Subgroups — D₁₃×#559 #13128420

Order 1664.

1240 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₃×#127	C₂	1	1	yes	maximal, characteristic
3	#11719418	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719386	C₂	1	4	yes	maximal, complemented
5	D₁₃×#124	C₂	1	4	yes	maximal, complemented
6	#11719369	C₂	1	4	yes	maximal, complemented
7	D₁₃×#123	C₂	1	4	yes	maximal, complemented
8	#11719290	C₂	1	2	yes	maximal, complemented
9	C₄×Q₈×D₁₃	C₂	1	2	yes	maximal, complemented
10	#11719264	C₂	1	4	yes	maximal, complemented
11	C₄×D₄×D₁₃	C₂	1	4	yes	maximal, complemented
12	C₂×Q₈×D₁₃	C₂²	1	2	yes	—
13	C₂×D₄×D₁₃	C₂²	1	4	yes	—
14	C₂×D₄×D₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₃	C₂²	1	4	yes	—
16	C₁₃×#127	C₂²	1	1	yes	characteristic
17	C₁₃×#124	C₂²	1	4	yes	complemented
18	C₁₃×#123	C₂²	1	4	yes	complemented
19	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
20	C₄×D₄×C₁₃	C₂²	1	4	yes	complemented
21	#88201	C₂²	1	4	yes	complemented
22	Q₈×Dic₁₃	C₂²	1	2	yes	complemented
23	#88194	C₂²	1	4	yes	—
24	#88194	C₂²	1	2	yes	—
25	#88190	C₂²	1	4	yes	complemented
26	#88189	C₂²	1	4	yes	complemented
27	D₄×Dic₁₃	C₂²	1	4	yes	complemented
28	#88183	C₂²	1	4	yes	complemented
29	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	complemented
30	#88148	C₂²	1	8	yes	complemented
31	#88147	C₂²	1	4	yes	complemented
32	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
33	C₄⋊C₄×D₁₃	C₂²	1	4	yes	—
34	#88141	C₂²	1	2	yes	complemented
35	#88141	C₂²	1	4	yes	complemented
36	#88140	C₂²	1	4	yes	complemented
37	#88139	C₂²	1	8	yes	complemented
38	#88139	C₂²	1	8	yes	complemented
39	#88137	C₂²	1	8	yes	complemented
40	#88135	C₂²	1	8	yes	complemented
41	C₂²⋊C₄×D₁₃	C₂²	1	8	yes	—
42	C₂²⋊C₄×D₁₃	C₂²	1	4	yes	—
43	#88129	C₂²	1	4	yes	complemented
44	#88128	C₂²	1	1	yes	characteristic
45	C₄×D₅₂	C₂²	1	4	yes	complemented
46	C₄²×D₁₃	C₂²	1	4	yes	—
47	C₄²×D₁₃	C₂²	1	1	yes	characteristic
48	C₄×Dic₂₆	C₂²	1	4	yes	complemented
49	C₄×Dic₂₆	C₂²	1	2	yes	complemented
50	C₂³×D₁₃	C₂³	1	4	yes	—
51	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
52	C₂×D₄×C₁₃	C₂³	1	4	yes	—
53	C₂×D₄×C₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
57	C₂²×Dic₁₃	C₂³	1	4	yes	—
58	Q₈×D₁₃	—	2	8	no	—
59	D₄×D₁₃	—	2	16	no	—
60	D₄×D₁₃	—	4	4	no	—
61	C₂×D₅₂	C₂³	1	4	yes	—
62	C₂×D₅₂	C₂³	1	2	yes	—
63	C₂×C₄×D₁₃	—	4	8	no	—
64	C₂×C₄×D₁₃	C₂³	1	4	yes	—
65	C₂×C₄×D₁₃	C₂³	1	4	yes	—
66	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂³	1	8	yes	—
