Subgroups — C₁₃×#7685 #15694024

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1271	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×M₁₆	—	2	1	no	—
6	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	—	2	1	no	—
11	C₁₃×#391	—	2	1	no	—
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#389	C₂²	1	2	yes	—
14	C₁₃×#384	C₄	1	1	yes	characteristic
15	C₁₃×#380	—	2	1	no	—
16	C₁₃×#380	C₄	1	2	yes	—
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#336	C₂²	1	2	yes	—
19	C₁₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	4	no	—
25	C₂×C₁₃×M₁₆	—	4	2	no	—
26	C₂×C₁₃×M₁₆	—	4	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	4	2	no	—
30	C₁₃×C₂²⋊Q₈	—	4	2	no	—
31	C₁₃×C₂²⋊Q₈	—	4	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
39	C₂×C₄²×C₁₃	—	4	2	no	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₂²⋊C₈	—	2	4	no	—
45	C₄×C₈×C₁₃	—	4	2	no	—
46	C₁₃×#95	—	2	2	no	—
47	C₁₃×#95	—	2	2	no	—
48	#7685	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	D₈	1	2	yes	—
67	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₁₃×M₁₆	—	4	8	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	2	4	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	4	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	4	1	no	—
89	C₄²×C₁₃	—	4	1	no	—
90	#2857	C₂×C₁₃	1	1	yes	characteristic
91	C₄×C₂²⋊C₈	C₂×C₁₃	1	1	yes	characteristic
92	#1728	C₂×C₁₃	1	1	yes	characteristic
93	#1271	C₂×C₁₃	1	4	yes	complemented
94	C₂³×C₁₃	—	2	1	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	2	1	no	—
98	C₂³×C₁₃	—	2	1	no	—
99	C₂³×C₁₃	—	2	1	no	—
100	C₂³×C₁₃	#102	1	2	yes	—
101	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₁₃	—	8	4	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	#99	1	2	yes	—
124	C₂×C₄×C₁₃	#101	1	2	yes	—
125	C₂×C₄×C₁₃	#102	1	2	yes	—
126	C₂×C₄×C₁₃	#104	1	2	yes	—
127	C₂×C₄×C₁₃	#104	1	2	yes	—
128	C₂×C₄×C₁₃	#131	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#137	1	2	yes	—
132	C₈×C₁₃	—	4	4	no	—
133	C₈×C₁₃	—	4	4	no	—
134	C₂²×M₁₆	—	2	1	no	—
135	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
136	C₂×C₂²⋊Q₈	—	2	1	no	—
137	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×C₂²⋊C₈	—	2	1	no	—
139	C₂×C₄×C₈	—	2	1	no	—
140	#391	—	2	1	no	—
141	#389	—	2	1	no	—
142	#389	C₂²×C₁₃	1	2	yes	—
143	#384	C₄×C₁₃	1	1	yes	characteristic
144	#380	—	2	1	no	—
145	#380	C₄×C₁₃	1	2	yes	—
146	#336	—	2	1	no	—
147	#336	C₂²×C₁₃	1	2	yes	—
148	#336	C₂²×C₁₃	1	2	yes	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	1	no	—
156	C₂²×C₁₃	#408	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#99	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
160	C₂²×C₁₃	#419	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	#327	1	4	yes	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	4	2	no	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₄⋊Q₈	—	4	2	no	—
182	C₂²⋊Q₈	—	4	2	no	—
183	C₂²⋊Q₈	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₄²	—	4	2	no	—
192	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²⋊C₈	—	2	4	no	—
197	C₄×C₈	—	4	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₂×C₁₃	—	4	2	no	—
201	C₂×C₁₃	#1472	1	1	yes	characteristic
202	C₂×C₁₃	#1485	1	1	yes	characteristic
203	C₂×C₁₃	#1492	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#327	1	1	yes	characteristic
205	C₂×C₁₃	#1508	1	1	yes	characteristic
206	C₂×C₁₃	#1526	1	1	yes	characteristic
207	C₂×C₁₃	#1546	1	1	yes	characteristic
208	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	C₁₃×D₈	1	2	yes	—
226	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	M₁₆	—	4	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₁₃	#7685	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₁₃×#102	1	2	yes	—
257	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
258	Q₈	—	8	4	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	C₁₃×#99	1	2	yes	—
280	C₂×C₄	C₁₃×#101	1	2	yes	—
281	C₂×C₄	C₁₃×#102	1	2	yes	—
282	C₂×C₄	C₁₃×#104	1	2	yes	—
283	C₂×C₄	C₁₃×#104	1	2	yes	—
284	C₂×C₄	C₁₃×#131	1	2	yes	—
285	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
286	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#137	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₈	—	4	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₁₃×#408	1	1	yes	characteristic
298	C₂²	C₂×C₁₃×#99	1	1	yes	characteristic
299	C₂²	C₂×C₁₃×#101	1	1	yes	characteristic
300	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
301	C₂²	C₁₃×#419	1	1	yes	characteristic
302	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
303	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₃×#327	1	4	yes	—
313	C₂	—	4	2	no	—
314	C₂	C₁₃×#1472	1	1	yes	characteristic
315	C₂	C₁₃×#1485	1	1	yes	characteristic
316	C₂	C₁₃×#1492	1	1	yes	characteristic
317	C₂	C₂×C₁₃×#327	1	1	yes	characteristic
318	C₂	C₁₃×#1508	1	1	yes	characteristic
319	C₂	C₁₃×#1526	1	1	yes	characteristic
320	C₂	C₁₃×#1546	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
4. C₁₃×#1271	C₂	semidirect
48. #7685	C₁₃	direct (N × H)
93. #1271	C₂×C₁₃	semidirect
249. C₁₃	#7685	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×#2857	2
C₄×C₁₃×C₂²⋊C₈	2
C₁₃×#1728	2
C₁₃×#1271	2
#7685	13

Subgroups — C13×#7685 #15694024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7685 #15694024