Subgroups — C₂³×C₂²⋊C₄×D₁₃ #16769469

Order 3328.

43678 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₁₃	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#6103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×D₁₃	C₂	1	56	yes	maximal, direct product factor
6	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁵×D₁₃	—	2	32	no	—
9	C₂⁵×D₁₃	C₄	1	32	yes	complemented
10	C₂⁵×D₁₃	C₂²	1	28	yes	—
11	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
12	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	complemented
14	C₂⁴×Dic₁₃	C₂²	1	2	yes	complemented
15	C₂³×C₄×D₁₃	—	2	32	no	—
16	C₂³×C₄×D₁₃	C₂²	1	56	yes	complemented
17	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
18	C₂²×C₂²⋊Dic₁₃	C₂²	1	28	yes	complemented
19	C₂²×#6103	C₂²	1	56	yes	complemented
20	C₂×C₂²⋊C₄×D₁₃	C₂²	1	448	yes	direct product factor
21	C₂⁵×C₁₃	—	2	16	no	—
22	C₂⁵×C₁₃	C₂×C₄	1	16	yes	complemented
23	C₂⁵×C₁₃	C₂³	1	14	yes	—
24	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
25	C₂⁴×D₁₃	—	2	16	no	—
26	C₂⁴×D₁₃	—	2	512	no	—
27	C₂⁴×D₁₃	—	2	448	no	—
28	C₂⁴×D₁₃	C₂×C₄	1	16	yes	—
29	C₂⁴×D₁₃	C₂×C₄	1	448	yes	complemented
30	C₂⁴×D₁₃	C₂×C₄	1	16	yes	complemented
31	C₂⁴×D₁₃	D₄	1	32	yes	—
32	C₂⁴×D₁₃	C₂³	1	112	yes	—
33	C₂⁴×D₁₃	C₂³	1	14	yes	—
34	C₂⁴×D₁₃	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₃	—	2	16	no	—
36	C₂³×C₄×C₁₃	C₂³	1	28	yes	complemented
37	C₂³×Dic₁₃	—	2	16	no	—
38	C₂³×Dic₁₃	C₂³	1	28	yes	complemented
39	C₂²×C₄×D₁₃	—	2	448	no	—
40	C₂²×C₄×D₁₃	C₂³	1	224	yes	complemented
41	C₂×C₁₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
42	C₂×C₂²⋊Dic₁₃	C₂³	1	112	yes	complemented
43	C₂×#6103	C₂³	1	224	yes	complemented
44	C₂²⋊C₄×D₁₃	C₂³	1	512	yes	direct product factor
45	C₂⁴×C₂²⋊C₄	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	—	2	128	no	—
47	C₂⁴×C₁₃	—	2	112	no	—
48	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
49	C₂⁴×C₁₃	C₂²×C₄	1	112	yes	complemented
50	C₂⁴×C₁₃	C₂×D₄	1	16	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
52	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
53	C₂³×D₁₃	—	2	128	no	—
54	C₂³×D₁₃	—	2	112	no	—
55	C₂³×D₁₃	—	2	3584	no	—
56	C₂³×D₁₃	—	2	896	no	—
57	C₂³×D₁₃	C₂²⋊C₄	1	256	yes	direct product factor
58	C₂³×D₁₃	C₂²×C₄	1	112	yes	—
59	C₂³×D₁₃	C₂²×C₄	1	896	yes	complemented
60	C₂³×D₁₃	C₂²×C₄	1	112	yes	complemented
61	C₂³×D₁₃	C₂×D₄	1	224	yes	—
62	C₂³×D₁₃	C₂⁴	1	64	yes	—
63	C₂³×D₁₃	C₂⁴	1	28	yes	—
64	C₂³×D₁₃	C₂⁴	1	28	yes	—
65	C₂²×C₄×C₁₃	—	2	112	no	—
66	C₂²×C₄×C₁₃	C₂⁴	1	56	yes	complemented
67	C₂²×Dic₁₃	—	2	112	no	—
68	C₂²×Dic₁₃	C₂⁴	1	56	yes	complemented
69	C₂×C₄×D₁₃	—	2	896	no	—
