Subgroups — C₂×C₁₃×#2258 #15720312

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2258	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₃×M₆₄	C₂	1	3	yes	maximal, complemented
4	C₂²×C₁₃×C₃₂	C₂	1	3	yes	maximal, complemented
5	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
6	C₁₃×#504	C₄	1	4	yes	—
7	C₁₃×#504	C₂²	1	4	yes	—
8	C₂×C₁₃×M₃₂	C₄	1	3	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	3	yes	—
10	C₁₃×M₆₄	C₂²	1	12	yes	complemented
11	C₂×C₁₃×C₃₂	C₂²	1	3	yes	—
12	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₃₂	C₂²	1	12	yes	complemented
14	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
15	C₁₃×#131	C₈	1	4	yes	—
16	C₁₃×#131	C₂×C₄	1	4	yes	—
17	C₂×C₁₃×M₁₆	C₈	1	3	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
19	C₁₃×M₃₂	C₂×C₄	1	12	yes	—
20	C₂×C₁₃×C₁₆	C₂×C₄	1	3	yes	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	6	yes	—
22	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	C₂³	1	6	yes	—
24	C₁₃×C₃₂	C₂³	1	6	yes	—
25	C₁₃×C₃₂	C₂³	1	2	yes	—
26	C₂×#2258	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₁₃×#41	C₁₆	1	4	yes	—
28	C₁₃×#41	C₂×C₈	1	4	yes	—
29	C₂×Q₈×C₁₃	C₁₆	1	1	yes	characteristic
30	C₂×D₄×C₁₃	C₁₆	1	3	yes	—
31	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
32	C₁₃×M₁₆	C₂×C₈	1	12	yes	—
33	C₂×C₈×C₁₃	C₂×C₈	1	3	yes	—
34	C₂×C₈×C₁₃	C₂×C₈	1	6	yes	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
37	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
38	C₁₃×C₁₆	C₂²×C₄	1	6	yes	—
39	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
40	C₂×#504	C₂×C₁₃	1	1	yes	characteristic
41	#2258	C₂×C₁₃	1	8	yes	direct product factor
42	C₂×M₆₄	C₂×C₁₃	1	3	yes	complemented
43	C₂²×C₃₂	C₂×C₁₃	1	3	yes	complemented
44	C₂³×C₁₃	C₂×C₁₆	1	3	yes	—
45	Q₈×C₁₃	C₂×C₁₆	1	4	yes	—
46	D₄×C₁₃	C₂×C₁₆	1	12	yes	—
47	C₂×C₄×C₁₃	C₂×C₁₆	1	3	yes	—
48	C₂×C₄×C₁₃	C₂×C₁₆	1	6	yes	—
49	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
51	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₁₃	C₂²×C₈	1	6	yes	—
53	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
54	C₂×#131	C₄×C₁₃	1	1	yes	characteristic
55	#504	C₄×C₁₃	1	4	yes	—
56	#504	C₂²×C₁₃	1	4	yes	—
57	C₂×M₃₂	C₄×C₁₃	1	3	yes	—
58	C₂²×C₁₆	C₂²×C₁₃	1	3	yes	—
59	M₆₄	C₂²×C₁₃	1	12	yes	complemented
60	C₂×C₃₂	C₂²×C₁₃	1	3	yes	—
61	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
62	C₂×C₃₂	C₂²×C₁₃	1	12	yes	complemented
63	C₂²×C₁₃	—	2	6	no	—
64	C₂²×C₁₃	C₂²×C₁₆	1	6	yes	—
65	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
66	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
67	C₄×C₁₃	C₂²×C₁₆	1	6	yes	—
68	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
