Subgroups — C₁₃×#29140 #15715479

Order 3328.

754 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2026	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2025	C₂	1	2	yes	maximal
5	C₁₃×#1958	C₂	1	2	yes	maximal
6	C₁₃×#1835	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1803	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1802	C₂	1	2	yes	maximal
9	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#125	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	C₂²	1	1	yes	characteristic
19	C₁₃×#390	C₂²	1	1	yes	characteristic
20	C₁₃×#389	—	2	2	no	—
21	C₁₃×#389	—	2	2	no	—
22	C₁₃×#388	—	2	2	no	—
23	C₁₃×#384	C₂²	1	2	yes	—
24	C₁₃×#382	C₂²	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	—	2	1	no	—
34	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
35	C₁₃×#125	—	4	4	no	—
36	C₁₃×#124	—	2	2	no	—
37	C₁₃×#124	Q₈	1	4	yes	—
38	C₄×Q₈×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₄²×C₁₃	—	2	1	no	—
51	C₂×C₄²×C₁₃	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃	D₄	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×C₄⋊C₈	—	2	4	no	—
57	C₁₃×#103	—	4	4	no	—
58	C₁₃×#102	—	4	4	no	—
59	C₄×C₈×C₁₃	—	2	4	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	4	no	—
63	#29140	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	—	2	1	no	—
65	C₂×Q₈×C₁₃	—	2	1	no	—
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	2	1	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	2	1	no	—
75	C₂²×C₄×C₁₃	—	2	1	no	—
76	C₂²×C₄×C₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₃	—	4	2	no	—
78	C₂²×C₄×C₁₃	—	4	4	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	8	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₂²⋊C₄	—	2	8	no	—
96	C₁₃×C₂²⋊C₄	—	4	4	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₄²×C₁₃	—	2	4	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	4	2	no	—
101	C₄²×C₁₃	—	4	2	no	—
102	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃	#41	1	2	yes	—
104	C₄²×C₁₃	#41	1	4	yes	—
105	#2859	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₄×#124	C₂×C₁₃	1	1	yes	characteristic
107	#2026	C₂×C₁₃	1	2	yes	complemented
108	#2025	C₂×C₁₃	1	2	yes	—
109	#1958	C₂×C₁₃	1	2	yes	—
110	#1835	C₂×C₁₃	1	2	yes	complemented
111	#1803	C₂×C₁₃	1	2	yes	complemented
112	#1802	C₂×C₁₃	1	2	yes	—
113	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	—	4	1	no	—
116	C₂³×C₁₃	—	4	1	no	—
117	C₂³×C₁₃	—	4	2	no	—
118	C₂³×C₁₃	—	4	2	no	—
119	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	Q₈×C₁₃	—	2	4	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	2	4	no	—
123	D₄×C₁₃	—	4	2	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	—	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	—	4	1	no	—
132	C₂×C₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	4	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	4	no	—
140	C₂×C₄×C₁₃	—	8	4	no	—
141	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
142	C₂×C₄×C₁₃	#124	1	2	yes	—
143	C₂×C₄×C₁₃	#124	1	2	yes	—
144	C₂×C₄×C₁₃	#135	1	4	yes	—
145	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
147	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
149	C₈×C₁₃	—	4	4	no	—
150	C₂×#125	C₂²×C₁₃	1	2	yes	—
151	C₂×#124	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
155	C₂×#103	C₂²×C₁₃	1	4	yes	—
156	C₂×#102	C₂²×C₁₃	1	4	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
158	#391	C₂²×C₁₃	1	1	yes	characteristic
159	#390	C₂²×C₁₃	1	1	yes	characteristic
160	#389	—	2	2	no	—
161	#389	—	2	2	no	—
162	#388	—	2	2	no	—
163	#384	C₂²×C₁₃	1	2	yes	—
164	#382	C₂²×C₁₃	1	1	yes	characteristic
