Subgroups — C₁₃×#13147 #15699486

Order 3328.

394 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1970	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#390	—	2	1	no	—
17	C₁₃×#389	—	2	1	no	—
18	C₁₃×#357	—	2	2	no	—
19	C₁₃×#357	C₄	1	4	yes	complemented
20	C₄×C₁₃×C₁₆	—	2	2	no	—
21	C₁₃×#340	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂×C₁₃×D₈	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	1	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
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	—	4	1	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₁₆	—	2	1	no	—
39	C₂×C₁₃×C₁₆	—	2	1	no	—
40	C₂×C₁₃×C₁₆	—	2	1	no	—
41	C₂×C₁₃×C₁₆	—	2	1	no	—
42	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
44	C₁₃×#107	—	2	1	no	—
45	C₁₃×#107	—	2	1	no	—
46	C₁₃×#107	—	2	1	no	—
47	C₁₃×#107	—	2	1	no	—
48	C₁₃×#107	—	2	1	no	—
49	C₁₃×#107	C₂×C₄	1	2	yes	—
50	C₁₃×#102	—	4	2	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₁₃×#95	—	4	1	no	—
54	#13147	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	—	4	1	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	1	no	—
59	C₂×D₄×C₁₃	—	4	1	no	—
60	C₂×D₄×C₁₃	—	4	1	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₁₃×D₈	—	2	2	no	—
69	C₁₃×D₈	—	2	2	no	—
70	C₁₃×D₈	—	4	1	no	—
71	C₁₃×D₈	—	4	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	4	1	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₄⋊C₄	—	4	1	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₄⋊C₄	—	4	1	no	—
87	C₁₃×C₄⋊C₄	—	8	2	no	—
88	C₁₃×C₂²⋊C₄	—	8	2	no	—
89	C₁₃×C₂²⋊C₄	—	8	2	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₁₃×C₁₆	—	2	2	no	—
93	C₁₃×C₁₆	—	2	2	no	—
94	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#357	C₂×C₁₃	1	1	yes	characteristic
96	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#1970	C₂×C₁₃	1	1	yes	characteristic
98	#1942	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#1380	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#1380	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	—	8	1	no	—
103	C₂³×C₁₃	—	8	1	no	—
104	C₂³×C₁₃	—	8	1	no	—
105	C₂³×C₁₃	—	8	1	no	—
106	C₂³×C₁₃	—	8	1	no	—
107	C₂³×C₁₃	—	8	1	no	—
108	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	8	1	no	—
112	D₄×C₁₃	—	8	1	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	—	4	2	no	—
118	C₂×C₄×C₁₃	—	8	1	no	—
119	C₂×C₄×C₁₃	—	8	1	no	—
120	C₂×C₄×C₁₃	—	8	1	no	—
121	C₂×C₄×C₁₃	—	8	1	no	—
122	C₂×C₄×C₁₃	—	8	1	no	—
123	C₂×C₄×C₁₃	—	8	1	no	—
124	C₂×C₄×C₁₃	—	8	1	no	—
125	C₂×C₄×C₁₃	—	8	1	no	—
126	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
127	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	#117	1	1	yes	characteristic
136	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
137	C₈×C₁₃	#123	1	1	yes	characteristic
138	C₈×C₁₃	#126	1	1	yes	characteristic
139	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
141	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×#102	—	2	1	no	—
146	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	#390	—	2	1	no	—
148	#389	—	2	1	no	—
149	#357	—	2	2	no	—
150	#357	C₄×C₁₃	1	4	yes	complemented
151	C₄×C₁₆	—	2	2	no	—
152	#340	—	2	1	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	2	no	—
155	C₂²×C₁₃	—	8	2	no	—
156	C₂²×C₁₃	—	16	1	no	—
157	C₂²×C₁₃	—	16	1	no	—
158	C₂²×C₁₃	—	16	1	no	—
159	C₂²×C₁₃	—	16	1	no	—
160	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
161	C₂²×C₁₃	#466	1	1	yes	characteristic
162	C₂²×C₁₃	#486	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
165	C₂²×C₁₃	#508	1	1	yes	characteristic
166	C₂²×C₁₃	#508	1	1	yes	characteristic
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	—	16	1	no	—
171	C₄×C₁₃	—	16	1	no	—
172	C₄×C₁₃	#388	1	1	yes	characteristic
173	C₄×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
174	C₄×C₁₃	#465	1	1	yes	characteristic
