Subgroups — C₁₃×#10536 #15696875

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 328 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
2	C₁₃×#1982	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1939	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	1	no	—
7	C₂²×C₁₃×D₈	—	2	1	no	—
8	C₂×C₁₃×#127	C₂²	1	2	yes	—
9	C₁₃×#493	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#102	—	2	2	no	—
14	C₂×C₁₃×#102	—	2	2	no	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₁₃×#336	—	2	1	no	—
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	C₂×C₁₃×D₈	—	2	4	no	—
23	C₂×C₁₃×D₈	—	2	4	no	—
24	C₂×C₁₃×D₈	—	4	2	no	—
25	C₂×C₁₃×D₈	—	4	2	no	—
26	C₂×C₁₃×D₈	—	4	2	no	—
27	C₂×C₁₃×D₈	—	4	2	no	—
28	C₂²×C₈×C₁₃	—	2	1	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₁₃×#127	—	2	2	no	—
33	C₁₃×#127	—	2	2	no	—
34	C₁₃×#127	—	2	2	no	—
35	C₁₃×#127	D₄	1	4	yes	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	4	2	no	—
44	C₁₃×#102	—	4	4	no	—
45	C₁₃×#102	—	4	4	no	—
46	C₄×C₈×C₁₃	—	2	2	no	—
47	C₁₃×#95	—	2	1	no	—
48	#10536	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	—	2	4	no	—
51	C₂×D₄×C₁₃	—	2	4	no	—
52	C₂×D₄×C₁₃	—	2	4	no	—
53	C₂×D₄×C₁₃	—	4	2	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	4	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	2	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	8	2	no	—
65	C₂×D₄×C₁₃	—	8	2	no	—
66	C₂²×C₄×C₁₃	—	2	1	no	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₁₃×D₈	—	4	8	no	—
72	C₁₃×D₈	—	4	8	no	—
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	D₈	1	2	yes	—
84	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
85	C₁₃×C₄⋊C₄	—	2	1	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	D₈	1	2	yes	complemented
88	C₄²×C₁₃	—	4	1	no	—
89	C₄²×C₁₃	—	4	1	no	—
90	C₄²×C₁₃	C₂×D₄	1	2	yes	—
91	C₄²×C₁₃	#41	1	2	yes	—
92	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
93	#1982	C₂×C₁₃	1	2	yes	complemented
94	#1939	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂×#331	C₂×C₁₃	1	2	yes	complemented
96	#1277	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂³×C₁₃	—	4	4	no	—
98	C₂³×C₁₃	—	8	2	no	—
99	C₂³×C₁₃	—	8	2	no	—
100	C₂³×C₁₃	—	8	2	no	—
101	C₂³×C₁₃	—	8	2	no	—
102	C₂³×C₁₃	—	8	2	no	—
103	C₂³×C₁₃	—	8	2	no	—
104	C₂³×C₁₃	#121	1	1	yes	characteristic
105	D₄×C₁₃	—	4	4	no	—
106	D₄×C₁₃	—	4	4	no	—
107	D₄×C₁₃	—	4	4	no	—
108	D₄×C₁₃	—	4	4	no	—
109	D₄×C₁₃	—	8	4	no	—
110	D₄×C₁₃	—	8	4	no	—
111	D₄×C₁₃	—	8	2	no	—
112	D₄×C₁₃	—	8	2	no	—
113	D₄×C₁₃	—	8	2	no	—
114	D₄×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#121	1	2	yes	—
125	C₂×C₄×C₁₃	#123	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
131	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	8	1	no	—
137	C₈×C₁₃	—	8	2	no	—
138	C₂²×D₈	—	2	1	no	—
139	C₂²×D₈	—	2	1	no	—
140	C₂×#127	C₂²×C₁₃	1	2	yes	—
141	#493	—	2	4	no	—
142	C₂×C₄⋊C₈	—	2	1	no	—
143	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
145	C₂×#102	—	2	2	no	—
146	C₂×#102	—	2	2	no	—
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
149	#336	—	2	1	no	—
150	#331	—	2	4	no	—
151	C₂²×C₁₃	—	8	4	no	—
152	C₂²×C₁₃	—	8	4	no	—
153	C₂²×C₁₃	—	8	4	no	—
154	C₂²×C₁₃	—	16	2	no	—
155	C₂²×C₁₃	—	16	2	no	—
156	C₂²×C₁₃	—	16	2	no	—
157	C₂²×C₁₃	—	16	2	no	—
158	C₂²×C₁₃	#394	1	1	yes	characteristic
159	C₂²×C₁₃	#459	1	1	yes	characteristic
160	C₂²×C₁₃	#460	1	1	yes	characteristic
161	C₂²×C₁₃	#466	1	1	yes	characteristic
162	C₂²×C₁₃	#469	1	1	yes	characteristic
163	C₂²×C₁₃	#472	1	1	yes	characteristic
164	C₂²×C₁₃	#472	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	—	8	1	no	—
168	C₄×C₁₃	#447	1	1	yes	characteristic
169	C₄×C₁₃	#450	1	1	yes	characteristic
170	C₄×C₁₃	#459	1	2	yes	—
171	C₄×C₁₃	#463	1	2	yes	—
172	C₄×C₁₃	#486	1	1	yes	characteristic
