Subgroups — C₁₃×#13549 #15699888

Order 3328.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1397	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×C₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₁₃×#405	C₄	1	1	yes	characteristic
9	C₁₃×M₆₄	C₄	1	4	yes	complemented
10	C₂×C₁₃×C₃₂	C₄	1	4	yes	complemented
11	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
14	C₁₃×#346	C₄	1	2	yes	—
15	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×M₁₆	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₁₃×#117	C₈	1	1	yes	characteristic
19	C₁₃×M₃₂	C₈	1	4	yes	—
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
25	C₁₃×#97	C₈	1	2	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₁₃×C₃₂	C₂×C₄	1	2	yes	—
28	C₁₃×C₃₂	C₂×C₄	1	2	yes	—
29	C₁₃×C₃₂	C₂×C₄	1	4	yes	—
30	#13549	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
32	C₁₃×M₁₆	C₁₆	1	4	yes	—
33	C₂×C₈×C₁₃	C₄²	1	2	yes	—
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₁₃×C₄⋊C₄	C₁₆	1	2	yes	—
39	C₁₃×C₂²⋊C₄	C₁₆	1	2	yes	—
40	C₄²×C₁₃	C₂×C₈	1	2	yes	—
41	C₁₃×C₁₆	C₄²	1	2	yes	—
42	C₁₃×C₁₆	C₂×C₈	1	4	yes	—
43	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
45	C₂×M₆₄	C₂×C₁₃	1	1	yes	characteristic
46	C₂²×C₃₂	C₂×C₁₃	1	1	yes	characteristic
47	#2108	C₂×C₁₃	1	1	yes	characteristic
48	#1397	C₂×C₁₃	1	2	yes	complemented
49	C₄×C₃₂	C₂×C₁₃	1	2	yes	complemented
50	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
53	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₁₃	C₄×C₈	1	2	yes	—
57	C₈×C₁₃	C₂×C₁₆	1	4	yes	—
58	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
60	C₂×M₃₂	C₄×C₁₃	1	1	yes	characteristic
61	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
62	#405	C₄×C₁₃	1	1	yes	characteristic
63	M₆₄	C₄×C₁₃	1	4	yes	complemented
64	C₂×C₃₂	C₄×C₁₃	1	4	yes	complemented
65	C₂×C₃₂	C₂²×C₁₃	1	2	yes	—
66	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
67	C₂×C₃₂	C₂²×C₁₃	1	1	yes	characteristic
68	#346	C₄×C₁₃	1	2	yes	—
69	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	C₄×C₁₆	1	2	yes	—
72	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
73	C₄×C₁₃	—	2	4	no	—
74	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
75	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
77	C₂×M₁₆	C₈×C₁₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
79	#117	C₈×C₁₃	1	1	yes	characteristic
80	M₃₂	C₈×C₁₃	1	4	yes	—
81	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
82	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
83	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
84	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
85	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
86	#97	C₈×C₁₃	1	2	yes	—
87	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
88	C₃₂	C₂×C₄×C₁₃	1	2	yes	—
89	C₃₂	C₂×C₄×C₁₃	1	2	yes	—
90	C₃₂	C₂×C₄×C₁₃	1	4	yes	—
91	C₂×C₁₃	—	2	2	no	—
92	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
93	C₂×C₁₃	#2258	1	2	yes	—
94	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
95	M₁₆	C₁₃×C₁₆	1	4	yes	—
96	C₂×C₈	C₄²×C₁₃	1	2	yes	—
97	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
98	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
99	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
100	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
101	C₄⋊C₄	C₁₃×C₁₆	1	2	yes	—
102	C₂²⋊C₄	C₁₃×C₁₆	1	2	yes	—
103	C₄²	C₂×C₈×C₁₃	1	2	yes	—
104	C₁₆	C₄²×C₁₃	1	2	yes	—
105	C₁₆	C₂×C₈×C₁₃	1	4	yes	—
106	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
107	C₁₆	C₂²×C₄×C₁₃	1	1	yes	characteristic
108	C₁₃	#13549	1	1	yes	characteristic, direct product factor, Sylow
109	C₂³	C₂×C₁₃×C₁₆	1	1	yes	characteristic
110	C₂×C₄	C₄×C₈×C₁₃	1	2	yes	—
111	C₂×C₄	C₂×C₁₃×C₁₆	1	4	yes	—
112	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
113	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
114	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
115	C₈	C₄×C₈×C₁₃	1	2	yes	—
116	C₈	C₂×C₁₃×C₁₆	1	4	yes	—
117	C₈	C₂×C₄²×C₁₃	1	1	yes	characteristic
118	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×C₁₃×C₁₆	1	2	yes	—
121	C₂²	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×C₁₃×C₁₆	1	2	yes	—
124	C₄	C₂×C₄×C₈×C₁₃	1	1	yes	characteristic
125	C₄	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×C₁₃×C₁₆	1	1	yes	characteristic
128	C₂	C₁₃×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₃×C₃₂	C₂²
commutator	C₂	C₂×C₄×C₁₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
socle	C₂²×C₁₃	C₂²×C₁₆
prime radical	C₂³×C₁₃	C₂×C₁₆
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
4. C₁₃×#1397	C₂	semidirect
5. C₄×C₁₃×C₃₂	C₂	semidirect
9. C₁₃×M₆₄	C₄	semidirect
10. C₂×C₁₃×C₃₂	C₄	semidirect
30. #13549	C₁₃	direct (N × H)
48. #1397	C₂×C₁₃	semidirect
49. C₄×C₃₂	C₂×C₁₃	semidirect
63. M₆₄	C₄×C₁₃	semidirect
64. C₂×C₃₂	C₄×C₁₃	semidirect
108. C₁₃	#13549	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×M₆₄	2
C₂²×C₁₃×C₃₂	2
C₁₃×#2108	2
C₁₃×#1397	2
C₄×C₁₃×C₃₂	2
#13549	13

Subgroups — C13×#13549 #15699888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13549 #15699888