68	C₂×C₄×D₁₃	C₂³	1	2	yes	—
69	C₂×Dic₂₆	C₂³	1	2	yes	—
70	C₂×Dic₂₆	C₂³	1	4	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
73	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
74	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	C₄²×C₁₃	C₂³	1	4	yes	—
77	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
78	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
79	#6103	C₂³	1	8	yes	—
80	#6103	C₂³	1	8	yes	—
81	#6103	C₂³	1	8	yes	—
82	C₄⋊Dic₁₃	C₂³	1	2	yes	—
83	C₄⋊Dic₁₃	C₂³	1	4	yes	—
84	#6101	C₂³	1	4	yes	—
85	#6101	C₂³	1	4	yes	—
86	#6101	C₂³	1	4	yes	—
87	C₄×Dic₁₃	C₂³	1	2	yes	—
88	C₄×Dic₁₃	C₂³	1	4	yes	—
89	C₄×Dic₁₃	C₂³	1	4	yes	—
90	C₄×Dic₁₃	C₂³	1	4	yes	—
91	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
92	C₂×#559	—	13	1	no	maximal, Sylow
93	C₂³×C₁₃	C₂⁴	1	4	yes	—
94	C₂²×D₁₃	—	2	16	no	—
95	C₂²×D₁₃	—	2	8	no	—
96	C₂²×D₁₃	C₂⁴	1	4	yes	—
97	C₂²×D₁₃	C₂⁴	1	1	yes	characteristic
98	Q₈×C₁₃	—	2	4	no	—
99	D₄×C₁₃	—	2	8	no	—
100	D₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
106	C₂²⋊D₁₃	—	2	16	no	—
107	C₂²⋊D₁₃	—	4	4	no	—
108	C₂×Dic₁₃	—	4	4	no	—
109	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₁₃	C₂⁴	1	4	yes	—
111	C₂×Dic₁₃	C₂⁴	1	4	yes	—
112	C₂×Dic₁₃	C₂⁴	1	2	yes	—
113	D₅₂	—	2	8	no	—
114	D₅₂	—	4	2	no	—
115	C₄×D₁₃	—	2	8	no	—
116	C₄×D₁₃	—	2	8	no	—
117	C₄×D₁₃	—	2	2	no	—
118	C₄×D₁₃	—	4	8	no	—
119	C₄×D₁₃	#41	1	8	yes	—
120	Dic₂₆	—	2	4	no	—
121	Dic₂₆	—	2	8	no	—
122	#559	—	13	1	no	—
123	#559	—	13	4	no	—
124	#559	—	13	4	no	—
125	#559	—	13	2	no	—
126	#559	—	13	4	no	—
127	#559	D₁₃	1	1	yes	characteristic, direct product factor
128	C₂×#127	—	13	1	no	—
129	C₂×#124	—	13	4	no	—
130	C₂×#123	—	13	4	no	—
131	C₂×C₄×Q₈	—	13	2	no	—
132	C₂×C₄×D₄	—	13	4	no	—
133	C₂²×C₁₃	—	2	8	no	—
134	C₂²×C₁₃	—	2	4	no	—
135	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
136	C₂×D₁₃	—	2	8	no	—
137	C₂×D₁₃	—	2	4	no	—
138	C₂×D₁₃	—	4	16	no	—
139	C₂×D₁₃	C₂×#41	1	4	yes	—
140	C₂×D₁₃	H(5,𝔽₂)	1	2	yes	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	C₂×#41	1	4	yes	—
145	Dic₁₃	—	2	4	no	—
146	Dic₁₃	—	2	4	no	—
147	Dic₁₃	—	2	1	no	—
148	Dic₁₃	C₂×#41	1	4	yes	—
149	C₂²×Q₈	—	13	2	no	—
150	C₂²×D₄	—	13	4	no	—
151	C₂²×D₄	—	13	2	no	—
152	C₂³×C₄	—	13	4	no	—
153	#127	—	13	2	no	—
154	#127	—	13	4	no	—
155	#127	—	13	1	no	—
156	#127	C₂×D₁₃	1	1	yes	characteristic
157	#124	—	13	4	no	—
158	#124	—	13	4	no	—
159	#124	—	13	8	no	—
160	#124	—	13	8	no	—