175	C₄×C₁₃	#488	1	1	yes	characteristic
176	C₂²×D₄	—	2	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
183	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	4	1	no	—
189	C₂×C₂²⋊C₄	—	4	1	no	—
190	C₂×C₂²⋊C₄	—	4	1	no	—
191	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
197	C₂×C₁₆	D₄×C₁₃	1	2	yes	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	2	1	no	—
201	#107	—	2	1	no	—
202	#107	—	2	1	no	—
203	#107	C₂×C₄×C₁₃	1	2	yes	—
204	#102	—	4	2	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	#95	—	4	1	no	—
208	C₂×C₁₃	—	16	2	no	—
209	C₂×C₁₃	#1934	1	1	yes	characteristic
210	C₂×C₁₃	C₄×D₁₆	1	1	yes	characteristic
211	C₂×C₁₃	C₄×SD₃₂	1	1	yes	characteristic
212	C₂×C₁₃	#2215	1	1	yes	characteristic
213	C₂×C₁₃	#2217	1	1	yes	characteristic
214	C₂×C₁₃	#2240	1	1	yes	characteristic
215	C₂×C₁₃	#2242	1	1	yes	characteristic
216	C₂⁴	—	4	1	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
227	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	D₈	—	2	2	no	—
230	D₈	—	2	2	no	—
231	D₈	—	4	1	no	—
232	D₈	—	4	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
239	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
240	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
241	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	8	2	no	—
249	C₂²⋊C₄	—	8	2	no	—
250	C₂²⋊C₄	—	8	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₃	#13147	1	1	yes	characteristic, direct product factor, Sylow
256	C₂³	—	4	2	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	8	1	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	—	8	1	no	—
262	C₂³	—	8	1	no	—
263	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
264	D₄	—	4	2	no	—
265	D₄	—	4	2	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	C₁₃×D₁₆	1	2	yes	—
282	C₂×C₄	C₁₃×SD₃₂	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
285	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
286	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
287	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	C₁₃×#117	1	1	yes	characteristic
291	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
292	C₈	C₁₃×#123	1	1	yes	characteristic
293	C₈	C₁₃×#126	1	1	yes	characteristic
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	16	1	no	—
298	C₂²	—	16	1	no	—
299	C₂²	—	16	1	no	—
300	C₂²	—	16	1	no	—
301	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
302	C₂²	C₁₃×#466	1	1	yes	characteristic
303	C₂²	C₁₃×#486	1	1	yes	characteristic
304	C₂²	C₂×C₁₃×D₁₆	1	1	yes	characteristic
305	C₂²	C₂×C₁₃×SD₃₂	1	1	yes	characteristic
306	C₂²	C₁₃×#508	1	1	yes	characteristic
307	C₂²	C₁₃×#508	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	16	1	no	—
312	C₄	—	16	1	no	—
313	C₄	C₁₃×#388	1	1	yes	characteristic
314	C₄	C₄×C₁₃×SD₁₆	1	1	yes	characteristic
315	C₄	C₁₃×#465	1	1	yes	characteristic
316	C₄	C₁₃×#488	1	1	yes	characteristic
317	C₂	—	16	2	no	—
318	C₂	C₁₃×#1934	1	1	yes	characteristic
319	C₂	C₄×C₁₃×D₁₆	1	1	yes	characteristic
320	C₂	C₄×C₁₃×SD₃₂	1	1	yes	characteristic
321	C₂	C₁₃×#2215	1	1	yes	characteristic
322	C₂	C₁₃×#2217	1	1	yes	characteristic
323	C₂	C₁₃×#2240	1	1	yes	characteristic
324	C₂	C₁₃×#2242	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
5. C₁₃×#1942	C₂	semidirect
6. C₁₃×#1380	C₂	semidirect
7. C₁₃×#1380	C₂	semidirect
19. C₁₃×#357	C₄	semidirect
54. #13147	C₁₃	direct (N × H)
96. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
98. #1942	C₂×C₁₃	semidirect
99. #1380	C₂×C₁₃	semidirect
100. #1380	C₂×C₁₃	semidirect
150. #357	C₄×C₁₃	semidirect
255. C₁₃	#13147	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×#357	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1970	2
C₁₃×#1942	2
C₁₃×#1380	2
#13147	13

Subgroups — C13×#13147 #15699486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13147 #15699486