Subgroups — C₂×C₁₃×#578 #13112641

Order 1664.

1408 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	C₁₃×#578	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#137	C₂	1	6	yes	maximal, complemented
4	C₂×C₁₃×#135	C₂	1	3	yes	maximal, complemented
5	C₂²×C₁₃×Q₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×2₋¹⁺⁴	—	2	8	no	—
8	C₁₃×2₋¹⁺⁴	C₂²	1	16	yes	complemented
9	C₂×C₁₃×#41	—	2	6	no	—
10	C₂×C₁₃×#41	C₂²	1	6	yes	—
11	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₁₃	—	2	2	no	—
13	C₂²×Q₈×C₁₃	C₂²	1	6	yes	—
14	C₁₃×#137	C₂²	1	48	yes	complemented
15	C₁₃×#135	C₂²	1	24	yes	complemented
16	C₂×C₁₃×Q₁₆	C₂²	1	3	yes	—
17	C₂×C₁₃×Q₁₆	C₂²	1	6	yes	—
18	C₂×C₁₃×Q₁₆	C₂²	1	24	yes	complemented
19	C₂×C₁₃×SD₁₆	C₂²	1	6	yes	—
20	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
21	C₁₃×#131	C₂²	1	8	yes	complemented
22	C₂×C₁₃×M₁₆	C₂²	1	3	yes	—
23	C₂²×C₈×C₁₃	C₂²	1	3	yes	—
24	C₁₃×#41	—	2	48	no	—
25	C₁₃×#41	—	2	12	no	—
26	C₁₃×#41	D₄	1	4	yes	—
27	C₁₃×#41	C₂³	1	24	yes	—
28	C₁₃×#41	C₂³	1	4	yes	—
29	C₂×Q₈×C₁₃	—	2	16	no	—
30	C₂×Q₈×C₁₃	—	2	6	no	—
31	C₂×Q₈×C₁₃	—	2	12	no	—
32	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
34	C₂×Q₈×C₁₃	C₂³	1	24	yes	—
35	C₂×D₄×C₁₃	—	2	6	no	—
36	C₂×D₄×C₁₃	D₄	1	3	yes	—
37	C₂×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	—	2	6	no	—
39	C₂²×C₄×C₁₃	—	2	6	no	—
40	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
41	C₁₃×Q₁₆	C₂³	1	12	yes	—
42	C₁₃×Q₁₆	C₂³	1	24	yes	—
43	C₁₃×SD₁₆	C₂³	1	24	yes	—
44	C₁₃×D₈	C₂³	1	4	yes	—
45	C₁₃×M₁₆	C₂³	1	12	yes	—
46	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂³	1	3	yes	—
48	C₂×C₈×C₁₃	C₂³	1	12	yes	—
49	C₂×#578	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	C₂×D₄	1	3	yes	—
52	Q₈×C₁₃	—	2	24	no	—
53	Q₈×C₁₃	—	2	6	no	—
54	Q₈×C₁₃	C₂×D₄	1	4	yes	—
55	Q₈×C₁₃	C₂⁴	1	12	yes	—
56	D₄×C₁₃	—	2	24	no	—
57	D₄×C₁₃	—	2	2	no	—
58	D₄×C₁₃	C₂×D₄	1	12	yes	—
59	D₄×C₁₃	C₂⁴	1	4	yes	—
60	C₂×C₄×C₁₃	—	2	24	no	—
61	C₂×C₄×C₁₃	—	2	6	no	—
62	C₂×C₄×C₁₃	—	2	24	no	—
63	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
64	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
65	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
66	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₁₃	C₂⁴	1	2	yes	—
68	C₈×C₁₃	C₂⁴	1	6	yes	—
69	C₂×2₋¹⁺⁴	C₂×C₁₃	1	2	yes	complemented
70	#578	C₂×C₁₃	1	16	yes	direct product factor
71	C₂×#137	C₂×C₁₃	1	6	yes	complemented
72	C₂×#135	C₂×C₁₃	1	3	yes	complemented
73	C₂²×Q₁₆	C₂×C₁₃	1	3	yes	complemented
74	C₂×#131	C₂×C₁₃	1	1	yes	characteristic, complemented
75	C₂²×C₁₃	—	2	6	no	—
76	C₂²×C₁₃	—	2	4	no	—
77	C₂²×C₁₃	—	4	4	no	—
78	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
79	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₁₃	—	2	12	no	—
81	C₄×C₁₃	C₂²×D₄	1	6	yes	—
82	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
84	2₋¹⁺⁴	—	2	8	no	—
85	2₋¹⁺⁴	C₂²×C₁₃	1	16	yes	complemented
86	C₂×#41	—	2	6	no	—
87	C₂×#41	C₂²×C₁₃	1	6	yes	—
88	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
89	C₂²×Q₈	—	2	2	no	—
90	C₂²×Q₈	C₂²×C₁₃	1	6	yes	—
91	#137	C₂²×C₁₃	1	48	yes	complemented
92	#135	C₂²×C₁₃	1	24	yes	complemented
93	C₂×Q₁₆	C₂²×C₁₃	1	3	yes	—
94	C₂×Q₁₆	C₂²×C₁₃	1	6	yes	—
95	C₂×Q₁₆	C₂²×C₁₃	1	24	yes	complemented
