Subgroups — C₁₃×#10260 #15696599

Order 3328.

794 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	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#459	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1954	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#331	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×D₈	—	2	2	no	—
7	C₂²×C₁₃×D₈	—	2	2	no	—
8	C₂×C₁₃×#127	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#493	—	2	4	no	—
11	C₁₃×#459	—	2	4	no	—
12	C₁₃×#459	—	2	4	no	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#102	—	2	2	no	—
16	C₂×C₁₃×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#331	—	2	4	no	—
19	C₂²×D₄×C₁₃	—	2	2	no	—
20	C₂²×D₄×C₁₃	—	2	2	no	—
21	C₂²×D₄×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₂×C₁₃×D₈	—	2	8	no	—
25	C₂×C₁₃×D₈	—	2	8	no	—
26	C₂×C₁₃×D₈	—	4	4	no	—
27	C₂×C₁₃×D₈	—	4	4	no	—
28	C₂×C₁₃×D₈	—	4	4	no	—
29	C₂×C₁₃×D₈	—	4	4	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₁₃×#127	—	2	2	no	—
34	C₁₃×#127	—	2	2	no	—
35	C₁₃×#127	—	2	2	no	—
36	C₁₃×#127	D₄	1	4	yes	—
37	C₄×D₄×C₁₃	—	2	2	no	—
38	C₄×D₄×C₁₃	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
45	C₁₃×#102	—	2	8	no	—
46	C₁₃×C₂²⋊C₈	—	4	4	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	#10260	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	2	1	no	—
50	C₂⁴×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	4	no	—
53	C₂×D₄×C₁₃	—	2	4	no	—
54	C₂×D₄×C₁₃	—	2	4	no	—
55	C₂×D₄×C₁₃	—	2	1	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂×D₄×C₁₃	—	4	2	no	—
61	C₂×D₄×C₁₃	—	4	4	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	—	4	2	no	—
66	C₂×D₄×C₁₃	—	8	4	no	—
67	C₂×D₄×C₁₃	D₈	1	2	yes	complemented
68	C₂²×C₄×C₁₃	—	2	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	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
76	C₁₃×D₈	—	4	16	no	—
77	C₁₃×D₈	—	4	16	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	—	4	1	no	—
84	C₂×C₈×C₁₃	—	4	2	no	—
85	C₂×C₈×C₁₃	D₈	1	4	yes	complemented
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₄²×C₁₃	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
91	C₂×#459	C₂×C₁₃	1	2	yes	complemented
92	#1954	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂×#331	C₂×C₁₃	1	2	yes	complemented
94	#1277	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂³×C₁₃	—	2	2	no	—
96	C₂³×C₁₃	—	4	1	no	—
97	C₂³×C₁₃	—	4	2	no	—
98	C₂³×C₁₃	—	4	1	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	—	4	4	no	—
101	C₂³×C₁₃	—	8	4	no	—
102	C₂³×C₁₃	—	8	2	no	—
103	C₂³×C₁₃	—	8	2	no	—
104	C₂³×C₁₃	—	8	4	no	—
105	C₂³×C₁₃	#121	1	1	yes	characteristic
106	D₄×C₁₃	—	2	4	no	—
107	D₄×C₁₃	—	4	4	no	—
108	D₄×C₁₃	—	4	8	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	4	no	—
111	D₄×C₁₃	—	8	4	no	—
112	D₄×C₁₃	—	8	4	no	—
113	D₄×C₁₃	—	8	2	no	—
114	D₄×C₁₃	—	8	2	no	—
115	D₄×C₁₃	—	8	2	no	—
116	D₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	8	2	no	—
121	C₂×C₄×C₁₃	—	8	1	no	—
122	C₂×C₄×C₁₃	—	8	1	no	—
123	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#120	1	2	yes	—
125	C₂×C₄×C₁₃	#121	1	2	yes	—
126	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
130	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
132	C₈×C₁₃	—	2	4	no	—
133	C₈×C₁₃	—	4	4	no	—
134	C₈×C₁₃	—	8	1	no	—
135	C₂²×D₈	—	2	2	no	—
136	C₂²×D₈	—	2	2	no	—
137	C₂×#127	C₂²×C₁₃	1	2	yes	—
138	C₂×C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
139	#493	—	2	4	no	—
140	#459	—	2	4	no	—
141	#459	—	2	4	no	—
142	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
144	C₂×#102	—	2	2	no	—
145	C₂×C₂²⋊C₈	—	2	2	no	—
146	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	#331	—	2	4	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	—	4	2	no	—
150	C₂²×C₁₃	—	8	2	no	—
151	C₂²×C₁₃	—	8	1	no	—
152	C₂²×C₁₃	—	8	1	no	—
153	C₂²×C₁₃	—	8	4	no	—
154	C₂²×C₁₃	—	8	4	no	—
155	C₂²×C₁₃	—	8	4	no	—
156	C₂²×C₁₃	—	16	4	no	—
157	C₂²×C₁₃	—	16	4	no	—
