Subgroups — #16769542

Order 3328.

10930 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₁₃	C₂	1	32	yes	maximal, complemented
2	#13128039	C₂	1	20	yes	maximal, complemented
3	C₂×#11719517	C₂	1	10	yes	maximal, complemented
4	C₂²×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×2₋¹⁺⁴	C₄	1	32	yes	complemented
6	C₂×C₁₃×2₋¹⁺⁴	C₂²	1	16	yes	—
7	C₂²×C₁₃×#41	C₂²	1	10	yes	—
8	C₂³×Q₈×C₁₃	C₂²	1	5	yes	—
9	#11719532	C₂²	1	160	yes	complemented
10	#41×Dic₁₃	C₂²	1	160	yes	complemented
11	#11719517	C₂²	1	40	yes	—
12	#11719517	C₂²	1	80	yes	complemented
13	C₂×Q₈×Dic₁₃	C₂²	1	80	yes	complemented
14	#11719503	C₂²	1	40	yes	—
15	C₂×#88180	C₂²	1	60	yes	—
16	C₁₃×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
17	C₂×C₁₃×#41	C₂×C₄	1	160	yes	complemented
18	C₂×C₁₃×#41	C₂³	1	80	yes	—
19	C₂²×Q₈×C₁₃	C₂×C₄	1	80	yes	complemented
20	C₂²×Q₈×C₁₃	C₂³	1	40	yes	—
21	C₂²×Q₈×C₁₃	C₂³	1	10	yes	—
22	C₂²×D₄×C₁₃	C₂³	1	10	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	15	yes	—
24	C₂×C₂²⋊Dic₁₃	C₂³	1	40	yes	—
25	Q₈×Dic₁₃	C₂³	1	320	yes	—
26	D₄×Dic₁₃	C₂³	1	320	yes	—
27	#88180	C₂³	1	80	yes	—
28	#88180	C₂³	1	240	yes	—
29	C₂×C₄⋊Dic₁₃	C₂³	1	120	yes	—
30	C₂×C₄×Dic₁₃	C₂³	1	120	yes	—
31	#62653	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₂⁴	1	5	yes	—
33	C₁₃×#41	C₂²×C₄	1	640	yes	—
34	C₂×Q₈×C₁₃	C₂²×C₄	1	80	yes	—
35	C₂×Q₈×C₁₃	C₂²×C₄	1	320	yes	—
36	C₂×Q₈×C₁₃	C₂⁴	1	40	yes	—
37	C₂×D₄×C₁₃	C₂²×C₄	1	80	yes	—
38	C₂×D₄×C₁₃	C₂⁴	1	40	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	120	yes	—
40	C₂²×C₄×C₁₃	C₂⁴	1	60	yes	—
41	C₂²×C₄×C₁₃	C₂⁴	1	10	yes	—
42	C₂²×Dic₁₃	C₂⁴	1	80	yes	—
43	C₂²⋊Dic₁₃	C₂⁴	1	16	yes	—
44	C₂²⋊Dic₁₃	C₂⁴	1	80	yes	—
45	C₄⋊Dic₁₃	C₂⁴	1	160	yes	—
46	C₄×Dic₁₃	C₂⁴	1	160	yes	—
47	C₂²×2₋¹⁺⁴	D₁₃	1	1	yes	characteristic
48	C₄×2₋¹⁺⁴	—	13	32	no	—
49	#3428	—	13	20	no	—
50	C₂×#519	—	13	10	no	—
51	C₂³×C₁₃	—	4	10	no	—
52	C₂³×C₁₃	C₂³×C₄	1	20	yes	—
53	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
54	C₂³×C₁₃	C₂⁵	1	10	yes	—
55	Q₈×C₁₃	C₂³×C₄	1	160	yes	—
56	D₄×C₁₃	C₂³×C₄	1	160	yes	—
57	C₂×C₄×C₁₃	C₂³×C₄	1	40	yes	—
58	C₂×C₄×C₁₃	C₂³×C₄	1	240	yes	—
59	C₂×C₄×C₁₃	C₂⁵	1	20	yes	—
60	C₂×Dic₁₃	—	4	80	no	—
61	C₂×Dic₁₃	C₂⁵	1	32	yes	—
62	C₂×2₋¹⁺⁴	Dic₁₃	1	32	yes	complemented
63	C₂×2₋¹⁺⁴	C₂×D₁₃	1	16	yes	—
64	C₂²×#41	C₂×D₁₃	1	10	yes	—
65	C₂³×Q₈	C₂×D₁₃	1	5	yes	—
66	#520	—	13	160	no	—
67	#519	—	13	40	no	—
68	#519	—	13	80	no	—
69	#518	—	13	40	no	—
70	C₄×#41	—	13	160	no	—
71	C₂×C₄×Q₈	—	13	80	no	—
72	C₂×#117	—	13	60	no	—
73	C₂²×C₁₃	—	2	20	no	—
74	C₂²×C₁₃	—	2	2	no	—
75	C₂²×C₁₃	—	4	20	no	—
76	C₂²×C₁₃	C₂⁴×C₄	1	2	yes	—
77	C₂²×C₁₃	C₂⁴×C₄	1	20	yes	—