96	C₂×SD₁₆	C₂²×C₁₃	1	6	yes	—
97	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
98	#131	C₂²×C₁₃	1	8	yes	complemented
99	C₂×M₁₆	C₂²×C₁₃	1	3	yes	—
100	C₂²×C₈	C₂²×C₁₃	1	3	yes	—
101	C₂×C₁₃	—	2	6	no	—
102	C₂×C₁₃	—	4	4	no	—
103	C₂×C₁₃	#578	1	2	yes	direct product factor
104	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
105	#41	—	2	48	no	—
106	#41	—	2	12	no	—
107	#41	D₄×C₁₃	1	4	yes	—
108	#41	C₂³×C₁₃	1	24	yes	—
109	#41	C₂³×C₁₃	1	4	yes	—
110	C₂×Q₈	—	2	16	no	—
111	C₂×Q₈	—	2	6	no	—
112	C₂×Q₈	—	2	12	no	—
113	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
114	C₂×Q₈	C₂³×C₁₃	1	6	yes	—
115	C₂×Q₈	C₂³×C₁₃	1	24	yes	—
116	C₂×D₄	—	2	6	no	—
117	C₂×D₄	D₄×C₁₃	1	3	yes	—
118	C₂×D₄	C₂³×C₁₃	1	2	yes	—
119	C₂²×C₄	—	2	6	no	—
120	C₂²×C₄	—	2	6	no	—
121	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
122	Q₁₆	C₂³×C₁₃	1	12	yes	—
123	Q₁₆	C₂³×C₁₃	1	24	yes	—
124	SD₁₆	C₂³×C₁₃	1	24	yes	—
125	D₈	C₂³×C₁₃	1	4	yes	—
126	M₁₆	C₂³×C₁₃	1	12	yes	—
127	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
128	C₂×C₈	C₂³×C₁₃	1	3	yes	—
129	C₂×C₈	C₂³×C₁₃	1	12	yes	—
130	C₁₃	C₂×#578	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³	—	2	2	no	—
132	C₂³	C₂×D₄×C₁₃	1	3	yes	—
133	Q₈	—	2	24	no	—
134	Q₈	—	2	6	no	—
135	Q₈	C₂×D₄×C₁₃	1	4	yes	—
136	Q₈	C₂⁴×C₁₃	1	12	yes	—
137	D₄	—	2	24	no	—
138	D₄	—	2	2	no	—
139	D₄	C₂×D₄×C₁₃	1	12	yes	—
140	D₄	C₂⁴×C₁₃	1	4	yes	—
141	C₂×C₄	—	2	6	no	—
142	C₂×C₄	—	2	24	no	—
143	C₂×C₄	—	2	24	no	—
144	C₂×C₄	C₂×D₄×C₁₃	1	3	yes	—
145	C₂×C₄	C₂×D₄×C₁₃	1	6	yes	—
146	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
147	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
148	C₈	C₂⁴×C₁₃	1	2	yes	—
149	C₈	C₂⁴×C₁₃	1	6	yes	—
150	C₂²	—	2	6	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	4	4	no	—
153	C₂²	C₂²×D₄×C₁₃	1	6	yes	—
154	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
155	C₄	—	2	12	no	—
156	C₄	C₂²×D₄×C₁₃	1	6	yes	—
157	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
158	C₄	C₂⁵×C₁₃	1	1	yes	characteristic
159	C₂	—	2	6	no	—
160	C₂	—	4	4	no	—
161	C₂	C₁₃×#578	1	2	yes	direct product factor
162	C₂	C₂³×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₁₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×2₋¹⁺⁴	C₂	semidirect
2. C₁₃×#578	C₂	direct (N × H)
3. C₂×C₁₃×#137	C₂	semidirect
4. C₂×C₁₃×#135	C₂	semidirect
5. C₂²×C₁₃×Q₁₆	C₂	semidirect
6. C₂×C₁₃×#131	C₂	semidirect
8. C₁₃×2₋¹⁺⁴	C₂²	semidirect
14. C₁₃×#137	C₂²	semidirect
15. C₁₃×#135	C₂²	semidirect
18. C₂×C₁₃×Q₁₆	C₂²	semidirect
21. C₁₃×#131	C₂²	semidirect
49. C₂×#578	C₁₃	direct (N × H)
69. C₂×2₋¹⁺⁴	C₂×C₁₃	semidirect
70. #578	C₂×C₁₃	direct (N × H)
71. C₂×#137	C₂×C₁₃	semidirect
72. C₂×#135	C₂×C₁₃	semidirect
73. C₂²×Q₁₆	C₂×C₁₃	semidirect
74. C₂×#131	C₂×C₁₃	semidirect
85. 2₋¹⁺⁴	C₂²×C₁₃	semidirect
91. #137	C₂²×C₁₃	semidirect
92. #135	C₂²×C₁₃	semidirect
95. C₂×Q₁₆	C₂²×C₁₃	semidirect
98. #131	C₂²×C₁₃	semidirect
103. C₂×C₁₃	#578	direct (N × H)
130. C₁₃	C₂×#578	direct (N × H)
161. C₂	C₁₃×#578	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×C₁₃×2₋¹⁺⁴	2
C₁₃×#578	2
C₂×C₁₃×#137	2
C₂×C₁₃×#135	2
C₂²×C₁₃×Q₁₆	2
C₂×C₁₃×#131	2
C₂×#578	13

Subgroups — C2×C13×#578 #13112641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#578 #13112641