161	#124	—	13	4	no	—
162	#124	C₂×D₁₃	1	4	yes	complemented
163	#123	—	13	4	no	—
164	#123	—	13	4	no	—
165	#123	—	13	8	no	—
166	#123	—	13	8	no	—
167	#123	—	13	4	no	—
168	#123	C₂×D₁₃	1	4	yes	complemented
169	C₄×Q₈	—	13	2	no	—
170	C₄×Q₈	—	13	4	no	—
171	C₄×Q₈	—	13	2	no	—
172	C₄×Q₈	—	13	2	no	—
173	C₄×Q₈	—	13	4	no	—
174	C₄×Q₈	C₂×D₁₃	1	2	yes	complemented
175	C₄×D₄	—	13	4	no	—
176	C₄×D₄	—	13	8	no	—
177	C₄×D₄	—	13	4	no	—
178	C₄×D₄	—	13	8	no	—
179	C₄×D₄	—	13	4	no	—
180	C₄×D₄	C₂×D₁₃	1	4	yes	complemented
181	C₂×C₄⋊C₄	—	13	2	no	—
182	C₂×C₄⋊C₄	—	13	4	no	—
183	C₂×C₂²⋊C₄	—	13	8	no	—
184	C₂×C₂²⋊C₄	—	13	4	no	—
185	C₂×C₄²	—	13	4	no	—
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₁₃	—	4	4	no	—
188	C₂×C₁₃	C₂²×#41	1	2	yes	—
189	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
190	D₁₃	—	4	4	no	—
191	D₁₃	#559	1	4	yes	direct product factor
192	C₂⁴	—	13	4	no	—
193	C₂×Q₈	—	13	4	no	—
194	C₂×Q₈	—	13	2	no	—
195	C₂×Q₈	—	26	8	no	—
196	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
197	C₂×D₄	—	13	8	no	—
198	C₂×D₄	—	13	4	no	—
199	C₂×D₄	—	13	4	no	—
200	C₂×D₄	—	13	2	no	—
201	C₂×D₄	—	26	16	no	—
202	C₂×D₄	—	52	4	no	—
203	C₂×D₄	C₂²×D₁₃	1	4	yes	—
204	C₂×D₄	C₂²×D₁₃	1	2	yes	—
205	C₂²×C₄	—	13	2	no	—
206	C₂²×C₄	—	13	4	no	—
207	C₂²×C₄	—	13	4	no	—
208	C₂²×C₄	—	13	4	no	—
209	C₂²×C₄	—	13	1	no	—
210	C₂²×C₄	—	13	8	no	—
211	C₂²×C₄	—	52	8	no	—
212	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
213	C₄⋊C₄	—	13	4	no	—
214	C₄⋊C₄	—	13	2	no	—
215	C₄⋊C₄	—	13	4	no	—
216	C₄⋊C₄	—	13	4	no	—
217	C₄⋊C₄	—	13	4	no	—
218	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
219	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
220	C₂²⋊C₄	—	13	8	no	—
221	C₂²⋊C₄	—	13	4	no	—
222	C₂²⋊C₄	—	13	8	no	—
223	C₂²⋊C₄	—	13	8	no	—
224	C₂²⋊C₄	—	13	8	no	—
225	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
226	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
227	C₄²	—	13	4	no	—
228	C₄²	—	13	4	no	—
229	C₄²	—	13	4	no	—
230	C₄²	—	13	1	no	—
231	C₄²	—	13	2	no	—
232	C₄²	C₂²×D₁₃	1	4	yes	—
233	C₄²	C₂²×D₁₃	1	1	yes	characteristic
234	C₁₃	C₂×#559	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	13	1	no	—
236	C₂³	—	13	4	no	—
237	C₂³	—	26	8	no	—
238	C₂³	—	26	16	no	—
239	C₂³	C₂³×D₁₃	1	4	yes	—
240	Q₈	—	2	4	no	—
241	Q₈	—	26	8	no	—
242	Q₈	—	26	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	26	16	no	—
246	D₄	—	26	8	no	—
247	D₄	—	52	2	no	—