165	#382	C₂²×C₁₃	1	2	yes	—
166	C₄×C₂²⋊C₄	—	2	2	no	—
167	C₄³	C₂²×C₁₃	1	1	yes	characteristic
168	#340	C₂²×C₁₃	1	4	yes	—
169	#340	C₂²×C₁₃	1	4	yes	—
170	#336	C₂²×C₁₃	1	2	yes	—
171	C₂²×C₁₃	—	4	2	no	—
172	C₂²×C₁₃	—	4	4	no	—
173	C₂²×C₁₃	—	8	1	no	—
174	C₂²×C₁₃	—	8	1	no	—
175	C₂²×C₁₃	—	8	2	no	—
176	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
178	C₂²×C₁₃	#535	1	1	yes	characteristic
179	C₂²×C₁₃	C₂×#135	1	2	yes	—
180	C₂²×C₁₃	#575	1	2	yes	—
181	C₄×C₁₃	—	2	4	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	—	8	1	no	—
186	C₄×C₁₃	—	8	1	no	—
187	C₄×C₁₃	—	8	4	no	—
188	C₄×C₁₃	#488	1	4	yes	—
189	C₄×C₁₃	#554	1	2	yes	—
190	C₄×C₁₃	#555	1	2	yes	—
191	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×D₄	C₂³×C₁₃	1	1	yes	characteristic
193	C₂³×C₄	—	2	1	no	—
194	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
195	#125	—	4	4	no	—
196	#124	—	2	2	no	—
197	#124	Q₈×C₁₃	1	4	yes	—
198	C₄×Q₈	—	2	4	no	—
199	C₄×D₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	D₄×C₁₃	1	2	yes	—
212	C₂×C₄²	D₄×C₁₃	1	2	yes	—
213	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	#103	—	4	4	no	—
218	#102	—	4	4	no	—
219	C₄×C₈	—	2	4	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	4	no	—
223	C₂×C₁₃	—	8	2	no	—
224	C₂×C₁₃	#2563	1	1	yes	characteristic
225	C₂×C₁₃	#3076	1	2	yes	—
226	C₂×C₁₃	C₂×#488	1	1	yes	characteristic
227	C₂×C₁₃	#3142	1	1	yes	characteristic
228	C₂×C₁₃	#3212	1	2	yes	—
229	C₂⁴	—	2	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	C₂×Q₈×C₁₃	1	2	yes	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₂×Q₈×C₁₃	1	2	yes	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	8	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	2	8	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
268	C₄²	C₁₃×#41	1	2	yes	—
269	C₄²	C₁₃×#41	1	4	yes	—
270	C₁₃	#29140	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³	—	2	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
277	Q₈	—	2	4	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
299	C₂×C₄	C₁₃×#124	1	2	yes	—
300	C₂×C₄	C₁₃×#124	1	2	yes	—
301	C₂×C₄	C₁₃×#135	1	4	yes	—
302	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
304	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
305	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
306	C₈	—	4	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
313	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
314	C₂²	C₁₃×#535	1	1	yes	characteristic
315	C₂²	C₂×C₁₃×#135	1	2	yes	—
316	C₂²	C₁₃×#575	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	4	no	—
324	C₄	C₁₃×#488	1	4	yes	—
325	C₄	C₁₃×#554	1	2	yes	—
326	C₄	C₁₃×#555	1	2	yes	—
327	C₂	—	8	2	no	—
328	C₂	C₁₃×#2563	1	1	yes	characteristic
329	C₂	C₁₃×#3076	1	2	yes	—
330	C₂	C₂×C₁₃×#488	1	1	yes	characteristic
331	C₂	C₁₃×#3142	1	1	yes	characteristic
332	C₂	C₁₃×#3212	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×#2859	C₂	semidirect
3. C₁₃×#2026	C₂	semidirect
6. C₁₃×#1835	C₂	semidirect
7. C₁₃×#1803	C₂	semidirect
9. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
63. #29140	C₁₃	direct (N × H)
105. #2859	C₂×C₁₃	semidirect
107. #2026	C₂×C₁₃	semidirect
110. #1835	C₂×C₁₃	semidirect
111. #1803	C₂×C₁₃	semidirect
113. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
270. C₁₃	#29140	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×#2859	2
C₄×C₁₃×#124	2
C₁₃×#2026	2
C₁₃×#2025	2
C₁₃×#1958	2
C₁₃×#1835	2
C₁₃×#1803	2
C₁₃×#1802	2
C₄×C₁₃×C₄⋊C₈	2
#29140	13

Subgroups — C13×#29140 #15715479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29140 #15715479