70	C₂×C₄×D₁₃	C₂⁴	1	128	yes	complemented
71	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
72	C₂²⋊Dic₁₃	C₂⁴	1	64	yes	complemented
73	#6103	C₂⁴	1	128	yes	complemented
74	C₂⁷	—	13	1	no	—
75	C₂⁵×C₄	—	13	2	no	—
76	C₂³×C₂²⋊C₄	—	13	1	no	—
77	C₂³×C₂²⋊C₄	—	13	2	no	—
78	C₂³×C₂²⋊C₄	—	13	56	no	—
79	C₂³×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, direct product factor
80	C₂³×C₁₃	—	2	448	no	—
81	C₂³×C₁₃	—	2	112	no	—
82	C₂³×C₁₃	C₂×C₂²⋊C₄	1	64	yes	complemented
83	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
84	C₂³×C₁₃	C₂³×C₄	1	112	yes	complemented
85	C₂³×C₁₃	C₂²×D₄	1	56	yes	—
86	C₂³×C₁₃	C₂⁵	1	8	yes	—
87	C₂³×C₁₃	C₂⁵	1	7	yes	—
88	C₂²×D₁₃	—	2	448	no	—
89	C₂²×D₁₃	—	2	112	no	—
90	C₂²×D₁₃	—	2	3584	no	—
91	C₂²×D₁₃	—	2	256	no	—
92	C₂²×D₁₃	C₂×C₂²⋊C₄	1	896	yes	direct product factor
93	C₂²×D₁₃	C₂³×C₄	1	112	yes	—
94	C₂²×D₁₃	C₂³×C₄	1	256	yes	complemented
95	C₂²×D₁₃	C₂³×C₄	1	112	yes	complemented
96	C₂²×D₁₃	C₂²×D₄	1	224	yes	—
97	C₂²×D₁₃	C₂⁵	1	8	yes	—
98	C₂²×D₁₃	C₂⁵	1	8	yes	—
99	C₂×C₄×C₁₃	—	2	112	no	—
100	C₂×C₄×C₁₃	C₂⁵	1	16	yes	complemented
101	C₂×Dic₁₃	—	2	112	no	—
102	C₂×Dic₁₃	C₂⁵	1	16	yes	complemented
103	C₄×D₁₃	—	2	256	no	—
104	C₂⁶	—	13	1	no	—
105	C₂⁶	—	13	32	no	—
106	C₂⁶	—	13	28	no	—
107	C₂⁶	—	13	1	no	—
108	C₂⁶	—	26	32	no	—
109	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
110	C₂⁴×C₄	—	13	2	no	—
111	C₂⁴×C₄	—	13	56	no	—
112	C₂⁴×C₄	—	26	32	no	—
113	C₂⁴×C₄	C₂×D₁₃	1	2	yes	complemented
114	C₂²×C₂²⋊C₄	—	13	28	no	—
115	C₂²×C₂²⋊C₄	—	13	56	no	—
116	C₂²×C₂²⋊C₄	—	13	448	no	—
117	C₂²×C₂²⋊C₄	C₂×D₁₃	1	28	yes	direct product factor
118	C₂²×C₁₃	—	2	16	no	—
119	C₂²×C₁₃	—	2	224	no	—
120	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	112	yes	complemented
121	C₂²×C₁₃	C₂⁴×C₄	1	14	yes	—
122	C₂²×C₁₃	C₂⁴×C₄	1	16	yes	complemented
123	C₂²×C₁₃	C₂³×D₄	1	28	yes	—
124	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
125	C₂×D₁₃	—	2	16	no	—
126	C₂×D₁₃	—	2	224	no	—
127	C₂×D₁₃	—	2	512	no	—
128	C₂×D₁₃	C₂²×C₂²⋊C₄	1	448	yes	direct product factor
129	C₂×D₁₃	C₂⁴×C₄	1	16	yes	—
130	C₂×D₁₃	C₂⁴×C₄	1	16	yes	complemented
131	C₂×D₁₃	C₂³×D₄	1	32	yes	—
132	C₄×C₁₃	—	2	16	no	—
133	Dic₁₃	—	2	16	no	—
134	C₂⁵	—	2	16	no	—
135	C₂⁵	—	13	16	no	—
136	C₂⁵	—	13	32	no	—
137	C₂⁵	—	13	14	no	—
138	C₂⁵	—	13	16	no	—
139	C₂⁵	—	13	448	no	—
140	C₂⁵	—	13	112	no	—
141	C₂⁵	—	13	14	no	—
142	C₂⁵	—	26	512	no	—
143	C₂⁵	—	26	448	no	—
144	C₂⁵	—	26	16	no	—
145	C₂⁵	C₄×D₁₃	1	16	yes	complemented
146	C₂⁵	C₂²×D₁₃	1	14	yes	—
147	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
148	C₂³×C₄	—	2	16	no	—
149	C₂³×C₄	—	13	28	no	—
150	C₂³×C₄	—	13	224	no	—