173	C₄×C₁₃	#490	1	1	yes	characteristic
174	C₂²×D₄	—	2	2	no	—
175	C₂²×D₄	—	2	2	no	—
176	C₂²×D₄	—	2	2	no	—
177	C₂×D₈	—	2	4	no	—
178	C₂×D₈	—	2	4	no	—
179	C₂×D₈	—	4	2	no	—
180	C₂×D₈	—	4	2	no	—
181	C₂×D₈	—	4	2	no	—
182	C₂×D₈	—	4	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
187	#127	—	2	2	no	—
188	#127	—	2	2	no	—
189	#127	—	2	2	no	—
190	#127	D₄×C₁₃	1	4	yes	—
191	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	4	2	no	—
199	#102	—	4	4	no	—
200	#102	—	4	4	no	—
201	C₄×C₈	—	2	2	no	—
202	#95	—	2	1	no	—
203	C₂×C₁₃	—	16	4	no	—
204	C₂×C₁₃	#1667	1	1	yes	characteristic
205	C₂×C₁₃	#1668	1	1	yes	characteristic
206	C₂×C₁₃	#1687	1	1	yes	characteristic
207	C₂×C₁₃	#1688	1	1	yes	characteristic
208	C₂×C₁₃	#2041	1	1	yes	characteristic
209	C₂×C₁₃	#2045	1	1	yes	characteristic
210	C₂×C₁₃	#2054	1	1	yes	characteristic
211	C₂⁴	—	4	2	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	8	2	no	—
227	C₂×D₄	—	8	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	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	8	no	—
234	D₈	—	4	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₁₃×D₈	1	2	yes	—
246	C₂×C₈	C₁₃×SD₁₆	1	2	yes	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	C₁₃×D₈	1	2	yes	complemented
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	C₂×D₄×C₁₃	1	2	yes	—
253	C₄²	C₁₃×#41	1	2	yes	—
254	C₁₃	#10536	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	—	4	4	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	8	2	no	—
261	C₂³	—	8	2	no	—
262	C₂³	C₁₃×#121	1	1	yes	characteristic
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
282	C₂×C₄	C₁₃×#121	1	2	yes	—
283	C₂×C₄	C₁₃×#123	1	2	yes	—
284	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
285	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
286	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
287	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
288	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
289	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
292	C₈	—	2	2	no	—
293	C₈	—	2	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	2	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	16	2	no	—
300	C₂²	—	16	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	C₁₃×#394	1	1	yes	characteristic
304	C₂²	C₁₃×#459	1	1	yes	characteristic
305	C₂²	C₁₃×#460	1	1	yes	characteristic
306	C₂²	C₁₃×#466	1	1	yes	characteristic
307	C₂²	C₁₃×#469	1	1	yes	characteristic
308	C₂²	C₁₃×#472	1	1	yes	characteristic
309	C₂²	C₁₃×#472	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	1	no	—
313	C₄	C₁₃×#447	1	1	yes	characteristic
314	C₄	C₁₃×#450	1	1	yes	characteristic
315	C₄	C₁₃×#459	1	2	yes	—
316	C₄	C₁₃×#463	1	2	yes	—
317	C₄	C₁₃×#486	1	1	yes	characteristic
318	C₄	C₁₃×#490	1	1	yes	characteristic
319	C₂	—	16	4	no	—
320	C₂	C₁₃×#1667	1	1	yes	characteristic
321	C₂	C₁₃×#1668	1	1	yes	characteristic
322	C₂	C₁₃×#1687	1	1	yes	characteristic
323	C₂	C₁₃×#1688	1	1	yes	characteristic
324	C₂	C₁₃×#2041	1	1	yes	characteristic
325	C₂	C₁₃×#2045	1	1	yes	characteristic
326	C₂	C₁₃×#2054	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#493	C₂
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
2. C₁₃×#1982	C₂	semidirect
3. C₁₃×#1939	C₂	semidirect
4. C₂×C₁₃×#331	C₂	semidirect
5. C₁₃×#1277	C₂	semidirect
48. #10536	C₁₃	direct (N × H)
87. C₁₃×C₄⋊C₄	D₈	semidirect
93. #1982	C₂×C₁₃	semidirect
94. #1939	C₂×C₁₃	semidirect
95. C₂×#331	C₂×C₁₃	semidirect
96. #1277	C₂×C₁₃	semidirect
249. C₄⋊C₄	C₁₃×D₈	semidirect
254. C₁₃	#10536	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
C₁₃×#1982	2
C₁₃×#1939	2
C₂×C₁₃×#331	2
C₁₃×#1277	2
#10536	13

Subgroups — C13×#10536 #15696875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10536 #15696875