Subgroups — C₁₃×#9878 #15696217

Order 3328.

680 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#391	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1280	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	—	2	1	no	—
12	C₁₃×#391	—	2	1	no	—
13	C₁₃×#391	—	2	1	no	—
14	C₁₃×#391	—	2	1	no	—
15	C₁₃×#391	—	2	1	no	—
16	C₂×C₁₃×#95	C₄	1	2	yes	—
17	C₂×C₁₃×#95	C₂²	1	2	yes	—
18	C₁₃×#336	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₃	—	2	1	no	—
22	C₂²×Q₈×C₁₃	—	2	2	no	—
23	C₂²×Q₈×C₁₃	—	4	2	no	—
24	C₂²×Q₈×C₁₃	—	4	2	no	—
25	C₂²×Q₈×C₁₃	C₈	1	4	yes	complemented
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₁₃	—	4	2	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₁₃×C₄⋊C₈	—	4	2	no	—
42	C₁₃×C₂²⋊C₈	—	2	2	no	—
43	C₁₃×C₂²⋊C₈	—	2	4	no	—
44	C₁₃×C₂²⋊C₈	Q₈	1	4	yes	complemented
45	C₄×C₈×C₁₃	—	4	2	no	—
46	C₁₃×#95	—	2	2	no	—
47	C₁₃×#95	—	2	2	no	—
48	C₁₃×#95	—	2	2	no	—
49	C₁₃×#95	—	2	2	no	—
50	#9878	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Q₈×C₁₃	—	2	4	no	—
53	C₂×Q₈×C₁₃	—	2	4	no	—
54	C₂×Q₈×C₁₃	—	4	8	no	—
55	C₂×Q₈×C₁₃	—	4	2	no	—
56	C₂×Q₈×C₁₃	—	4	2	no	—
57	C₂×Q₈×C₁₃	—	4	8	no	—
58	C₂×Q₈×C₁₃	—	4	2	no	—
59	C₂×Q₈×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
81	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₁₃	#41	1	2	yes	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₂×C₈×C₁₃	—	4	2	no	—
90	C₂×C₈×C₁₃	—	4	2	no	—
91	C₂×C₈×C₁₃	—	4	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	4	1	no	—
97	C₄²×C₁₃	—	4	1	no	—
98	C₂×#391	C₂×C₁₃	1	1	yes	characteristic
99	#1797	C₂×C₁₃	1	1	yes	characteristic
100	C₄×C₂²⋊C₈	C₂×C₁₃	1	1	yes	characteristic
101	#1280	C₂×C₁₃	1	4	yes	—
102	C₂³×C₁₃	—	2	1	no	—
103	C₂³×C₁₃	—	2	1	no	—
104	C₂³×C₁₃	—	2	1	no	—
105	C₂³×C₁₃	—	2	1	no	—
106	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
109	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂³×C₁₃	#131	1	2	yes	—
111	Q₈×C₁₃	—	4	8	no	—
112	Q₈×C₁₃	—	4	8	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	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	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	4	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	—	8	4	no	—
138	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
140	C₂×C₄×C₁₃	#99	1	2	yes	—
141	C₂×C₄×C₁₃	#100	1	2	yes	—
142	C₂×C₄×C₁₃	#101	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₃	—	4	4	no	—
147	C₈×C₁₃	—	8	2	no	—
148	C₂³×Q₈	C₄×C₁₃	1	1	yes	characteristic
149	C₂²×C₄⋊C₄	C₄×C₁₃	1	1	yes	characteristic
150	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄×C₈	—	2	1	no	—
155	#391	—	2	1	no	—
156	#391	—	2	1	no	—
157	#391	—	2	1	no	—
158	#391	—	2	1	no	—
159	C₂×#95	C₄×C₁₃	1	2	yes	—
160	C₂×#95	C₂²×C₁₃	1	2	yes	—
161	#336	—	2	1	no	—
162	#336	—	2	1	no	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	2	2	no	—
171	C₂²×C₁₃	#391	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
173	C₂²×C₁₃	#408	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#99	1	1	yes	characteristic
175	C₂²×C₁₃	#410	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
178	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
179	C₂²×C₁₃	#446	1	2	yes	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	2	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	4	4	no	—
186	C₄×C₁₃	—	4	1	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	#323	1	4	yes	—