151	C₂³×C₄	—	26	448	no	—
152	C₂³×C₄	—	26	16	no	—
153	C₂³×C₄	C₂²×D₁₃	1	28	yes	complemented
154	C₂×C₂²⋊C₄	—	13	112	no	—
155	C₂×C₂²⋊C₄	—	13	224	no	—
156	C₂×C₂²⋊C₄	—	13	512	no	—
157	C₂×C₂²⋊C₄	C₂²×D₁₃	1	112	yes	direct product factor
158	C₂×C₁₃	—	2	16	no	—
159	C₂×C₁₃	C₂³×C₂²⋊C₄	1	28	yes	complemented
160	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
161	C₂×C₁₃	C₂⁴×D₄	1	2	yes	—
162	D₁₃	—	2	16	no	—
163	D₁₃	C₂³×C₂²⋊C₄	1	32	yes	direct product factor
164	C₂⁴	—	2	128	no	—
165	C₂⁴	—	2	112	no	—
166	C₂⁴	—	13	256	no	—
167	C₂⁴	—	13	112	no	—
168	C₂⁴	—	13	224	no	—
169	C₂⁴	—	13	28	no	—
170	C₂⁴	—	13	112	no	—
171	C₂⁴	—	13	896	no	—
172	C₂⁴	—	13	64	no	—
173	C₂⁴	—	13	28	no	—
174	C₂⁴	—	26	3584	no	—
175	C₂⁴	—	26	896	no	—
176	C₂⁴	—	26	128	no	—
177	C₂⁴	—	26	112	no	—
178	C₂⁴	C₂×C₄×D₁₃	1	8	yes	—
179	C₂⁴	C₂×C₄×D₁₃	1	112	yes	complemented
180	C₂⁴	D₄×D₁₃	1	16	yes	—
181	C₂⁴	C₂³×D₁₃	1	28	yes	—
182	C₂⁴	C₂³×D₁₃	1	7	yes	—
183	C₂²×C₄	—	2	112	no	—
184	C₂²×C₄	—	13	56	no	—
185	C₂²×C₄	—	13	128	no	—
186	C₂²×C₄	—	26	896	no	—
187	C₂²×C₄	—	26	112	no	—
188	C₂²×C₄	C₂³×D₁₃	1	56	yes	complemented
189	C₂²⋊C₄	—	13	64	no	—
190	C₂²⋊C₄	—	13	128	no	—
191	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	direct product factor
192	C₁₃	C₂⁴×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	448	no	—
194	C₂³	—	2	112	no	—
195	C₂³	—	13	896	no	—
196	C₂³	—	13	112	no	—
197	C₂³	—	13	224	no	—
198	C₂³	—	13	8	no	—
199	C₂³	—	13	112	no	—
200	C₂³	—	13	256	no	—
201	C₂³	—	13	8	no	—
202	C₂³	—	26	3584	no	—
203	C₂³	—	26	256	no	—
204	C₂³	—	26	448	no	—
205	C₂³	—	26	112	no	—
206	C₂³	C₂²⋊C₄×D₁₃	1	64	yes	direct product factor
207	C₂³	C₂²×C₄×D₁₃	1	28	yes	—
208	C₂³	C₂²×C₄×D₁₃	1	112	yes	complemented
209	C₂³	C₂×D₄×D₁₃	1	56	yes	—
210	C₂³	C₂⁴×D₁₃	1	8	yes	—
211	C₂³	C₂⁴×D₁₃	1	7	yes	—
212	C₂×C₄	—	2	112	no	—
213	C₂×C₄	—	13	16	no	—
214	C₂×C₄	—	26	256	no	—
215	C₂×C₄	—	26	112	no	—
216	C₂×C₄	C₂⁴×D₁₃	1	16	yes	complemented
217	C₂²	—	2	224	no	—
218	C₂²	—	2	16	no	—
219	C₂²	—	13	448	no	—
220	C₂²	—	13	16	no	—
221	C₂²	—	13	32	no	—
222	C₂²	—	13	16	no	—
223	C₂²	—	26	512	no	—
224	C₂²	—	26	224	no	—
225	C₂²	—	26	16	no	—
226	C₂²	C₂×C₂²⋊C₄×D₁₃	1	112	yes	direct product factor
227	C₂²	C₂³×C₄×D₁₃	1	14	yes	—
228	C₂²	C₂³×C₄×D₁₃	1	16	yes	complemented
229	C₂²	C₂²×D₄×D₁₃	1	28	yes	—
230	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
231	C₄	—	2	16	no	—
232	C₄	—	26	16	no	—
233	C₂	—	2	16	no	—
234	C₂	—	13	32	no	—
235	C₂	—	26	16	no	—
236	C₂	C₂²×C₂²⋊C₄×D₁₃	1	28	yes	direct product factor
237	C₂	C₂⁴×C₄×D₁₃	1	1	yes	characteristic
238	C₂	C₂³×D₄×D₁₃	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₂²⋊C₄	C₂