78	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
79	C₄×C₁₃	C₂⁴×C₄	1	40	yes	—
80	Dic₁₃	—	2	32	no	—
81	2₋¹⁺⁴	C₂×Dic₁₃	1	256	yes	complemented
82	C₂×#41	C₂×Dic₁₃	1	160	yes	complemented
83	C₂×#41	C₂²×D₁₃	1	80	yes	—
84	C₂²×Q₈	C₂×Dic₁₃	1	80	yes	complemented
85	C₂²×Q₈	C₂²×D₁₃	1	40	yes	—
86	C₂²×Q₈	C₂²×D₁₃	1	10	yes	—
87	C₂²×D₄	C₂²×D₁₃	1	10	yes	—
88	C₂³×C₄	C₂²×D₁₃	1	15	yes	—
89	C₄×Q₈	—	13	320	no	—
90	C₄×D₄	—	13	320	no	—
91	#117	—	13	80	no	—
92	#117	—	13	240	no	—
93	C₂×C₄⋊C₄	—	13	120	no	—
94	C₂×C₂²⋊C₄	—	13	40	no	—
95	C₂×C₄²	—	13	120	no	—
96	C₂×C₁₃	—	2	2	no	—
97	C₂×C₁₃	—	2	20	no	—
98	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
99	C₂×C₁₃	2₋¹⁺⁶	1	2	yes	—
100	C₂⁴	C₂³×D₁₃	1	5	yes	—
101	#41	C₂²×Dic₁₃	1	640	yes	—
102	C₂×Q₈	C₂²×Dic₁₃	1	80	yes	—
103	C₂×Q₈	C₂²×Dic₁₃	1	320	yes	—
104	C₂×Q₈	C₂³×D₁₃	1	40	yes	—
105	C₂×D₄	C₂²×Dic₁₃	1	80	yes	—
106	C₂×D₄	C₂³×D₁₃	1	40	yes	—
107	C₂²×C₄	—	13	80	no	—
108	C₂²×C₄	C₂²×Dic₁₃	1	120	yes	—
109	C₂²×C₄	C₂³×D₁₃	1	60	yes	—
110	C₂²×C₄	C₂³×D₁₃	1	10	yes	—
111	C₄⋊C₄	—	13	160	no	—
112	C₂²⋊C₄	—	13	16	no	—
113	C₂²⋊C₄	—	13	80	no	—
114	C₄²	—	13	160	no	—
115	C₁₃	#62653	1	1	yes	characteristic, complemented, Sylow
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×Dic₁₃	1	20	yes	—
118	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
119	C₂³	C₂⁴×D₁₃	1	10	yes	—
120	Q₈	C₂³×Dic₁₃	1	160	yes	—
121	D₄	C₂³×Dic₁₃	1	160	yes	—
122	C₂×C₄	—	13	32	no	—
123	C₂×C₄	—	52	80	no	—
124	C₂×C₄	C₂³×Dic₁₃	1	40	yes	—
125	C₂×C₄	C₂³×Dic₁₃	1	240	yes	—
126	C₂×C₄	C₂⁴×D₁₃	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×Dic₁₃	1	2	yes	—
131	C₂²	C₂⁴×Dic₁₃	1	20	yes	—
132	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
133	C₄	—	26	32	no	—
134	C₄	C₂⁴×Dic₁₃	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×Dic₁₃	1	1	yes	characteristic
138	C₂	#13129648	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Fitting, prime radical	C₂²×C₁₃×2₋¹⁺⁴	C₂
socle, cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₁₃
hypocenter	C₁₃	#62653
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₁₃	C₂	semidirect
2. #13128039	C₂	semidirect
3. C₂×#11719517	C₂	semidirect
5. C₂×C₁₃×2₋¹⁺⁴	C₄	semidirect
9. #11719532	C₂²	semidirect
10. #41×Dic₁₃	C₂²	semidirect
12. #11719517	C₂²	semidirect
13. C₂×Q₈×Dic₁₃	C₂²	semidirect
16. C₁₃×2₋¹⁺⁴	C₂×C₄	semidirect
17. C₂×C₁₃×#41	C₂×C₄	semidirect
19. C₂²×Q₈×C₁₃	C₂×C₄	semidirect
62. C₂×2₋¹⁺⁴	Dic₁₃	semidirect
81. 2₋¹⁺⁴	C₂×Dic₁₃	semidirect
82. C₂×#41	C₂×Dic₁₃	semidirect
84. C₂²×Q₈	C₂×Dic₁₃	semidirect
115. C₁₃	#62653	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₁₃	2
#13128039	2
C₂×#11719517	2
C₂²×C₁₃×2₋¹⁺⁴	2
#62653	13