189	C₂²×Q₈	—	2	1	no	—
190	C₂²×Q₈	—	2	1	no	—
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×Q₈	—	4	2	no	—
193	C₂²×Q₈	—	4	2	no	—
194	C₂²×Q₈	C₈×C₁₃	1	4	yes	complemented
195	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
196	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
197	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
198	C₂³×C₄	C₂×C₄×C₁₃	1	2	yes	—
199	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	4	2	no	—
205	C₂×C₄²	—	4	2	no	—
206	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
210	C₄⋊C₈	—	4	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	4	no	—
213	C₂²⋊C₈	Q₈×C₁₃	1	4	yes	complemented
214	C₄×C₈	—	4	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	2	2	no	—
221	C₂×C₁₃	C₂×#323	1	1	yes	characteristic
222	C₂×C₁₃	#1461	1	1	yes	characteristic
223	C₂×C₁₃	#1469	1	1	yes	characteristic
224	C₂×C₁₃	#1472	1	1	yes	characteristic
225	C₂×C₁₃	#1984	1	1	yes	characteristic
226	C₂×C₁₃	#1996	1	1	yes	characteristic
227	C₂×C₁₃	#1997	1	1	yes	characteristic
228	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	8	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
257	C₂²×C₄	C₂×C₈×C₁₃	1	4	yes	—
258	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
262	C₂²×C₄	C₁₃×#41	1	2	yes	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₁₃	#9878	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
281	C₂³	C₄×Q₈×C₁₃	1	2	yes	—
282	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
283	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
284	C₂³	C₁₃×#131	1	2	yes	—
285	Q₈	—	4	8	no	—
286	Q₈	—	4	8	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
313	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
314	C₂×C₄	C₁₃×#99	1	2	yes	—
315	C₂×C₄	C₁₃×#100	1	2	yes	—
316	C₂×C₄	C₁₃×#101	1	2	yes	—
317	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
318	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
319	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
320	C₈	—	4	4	no	—
321	C₈	—	8	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	C₁₃×#391	1	1	yes	characteristic
331	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
332	C₂²	C₁₃×#408	1	1	yes	characteristic
333	C₂²	C₂×C₁₃×#99	1	1	yes	characteristic
334	C₂²	C₁₃×#410	1	1	yes	characteristic
335	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
336	C₂²	C₂×C₁₃×#101	1	1	yes	characteristic
337	C₂²	C₈×Q₈×C₁₃	1	2	yes	—
338	C₂²	C₁₃×#446	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	4	no	—
346	C₄	—	8	2	no	—
347	C₄	C₁₃×#323	1	4	yes	—
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	C₂×C₁₃×#323	1	1	yes	characteristic
351	C₂	C₁₃×#1461	1	1	yes	characteristic
352	C₂	C₁₃×#1469	1	1	yes	characteristic
353	C₂	C₁₃×#1472	1	1	yes	characteristic
354	C₂	C₁₃×#1984	1	1	yes	characteristic
355	C₂	C₁₃×#1996	1	1	yes	characteristic
356	C₂	C₁₃×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₂²×Q₈×C₁₃	C₈	semidirect
44. C₁₃×C₂²⋊C₈	Q₈	semidirect
50. #9878	C₁₃	direct (N × H)
194. C₂²×Q₈	C₈×C₁₃	semidirect
213. C₂²⋊C₈	Q₈×C₁₃	semidirect
275. C₁₃	#9878	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×#391	2
C₁₃×#1797	2
C₄×C₁₃×C₂²⋊C₈	2
C₁₃×#1280	2
#9878	13

Subgroups — C13×#9878 #15696217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9878 #15696217