socle	C₂⁵×C₁₃	C₂³
prime radical	C₂⁶×D₁₃	C₂
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₂⁴×C₂²⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₁₃	C₂	semidirect
3. C₂³×C₂²⋊Dic₁₃	C₂	semidirect
4. C₂³×#6103	C₂	semidirect
5. C₂²×C₂²⋊C₄×D₁₃	C₂	direct (N × H)
6. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
9. C₂⁵×D₁₃	C₄	semidirect
13. C₂⁴×C₄×C₁₃	C₂²	semidirect
14. C₂⁴×Dic₁₃	C₂²	semidirect
16. C₂³×C₄×D₁₃	C₂²	semidirect
17. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
18. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
19. C₂²×#6103	C₂²	semidirect
20. C₂×C₂²⋊C₄×D₁₃	C₂²	direct (N × H)
22. C₂⁵×C₁₃	C₂×C₄	semidirect
29. C₂⁴×D₁₃	C₂×C₄	semidirect
30. C₂⁴×D₁₃	C₂×C₄	semidirect
36. C₂³×C₄×C₁₃	C₂³	semidirect
38. C₂³×Dic₁₃	C₂³	semidirect
40. C₂²×C₄×D₁₃	C₂³	semidirect
41. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
42. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
43. C₂×#6103	C₂³	semidirect
44. C₂²⋊C₄×D₁₃	C₂³	direct (N × H)
49. C₂⁴×C₁₃	C₂²×C₄	semidirect
57. C₂³×D₁₃	C₂²⋊C₄	direct (N × H)
59. C₂³×D₁₃	C₂²×C₄	semidirect
60. C₂³×D₁₃	C₂²×C₄	semidirect
66. C₂²×C₄×C₁₃	C₂⁴	semidirect
68. C₂²×Dic₁₃	C₂⁴	semidirect
70. C₂×C₄×D₁₃	C₂⁴	semidirect
71. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
72. C₂²⋊Dic₁₃	C₂⁴	semidirect
73. #6103	C₂⁴	semidirect
79. C₂³×C₂²⋊C₄	D₁₃	direct (N × H)
82. C₂³×C₁₃	C₂×C₂²⋊C₄	semidirect
84. C₂³×C₁₃	C₂³×C₄	semidirect
92. C₂²×D₁₃	C₂×C₂²⋊C₄	direct (N × H)
94. C₂²×D₁₃	C₂³×C₄	semidirect
95. C₂²×D₁₃	C₂³×C₄	semidirect
100. C₂×C₄×C₁₃	C₂⁵	semidirect
102. C₂×Dic₁₃	C₂⁵	semidirect
113. C₂⁴×C₄	C₂×D₁₃	semidirect
117. C₂²×C₂²⋊C₄	C₂×D₁₃	direct (N × H)
120. C₂²×C₁₃	C₂²×C₂²⋊C₄	semidirect
122. C₂²×C₁₃	C₂⁴×C₄	semidirect
128. C₂×D₁₃	C₂²×C₂²⋊C₄	direct (N × H)
130. C₂×D₁₃	C₂⁴×C₄	semidirect
145. C₂⁵	C₄×D₁₃	semidirect
153. C₂³×C₄	C₂²×D₁₃	semidirect
157. C₂×C₂²⋊C₄	C₂²×D₁₃	direct (N × H)
159. C₂×C₁₃	C₂³×C₂²⋊C₄	semidirect
163. D₁₃	C₂³×C₂²⋊C₄	direct (N × H)
179. C₂⁴	C₂×C₄×D₁₃	semidirect
188. C₂²×C₄	C₂³×D₁₃	semidirect
191. C₂²⋊C₄	C₂³×D₁₃	direct (N × H)
192. C₁₃	C₂⁴×C₂²⋊C₄	semidirect
206. C₂³	C₂²⋊C₄×D₁₃	direct (N × H)
208. C₂³	C₂²×C₄×D₁₃	semidirect
216. C₂×C₄	C₂⁴×D₁₃	semidirect
226. C₂²	C₂×C₂²⋊C₄×D₁₃	direct (N × H)
228. C₂²	C₂³×C₄×D₁₃	semidirect
236. C₂	C₂²×C₂²⋊C₄×D₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×D₁₃	2
C₂⁴×C₄×D₁₃	2
C₂³×C₂²⋊Dic₁₃	2
C₂³×#6103	2
C₂²×C₂²⋊C₄×D₁₃	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂⁴×C₂²⋊C₄	13

Subgroups — C23×C22⋊C4×D13 #16769469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊C₄×D₁₃ #16769469