158	C₂²×C₁₃	#392	1	1	yes	characteristic
159	C₂²×C₁₃	#459	1	2	yes	—
160	C₂²×C₁₃	#466	1	1	yes	characteristic
161	C₂²×C₁₃	#469	1	1	yes	characteristic
162	C₂²×C₁₃	#472	1	2	yes	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	8	1	no	—
165	C₄×C₁₃	#447	1	2	yes	—
166	C₄×C₁₃	#447	1	2	yes	—
167	C₄×C₁₃	#449	1	2	yes	—
168	C₄×C₁₃	#493	1	1	yes	characteristic
169	C₄×C₁₃	#496	1	1	yes	characteristic
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	D₄×C₁₃	1	1	yes	characteristic
174	C₂³×C₄	—	2	1	no	—
175	C₂×D₈	—	2	8	no	—
176	C₂×D₈	—	2	8	no	—
177	C₂×D₈	—	4	4	no	—
178	C₂×D₈	—	4	4	no	—
179	C₂×D₈	—	4	4	no	—
180	C₂×D₈	—	4	4	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	D₄×C₁₃	1	2	yes	—
184	#127	—	2	2	no	—
185	#127	—	2	2	no	—
186	#127	—	2	2	no	—
187	#127	D₄×C₁₃	1	4	yes	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	2	2	no	—
190	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
196	#102	—	2	8	no	—
197	C₂²⋊C₈	—	4	4	no	—
198	C₄×C₈	—	2	2	no	—
199	C₂×C₁₃	—	8	2	no	—
200	C₂×C₁₃	—	16	4	no	—
201	C₂×C₁₃	#1667	1	2	yes	—
202	C₂×C₁₃	#1687	1	2	yes	—
203	C₂×C₁₃	#1999	1	1	yes	characteristic
204	C₂×C₁₃	#2011	1	1	yes	characteristic
205	C₂×C₁₃	#2018	1	1	yes	characteristic
206	C₂⁴	—	2	1	no	—
207	C₂⁴	—	4	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	8	4	no	—
224	C₂×D₄	C₁₃×D₈	1	2	yes	complemented
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
233	D₈	—	4	16	no	—
234	D₈	—	4	16	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₁₃×D₈	1	4	yes	complemented
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	C₂×D₄×C₁₃	1	2	yes	—
246	C₄²	C₂×D₄×C₁₃	1	2	yes	—
247	C₁₃	#10260	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³	—	2	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	8	4	no	—
255	C₂³	—	8	2	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	4	no	—
258	C₂³	C₁₃×#121	1	1	yes	characteristic
259	D₄	—	2	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	8	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	4	no	—
264	D₄	—	8	4	no	—
265	D₄	—	8	4	no	—
266	D₄	—	8	2	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	8	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
277	C₂×C₄	C₁₃×#120	1	2	yes	—
278	C₂×C₄	C₁₃×#121	1	2	yes	—
279	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
282	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
283	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
284	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
285	C₈	—	2	4	no	—
286	C₈	—	4	4	no	—
287	C₈	—	8	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	4	no	—
297	C₂²	—	16	4	no	—
298	C₂²	C₁₃×#392	1	1	yes	characteristic
299	C₂²	C₁₃×#459	1	2	yes	—
300	C₂²	C₁₃×#466	1	1	yes	characteristic
301	C₂²	C₁₃×#469	1	1	yes	characteristic
302	C₂²	C₁₃×#472	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	8	1	no	—
305	C₄	C₁₃×#447	1	2	yes	—
306	C₄	C₁₃×#447	1	2	yes	—
307	C₄	C₁₃×#449	1	2	yes	—
308	C₄	C₁₃×#493	1	1	yes	characteristic
309	C₄	C₁₃×#496	1	1	yes	characteristic
310	C₂	—	8	2	no	—
311	C₂	—	16	4	no	—
312	C₂	C₁₃×#1667	1	2	yes	—
313	C₂	C₁₃×#1687	1	2	yes	—
314	C₂	C₁₃×#1999	1	1	yes	characteristic
315	C₂	C₁₃×#2011	1	1	yes	characteristic
316	C₂	C₁₃×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#493	C₂	semidirect
2. C₂×C₁₃×#459	C₂	semidirect
3. C₁₃×#1954	C₂	semidirect
4. C₂×C₁₃×#331	C₂	semidirect
5. C₁₃×#1277	C₂	semidirect
48. #10260	C₁₃	direct (N × H)
67. C₂×D₄×C₁₃	D₈	semidirect
85. C₂×C₈×C₁₃	D₈	semidirect
90. C₂×#493	C₂×C₁₃	semidirect
91. C₂×#459	C₂×C₁₃	semidirect
92. #1954	C₂×C₁₃	semidirect
93. C₂×#331	C₂×C₁₃	semidirect
94. #1277	C₂×C₁₃	semidirect
224. C₂×D₄	C₁₃×D₈	semidirect
242. C₂×C₈	C₁₃×D₈	semidirect
247. C₁₃	#10260	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×#493	2
C₂×C₁₃×#459	2
C₁₃×#1954	2
C₂×C₁₃×#331	2
C₁₃×#1277	2
#10260	13

Subgroups — C13×#10260 #15696599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10260 #15696599