Subgroups — C₂²×C₄×#6090 #16015542

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6090	C₂	1	2	yes	maximal
2	C₂×C₄×#6090	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	6	yes	—
9	C₂²×#6090	C₄	1	16	yes	direct product factor
10	C₂²×#6090	C₂²	1	12	yes	—
11	C₄×#6090	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₁₃	C₈	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₈	1	8	yes	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
19	C₂×C₄²×C₁₃	C₈	1	6	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
22	C₄×C₈×C₁₃	C₂³	1	4	yes	—
23	C₂×#6090	C₂×C₄	1	48	yes	direct product factor
24	C₂×#6090	C₂³	1	8	yes	—
25	C₂²×C₄×C₁₆	—	13	1	no	maximal, Sylow
26	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₁₆	1	16	yes	complemented
28	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
29	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
30	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
32	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
33	C₂×C₈×C₁₃	C₄²	1	12	yes	—
34	C₂×C₈×C₁₃	C₂×C₈	1	24	yes	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
36	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
37	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
38	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×C₈	1	12	yes	—
40	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
41	#6090	C₂²×C₄	1	16	yes	direct product factor
42	C₂³×C₁₆	—	13	2	no	—
43	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
44	C₂×C₄×C₁₆	—	13	12	no	—
45	C₂³×C₁₃	C₄×C₈	1	4	yes	—
46	C₂³×C₁₃	C₂×C₁₆	1	8	yes	—
47	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
48	C₂×C₄×C₁₃	C₄×C₈	1	12	yes	—
49	C₂×C₄×C₁₃	C₂×C₁₆	1	48	yes	complemented
50	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
51	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
52	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
53	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
54	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
55	C₈×C₁₃	C₂×C₄²	1	6	yes	—
56	C₈×C₁₃	C₂²×C₈	1	8	yes	—
57	C₈×C₁₃	C₂³×C₄	1	2	yes	—
58	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
59	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
60	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
61	C₂²×C₁₆	—	13	16	no	—
62	C₂²×C₁₆	—	13	12	no	—
63	C₂×C₄×C₈	Dic₁₃	1	6	yes	—
64	C₂×C₄×C₈	C₂×D₁₃	1	6	yes	—
65	C₄×C₁₆	—	13	16	no	—
66	C₂²×C₁₃	C₄×C₁₆	1	16	yes	complemented
67	C₂²×C₁₃	C₂×C₄×C₈	1	6	yes	—
68	C₂²×C₁₃	C₂²×C₁₆	1	12	yes	—
69	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
70	C₄×C₁₃	C₂×C₄×C₈	1	6	yes	—
71	C₄×C₁₃	C₂²×C₁₆	1	16	yes	complemented
72	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
73	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
74	C₂³×C₄	#1055	1	2	yes	—
75	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
76	C₂²×C₈	#1055	1	8	yes	—
77	C₂²×C₈	C₄×D₁₃	1	8	yes	—
78	C₂²×C₈	C₂×Dic₁₃	1	6	yes	—
79	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
80	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
81	C₂×C₄²	#1055	1	6	yes	—
82	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
83	C₂×C₁₆	—	13	48	no	—
84	C₂×C₁₆	—	13	8	no	—
85	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
86	C₄×C₈	C₂²×D₁₃	1	4	yes	—
87	C₂×C₁₃	C₂×C₄×C₁₆	1	12	yes	complemented
88	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
89	C₂×C₁₃	C₂³×C₁₆	1	2	yes	—
90	C₂⁴	C₂×#1055	1	1	yes	characteristic
91	C₂²×C₄	#6090	1	16	yes	direct product factor
92	C₂²×C₄	C₂×#1055	1	4	yes	—
93	C₂²×C₄	C₂×#1055	1	3	yes	—
94	C₂²×C₄	C₂×#1055	1	12	yes	—
95	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
96	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
97	C₂×C₈	C₂×#1055	1	24	yes	—
98	C₂×C₈	C₄×Dic₁₃	1	12	yes	—
99	C₂×C₈	C₂×C₄×D₁₃	1	12	yes	—
100	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
101	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
102	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
103	C₄²	C₂×#1055	1	12	yes	—
104	C₄²	C₂²×Dic₁₃	1	4	yes	—
105	C₁₆	—	13	16	no	—
106	C₁₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₄×#1055	1	4	yes	—
108	C₂³	C₂×#6090	1	8	yes	—
109	C₂³	C₂²×#1055	1	3	yes	—
110	C₂×C₄	C₄×#1055	1	12	yes	—
111	C₂×C₄	C₂×#6090	1	48	yes	direct product factor
112	C₂×C₄	C₂²×#1055	1	6	yes	—
113	C₂×C₄	C₂²×#1055	1	8	yes	—
114	C₂×C₄	C₂²×#1055	1	3	yes	—
115	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
116	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
117	C₈	C₂²×#1055	1	8	yes	—
118	C₈	C₂×C₄×Dic₁₃	1	6	yes	—
119	C₈	C₂²×C₄×D₁₃	1	2	yes	—
120	C₂²	C₄×#6090	1	16	yes	direct product factor
121	C₂²	C₂×C₄×#1055	1	6	yes	—
122	C₂²	C₂²×#6090	1	12	yes	—
123	C₂²	C₂³×#1055	1	1	yes	characteristic
124	C₄	C₂×C₄×#1055	1	6	yes	—
125	C₄	C₂²×#6090	1	16	yes	direct product factor
126	C₄	C₂³×#1055	1	1	yes	characteristic
127	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
128	C₂	C₂×C₄×#6090	1	12	yes	direct product factor
129	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
130	C₂	C₂³×#6090	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₈	D₁₃
commutator, hypocenter	C₁₃	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#6090	C₂	direct (N × H)
9. C₂²×#6090	C₄	direct (N × H)
11. C₄×#6090	C₂²	direct (N × H)
23. C₂×#6090	C₂×C₄	direct (N × H)
27. C₂²×C₄×C₁₃	C₁₆	semidirect
41. #6090	C₂²×C₄	direct (N × H)
49. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
66. C₂²×C₁₃	C₄×C₁₆	semidirect
71. C₄×C₁₃	C₂²×C₁₆	semidirect
87. C₂×C₁₃	C₂×C₄×C₁₆	semidirect
91. C₂²×C₄	#6090	direct (N × H)
106. C₁₃	C₂²×C₄×C₁₆	semidirect
111. C₂×C₄	C₂×#6090	direct (N × H)
120. C₂²	C₄×#6090	direct (N × H)
125. C₄	C₂²×#6090	direct (N × H)
128. C₂	C₂×C₄×#6090	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6090	2
C₂×C₄×#6090	2
C₂²×C₄×C₈×C₁₃	2
C₂²×C₄×C₁₆	13

Subgroups — C22×C4×#6090 #16015542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#6090 #16015542