Subgroups — C₁₃×#7687 #15694026

Order 3328.

562 conjugacy classes of subgroups.

The subgroup lattice has 318 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×#1272	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	C₄	1	2	yes	—
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	C₂²	1	2	yes	—
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#389	—	2	1	no	—
14	C₁₃×#384	C₄	1	1	yes	characteristic
15	C₁₃×#380	—	2	1	no	—
16	C₁₃×#380	—	2	1	no	—
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#336	C₂²	1	2	yes	—
19	C₁₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
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	4	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	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×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₁₃	—	4	2	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₂²⋊C₈	—	2	4	no	—
44	C₄×C₈×C₁₃	—	4	2	no	—
45	C₁₃×#95	—	2	2	no	—
46	C₁₃×#95	—	2	2	no	—
47	#7687	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	—	2	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	—	4	1	no	—
61	C₂²×C₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₁₃	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₁₃×M₁₆	—	4	8	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₄²×C₁₃	—	2	2	no	—
85	C₄²×C₁₃	—	2	2	no	—
86	C₄²×C₁₃	—	4	1	no	—
87	C₄²×C₁₃	—	4	1	no	—
88	#2857	C₂×C₁₃	1	1	yes	characteristic
89	C₄×C₂²⋊C₈	C₂×C₁₃	1	1	yes	characteristic
90	#1728	C₂×C₁₃	1	1	yes	characteristic
91	#1272	C₂×C₁₃	1	4	yes	complemented
92	C₂³×C₁₃	—	2	1	no	—
93	C₂³×C₁₃	—	2	1	no	—
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	#103	1	2	yes	—
99	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	8	4	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	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	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	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	#99	1	2	yes	—
122	C₂×C₄×C₁₃	#100	1	2	yes	—
123	C₂×C₄×C₁₃	#103	1	2	yes	—
124	C₂×C₄×C₁₃	#104	1	2	yes	—
125	C₂×C₄×C₁₃	#104	1	2	yes	—
126	C₂×C₄×C₁₃	#131	1	2	yes	—
127	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#137	1	2	yes	—
130	C₈×C₁₃	—	4	4	no	—
131	C₈×C₁₃	—	4	4	no	—
132	C₂²×M₁₆	—	2	1	no	—
133	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
134	C₂×C₂²⋊Q₈	C₄×C₁₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₈	—	2	1	no	—
137	C₂×C₄×C₈	—	2	1	no	—
138	#391	C₂²×C₁₃	1	2	yes	—
139	#389	—	2	1	no	—
140	#389	—	2	1	no	—
141	#384	C₄×C₁₃	1	1	yes	characteristic
142	#380	—	2	1	no	—
143	#380	—	2	1	no	—
144	#336	—	2	1	no	—
145	#336	C₂²×C₁₃	1	2	yes	—
146	#336	C₂²×C₁₃	1	2	yes	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	4	1	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	#408	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#99	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
158	C₂²×C₁₃	#419	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	—	8	2	no	—
169	C₄×C₁₃	#328	1	4	yes	—
170	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
171	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	4	2	no	—
176	C₂×M₁₆	—	4	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₄⋊Q₈	—	4	2	no	—
180	C₂²⋊Q₈	—	4	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²⋊C₈	—	2	4	no	—
194	C₄×C₈	—	4	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	C₂×C₁₃	—	4	2	no	—
198	C₂×C₁₃	#1471	1	1	yes	characteristic
199	C₂×C₁₃	#1485	1	1	yes	characteristic
200	C₂×C₁₃	#1493	1	1	yes	characteristic
201	C₂×C₁₃	C₂×#328	1	1	yes	characteristic
202	C₂×C₁₃	#1508	1	1	yes	characteristic
203	C₂×C₁₃	#1525	1	1	yes	characteristic
204	C₂×C₁₃	#1548	1	1	yes	characteristic
205	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	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	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
223	C₂²×C₄	C₁₃×Q₁₆	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	M₁₆	—	4	8	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₁₃	#7687	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	C₁₃×#103	1	2	yes	—
253	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
254	Q₈	—	8	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	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	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₁₃×#99	1	2	yes	—
276	C₂×C₄	C₁₃×#100	1	2	yes	—
277	C₂×C₄	C₁₃×#103	1	2	yes	—
278	C₂×C₄	C₁₃×#104	1	2	yes	—
279	C₂×C₄	C₁₃×#104	1	2	yes	—
280	C₂×C₄	C₁₃×#131	1	2	yes	—
281	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
282	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
283	C₂×C₄	C₁₃×#137	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	4	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	C₁₃×#408	1	1	yes	characteristic
294	C₂²	C₂×C₁₃×#99	1	1	yes	characteristic
295	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
296	C₂²	C₂×C₁₃×#103	1	1	yes	characteristic
297	C₂²	C₁₃×#419	1	1	yes	characteristic
298	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
299	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	C₁₃×#328	1	4	yes	—
309	C₂	—	4	2	no	—
310	C₂	C₁₃×#1471	1	1	yes	characteristic
311	C₂	C₁₃×#1485	1	1	yes	characteristic
312	C₂	C₁₃×#1493	1	1	yes	characteristic
313	C₂	C₂×C₁₃×#328	1	1	yes	characteristic
314	C₂	C₁₃×#1508	1	1	yes	characteristic
315	C₂	C₁₃×#1525	1	1	yes	characteristic
316	C₂	C₁₃×#1548	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×#1272	C₂	semidirect
47. #7687	C₁₃	direct (N × H)
91. #1272	C₂×C₁₃	semidirect
245. C₁₃	#7687	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×#1272	2
#7687	13

Subgroups — C13×#7687 #15694026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#7687 #15694026