Subgroups — C₂³×#88052 #16769380

Order 3328.

2334 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#6090	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₄	1	14	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	14	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88052	C₂²	1	112	yes	direct product factor
10	C₂²×#6090	C₂²	1	28	yes	complemented
11	C₂⁵×C₁₃	C₈	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₈	1	14	yes	—
13	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂×C₄	1	14	yes	—
15	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂×C₄	1	84	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	14	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	7	yes	—
19	C₂²×C₈×C₁₃	C₂³	1	7	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	28	yes	—
21	#88052	C₂³	1	64	yes	direct product factor
22	C₂×#6090	C₂³	1	56	yes	complemented
23	C₂³×M₃₂	—	13	1	no	maximal, Sylow
24	C₂⁴×C₁₃	—	2	8	no	—
25	C₂⁴×C₁₃	C₂×C₈	1	14	yes	—
26	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₂×C₈	1	84	yes	—
28	C₂²×C₄×C₁₃	C₂×C₈	1	14	yes	—
29	C₂²×C₄×C₁₃	C₂×C₈	1	7	yes	—
30	C₂²×C₄×C₁₃	C₂²×C₄	1	28	yes	—
31	C₂²×C₄×C₁₃	C₂²×C₄	1	7	yes	—
32	C₂×C₈×C₁₃	C₂²×C₄	1	21	yes	—
33	C₂×C₈×C₁₃	C₂²×C₄	1	56	yes	—
34	C₂×C₈×C₁₃	C₂²×C₄	1	28	yes	—
35	C₂×C₈×C₁₃	C₂⁴	1	7	yes	—
36	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
37	#6090	C₂⁴	1	16	yes	complemented
38	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
39	C₂²×M₃₂	—	13	28	no	—
40	C₂³×C₁₆	—	13	2	no	—
41	C₂³×C₁₃	—	2	56	no	—
42	C₂³×C₁₃	M₃₂	1	8	yes	complemented
43	C₂³×C₁₃	C₂²×C₈	1	28	yes	—
44	C₂³×C₁₃	C₂²×C₈	1	7	yes	—
45	C₂×C₄×C₁₃	C₂²×C₈	1	56	yes	—
46	C₂×C₄×C₁₃	C₂²×C₈	1	21	yes	—
47	C₂×C₄×C₁₃	C₂²×C₈	1	28	yes	—
48	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
49	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
50	C₈×C₁₃	C₂³×C₄	1	7	yes	—
51	C₈×C₁₃	C₂³×C₄	1	8	yes	—
52	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
53	C₂⁴×C₄	Dic₁₃	1	1	yes	characteristic
54	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
55	C₂³×C₈	Dic₁₃	1	14	yes	—
56	C₂³×C₈	C₂×D₁₃	1	14	yes	—
57	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
58	C₂×M₃₂	—	13	112	no	—
59	C₂²×C₁₆	—	13	28	no	—
60	C₂²×C₁₃	—	2	56	no	—
61	C₂²×C₁₃	C₂×M₃₂	1	28	yes	complemented
62	C₂²×C₁₃	C₂³×C₈	1	8	yes	—
63	C₂²×C₁₃	C₂³×C₈	1	7	yes	—
64	C₄×C₁₃	C₂³×C₈	1	7	yes	—
65	C₄×C₁₃	C₂³×C₈	1	8	yes	—
66	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
67	C₂⁵	#1055	1	1	yes	characteristic
68	C₂³×C₄	#1055	1	14	yes	—
69	C₂³×C₄	#1055	1	1	yes	characteristic
70	C₂³×C₄	C₂×Dic₁₃	1	14	yes	—
71	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₁₃	1	14	yes	—
73	C₂²×C₈	C₂×Dic₁₃	1	84	yes	—
74	C₂²×C₈	C₂×Dic₁₃	1	7	yes	—
75	C₂²×C₈	C₂²×D₁₃	1	28	yes	—
76	C₂²×C₈	C₂²×D₁₃	1	7	yes	—
77	M₃₂	—	13	64	no	—
78	C₂×C₁₆	—	13	56	no	—
79	C₂×C₁₃	—	2	8	no	—
80	C₂×C₁₃	C₂²×M₃₂	1	14	yes	complemented
81	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
82	C₂⁴	—	2	8	no	—
83	C₂⁴	C₂×#1055	1	14	yes	—
84	C₂⁴	C₂×#1055	1	1	yes	characteristic
85	C₂²×C₄	C₂×#1055	1	84	yes	—
86	C₂²×C₄	C₂×#1055	1	14	yes	—
87	C₂²×C₄	C₂×#1055	1	7	yes	—
88	C₂²×C₄	C₂²×Dic₁₃	1	28	yes	—
89	C₂²×C₄	C₂²×Dic₁₃	1	7	yes	—
90	C₂×C₈	C₂²×Dic₁₃	1	56	yes	—
91	C₂×C₈	C₂²×Dic₁₃	1	28	yes	—
92	C₂×C₈	C₂²×Dic₁₃	1	21	yes	—
93	C₂×C₈	C₂³×D₁₃	1	7	yes	—
94	C₂×C₈	C₂³×D₁₃	1	8	yes	—
95	C₁₆	—	13	16	no	—
96	C₁₃	C₂³×M₃₂	1	1	yes	characteristic, complemented, Sylow
97	C₂³	—	2	56	no	—
98	C₂³	#88052	1	8	yes	direct product factor
99	C₂³	C₂²×#1055	1	28	yes	—
100	C₂³	C₂²×#1055	1	7	yes	—
101	C₂×C₄	C₂²×#1055	1	56	yes	—
102	C₂×C₄	C₂²×#1055	1	21	yes	—
103	C₂×C₄	C₂²×#1055	1	28	yes	—
104	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
105	C₂×C₄	C₂³×Dic₁₃	1	7	yes	—
106	C₈	C₂³×Dic₁₃	1	7	yes	—
107	C₈	C₂³×Dic₁₃	1	8	yes	—
108	C₈	C₂⁴×D₁₃	1	1	yes	characteristic
109	C₂²	—	2	56	no	—
110	C₂²	C₂×#88052	1	28	yes	direct product factor
111	C₂²	C₂³×#1055	1	8	yes	—
112	C₂²	C₂³×#1055	1	7	yes	—
113	C₄	C₂³×#1055	1	7	yes	—
114	C₄	C₂³×#1055	1	8	yes	—
115	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
116	C₂	—	2	8	no	—
117	C₂	C₂²×#88052	1	14	yes	direct product factor
118	C₂	C₂⁴×#1055	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×C₈
prime radical	C₂⁵×C₁₃	C₈
cosocle	C₈×C₁₃	C₂⁵
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂³×M₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#88052	C₂	direct (N × H)
2. C₂³×#6090	C₂	semidirect
9. C₂×#88052	C₂²	direct (N × H)
10. C₂²×#6090	C₂²	semidirect
21. #88052	C₂³	direct (N × H)
22. C₂×#6090	C₂³	semidirect
37. #6090	C₂⁴	semidirect
42. C₂³×C₁₃	M₃₂	semidirect
61. C₂²×C₁₃	C₂×M₃₂	semidirect
80. C₂×C₁₃	C₂²×M₃₂	semidirect
96. C₁₃	C₂³×M₃₂	semidirect
98. C₂³	#88052	direct (N × H)
110. C₂²	C₂×#88052	direct (N × H)
117. C₂	C₂²×#88052	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
C₂³×#6090	2
C₂⁴×C₈×C₁₃	2
C₂³×M₃₂	13

Subgroups — C23×#88052 #16769380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88052 #16769380