69	C₂×#41	C₈×C₁₃	1	1	yes	characteristic
70	#131	C₈×C₁₃	1	4	yes	—
71	#131	C₂×C₄×C₁₃	1	4	yes	—
72	C₂×M₁₆	C₈×C₁₃	1	3	yes	—
73	C₂²×C₈	C₂×C₄×C₁₃	1	3	yes	—
74	M₃₂	C₂×C₄×C₁₃	1	12	yes	—
75	C₂×C₁₆	C₂×C₄×C₁₃	1	3	yes	—
76	C₂×C₁₆	C₂×C₄×C₁₃	1	6	yes	—
77	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
78	C₂×C₁₆	C₂³×C₁₃	1	6	yes	—
79	C₃₂	C₂³×C₁₃	1	2	yes	—
80	C₃₂	C₂³×C₁₃	1	6	yes	—
81	C₂×C₁₃	—	2	6	no	—
82	C₂×C₁₃	#2258	1	2	yes	direct product factor
83	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
84	#41	C₁₃×C₁₆	1	4	yes	—
85	#41	C₂×C₈×C₁₃	1	4	yes	—
86	C₂×Q₈	C₁₃×C₁₆	1	1	yes	characteristic
87	C₂×D₄	C₁₃×C₁₆	1	3	yes	—
88	C₂²×C₄	C₂×C₈×C₁₃	1	3	yes	—
89	M₁₆	C₂×C₈×C₁₃	1	12	yes	—
90	C₂×C₈	C₂×C₈×C₁₃	1	3	yes	—
91	C₂×C₈	C₂×C₈×C₁₃	1	6	yes	—
92	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
93	C₂×C₈	C₂²×C₄×C₁₃	1	6	yes	—
94	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
95	C₁₆	C₂²×C₄×C₁₃	1	6	yes	—
96	C₁₆	C₂⁴×C₁₃	1	1	yes	characteristic
97	C₁₃	C₂×#2258	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³	C₂×C₁₃×C₁₆	1	3	yes	—
99	Q₈	C₂×C₁₃×C₁₆	1	4	yes	—
100	D₄	C₂×C₁₃×C₁₆	1	12	yes	—
101	C₂×C₄	C₂×C₁₃×C₁₆	1	3	yes	—
102	C₂×C₄	C₂×C₁₃×C₁₆	1	6	yes	—
103	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈×C₁₃	1	6	yes	—
105	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
106	C₈	C₂²×C₈×C₁₃	1	6	yes	—
107	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×C₁₃×C₁₆	1	6	yes	—
110	C₂²	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
111	C₄	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
112	C₄	C₂²×C₁₃×C₁₆	1	6	yes	—
113	C₄	C₂³×C₈×C₁₃	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	C₁₃×#2258	1	2	yes	direct product factor
116	C₂	C₂³×C₁₃×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₃×C₃₂	C₂²
commutator	C₂	C₂³×C₁₃×C₁₆
Frattini, cosocle	C₁₆	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×C₁₆
prime radical	C₂×C₁₃×#41	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2258	C₂	direct (N × H)
3. C₂×C₁₃×M₆₄	C₂	semidirect
4. C₂²×C₁₃×C₃₂	C₂	semidirect
10. C₁₃×M₆₄	C₂²	semidirect
13. C₂×C₁₃×C₃₂	C₂²	semidirect
26. C₂×#2258	C₁₃	direct (N × H)
41. #2258	C₂×C₁₃	direct (N × H)
42. C₂×M₆₄	C₂×C₁₃	semidirect
43. C₂²×C₃₂	C₂×C₁₃	semidirect
59. M₆₄	C₂²×C₁₃	semidirect
62. C₂×C₃₂	C₂²×C₁₃	semidirect
82. C₂×C₁₃	#2258	direct (N × H)
97. C₁₃	C₂×#2258	direct (N × H)
115. C₂	C₁₃×#2258	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×#504	2
C₁₃×#2258	2
C₂×C₁₃×M₆₄	2
C₂²×C₁₃×C₃₂	2
C₂×#2258	13

Subgroups — C2×C13×#2258 #15720312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2258 #15720312