Subgroups — C₁₃×#9646 #15695985

Order 3328.

610 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	C₄×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×D₄⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₁₃×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	—	2	2	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₁₃×#391	C₄	1	1	yes	characteristic
14	C₁₃×#390	C₄	1	1	yes	characteristic
15	C₁₃×#388	—	2	2	no	—
16	C₁₃×#382	C₄	1	1	yes	characteristic
17	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
18	C₄³×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×Q₈⋊C₈	—	2	4	no	—
20	C₁₃×D₄⋊C₈	—	2	4	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₁₃×#124	—	2	2	no	—
27	C₁₃×#124	—	2	2	no	—
28	C₄×Q₈×C₁₃	—	2	2	no	—
29	C₄×Q₈×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊C₈	—	2	2	no	—
45	C₁₃×C₄⋊C₈	—	2	2	no	—
46	C₁₃×C₄⋊C₈	—	4	4	no	—
47	C₁₃×#97	—	2	8	no	—
48	C₄×C₈×C₁₃	—	2	2	no	—
49	C₄×C₈×C₁₃	D₄	1	4	yes	—
50	C₁₃×#95	—	2	2	no	—
51	C₁₃×#95	—	2	2	no	—
52	#9646	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	—	2	1	no	—
54	C₂×Q₈×C₁₃	—	2	1	no	—
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×Q₈×C₁₃	M₁₆	1	2	yes	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	1	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	—	2	8	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	4	4	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	2	8	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₁₃×C₂²⋊C₄	—	4	2	no	—
84	C₄²×C₁₃	—	2	4	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	4	1	no	—
91	C₄²×C₁₃	—	4	1	no	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	C₄×#124	C₂×C₁₃	1	1	yes	characteristic
95	#1842	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂×Q₈⋊C₈	C₂×C₁₃	1	2	yes	complemented
98	C₂×D₄⋊C₈	C₂×C₁₃	1	2	yes	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	—	4	1	no	—
101	C₂³×C₁₃	—	4	1	no	—
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	2	4	no	—
108	D₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	—	8	1	no	—
124	C₂×C₄×C₁₃	—	8	1	no	—
125	C₂×C₄×C₁₃	#104	1	4	yes	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	#121	1	2	yes	—
128	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#131	1	2	yes	—
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₃	#137	1	2	yes	—
135	C₈×C₁₃	—	2	8	no	—
136	C₈×C₁₃	—	8	2	no	—
137	C₂×#124	C₄×C₁₃	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×C₄⋊C₈	—	2	2	no	—
141	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
142	C₂×#97	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
144	#391	C₄×C₁₃	1	1	yes	characteristic
145	#390	C₄×C₁₃	1	1	yes	characteristic
146	#388	—	2	2	no	—
147	#382	C₄×C₁₃	1	1	yes	characteristic
148	C₄×C₂²⋊C₄	—	2	2	no	—
149	C₄³	C₂²×C₁₃	1	1	yes	characteristic
150	Q₈⋊C₈	—	2	4	no	—
151	D₄⋊C₈	—	2	4	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	1	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	#390	1	1	yes	characteristic
158	C₂²×C₁₃	#407	1	1	yes	characteristic
159	C₂²×C₁₃	#408	1	1	yes	characteristic
160	C₂²×C₁₃	#417	1	2	yes	—
161	C₂²×C₁₃	C₂×#104	1	2	yes	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	4	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	—	4	2	no	—
166	C₄×C₁₃	—	8	1	no	—
167	C₄×C₁₃	—	8	1	no	—
168	C₄×C₁₃	#435	1	2	yes	—
169	C₄×C₁₃	#435	1	2	yes	—
170	C₄×C₁₃	#489	1	2	yes	—
171	C₄×C₁₃	#496	1	1	yes	characteristic
172	C₄×C₁₃	#497	1	1	yes	characteristic
173	C₂²×Q₈	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂²×D₄	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	#124	—	2	2	no	—
179	#124	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₁₃	1	2	yes	—
194	C₂×C₄²	D₄×C₁₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	4	4	no	—
199	#97	—	2	8	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	D₄×C₁₃	1	4	yes	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₂×C₁₃	—	8	2	no	—
205	C₂×C₁₃	#1483	1	2	yes	—
206	C₂×C₁₃	#1489	1	2	yes	—
207	C₂×C₁₃	#1955	1	1	yes	characteristic
208	C₂×C₁₃	#1959	1	1	yes	characteristic
209	C₂×C₁₃	#1962	1	1	yes	characteristic
210	C₂⁴	—	2	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₁₃×M₁₆	1	2	yes	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₁₃×M₁₆	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
227	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂×C₈	—	2	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	2	8	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₄²	C₂×D₄×C₁₃	1	2	yes	—
250	C₄²	C₁₃×#41	1	2	yes	—
251	C₁₃	#9646	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³	—	2	2	no	—
253	C₂³	—	4	1	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
258	Q₈	—	2	4	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	C₁₃×#104	1	4	yes	—
279	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
280	C₂×C₄	C₁₃×#121	1	2	yes	—
281	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
282	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
285	C₂×C₄	C₁₃×#131	1	2	yes	—
286	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
287	C₂×C₄	C₁₃×#137	1	2	yes	—
288	C₈	—	2	8	no	—
289	C₈	—	8	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	1	no	—
294	C₂²	—	8	1	no	—
295	C₂²	C₁₃×#390	1	1	yes	characteristic
296	C₂²	C₁₃×#407	1	1	yes	characteristic
297	C₂²	C₁₃×#408	1	1	yes	characteristic
298	C₂²	C₁₃×#417	1	2	yes	—
299	C₂²	C₂×C₁₃×#104	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	C₁₃×#435	1	2	yes	—
307	C₄	C₁₃×#435	1	2	yes	—
308	C₄	C₁₃×#489	1	2	yes	—
309	C₄	C₁₃×#496	1	1	yes	characteristic
310	C₄	C₁₃×#497	1	1	yes	characteristic
311	C₂	—	8	2	no	—
312	C₂	C₁₃×#1483	1	2	yes	—
313	C₂	C₁₃×#1489	1	2	yes	—
314	C₂	C₁₃×#1955	1	1	yes	characteristic
315	C₂	C₁₃×#1959	1	1	yes	characteristic
316	C₂	C₁₃×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1842	C₂	semidirect
3. C₄×C₁₃×#97	C₂	semidirect
4. C₂×C₁₃×Q₈⋊C₈	C₂	semidirect
52. #9646	C₁₃	direct (N × H)
95. #1842	C₂×C₁₃	semidirect
96. C₄×#97	C₂×C₁₃	semidirect
97. C₂×Q₈⋊C₈	C₂×C₁₃	semidirect
251. C₁₃	#9646	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#124	2
C₁₃×#1842	2
C₄×C₁₃×#97	2
C₂×C₁₃×Q₈⋊C₈	2
C₂×C₁₃×D₄⋊C₈	2
#9646	13

Subgroups — C13×#9646 #15695985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9646 #15695985