃	—	2	4	no	—
59	C₁₃×#95	—	2	2	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	4	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	#29142	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	—	2	1	no	—
66	C₂×Q₈×C₁₃	—	2	1	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	—	4	4	no	—
70	C₂×Q₈×C₁₃	—	4	4	no	—
71	C₂×Q₈×C₁₃	—	4	4	no	—
72	C₂×Q₈×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	2	1	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	—	2	1	no	—
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₃	—	2	1	no	—
81	C₂²×C₄×C₁₃	—	4	4	no	—
82	C₂²×C₄×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₂²⋊C₄	—	2	8	no	—
98	C₁₃×C₂²⋊C₄	—	4	2	no	—
99	C₁₃×C₂²⋊C₄	—	4	4	no	—
100	C₄²×C₁₃	—	2	4	no	—
101	C₄²×C₁₃	—	2	2	no	—
102	C₄²×C₁₃	—	4	2	no	—
103	C₄²×C₁₃	—	4	2	no	—
104	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
105	C₄²×C₁₃	#41	1	2	yes	—
106	C₄²×C₁₃	#41	1	4	yes	—
107	#2862	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₄×#124	C₂×C₁₃	1	1	yes	characteristic
109	#2024	C₂×C₁₃	1	2	yes	complemented
110	#2023	C₂×C₁₃	1	2	yes	—
111	#1958	C₂×C₁₃	1	2	yes	—
112	#1835	C₂×C₁₃	1	2	yes	complemented
113	#1805	C₂×C₁₃	1	2	yes	complemented
114	#1804	C₂×C₁₃	1	2	yes	—
115	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	2	2	no	—
117	C₂³×C₁₃	—	4	1	no	—
118	C₂³×C₁₃	—	4	1	no	—
119	C₂³×C₁₃	—	4	2	no	—
120	C₂³×C₁₃	—	4	2	no	—
121	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	4	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	8	8	no	—
125	D₄×C₁₃	—	2	4	no	—
126	D₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	4	no	—
143	C₂×C₄×C₁₃	—	8	4	no	—
144	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
145	C₂×C₄×C₁₃	#124	1	2	yes	—
146	C₂×C₄×C₁₃	#124	1	2	yes	—
147	C₂×C₄×C₁₃	#135	1	4	yes	—
148	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
152	C₈×C₁₃	—	4	4	no	—
153	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
154	C₂×#124	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
158	C₂×#103	C₂²×C₁₃	1	4	yes	—
159	C₂×#102	C₂²×C₁₃	1	4	yes	—
160	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
161	#391	—	2	2	no	—
162	#391	—	2	2	no	—
163	#391	C₂²×C₁₃	1	1	yes	characteristic
164	#390	C₂²×C₁₃	1	1	yes	characteristic
165	#388	—	2	2	no	—
166	#382	C₂²×C₁₃	1	2	yes	—
167	#382	C₂²×C₁₃	1	1	yes	characteristic
168	#382	C₂²×C₁₃	1	2	yes	—
169	C₄×C₂²⋊C₄	—	2	2	no	—
170	C₄³	C₂²×C₁₃	1	1	yes	characteristic
171	#340	C₂²×C₁₃	1	4	yes	—
172	#340	C₂²×C₁₃	1	4	yes	—
173	#336	C₂²×C₁₃	1	2	yes	—
174	C₂²×C₁₃	—	4	2	no	—
175	C₂²×C₁₃	—	4	4	no	—
176	C₂²×C₁₃	—	8	1	no	—
177	C₂²×C₁₃	—	8	1	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
181	C₂²×C₁₃	#535	1	1	yes	characteristic
182	C₂²×C₁₃	C₂×#135	1	2	yes	—
183	C₂²×C₁₃	#575	1	2	yes	—
184	C₄×C₁₃	—	2	4	no	—
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	2	4	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	—	8	1	no	—
190	C₄×C₁₃	—	8	1	no	—
191	C₄×C₁₃	—	8	4	no	—
192	C₄×C₁₃	#554	1	2	yes	—
193	C₄×C₁₃	#555	1	2	yes	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×D₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
199	C₄⋊Q₈	—	4	4	no	—
200	#124	—	2	2	no	—
201	#124	Q₈×C₁₃	1	4	yes	complemented
202	C₄×Q₈	—	2	4	no	—
203	C₄×D₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	D₄×C₁₃	1	2	yes	—
214	C₂×C₄²	D₄×C₁₃	1	2	yes	—
215	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
218	C₄⋊C₈	—	2	4	no	—
219	#103	—	4	4	no	—
220	#102	—	4	4	no	—
221	C₄×C₈	—	2	4	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	4	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	C₂×C₁₃	—	8	2	no	—
228	C₂×C₁₃	#2563	1	1	yes	characteristic
229	C₂×C₁₃	#3076	1	2	yes	—
230	C₂×C₁₃	#3131	1	1	yes	characteristic
231	C₂×C₁₃	#3140	1	1	yes	characteristic
232	C₂×C₁₃	#3211	1	2	yes	—
233	C₂⁴	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	—	4	4	no	—
240	C₂×Q₈	C₂×Q₈×C₁₃	1	2	yes	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	C₂×Q₈×C₁₃	1	2	yes	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
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	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
273	C₄²	C₁₃×#41	1	2	yes	—
274	C₄²	C₁₃×#41	1	4	yes	—
275	C₁₃	#29142	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	8	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
305	C₂×C₄	C₁₃×#124	1	2	yes	—
306	C₂×C₄	C₁₃×#124	1	2	yes	—
307	C₂×C₄	C₁₃×#135	1	4	yes	—
308	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
310	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
311	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
319	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
320	C₂²	C₁₃×#535	1	1	yes	characteristic
321	C₂²	C₂×C₁₃×#135	1	2	yes	—
322	C₂²	C₁₃×#575	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	4	no	—
331	C₄	C₁₃×#554	1	2	yes	—
332	C₄	C₁₃×#555	1	2	yes	—
333	C₂	—	8	2	no	—
334	C₂	C₁₃×#2563	1	1	yes	characteristic
335	C₂	C₁₃×#3076	1	2	yes	—
336	C₂	C₁₃×#3131	1	1	yes	characteristic
337	C₂	C₁₃×#3140	1	1	yes	characteristic
338	C₂	C₁₃×#3211	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2862	C₂	semidirect
3. C₁₃×#2024	C₂	semidirect
6. C₁₃×#1835	C₂	semidirect
7. C₁₃×#1805	C₂	semidirect
9. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
38. C₁₃×#124	Q₈	semidirect
64. #29142	C₁₃	direct (N × H)
107. #2862	C₂×C₁₃	semidirect
109. #2024	C₂×C₁₃	semidirect
112. #1835	C₂×C₁₃	semidirect
113. #1805	C₂×C₁₃	semidirect
115. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
201. #124	Q₈×C₁₃	semidirect
275. C₁₃	#29142	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2862	2
C₄×C₁₃×#124	2
C₁₃×#2024	2
C₁₃×#2023	2
C₁₃×#1958	2
C₁₃×#1835	2
C₁₃×#1805	2
C₁₃×#1804	2
C₄×C₁₃×C₄⋊C₈	2
#29142	13

Subgroups — C13×#29142 #15715481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29142 #15715481