Subgroups — C₂×D₁₃×#124 #13128081

Order 1664.

1880 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#124	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₄×D₁₃	C₂
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	C₂²×#124
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₃	C₂	semidirect
3. C₂×C₁₃×#124	C₂	semidirect
4. C₂×#88201	C₂	semidirect
5. C₂×#88190	C₂	semidirect
6. D₁₃×#124	C₂	direct (N × H)
7. C₂×#88139	C₂	semidirect
8. C₂×C₂²⋊C₄×D₁₃	C₂	semidirect
9. C₂×#88129	C₂	semidirect
10. C₂×C₄²×D₁₃	C₂	semidirect
12. C₂²×Q₈×C₁₃	C₂²	semidirect
16. C₂×Q₈×D₁₃	C₂²	semidirect
23. C₂²×Dic₂₆	C₂²	semidirect
24. C₂²×Dic₂₆	C₂²	semidirect
25. C₁₃×#124	C₂²	semidirect
26. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
27. C₂×C₄²×C₁₃	C₂²	semidirect
28. C₂×C₂²⋊Dic₁₃	C₂²	semidirect
29. #88201	C₂²	semidirect
30. #88190	C₂²	semidirect
31. C₂×#6103	C₂²	semidirect
32. C₂×#6103	C₂²	semidirect
33. C₂×C₄×Dic₁₃	C₂²	semidirect
34. C₂×C₄×Dic₁₃	C₂²	semidirect
35. #88139	C₂²	semidirect
36. C₂²⋊C₄×D₁₃	C₂²	semidirect
37. #88129	C₂²	semidirect
38. C₄²×D₁₃	C₂²	semidirect
46. C₂×Q₈×C₁₃	C₂³	semidirect
69. C₂×Dic₂₆	C₂³	semidirect
70. C₂×Dic₂₆	C₂³	semidirect
71. C₁₃×C₂²⋊C₄	C₂³	semidirect
72. C₄²×C₁₃	C₂³	semidirect
73. C₂²⋊Dic₁₃	C₂³	semidirect
74. #6103	C₂³	semidirect
75. #6103	C₂³	semidirect
76. C₄×Dic₁₃	C₂³	semidirect
77. C₄×Dic₁₃	C₂³	semidirect
120. C₂×#124	D₁₃	direct (N × H)
133. C₂×D₁₃	#124	direct (N × H)
147. C₂²×Q₈	C₂×D₁₃	semidirect
160. #124	C₂×D₁₃	direct (N × H)
165. C₂×C₂²⋊C₄	C₂×D₁₃	semidirect
169. C₂×C₄²	C₂×D₁₃	semidirect
171. C₂×C₁₃	C₂×#124	semidirect
175. D₁₃	C₂×#124	direct (N × H)
188. C₂×Q₈	C₂²×D₁₃	semidirect
211. C₂²⋊C₄	C₂²×D₁₃	semidirect
214. C₄²	C₂²×D₁₃	semidirect
215. C₁₃	C₂²×#124	semidirect
272. C₂	D₁₃×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₃	2
C₂²×D₄×D₁₃	2
C₂×C₁₃×#124	2
C₂×#88201	2
C₂×#88190	2
D₁₃×#124	2
C₂×#88139	2
C₂×C₂²⋊C₄×D₁₃	2
C₂×#88129	2
C₂×C₄²×D₁₃	2
C₂²×#124	13

Subgroups — C2×D13×#124 #13128081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃×#124 #13128081