Subgroups — C₁₃×#2562 #13111620

Order 1664.

532 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#400	C₂	1	2	yes	maximal, complemented
4	C₁₃×#394	C₂	1	2	yes	maximal
5	C₁₃×#388	C₂	1	4	yes	maximal, complemented
6	C₁₃×#380	C₂	1	2	yes	maximal, complemented
7	C₁₃×#380	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
11	C₁₃×#125	—	2	4	no	—
12	C₄×D₄×C₁₃	—	2	4	no	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#95	C₂²	1	2	yes	—
25	C₁₃×#95	C₂²	1	4	yes	—
26	C₁₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₁₃	C₂³	1	2	yes	—
28	C₂×D₄×C₁₃	—	2	4	no	—
29	C₂×D₄×C₁₃	—	2	2	no	—
30	C₂²×C₄×C₁₃	—	2	4	no	—
31	C₂²×C₄×C₁₃	—	2	4	no	—
32	C₂²×C₄×C₁₃	—	2	4	no	—
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₁₃×C₄⋊C₄	—	2	4	no	—
43	C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₁₃×C₂²⋊C₄	—	2	8	no	—
47	C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₁₃×C₂²⋊C₄	—	4	4	no	—
49	C₄²×C₁₃	—	4	2	no	—
50	C₄²×C₁₃	D₄	1	4	yes	—
51	#2562	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	—	2	2	no	—
54	C₂³×C₁₃	—	2	2	no	—
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	2	2	no	—
57	C₂³×C₁₃	—	2	2	no	—
58	C₂³×C₁₃	#41	1	4	yes	—
59	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
60	D₄×C₁₃	—	2	8	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	4	8	no	—
76	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	#41	1	4	yes	—
80	C₂×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
81	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic
82	#400	C₂×C₁₃	1	2	yes	complemented
83	#394	C₂×C₁₃	1	2	yes	—
84	#388	C₂×C₁₃	1	4	yes	complemented
85	#380	C₂×C₁₃	1	2	yes	complemented
86	#380	C₂×C₁₃	1	2	yes	complemented
87	C₄×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂²×C₁₃	—	2	4	no	—
89	C₂²×C₁₃	—	2	4	no	—
90	C₂²×C₁₃	—	2	4	no	—
91	C₂²×C₁₃	—	4	2	no	—
92	C₂²×C₁₃	—	4	2	no	—
93	C₂²×C₁₃	—	4	2	no	—
94	C₂²×C₁₃	—	4	2	no	—
95	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
96	C₂²×C₁₃	C₂×#41	1	2	yes	—
97	C₂²×C₁₃	C₂×#41	1	2	yes	—
98	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	4	2	no	—
103	C₄×C₁₃	—	4	2	no	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	#124	1	4	yes	—
107	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
109	#125	—	2	4	no	—
110	C₄×D₄	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
117	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
120	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
121	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
122	#95	C₂²×C₁₃	1	2	yes	—
123	#95	C₂²×C₁₃	1	4	yes	—
124	#95	C₂²×C₁₃	1	4	yes	—
125	C₂×C₁₃	—	4	4	no	—
126	C₂×C₁₃	#525	1	1	yes	characteristic
127	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
128	C₂×C₁₃	#537	1	1	yes	characteristic
129	C₂×C₁₃	#552	1	2	yes	—
130	C₂×C₁₃	#553	1	2	yes	—
131	C₂⁴	C₂³×C₁₃	1	2	yes	—
132	C₂×D₄	—	2	4	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
139	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	2	8	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	4	2	no	—
154	C₄²	D₄×C₁₃	1	4	yes	—
155	C₁₃	#2562	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	C₁₃×#41	1	4	yes	—
163	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
164	D₄	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
181	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
183	C₂×C₄	C₁₃×#41	1	4	yes	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
192	C₂²	C₂×C₁₃×#41	1	2	yes	—
193	C₂²	C₂×C₁₃×#41	1	2	yes	—
194	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
195	C₂²	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	4	no	—
202	C₄	C₁₃×#124	1	4	yes	—
203	C₂	—	4	4	no	—
204	C₂	C₁₃×#525	1	1	yes	characteristic
205	C₂	C₂×C₁₃×#124	1	1	yes	characteristic
206	C₂	C₁₃×#537	1	1	yes	characteristic
207	C₂	C₁₃×#552	1	2	yes	—
208	C₂	C₁₃×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
3. C₁₃×#400	C₂	semidirect
5. C₁₃×#388	C₂	semidirect
6. C₁₃×#380	C₂	semidirect
7. C₁₃×#380	C₂	semidirect
8. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
51. #2562	C₁₃	direct (N × H)
80. C₂×#125	C₂×C₁₃	semidirect
82. #400	C₂×C₁₃	semidirect
84. #388	C₂×C₁₃	semidirect
85. #380	C₂×C₁₃	semidirect
86. #380	C₂×C₁₃	semidirect
87. C₄×C₄⋊C₄	C₂×C₁₃	semidirect
155. C₁₃	#2562	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×#125	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#400	2
C₁₃×#394	2
C₁₃×#388	2
C₁₃×#380	2
C₄×C₁₃×C₄⋊C₄	2
#2562	13

Subgroups — C13×#2562 #13111620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2562 #13111620