248	D₄	—	52	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	8	no	—
252	C₂×C₄	—	13	4	no	—
253	C₂×C₄	—	13	4	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	26	8	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	8	no	—
258	C₂×C₄	—	52	8	no	—
259	C₂×C₄	—	52	4	no	—
260	C₂×C₄	C₂³×D₁₃	1	4	yes	—
261	C₂×C₄	C₂³×D₁₃	1	4	yes	—
262	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂³×D₁₃	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	13	4	no	—
267	C₂²	—	13	2	no	—
268	C₂²	—	26	8	no	—
269	C₂²	—	26	4	no	—
270	C₂²	—	52	16	no	—
271	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	1	no	—
275	C₄	—	13	4	no	—
276	C₄	—	26	4	no	—
277	C₄	—	26	4	no	—
278	C₄	—	26	1	no	—
279	C₄	#41×D₁₃	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	13	4	no	—
282	C₂	—	52	4	no	—
283	C₂	C₂×#41×D₁₃	1	2	yes	—
284	C₂	D₁₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#559	C₂
prime radical	D₁₃×#127	C₂
hypercenter	#559	D₁₃
hypocenter	C₁₃	C₂×#559
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. C₁₃×#559	C₂	semidirect
3. #11719418	C₂	semidirect
4. #11719386	C₂	semidirect
5. D₁₃×#124	C₂	semidirect
6. #11719369	C₂	semidirect
7. D₁₃×#123	C₂	semidirect
8. #11719290	C₂	semidirect
9. C₄×Q₈×D₁₃	C₂	semidirect
10. #11719264	C₂	semidirect
11. C₄×D₄×D₁₃	C₂	semidirect
17. C₁₃×#124	C₂²	semidirect
18. C₁₃×#123	C₂²	semidirect
19. C₄×Q₈×C₁₃	C₂²	semidirect
20. C₄×D₄×C₁₃	C₂²	semidirect
21. #88201	C₂²	semidirect
22. Q₈×Dic₁₃	C₂²	semidirect
25. #88190	C₂²	semidirect
26. #88189	C₂²	semidirect
27. D₄×Dic₁₃	C₂²	semidirect
28. #88183	C₂²	semidirect
29. C₄×C₂²⋊D₁₃	C₂²	semidirect
30. #88148	C₂²	semidirect
31. #88147	C₂²	semidirect
34. #88141	C₂²	semidirect
35. #88141	C₂²	semidirect
36. #88140	C₂²	semidirect
37. #88139	C₂²	semidirect
38. #88139	C₂²	semidirect
39. #88137	C₂²	semidirect
40. #88135	C₂²	semidirect
43. #88129	C₂²	semidirect
45. C₄×D₅₂	C₂²	semidirect
48. C₄×Dic₂₆	C₂²	semidirect
49. C₄×Dic₂₆	C₂²	semidirect
127. #559	D₁₃	direct (N × H)
162. #124	C₂×D₁₃	semidirect
168. #123	C₂×D₁₃	semidirect
174. C₄×Q₈	C₂×D₁₃	semidirect
180. C₄×D₄	C₂×D₁₃	semidirect
191. D₁₃	#559	direct (N × H)
234. C₁₃	C₂×#559	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#559	2
D₁₃×#127	2
#11719418	2
#11719386	2
D₁₃×#124	2
#11719369	2
D₁₃×#123	2
#11719290	2
C₄×Q₈×D₁₃	2
#11719264	2
C₄×D₄×D₁₃	2
C₂×#559	13

Subgroups — D13×#559 #13128420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×#559 #13128420