Subgroups — C₁₃×#3256 #13112314

Order 1664.

354 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#529	C₂	1	2	yes	maximal, complemented
2	C₁₃×#479	C₂	1	2	yes	maximal, complemented
3	C₁₃×#478	C₂	1	2	yes	maximal, complemented
4	C₁₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#469	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#468	C₂	1	2	yes	maximal, complemented
7	C₁₃×#464	C₂	1	2	yes	maximal, complemented
8	C₁₃×#463	C₂	1	2	yes	maximal, complemented
9	C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#126	—	2	2	no	—
16	C₁₃×#125	C₂²	1	2	yes	—
17	C₁₃×#124	C₂²	1	2	yes	—
18	C₁₃×#123	—	2	2	no	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₁₃×#121	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
22	C₄×D₄×C₁₃	—	2	2	no	—
23	C₄×D₄×C₁₃	C₂²	1	2	yes	—
24	C₁₃×#117	—	2	2	no	—
25	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#107	C₂²	1	2	yes	—
27	C₁₃×#106	C₂²	1	1	yes	characteristic
28	C₁₃×#106	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₁₃×#103	C₂²	1	2	yes	—
32	C₁₃×#103	C₂²	1	2	yes	—
33	C₁₃×#102	C₂²	1	2	yes	—
34	C₁₃×#102	C₂²	1	2	yes	—
35	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂×D₄×C₁₃	—	2	2	no	—
37	C₂×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	D₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×Q₁₆	—	4	1	no	—
42	C₁₃×SD₁₆	—	4	2	no	—
43	C₁₃×D₈	—	4	1	no	—
44	C₁₃×M₁₆	—	4	1	no	—
45	C₁₃×M₁₆	—	4	1	no	—
46	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂³	1	2	yes	—
48	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₁₃×C₂²⋊C₄	—	2	2	no	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₁₃×C₂²⋊C₄	—	2	2	no	—
59	C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	D₄	1	1	yes	characteristic
62	C₄²×C₁₃	D₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	2	yes	—
64	#3256	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₁₃	—	2	2	no	—
68	Q₈×C₁₃	—	2	2	no	—
69	D₄×C₁₃	—	2	2	no	—
70	D₄×C₁₃	—	2	2	no	—
71	D₄×C₁₃	—	4	4	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₈×C₁₃	—	2	1	no	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	—	2	1	no	—
86	#529	C₂×C₁₃	1	2	yes	complemented
87	#479	C₂×C₁₃	1	2	yes	complemented
88	#478	C₂×C₁₃	1	2	yes	complemented
89	#470	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#469	C₂×C₁₃	1	1	yes	characteristic
91	#468	C₂×C₁₃	1	2	yes	complemented
92	#464	C₂×C₁₃	1	2	yes	complemented
93	#463	C₂×C₁₃	1	2	yes	complemented
94	#423	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	2	no	—
97	C₂²×C₁₃	—	4	2	no	—
98	C₂²×C₁₃	—	4	2	no	—
99	C₂²×C₁₃	—	4	2	no	—
100	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	4	2	no	—
107	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
110	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
111	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
112	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
114	#126	—	2	2	no	—
115	#125	C₂²×C₁₃	1	2	yes	—
116	#124	C₂²×C₁₃	1	2	yes	—
117	#123	—	2	2	no	—
118	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
119	#121	C₂²×C₁₃	1	2	yes	—
120	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
121	C₄×D₄	—	2	2	no	—
122	C₄×D₄	C₂²×C₁₃	1	2	yes	—
123	#117	—	2	2	no	—
124	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
125	#107	C₂²×C₁₃	1	2	yes	—
126	#106	C₂²×C₁₃	1	1	yes	characteristic
127	#106	C₂²×C₁₃	1	1	yes	characteristic
128	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
129	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	#103	C₂²×C₁₃	1	2	yes	—
131	#103	C₂²×C₁₃	1	2	yes	—
132	#102	C₂²×C₁₃	1	2	yes	—
133	#102	C₂²×C₁₃	1	2	yes	—
134	C₂×C₁₃	—	4	1	no	—
135	C₂×C₁₃	—	8	2	no	—
136	C₂×C₁₃	#537	1	1	yes	characteristic
137	C₂×C₁₃	#575	1	2	yes	—
138	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	C₂³×C₁₃	1	2	yes	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	D₄×C₁₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
144	Q₁₆	—	4	1	no	—
145	SD₁₆	—	4	2	no	—
146	D₈	—	4	1	no	—
147	M₁₆	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
150	C₂×C₈	C₂³×C₁₃	1	2	yes	—
151	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
157	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	D₄×C₁₃	1	1	yes	characteristic
165	C₄²	D₄×C₁₃	1	1	yes	characteristic
166	C₄²	C₂³×C₁₃	1	2	yes	—
167	C₁₃	#3256	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³	—	2	2	no	—
169	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
185	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
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	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
202	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	C₁₃×#537	1	1	yes	characteristic
206	C₂	C₁₃×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#529	C₂	semidirect
2. C₁₃×#479	C₂	semidirect
3. C₁₃×#478	C₂	semidirect
4. C₁₃×#470	C₂	semidirect
6. C₁₃×#468	C₂	semidirect
7. C₁₃×#464	C₂	semidirect
8. C₁₃×#463	C₂	semidirect
9. C₁₃×#423	C₂	semidirect
64. #3256	C₁₃	direct (N × H)
86. #529	C₂×C₁₃	semidirect
87. #479	C₂×C₁₃	semidirect
88. #478	C₂×C₁₃	semidirect
89. #470	C₂×C₁₃	semidirect
91. #468	C₂×C₁₃	semidirect
92. #464	C₂×C₁₃	semidirect
93. #463	C₂×C₁₃	semidirect
94. #423	C₂×C₁₃	semidirect
167. C₁₃	#3256	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#529	2
C₁₃×#479	2
C₁₃×#478	2
C₁₃×#470	2
C₁₃×#469	2
C₁₃×#468	2
C₁₃×#464	2
C₁₃×#463	2
C₁₃×#423	2
#3256	13

Subgroups — C13×#3256 #13112314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3256 #13112314